Research ArticlePerihelion Precession and Deflection of Light in the GeneralSpherically Symmetric Spacetime

Ya-Peng Hu12 Hongsheng Zhang3 Jun-Peng Hou1 and Liang-Zun Tang1

1 College of Science Nanjing University of Aeronautics and Astronautics Nanjing 210016 China2 INPAC Department of Physics and Shanghai Key Laboratory of Particle Physics and Cosmology Shanghai Jiao Tong UniversityShanghai 200240 China

3 Shanghai United Center for Astrophysics (SUCA) Shanghai Normal University 100 Guilin Road Shanghai 200234 China

Correspondence should be addressed to Ya-Peng Hu huyppkueducn

Received 24 September 2014 Accepted 10 November 2014 Published 24 November 2014

Academic Editor Rong-Gen Cai

Copyright copy 2014 Ya-Peng Hu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

The perihelion precession and deflection of light have been investigated in the 4-dimensional general spherically symmetricspacetime and the master equation is obtained As the application of this master equation the Reissner-Nordstorm-AdS solutionandClifton-Barrow solution in119891(119877) gravity have been taken as examplesWe find that both the electric charge and119891(119877) gravity canaffect the perihelion precession and deflection of light while the cosmological constant can only effect the perihelion precessionMoreover we clarify a subtlety in the deflection of light in the solar system that the possible sunrsquos electric charge is usually used tointerpret the gap between the experiment data and theoretical result However after also considering the effect from the sunrsquos sameelectric charge on the perihelion precession of Mercury we can find that it is not the truth

1 Introduction

In the history the perihelion precession of Mercury anddeflection of light in the solar system are two well-knownphenomena to check the correctness of general relativity[1ndash5] Nowadays we know that the foundation of generalrelativity is a very significant event in modern physics Notonly can general relativity give new insights into our under-standing of gravity but also it has been the basic theory in ourmodern cosmology [1] Since the perihelion precession anddeflection of light are usually constrained in the solar systemor some planet such as Mercury [1ndash3] therefore it will beworthy to investigate the perihelion precession and deflectionof light in the more general case In this paper the perihelionprecession and deflection of light have been considered inthe 4-dimensional general spherically symmetric spacetimeSince the perihelion precession and deflection of light can betreated as the time-like and null geodesic in spacetime [3]thus we obtain the corresponding main equation in the 4-dimensional general spherically symmetric spacetime

Note that due to the Birkhoff theorem or the gener-alization of the Birkhoff theorem in Einstein gravity thegeneral spherically symmetric spacetimes in Einstein gravityare very limited However the Birkhoff theorem can beinvalid in some modified gravities that is 119891(119877) gravitywhich is a kind of a higher derivative gravity theory [6 7]Therefore as the application of the master equation in the4-dimensional general spherically symmetric spacetime wetake theReissner-Nordstorm-AdS solution inEinstein gravityand Clifton-Barrow solution in 119891(119877) gravity [8 9] as the twoexamples to discuss the corresponding perihelion precessionand deflection of light For the Reissner-Nordstorm-AdSsolution with the electric charge and cosmological constantwe find that the electric charge can affect both the perihelionprecession and deflection of light while the cosmologicalconstant can only affect the perihelion precession whichare consistent with the results in the previous work [10ndash14] It should be emphasized that there is a subtlety in theprevious work to discuss the well-known deflection of lightin the solar system The subtlety is that the possible sunrsquos

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 604321 7 pageshttpdxdoiorg1011552014604321

2 Advances in High Energy Physics

electric charge is usually used to interpret the gap betweenthe experiment data and theoretical result in the deflection oflight in the solar system because the electric charge can affectthe deflection of light [15ndash23]However after using our resultsand also considering the effect from the sunrsquos same electriccharge on the perihelion precession of Mercury [24 25] wecan find that it is not the truth that is the gap betweenthe experiment data and theoretical result for the deflectionof light in solar system cannot completely come from thepossible sunrsquos electric charge For the Clifton-Barrow solutionin 119891(119877) gravity [8 9] there is a parameter 119889 which is thepower of Ricci scalar 119877 in one kind of 119891(119877) gravity in thissolution It is obvious that the Birkhoff theorem is invalid forthis kind of 119891(119877) gravity since the Schwarzschild metric isalso another static vacuum solution For the simplicity andmaking some explicit comparison with other results we justconsider the case with small 119889 in our paper because 119889 =0 is just the Einstein gravity in our setting Therefore theparameter 119889 can represent the derivation of 119891(119877) gravity toEinstein gravity From our results we can easily find that the119891(119877) gravity can affect both the perihelion precession and thedeflection of light

The rest of the paper is organized as follows In Section 2after investigating the perihelion precession and deflectionof light in the 4-dimensional general spherically symmetricspacetime the master equation is obtained In Section 3the Reissner-Nordstorm-AdS solution in Einstein gravity andClifton-Barrow solution in 119891(119877) gravity are taken as the twoexamples for the application of master equation and theeffects from the cosmological constant electric charge and119891(119877) gravity on the perihelion precession and deflection oflight are investigated Finally besides a simple conclusion wealso make several discussions according to the experimentdata in Section 4

2 Perihelion Precession and Deflectionof Light in the 4-Dimensional SphericallySymmetric Spacetime General Case

For the 4-dimensional general spherically symmetric space-time its line element can be

1198891199042 = minus119891 (119903) 1198891199052 +

1198891199032

ℎ (119903)+ 1199032 (1198891205792 + sin21205791198891205932) (1)

Since the perihelion precession and deflection of light areusually treated as the time-like and null geodesic in space-time respectively therefore we first discuss the geodesics 120574(120591)in the above general spherically symmetric spacetimeWe setthe geodesic 120574(120591) expressed in the above coordinates 119909120583 =(119905 119903 120579 120593) as 119909120583(120591) which are satisfied

1198892119909120583

1198891205912+ Γ120583]120590

119889119909]

119889120591

119889119909120590

119889120591= 0 (2)

Generally after the above equation is solved the geodesic120574(120591) is obtained However considering the symmetry ofspacetime (1) we could find in the following that there is amore simple way to obtain the geodesic 120574(120591) First we can

find that one component of the geodesic 120574(120591) can alwaysbe chosen as 120579(120591) = 1205872 which means that the geodesiccan always be chosen to lay in the equatorial plane of thespherically symmetric spacetimeTherefore the geodesic canbe simplified

119905 = 119905 (120591) 119903 = 119903 (120591) 120579 =120587

2 120593 = 120593 (120591) (3)

If we let 119880119886 equiv (120597120597120591)119886 be the tangent vector of geodesic 120574(120591)we could define

120581 = minus119892119886119887119880119886119880119887 = minus119892

119886119887(120597

120597120591)119886

(120597

120597120591)119887

(4)

Thus 120581 = 1 corresponds to the time-like geodesic while120581 = 0 is the null geodesic After inserting (120597120597120591)119886 =(119889119909120583119889120591)(120597120597119909120583)

119886 and the metric (1) we could obtain

minus120581 = 119892119886119887(120597

120597120591)119886

(120597

120597120591)119887

= minus119891 (119903) (119889119905

119889120591)2

+ ℎ (119903)minus1 (

119889119903

119889120591)2

+ 1199032 (119889120593

119889120591)2

(5)

where we have used 120579 = 1205872 Second note that (120597120597119905)119886 and(120597120597120593)119886 are two killing vectors in the spherically symmetricspacetime (1) Therefore there are two conserved quantitiesalong the geodesic 120574(120591)

119864 = minus 119892119886119887(120597

120597119905)119886

(120597

120597120591)119887

= 119891 (119903)119889119905

119889120591 (6)

119871 = 119892119886119887(120597

120597120593)119886

(120597

120597120591)119887

= 1199032119889120593

119889120591 (7)

where the physical meanings of 119864 and 119871 can be found indetail in [3] that is 119864 can be interpreted as the total energy(including gravitational potential energy) per unit rest massof a particle in the time-like case while 119871 can be interpretedas the angular momentum per unit rest mass of a particle Inaddition ℏ119864 and ℏ119871 can be interpreted as the total energy andangular momentum of a photon in the null case respectively

After inserting (6) and (7) into (5) we could obtain

(119889119903

119889120591)2

=ℎ (119903)

119891 (119903)1198642 minus ℎ (119903) (120581 +

1198712

1199032) (8)

Obviously the above equation contains only one function119903(120591) which could be solved in principle Hence after insert-ing the solved 119903(120591) into (6) and (7) the remaining compo-nents 119905(120591) and 120593(120591) of geodesic could be finally obtained

It should be pointed out that perihelion precession anddeflection of light are usually related to the orbit of geodesicthat is 119903(120593) Therefore it is convenient to change (8) as

(119889119903

119889120593)2

(119871

1199032)2

=ℎ (119903)

119891 (119903)1198642 minus ℎ (119903) (120581 +

1198712

1199032) (9)

where we have used (7) In addition it has been found thatthe coordinate 120583 equiv 1119903 is more convenient than 119903 to discuss

Advances in High Energy Physics 3

the perihelion precession and deflection of light Thus themaster equation investigated in our paper could be simplyobtained from (9) by changing 119903 into 120583

(119889120583

119889120593)2

=ℎ (120583)

119891 (120583)(119864

119871)2

minus ℎ (120583) (120581

1198712+ 1205832) (10)

3 Perihelion Precession and Deflectionof Light in the 4-Dimensional SphericallySymmetric Spacetime Special Case

Since functions ℎ(119903) and 119891(119903) are usually different for eachspherically symmetric spacetime thus the perihelion pre-cession and deflection of light may be different in differentspherically symmetric spacetime Therefore we can use thedifferences in perihelion precession and deflection of lightto extract the information in ℎ(119903) and 119891(119903) In this sectionwe will take two spherically symmetric solutions as examplesfor the application of master equation (10) to discuss thecorresponding perihelion precession and deflection of lightThe first solution is the Reissner-Nordstorm-AdS solutionwhich is a well-known solution in Einstein gravity with thecosmological constant and electric charge while the othersolution is the Clifton-Barrow solution in 119891(119877) gravity The119891(119877) gravity is a kind of higher derivative gravity theoryand the Clifton-Barrow solution can be considered as ageneralization of Schwarzschild solution in Einstein gravityFor this solution the advantages are that not only it is thespherically symmetric spacetime in119891(119877) gravity but also ℎ(119903)and 119891(119903) are different

31 Reissner-Nordstorm-AdS Solution For the Reissner-Nordstorm-AdS solution the two functions ℎ(119903) and119891(119903) are

119891 (119903) = ℎ (119903) = 1 minus2119872

119903+Λ

31199032 +

1198762

1199032 (11)

where Λ is the cosmological constant and 119876 is the electriccharge Therefore (10) becomes

(119889120583

119889120593)2

= (119864

119871)2

minus (1 minus 2119872120583 +Λ

31205832+ 12058321198762)(

120581

1198712+ 1205832)

(12)

For the perihelion precession one usually considers the time-like geodesic that is 120581 = 1 Therefore (12) for the time-likegeodesic can be

1198892120583

1198891205932+ (1 +

1198762

1198712)120583 =

119872

1198712+ 31205832119872+

Λ

312058331198712minus 211987621205833 (13)

Obviously compared with the case in Newtonrsquos gravity11988921205831198891205932 + 120583 = 1198721198712 the term 31205832119872 comes from thecorrection of general relativity while the last two terms arefrom the cosmological constant and electric charge and theabove equation can return to the well-known Schwarzschildcase whenΛ = 119876 = 0 Note that the analytical solution of (13)

is absent like in the Schwarzschild case However there is anapproximation solution of (13)

120583 (120593) =119872

1198712(1 + 119890 cos120593)

+31198723

1198714(1 + 119890120593 sin120593 + 1198902 (1

2minus1

6cos 2120593))

+Λ1198714

31198723(1 minus

3

2119890120593 sin120593) minus 1198721198762

1198714(1 +

1

2119890120593 sin120593)

minus211987621198723

1198716(1 +

3

2119890120593 sin120593 + 31198902 (1

2minus1

6cos 2120593))

(14)

in the following conditions

31198721205832 ≪ 120583Λ

312058331198712≪ 120583

1198762

1198712≪ 1 211987621205832 ≪ 1 (15)

where 120583(120593) = (1198721198712)(1 + 119890 cos120593) is the analytical ellipticalsolution which has already been found in Newtonrsquos gravityand 119890 is the orbital eccentricity which has been considered asa small constant Therefore (14) can be further reduced afterneglecting the high order terms

120583 (120593)

=119872

11987121 + 119890 [cos120593 + (3119872

2

1198712minusΛ1198716

21198724minus1198762

21198712)120593 sin120593]

(16)

where the conditions 311987221198712 ≪ 1 and Λ119871631198724 ≪ 1 havealso been assumed Note that the above equation could befurther simplified as

120583 (120593) =119872

1198712[1 + 119890 cos (120593 minus 120576120593)] (17)

where we have set

120576 = (31198722

1198712minusΛ1198716

21198724minus1198762

21198712) (18)

For the perihelion of orbit 119903(120593) it satisfies cos(120593 minus 120576120593) = 1and hence 120593 = 2120587 + 2120587120576 Therefore the precession angle ofperihelion is

Δ120593 = 2120587120576 = 2120587(31198722

1198712minusΛ1198716

21198724minus1198762

21198712) (19)

Note that here 119871 is the angular momentum per unit restmass of the particle along the time-like geodesic Obviouslyboth the cosmological constant and the electric charge canaffect the precession angle of perihelion When both thecosmological constant and electric charge disappear in (19)the result recovers the standard general relativity result withSchwarzschild solution Δ120593 = 612058711987221198712 asymp 6120587119872119886 where 119886is the semimajor axis of the ellipse and 119886 asymp 1198712119872 when theeccentricity 119890 is small

4 Advances in High Energy Physics

Next we will discuss the deflection of light in theReissner-Nordstorm-AdS spacetime In this case the corre-sponding geodesic is the null geodesic that is 120581 = 0 Similarto the procedure dealt with in the perihelion precession weneed to find out the approximation solution of the orbit ofdeflection of light The master equation is

(119889120583

119889120593)2

+ 1205832 =1198642

1198712+ 21198721205833 minus

Λ

3minus 11987621205834 (20)

Note that the cosmological constant term is just a constantterm like11986421198712Therefore the cosmological constant will notaffect the deflection angle of light In fact (20) can be furthersimplified as

1198892120583

1198891205932+ 120583 = 31198721205832 minus 211987621205833 (21)

which shows more clearly that the cosmological constantdoes not affect the deflection angle Note that the approxi-mation solution of (21) is

120583 (120593) =1

119897sin120593 + 119872

1198972(1 minus cos120593)2

minus21198762

1198973(minus

3

8120593 cos120593 + 1

32sin 3120593)

(22)

where 119897 is a constant 120583(120593) = (1119897) sin120593 is in fact a straightline expressed in polar coordinates (120583 120593) and we have usedthe condition120583(0) = 0Therefore the deflection angle of light120573 can be obtained from the equation

120583 (120587 + 120573) = 0 (23)

After using the approximation

sin (120587 + 120573) asymp minus120573 cos (120587 + 120573) asymp minus1 (24)

the angle is

120573 =4119872

119897minus31205871198762

41198972 (25)

where the first term is just the well-known deflection angleof light in Schwarzschild spacetime while the second term isfrom the effect of the electric charge

32 Clifton-Barrow Solution In this subsection we willinvestigate the perihelion precession and deflection of lightin the Clifton-Barrow solution in 119891(119877) gravity In a general119891(119877) theory the uniqueness theorem of the sphericallysymmetric space becomes invalid 119877 may be not zero evenfor vacuum solutions Considering the 4-dimensional actionin the following form

119878 =1

16120587119866(int

M

1198894119909radicminus det (119892)119877119889+1

+int120597M

1198893119909radicminus det (ℎ)2119861) (26)

where 119889 is a constant 119861 represents the correspondingboundary term for the 119891(119877) term 119892 is the 4-dimensionalmetric ℎ denotes the induced metric on the boundary thecorresponding field equation reads

(1 + 119889) 119877119889119877120583] minus

1

2119877119889+1119892

120583] minus (1 + 119889) nabla120583nabla]119877119889

+ 119892120583] (1 + 119889)r119877119889 = 0

(27)

It is easy to see that the Schwarzschild metric is a solutionfor the above equation since Schwarzschild metric has avanishing Ricci scalar119877 In addition the119891(119877) gravity permitsa nontrivial spherically symmetric solution other than theSchwarzschildmetric and the called Clifton-Barrow solutionalso solves the field equation [8]

1198891199042 = minus119891 (119903) 1198891199052 +

1198891199032

ℎ (119903)+ 1199032 (1198891205792 + sin21205791198891206012) (28)

where

119891 (119903) = 1199032119889((1+2119889)(1minus119889)) +119888

119903(1minus4119889)(1minus119889)

ℎ (119903) =(1 minus 119889)2

(1 minus 2119889 + 41198892) (1 minus 2119889 (1 + 119889))

times (1 +119888

119903(1minus2119889+41198892)(1minus119889)

)

(29)

Here 119888 is a constant which reduces to the Schwarzschildmassparameter 119888 = minus2119872 in Einstein gravity

Therefore one can directly insert (29) into the masterequation (10) to discuss its perihelion precession and deflec-tion of light Note that after inserting the two functions aboveinto (10) it will be found that the equations are difficult tobe solved directly In order to explicitly show the effects onperihelion precession and deflection of light from the 119891(119877)gravity here we just consider the case that the constant 119889 isvery small since 119889 = 0 is just the Einstein gravity In this casethe master equation (10) turns out to be

1198892120583

1198891205932+ 120583 =

120581119872

1198712+ 31198721205832

+ [120581119872

1198712+1198642

11987121

120583minus 2120583 + 51198721205832

+(120581119872

1198712+ 31198721205832) ln 1

120583] 119889

(30)

where we have kept the linear terms of 119889 and neglected thehigher order terms

For the perihelion precession that is 120581 = 1 the aboveequation is still a little complicated which can be furthersimplified in the larger radius case that is 120583 = 1119903 sim 0 Thusthe above equation is

1198892120583

1198891205932+ 120583 =

120581119872

1198712+ 31198721205832 +

1198642

1198712119889

120583 (31)

Advances in High Energy Physics 5

and the approximate solution can be easily obtained

120583 (120593) =119872

1198712(1 + 119890 cos120593)

+31198723

1198714(1 + 119890120593 sin120593 + 1198902 (1

2minus1

6cos 2120593))

+1198642

119872(1 minus

1

2119890120593 sin120593)119889

(32)

which can be further simplified

120583 (120593) =119872

11987121 + 119890 [cos120593 + (3119872

2

1198712minus11986421198712

21198722119889)120593 sin120593]

(33)

Therefore the angle of perihelion precession is

Δ120593 = 2120587120576 = 2120587(31198722

1198712minus11986421198712

21198722119889) (34)

From the above simple discussion we can easily find thatthere can be indeed an effect from the 119891(119877) gravity on theperihelion precession

For the deflection of light that is 120581 = 0 the masterequation becomes

1198892120583

1198891205932+ 120583 = 31198721205832 + (

1198642

11987121

120583minus 2120583 + 51198721205832 + 31198721205832 ln 1

120583)119889

(35)

which can also be simplified in the larger radius case that is120583 = 1119903 sim 0 as

1198892120583

1198891205932+ 120583 = 31198721205832 +

1198642

1198712119889

120583 (36)

If we solve this equation in the exact same way used beforewe obtain

120583 (120593) =1

119897sin120593

+119872

1198972(1 minus cos120593)2 + 1198642119897

1198712(minus120593 cos120593 + sin120593 ln sin120593) 119889

(37)

which can be further simplified near 120593 = 120587

120583 (120593) =1

119897sin120593 + 119872

1198972(1 minus cos120593)2 + 1198642119897

1198712(minus120593 cos120593) 119889

(38)

Therefore the angle can be obtained

120573 =4119872

119897+1198642119897 (4119872 + 120587119897)

1198712119889 (39)

4 Conclusion and Discussion

In this paper the perihelion precession and deflection of lighthave been investigated in the 4-dimensional general spher-ically symmetric spacetime Since the perihelion precessionand deflection of light can be treated respectively as thetime-like and null geodesic in spacetime themaster equationof perihelion precession and deflection of light is obtainedMoreover the Reissner-Nordstorm-AdS solution in Einsteingravity and Clifton-Barrow solution in 119891(119877) gravity aretaken as the two examples for the application of this masterequation and the effects from the cosmological constantelectric charge and119891(119877) gravity on the perihelion precessionand deflection of light are investigated We find that theelectric charge can affect both the perihelion precession andthe deflection of light while the cosmological constant canonly affect the perihelion precession which are consistentwith the results in the previous works In addition afterconsidering the case with small 119889 we easily find that the119891(119877)gravity can also affect both the perihelion precession and thedeflection of light Several discussions related to our resultsare in order

(i) During investigating the effects from the electriccharge cosmological constant and 119891(119877) gravity on the peri-helion precession and deflection of light we have assumed theapproximate conditions like (15) and small 119889 Therefore theresults under other approximate conditions are interestingopen issues

(ii) The perihelion precession and deflection of lightcan contain the information of the spacetime that is thecosmological constant electric charge and small 119889 Partic-ularly if we just consider them in the solar system sincethe angle of perihelion precession and deflection of lightin the solar system can be detected by the experimentsthus the information like cosmological constant and electriccharge can be extracted from the angles by the experimentsTherefore we may also extract the information of darkmatter such as its distribution of energy density 120588(119903) throughdetecting the perihelion precession and deflection of lightby the experiments if we can first find the static solutioncontaining the dark matter which will be further studied inthe future work

(iii) A simple constraint on the cosmological constantAlthough the cosmological constant does not affect thedeflection of light it can affect the perihelion precessionFrom our results if the cosmological constant indeed existsin our universe it can also give effects in the solar systemTherefore we can give a simple constraint on the cosmologi-cal constant from the well-known phenomenonmdashperihelionprecession of Mercury in the solar system The experimentdata of anomalous precession angle is (4256 plusmn 094)10158401015840 percentury [26] Therefore the theoretical result should be in(4256 plusmn 094)10158401015840 Considering the simple case 119876 = 0 in (19)and recovering the constants 119866 and 119888 (19) becomes

Δ120593 = 2120587(311987221198662

11987121198882minus

Λ1198716

211987241198663) (40)

6 Advances in High Energy Physics

Since the theoretical result of the first term is 430310158401015840 percentury thus the possible contribution from cosmologicalconstant 120587Λ119871611987241198663 must be less than 14110158401015840 per centurywhere we just consider the positive cosmological constantwhich can accelerate our universe After inserting the con-stants the mass of the sun 119872 = 1989 times 1030 kg and theangular momentum of unit mass of Mercury 119871 = 272 times

1016m2sminus1 we can constrain the cosmological constant Λ lt

589times10minus11 kgm3 which is consistent with observation datain our universe Λ = 19 times 10minus25 kgm3

(iv) The subtlety in the deflection of light in the solarsystem Note that there is a gap between the experiment data(161 plusmn 04)10158401015840 and theoretical result 17510158401015840 in Schwarzschildspacetime [24 25] From the result in (25) one may considerthat this difference may be from the sunrsquos possible electriccharge effect However we will give a simple proof in thefollowing that it is not true After recovering the constantsthe angle of starlight deflection is

120573 =4119872119866

1198882119897minus31205871198762119866

411989721198884 (41)

where we have used the Gauss unit of119876 that is kg12m32sminus1fromwhichwe canfind that the charge could indeedmake theangle smaller and hence make the gap smaller If the chargecould make the angle smaller 110158401015840 that is (31205871198762119866411989721198884) times180120587times60times60 = 1 we can obtain the sunwith possible charge119876 = 11 times 1028 kg12m32sminus1 However note that the sunrsquoscharge can also affect the perihelion precession of Mercuryin (19) which can be recovered

Δ120593 = 2120587(311987221198662

11987121198882minus

1198762119866

211987121198882) (42)

where we have neglected the effect of cosmological constantsince the observation data in our universe Λ = 19 times10minus25 kgm3 is very smallTherefore after the simple calcula-tion we can obtain that the sunrsquos charge effect on the angle ofperihelion precession ofMercury will be120587119876211986611987121198882 = 0038which is larger than the first term 61205871198722119866211987121198882 = 497times10minus7This is obviously opposite to the experiment observationThus the gap between the experiment data and theoreticalresult for the starlight deflection in solar system could notcompletely come from the sunrsquos electric charge and a moresuitable explanation of this difference will still be an openissue

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (NSFC) under Grant no 11105004 andShanghai Key Laboratory of Particle Physics and Cosmologyunder Grant no 11DZ2260700 and partially by Grants from

NSFC (nos 10821504 10975168 and 11035008) and theMinistry of Science andTechnology ofChina underGrant no2010CB833004 Professor Hongsheng Zhang is supported bythe Program for Professor of Special Appointment (EasternScholar) at Shanghai Institutions of Higher Learning andNational Natural Science Foundation of China under Grantnos 11075106 and 11275128

References

[1] S Weinberg Gravitation and Cosmology Principles and Appli-cations of the General Theory of Relativity John Wiley amp SonsNew York NY USA 1972

[2] C W Misner K S Thorne and J A Wheeler Gravitation WH Freeman and Co San Francisco Calif USA 1973

[3] R M Wald General Relativity University of Chicago PressChicago Ill USA 1984

[4] S G Turyshev U E Israelsson M Shao et al ldquoSpace-basedresearch in fundamental physics and quantum technologiesrdquoInternational Journal of Modern Physics D vol 16 no 12 pp1879ndash1925 2008

[5] S G Turyshev ldquoExperimental tests of general relativityrdquoAnnualReview of Nuclear and Particle Science vol 58 pp 207ndash2482008

[6] T P Sotiriou and V Faraoni ldquo119891(119877) theories of gravityrdquo Reviewsof Modern Physics vol 82 no 1 pp 451ndash497 2010

[7] R G Cai L M Cao Y P Hu and N Ohta ldquoGeneralizedMisner-Sharp energy in f (R) gravityrdquo Physical Review D vol80 Article ID 104016 2009

[8] T Clifton and J D Barrow ldquoThe power of general relativityrdquoPhysical Review D Third Series vol 72 no 10 Article ID103005 2005 Erratum in Physical Review D Third Series vol90 Article ID 029902 2014

[9] H Zhang Y Hu and X Z Li ldquoMisner-Sharp mass in n-dimensional f(R) gravityrdquo Physical Review D vol 90 Article ID024062 2014

[10] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 1989

[11] P J Peebles and B Ratra ldquoThe cosmological constant and darkenergyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash6062003

[12] A Albrecht and C Skordis ldquoPhenomenology of a realisticaccelerating universe using only Planck-scale physicsrdquo PhysicalReview Letters vol 84 p 2076 2000

[13] M Li X D Li S Wang and Y Wang ldquoDark energyrdquo Commu-nications in Theoretical Physics vol 56 no 3 p 525 2011

[14] V Kagramanova J Kunz and C Lammerzahl ldquoSolar systemeffects in Schwarzschild-de Sitter space-timerdquo Physics Letters Bvol 634 pp 465ndash470 2006

[15] P Jetzer and M Sereno ldquoTwo-body problem with the cosmo-logical constant and observational constraintsrdquo Physical ReviewD vol 73 no 4 Article ID 044015 6 pages 2006

[16] L Iorio ldquoSolar System motions and the cosmological constanta new approachrdquo Advances in Astronomy vol 2012 Article ID268647 9 pages 2012

[17] L Iorio ldquoConstraining the angular momentum of the sun withplanetary orbital motions and general relativityrdquo Solar Physicsvol 281 no 2 pp 815ndash826 2012

[18] L Iorio ldquoConstraining the electric charges of some astro-nomical bodies in Reissner-Nordstrom spacetimes and generic

Advances in High Energy Physics 7

119903minus2-type power-law potentials from orbital motionsrdquo GeneralRelativity and Gravitation vol 44 no 7 pp 1753ndash1767 2012

[19] M Sereno and P Jetzer ldquoDark matter versus modificationsof the gravitational inverse-square law results from planetarymotion in the Solar systemrdquo Monthly Notices of the RoyalAstronomical Society vol 371 no 2 pp 626ndash632 2006

[20] M Sereno ldquoWeak field limit of Reissner-Nordstrom black holelensingrdquo Physical Review D vol 69 no 2 Article ID 023002 4pages 2004

[21] M Sereno and P Jetzer ldquoSolar and stellar system tests of thecosmological constantrdquo Physical Review D vol 73 Article ID063004 2006

[22] M Sereno ldquoInfluence of the cosmological constant on gravi-tational lensing in small systemsrdquo Physical Review D vol 77Article ID 043004 2008

[23] M Sereno ldquoRole of Λ in the cosmological lens equationrdquoPhysical Review Letters vol 102 no 2 Article ID 021301 2009

[24] F W Dyson A S Eddington and C Davidson ldquoA determi-nation of the deflection of light by the sunrsquos gravitational fieldfrom observations made at the total eclipse of May 29 1919rdquoPhilosophical Transactions of the Royal Society A vol 220 no571ndash581 pp 291ndash333 1920

[25] D Kennefick ldquoTesting relativity from the 1919 eclipsemdasha ques-tion of biasrdquo Physics Today vol 62 no 3 pp 37ndash42 2009

[26] G M Clemence ldquoThe relativity effect in planetary motionsrdquoReviews of Modern Physics vol 19 p 361 1947

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ThermodynamicsJournal of

2 Advances in High Energy Physics

electric charge is usually used to interpret the gap betweenthe experiment data and theoretical result in the deflection oflight in the solar system because the electric charge can affectthe deflection of light [15ndash23]However after using our resultsand also considering the effect from the sunrsquos same electriccharge on the perihelion precession of Mercury [24 25] wecan find that it is not the truth that is the gap betweenthe experiment data and theoretical result for the deflectionof light in solar system cannot completely come from thepossible sunrsquos electric charge For the Clifton-Barrow solutionin 119891(119877) gravity [8 9] there is a parameter 119889 which is thepower of Ricci scalar 119877 in one kind of 119891(119877) gravity in thissolution It is obvious that the Birkhoff theorem is invalid forthis kind of 119891(119877) gravity since the Schwarzschild metric isalso another static vacuum solution For the simplicity andmaking some explicit comparison with other results we justconsider the case with small 119889 in our paper because 119889 =0 is just the Einstein gravity in our setting Therefore theparameter 119889 can represent the derivation of 119891(119877) gravity toEinstein gravity From our results we can easily find that the119891(119877) gravity can affect both the perihelion precession and thedeflection of light

The rest of the paper is organized as follows In Section 2after investigating the perihelion precession and deflectionof light in the 4-dimensional general spherically symmetricspacetime the master equation is obtained In Section 3the Reissner-Nordstorm-AdS solution in Einstein gravity andClifton-Barrow solution in 119891(119877) gravity are taken as the twoexamples for the application of master equation and theeffects from the cosmological constant electric charge and119891(119877) gravity on the perihelion precession and deflection oflight are investigated Finally besides a simple conclusion wealso make several discussions according to the experimentdata in Section 4

2 Perihelion Precession and Deflectionof Light in the 4-Dimensional SphericallySymmetric Spacetime General Case

For the 4-dimensional general spherically symmetric space-time its line element can be

1198891199042 = minus119891 (119903) 1198891199052 +

1198891199032

ℎ (119903)+ 1199032 (1198891205792 + sin21205791198891205932) (1)

Since the perihelion precession and deflection of light areusually treated as the time-like and null geodesic in space-time respectively therefore we first discuss the geodesics 120574(120591)in the above general spherically symmetric spacetimeWe setthe geodesic 120574(120591) expressed in the above coordinates 119909120583 =(119905 119903 120579 120593) as 119909120583(120591) which are satisfied

1198892119909120583

1198891205912+ Γ120583]120590

119889119909]

119889120591

119889119909120590

119889120591= 0 (2)

Generally after the above equation is solved the geodesic120574(120591) is obtained However considering the symmetry ofspacetime (1) we could find in the following that there is amore simple way to obtain the geodesic 120574(120591) First we can

find that one component of the geodesic 120574(120591) can alwaysbe chosen as 120579(120591) = 1205872 which means that the geodesiccan always be chosen to lay in the equatorial plane of thespherically symmetric spacetimeTherefore the geodesic canbe simplified

119905 = 119905 (120591) 119903 = 119903 (120591) 120579 =120587

2 120593 = 120593 (120591) (3)

If we let 119880119886 equiv (120597120597120591)119886 be the tangent vector of geodesic 120574(120591)we could define

120581 = minus119892119886119887119880119886119880119887 = minus119892

119886119887(120597

120597120591)119886

(120597

120597120591)119887

(4)

Thus 120581 = 1 corresponds to the time-like geodesic while120581 = 0 is the null geodesic After inserting (120597120597120591)119886 =(119889119909120583119889120591)(120597120597119909120583)

119886 and the metric (1) we could obtain

minus120581 = 119892119886119887(120597

120597120591)119886

(120597

120597120591)119887

= minus119891 (119903) (119889119905

119889120591)2

+ ℎ (119903)minus1 (

119889119903

119889120591)2

+ 1199032 (119889120593

119889120591)2

(5)

where we have used 120579 = 1205872 Second note that (120597120597119905)119886 and(120597120597120593)119886 are two killing vectors in the spherically symmetricspacetime (1) Therefore there are two conserved quantitiesalong the geodesic 120574(120591)

119864 = minus 119892119886119887(120597

120597119905)119886

(120597

120597120591)119887

= 119891 (119903)119889119905

119889120591 (6)

119871 = 119892119886119887(120597

120597120593)119886

(120597

120597120591)119887

= 1199032119889120593

119889120591 (7)

where the physical meanings of 119864 and 119871 can be found indetail in [3] that is 119864 can be interpreted as the total energy(including gravitational potential energy) per unit rest massof a particle in the time-like case while 119871 can be interpretedas the angular momentum per unit rest mass of a particle Inaddition ℏ119864 and ℏ119871 can be interpreted as the total energy andangular momentum of a photon in the null case respectively

After inserting (6) and (7) into (5) we could obtain

(119889119903

119889120591)2

=ℎ (119903)

119891 (119903)1198642 minus ℎ (119903) (120581 +

1198712

1199032) (8)

Obviously the above equation contains only one function119903(120591) which could be solved in principle Hence after insert-ing the solved 119903(120591) into (6) and (7) the remaining compo-nents 119905(120591) and 120593(120591) of geodesic could be finally obtained

It should be pointed out that perihelion precession anddeflection of light are usually related to the orbit of geodesicthat is 119903(120593) Therefore it is convenient to change (8) as

(119889119903

119889120593)2

(119871

1199032)2

=ℎ (119903)

119891 (119903)1198642 minus ℎ (119903) (120581 +

1198712

1199032) (9)

where we have used (7) In addition it has been found thatthe coordinate 120583 equiv 1119903 is more convenient than 119903 to discuss

Advances in High Energy Physics 3

the perihelion precession and deflection of light Thus themaster equation investigated in our paper could be simplyobtained from (9) by changing 119903 into 120583

(119889120583

119889120593)2

=ℎ (120583)

119891 (120583)(119864

119871)2

minus ℎ (120583) (120581

1198712+ 1205832) (10)

3 Perihelion Precession and Deflectionof Light in the 4-Dimensional SphericallySymmetric Spacetime Special Case

Since functions ℎ(119903) and 119891(119903) are usually different for eachspherically symmetric spacetime thus the perihelion pre-cession and deflection of light may be different in differentspherically symmetric spacetime Therefore we can use thedifferences in perihelion precession and deflection of lightto extract the information in ℎ(119903) and 119891(119903) In this sectionwe will take two spherically symmetric solutions as examplesfor the application of master equation (10) to discuss thecorresponding perihelion precession and deflection of lightThe first solution is the Reissner-Nordstorm-AdS solutionwhich is a well-known solution in Einstein gravity with thecosmological constant and electric charge while the othersolution is the Clifton-Barrow solution in 119891(119877) gravity The119891(119877) gravity is a kind of higher derivative gravity theoryand the Clifton-Barrow solution can be considered as ageneralization of Schwarzschild solution in Einstein gravityFor this solution the advantages are that not only it is thespherically symmetric spacetime in119891(119877) gravity but also ℎ(119903)and 119891(119903) are different

31 Reissner-Nordstorm-AdS Solution For the Reissner-Nordstorm-AdS solution the two functions ℎ(119903) and119891(119903) are

119891 (119903) = ℎ (119903) = 1 minus2119872

119903+Λ

31199032 +

1198762

1199032 (11)

where Λ is the cosmological constant and 119876 is the electriccharge Therefore (10) becomes

(119889120583

119889120593)2

= (119864

119871)2

minus (1 minus 2119872120583 +Λ

31205832+ 12058321198762)(

120581

1198712+ 1205832)

(12)

For the perihelion precession one usually considers the time-like geodesic that is 120581 = 1 Therefore (12) for the time-likegeodesic can be

1198892120583

1198891205932+ (1 +

1198762

1198712)120583 =

119872

1198712+ 31205832119872+

Λ

312058331198712minus 211987621205833 (13)

Obviously compared with the case in Newtonrsquos gravity11988921205831198891205932 + 120583 = 1198721198712 the term 31205832119872 comes from thecorrection of general relativity while the last two terms arefrom the cosmological constant and electric charge and theabove equation can return to the well-known Schwarzschildcase whenΛ = 119876 = 0 Note that the analytical solution of (13)

is absent like in the Schwarzschild case However there is anapproximation solution of (13)

120583 (120593) =119872

1198712(1 + 119890 cos120593)

+31198723

1198714(1 + 119890120593 sin120593 + 1198902 (1

2minus1

6cos 2120593))

+Λ1198714

31198723(1 minus

3

2119890120593 sin120593) minus 1198721198762

1198714(1 +

1

2119890120593 sin120593)

minus211987621198723

1198716(1 +

3

2119890120593 sin120593 + 31198902 (1

2minus1

6cos 2120593))

(14)

in the following conditions

31198721205832 ≪ 120583Λ

312058331198712≪ 120583

1198762

1198712≪ 1 211987621205832 ≪ 1 (15)

where 120583(120593) = (1198721198712)(1 + 119890 cos120593) is the analytical ellipticalsolution which has already been found in Newtonrsquos gravityand 119890 is the orbital eccentricity which has been considered asa small constant Therefore (14) can be further reduced afterneglecting the high order terms

120583 (120593)

=119872

11987121 + 119890 [cos120593 + (3119872

2

1198712minusΛ1198716

21198724minus1198762

21198712)120593 sin120593]

(16)

where the conditions 311987221198712 ≪ 1 and Λ119871631198724 ≪ 1 havealso been assumed Note that the above equation could befurther simplified as

120583 (120593) =119872

1198712[1 + 119890 cos (120593 minus 120576120593)] (17)

where we have set

120576 = (31198722

1198712minusΛ1198716

21198724minus1198762

21198712) (18)

For the perihelion of orbit 119903(120593) it satisfies cos(120593 minus 120576120593) = 1and hence 120593 = 2120587 + 2120587120576 Therefore the precession angle ofperihelion is

Δ120593 = 2120587120576 = 2120587(31198722

1198712minusΛ1198716

21198724minus1198762

21198712) (19)

Note that here 119871 is the angular momentum per unit restmass of the particle along the time-like geodesic Obviouslyboth the cosmological constant and the electric charge canaffect the precession angle of perihelion When both thecosmological constant and electric charge disappear in (19)the result recovers the standard general relativity result withSchwarzschild solution Δ120593 = 612058711987221198712 asymp 6120587119872119886 where 119886is the semimajor axis of the ellipse and 119886 asymp 1198712119872 when theeccentricity 119890 is small

4 Advances in High Energy Physics

Next we will discuss the deflection of light in theReissner-Nordstorm-AdS spacetime In this case the corre-sponding geodesic is the null geodesic that is 120581 = 0 Similarto the procedure dealt with in the perihelion precession weneed to find out the approximation solution of the orbit ofdeflection of light The master equation is

(119889120583

119889120593)2

+ 1205832 =1198642

1198712+ 21198721205833 minus

Λ

3minus 11987621205834 (20)

Note that the cosmological constant term is just a constantterm like11986421198712Therefore the cosmological constant will notaffect the deflection angle of light In fact (20) can be furthersimplified as

1198892120583

1198891205932+ 120583 = 31198721205832 minus 211987621205833 (21)

which shows more clearly that the cosmological constantdoes not affect the deflection angle Note that the approxi-mation solution of (21) is

120583 (120593) =1

119897sin120593 + 119872

1198972(1 minus cos120593)2

minus21198762

1198973(minus

3

8120593 cos120593 + 1

32sin 3120593)

(22)

where 119897 is a constant 120583(120593) = (1119897) sin120593 is in fact a straightline expressed in polar coordinates (120583 120593) and we have usedthe condition120583(0) = 0Therefore the deflection angle of light120573 can be obtained from the equation

120583 (120587 + 120573) = 0 (23)

After using the approximation

sin (120587 + 120573) asymp minus120573 cos (120587 + 120573) asymp minus1 (24)

the angle is

120573 =4119872

119897minus31205871198762

41198972 (25)

where the first term is just the well-known deflection angleof light in Schwarzschild spacetime while the second term isfrom the effect of the electric charge

32 Clifton-Barrow Solution In this subsection we willinvestigate the perihelion precession and deflection of lightin the Clifton-Barrow solution in 119891(119877) gravity In a general119891(119877) theory the uniqueness theorem of the sphericallysymmetric space becomes invalid 119877 may be not zero evenfor vacuum solutions Considering the 4-dimensional actionin the following form

119878 =1

16120587119866(int

M

1198894119909radicminus det (119892)119877119889+1

+int120597M

1198893119909radicminus det (ℎ)2119861) (26)

where 119889 is a constant 119861 represents the correspondingboundary term for the 119891(119877) term 119892 is the 4-dimensionalmetric ℎ denotes the induced metric on the boundary thecorresponding field equation reads

(1 + 119889) 119877119889119877120583] minus

1

2119877119889+1119892

120583] minus (1 + 119889) nabla120583nabla]119877119889

+ 119892120583] (1 + 119889)r119877119889 = 0

(27)

It is easy to see that the Schwarzschild metric is a solutionfor the above equation since Schwarzschild metric has avanishing Ricci scalar119877 In addition the119891(119877) gravity permitsa nontrivial spherically symmetric solution other than theSchwarzschildmetric and the called Clifton-Barrow solutionalso solves the field equation [8]

1198891199042 = minus119891 (119903) 1198891199052 +

1198891199032

ℎ (119903)+ 1199032 (1198891205792 + sin21205791198891206012) (28)

where

119891 (119903) = 1199032119889((1+2119889)(1minus119889)) +119888

119903(1minus4119889)(1minus119889)

ℎ (119903) =(1 minus 119889)2

(1 minus 2119889 + 41198892) (1 minus 2119889 (1 + 119889))

times (1 +119888

119903(1minus2119889+41198892)(1minus119889)

)

(29)

Here 119888 is a constant which reduces to the Schwarzschildmassparameter 119888 = minus2119872 in Einstein gravity

Therefore one can directly insert (29) into the masterequation (10) to discuss its perihelion precession and deflec-tion of light Note that after inserting the two functions aboveinto (10) it will be found that the equations are difficult tobe solved directly In order to explicitly show the effects onperihelion precession and deflection of light from the 119891(119877)gravity here we just consider the case that the constant 119889 isvery small since 119889 = 0 is just the Einstein gravity In this casethe master equation (10) turns out to be

1198892120583

1198891205932+ 120583 =

120581119872

1198712+ 31198721205832

+ [120581119872

1198712+1198642

11987121

120583minus 2120583 + 51198721205832

+(120581119872

1198712+ 31198721205832) ln 1

120583] 119889

(30)

where we have kept the linear terms of 119889 and neglected thehigher order terms

For the perihelion precession that is 120581 = 1 the aboveequation is still a little complicated which can be furthersimplified in the larger radius case that is 120583 = 1119903 sim 0 Thusthe above equation is

1198892120583

1198891205932+ 120583 =

120581119872

1198712+ 31198721205832 +

1198642

1198712119889

120583 (31)

Advances in High Energy Physics 5

and the approximate solution can be easily obtained

120583 (120593) =119872

1198712(1 + 119890 cos120593)

+31198723

1198714(1 + 119890120593 sin120593 + 1198902 (1

2minus1

6cos 2120593))

+1198642

119872(1 minus

1

2119890120593 sin120593)119889

(32)

which can be further simplified

120583 (120593) =119872

11987121 + 119890 [cos120593 + (3119872

2

1198712minus11986421198712

21198722119889)120593 sin120593]

(33)

Therefore the angle of perihelion precession is

Δ120593 = 2120587120576 = 2120587(31198722

1198712minus11986421198712

21198722119889) (34)

From the above simple discussion we can easily find thatthere can be indeed an effect from the 119891(119877) gravity on theperihelion precession

For the deflection of light that is 120581 = 0 the masterequation becomes

1198892120583

1198891205932+ 120583 = 31198721205832 + (

1198642

11987121

120583minus 2120583 + 51198721205832 + 31198721205832 ln 1

120583)119889

(35)

which can also be simplified in the larger radius case that is120583 = 1119903 sim 0 as

1198892120583

1198891205932+ 120583 = 31198721205832 +

1198642

1198712119889

120583 (36)

If we solve this equation in the exact same way used beforewe obtain

120583 (120593) =1

119897sin120593

+119872

1198972(1 minus cos120593)2 + 1198642119897

1198712(minus120593 cos120593 + sin120593 ln sin120593) 119889

(37)

which can be further simplified near 120593 = 120587

120583 (120593) =1

119897sin120593 + 119872

1198972(1 minus cos120593)2 + 1198642119897

1198712(minus120593 cos120593) 119889

(38)

Therefore the angle can be obtained

120573 =4119872

119897+1198642119897 (4119872 + 120587119897)

1198712119889 (39)

4 Conclusion and Discussion

In this paper the perihelion precession and deflection of lighthave been investigated in the 4-dimensional general spher-ically symmetric spacetime Since the perihelion precessionand deflection of light can be treated respectively as thetime-like and null geodesic in spacetime themaster equationof perihelion precession and deflection of light is obtainedMoreover the Reissner-Nordstorm-AdS solution in Einsteingravity and Clifton-Barrow solution in 119891(119877) gravity aretaken as the two examples for the application of this masterequation and the effects from the cosmological constantelectric charge and119891(119877) gravity on the perihelion precessionand deflection of light are investigated We find that theelectric charge can affect both the perihelion precession andthe deflection of light while the cosmological constant canonly affect the perihelion precession which are consistentwith the results in the previous works In addition afterconsidering the case with small 119889 we easily find that the119891(119877)gravity can also affect both the perihelion precession and thedeflection of light Several discussions related to our resultsare in order

(i) During investigating the effects from the electriccharge cosmological constant and 119891(119877) gravity on the peri-helion precession and deflection of light we have assumed theapproximate conditions like (15) and small 119889 Therefore theresults under other approximate conditions are interestingopen issues

(ii) The perihelion precession and deflection of lightcan contain the information of the spacetime that is thecosmological constant electric charge and small 119889 Partic-ularly if we just consider them in the solar system sincethe angle of perihelion precession and deflection of lightin the solar system can be detected by the experimentsthus the information like cosmological constant and electriccharge can be extracted from the angles by the experimentsTherefore we may also extract the information of darkmatter such as its distribution of energy density 120588(119903) throughdetecting the perihelion precession and deflection of lightby the experiments if we can first find the static solutioncontaining the dark matter which will be further studied inthe future work

(iii) A simple constraint on the cosmological constantAlthough the cosmological constant does not affect thedeflection of light it can affect the perihelion precessionFrom our results if the cosmological constant indeed existsin our universe it can also give effects in the solar systemTherefore we can give a simple constraint on the cosmologi-cal constant from the well-known phenomenonmdashperihelionprecession of Mercury in the solar system The experimentdata of anomalous precession angle is (4256 plusmn 094)10158401015840 percentury [26] Therefore the theoretical result should be in(4256 plusmn 094)10158401015840 Considering the simple case 119876 = 0 in (19)and recovering the constants 119866 and 119888 (19) becomes

Δ120593 = 2120587(311987221198662

11987121198882minus

Λ1198716

211987241198663) (40)

6 Advances in High Energy Physics

Since the theoretical result of the first term is 430310158401015840 percentury thus the possible contribution from cosmologicalconstant 120587Λ119871611987241198663 must be less than 14110158401015840 per centurywhere we just consider the positive cosmological constantwhich can accelerate our universe After inserting the con-stants the mass of the sun 119872 = 1989 times 1030 kg and theangular momentum of unit mass of Mercury 119871 = 272 times

1016m2sminus1 we can constrain the cosmological constant Λ lt

589times10minus11 kgm3 which is consistent with observation datain our universe Λ = 19 times 10minus25 kgm3

(iv) The subtlety in the deflection of light in the solarsystem Note that there is a gap between the experiment data(161 plusmn 04)10158401015840 and theoretical result 17510158401015840 in Schwarzschildspacetime [24 25] From the result in (25) one may considerthat this difference may be from the sunrsquos possible electriccharge effect However we will give a simple proof in thefollowing that it is not true After recovering the constantsthe angle of starlight deflection is

120573 =4119872119866

1198882119897minus31205871198762119866

411989721198884 (41)

where we have used the Gauss unit of119876 that is kg12m32sminus1fromwhichwe canfind that the charge could indeedmake theangle smaller and hence make the gap smaller If the chargecould make the angle smaller 110158401015840 that is (31205871198762119866411989721198884) times180120587times60times60 = 1 we can obtain the sunwith possible charge119876 = 11 times 1028 kg12m32sminus1 However note that the sunrsquoscharge can also affect the perihelion precession of Mercuryin (19) which can be recovered

Δ120593 = 2120587(311987221198662

11987121198882minus

1198762119866

211987121198882) (42)

where we have neglected the effect of cosmological constantsince the observation data in our universe Λ = 19 times10minus25 kgm3 is very smallTherefore after the simple calcula-tion we can obtain that the sunrsquos charge effect on the angle ofperihelion precession ofMercury will be120587119876211986611987121198882 = 0038which is larger than the first term 61205871198722119866211987121198882 = 497times10minus7This is obviously opposite to the experiment observationThus the gap between the experiment data and theoreticalresult for the starlight deflection in solar system could notcompletely come from the sunrsquos electric charge and a moresuitable explanation of this difference will still be an openissue

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (NSFC) under Grant no 11105004 andShanghai Key Laboratory of Particle Physics and Cosmologyunder Grant no 11DZ2260700 and partially by Grants from

NSFC (nos 10821504 10975168 and 11035008) and theMinistry of Science andTechnology ofChina underGrant no2010CB833004 Professor Hongsheng Zhang is supported bythe Program for Professor of Special Appointment (EasternScholar) at Shanghai Institutions of Higher Learning andNational Natural Science Foundation of China under Grantnos 11075106 and 11275128

References

[1] S Weinberg Gravitation and Cosmology Principles and Appli-cations of the General Theory of Relativity John Wiley amp SonsNew York NY USA 1972

[2] C W Misner K S Thorne and J A Wheeler Gravitation WH Freeman and Co San Francisco Calif USA 1973

[3] R M Wald General Relativity University of Chicago PressChicago Ill USA 1984

[4] S G Turyshev U E Israelsson M Shao et al ldquoSpace-basedresearch in fundamental physics and quantum technologiesrdquoInternational Journal of Modern Physics D vol 16 no 12 pp1879ndash1925 2008

[5] S G Turyshev ldquoExperimental tests of general relativityrdquoAnnualReview of Nuclear and Particle Science vol 58 pp 207ndash2482008

[6] T P Sotiriou and V Faraoni ldquo119891(119877) theories of gravityrdquo Reviewsof Modern Physics vol 82 no 1 pp 451ndash497 2010

[7] R G Cai L M Cao Y P Hu and N Ohta ldquoGeneralizedMisner-Sharp energy in f (R) gravityrdquo Physical Review D vol80 Article ID 104016 2009

[8] T Clifton and J D Barrow ldquoThe power of general relativityrdquoPhysical Review D Third Series vol 72 no 10 Article ID103005 2005 Erratum in Physical Review D Third Series vol90 Article ID 029902 2014

[9] H Zhang Y Hu and X Z Li ldquoMisner-Sharp mass in n-dimensional f(R) gravityrdquo Physical Review D vol 90 Article ID024062 2014

[10] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 1989

[11] P J Peebles and B Ratra ldquoThe cosmological constant and darkenergyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash6062003

[12] A Albrecht and C Skordis ldquoPhenomenology of a realisticaccelerating universe using only Planck-scale physicsrdquo PhysicalReview Letters vol 84 p 2076 2000

[13] M Li X D Li S Wang and Y Wang ldquoDark energyrdquo Commu-nications in Theoretical Physics vol 56 no 3 p 525 2011

[14] V Kagramanova J Kunz and C Lammerzahl ldquoSolar systemeffects in Schwarzschild-de Sitter space-timerdquo Physics Letters Bvol 634 pp 465ndash470 2006

[15] P Jetzer and M Sereno ldquoTwo-body problem with the cosmo-logical constant and observational constraintsrdquo Physical ReviewD vol 73 no 4 Article ID 044015 6 pages 2006

[16] L Iorio ldquoSolar System motions and the cosmological constanta new approachrdquo Advances in Astronomy vol 2012 Article ID268647 9 pages 2012

[17] L Iorio ldquoConstraining the angular momentum of the sun withplanetary orbital motions and general relativityrdquo Solar Physicsvol 281 no 2 pp 815ndash826 2012

[18] L Iorio ldquoConstraining the electric charges of some astro-nomical bodies in Reissner-Nordstrom spacetimes and generic

Advances in High Energy Physics 7

119903minus2-type power-law potentials from orbital motionsrdquo GeneralRelativity and Gravitation vol 44 no 7 pp 1753ndash1767 2012

[19] M Sereno and P Jetzer ldquoDark matter versus modificationsof the gravitational inverse-square law results from planetarymotion in the Solar systemrdquo Monthly Notices of the RoyalAstronomical Society vol 371 no 2 pp 626ndash632 2006

[20] M Sereno ldquoWeak field limit of Reissner-Nordstrom black holelensingrdquo Physical Review D vol 69 no 2 Article ID 023002 4pages 2004

[21] M Sereno and P Jetzer ldquoSolar and stellar system tests of thecosmological constantrdquo Physical Review D vol 73 Article ID063004 2006

[22] M Sereno ldquoInfluence of the cosmological constant on gravi-tational lensing in small systemsrdquo Physical Review D vol 77Article ID 043004 2008

[23] M Sereno ldquoRole of Λ in the cosmological lens equationrdquoPhysical Review Letters vol 102 no 2 Article ID 021301 2009

[24] F W Dyson A S Eddington and C Davidson ldquoA determi-nation of the deflection of light by the sunrsquos gravitational fieldfrom observations made at the total eclipse of May 29 1919rdquoPhilosophical Transactions of the Royal Society A vol 220 no571ndash581 pp 291ndash333 1920

[25] D Kennefick ldquoTesting relativity from the 1919 eclipsemdasha ques-tion of biasrdquo Physics Today vol 62 no 3 pp 37ndash42 2009

[26] G M Clemence ldquoThe relativity effect in planetary motionsrdquoReviews of Modern Physics vol 19 p 361 1947

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

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AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 3

the perihelion precession and deflection of light Thus themaster equation investigated in our paper could be simplyobtained from (9) by changing 119903 into 120583

(119889120583

119889120593)2

=ℎ (120583)

119891 (120583)(119864

119871)2

minus ℎ (120583) (120581

1198712+ 1205832) (10)

3 Perihelion Precession and Deflectionof Light in the 4-Dimensional SphericallySymmetric Spacetime Special Case

Since functions ℎ(119903) and 119891(119903) are usually different for eachspherically symmetric spacetime thus the perihelion pre-cession and deflection of light may be different in differentspherically symmetric spacetime Therefore we can use thedifferences in perihelion precession and deflection of lightto extract the information in ℎ(119903) and 119891(119903) In this sectionwe will take two spherically symmetric solutions as examplesfor the application of master equation (10) to discuss thecorresponding perihelion precession and deflection of lightThe first solution is the Reissner-Nordstorm-AdS solutionwhich is a well-known solution in Einstein gravity with thecosmological constant and electric charge while the othersolution is the Clifton-Barrow solution in 119891(119877) gravity The119891(119877) gravity is a kind of higher derivative gravity theoryand the Clifton-Barrow solution can be considered as ageneralization of Schwarzschild solution in Einstein gravityFor this solution the advantages are that not only it is thespherically symmetric spacetime in119891(119877) gravity but also ℎ(119903)and 119891(119903) are different

31 Reissner-Nordstorm-AdS Solution For the Reissner-Nordstorm-AdS solution the two functions ℎ(119903) and119891(119903) are

119891 (119903) = ℎ (119903) = 1 minus2119872

119903+Λ

31199032 +

1198762

1199032 (11)

where Λ is the cosmological constant and 119876 is the electriccharge Therefore (10) becomes

(119889120583

119889120593)2

= (119864

119871)2

minus (1 minus 2119872120583 +Λ

31205832+ 12058321198762)(

120581

1198712+ 1205832)

(12)

For the perihelion precession one usually considers the time-like geodesic that is 120581 = 1 Therefore (12) for the time-likegeodesic can be

1198892120583

1198891205932+ (1 +

1198762

1198712)120583 =

119872

1198712+ 31205832119872+

Λ

312058331198712minus 211987621205833 (13)

Obviously compared with the case in Newtonrsquos gravity11988921205831198891205932 + 120583 = 1198721198712 the term 31205832119872 comes from thecorrection of general relativity while the last two terms arefrom the cosmological constant and electric charge and theabove equation can return to the well-known Schwarzschildcase whenΛ = 119876 = 0 Note that the analytical solution of (13)

is absent like in the Schwarzschild case However there is anapproximation solution of (13)

120583 (120593) =119872

1198712(1 + 119890 cos120593)

+31198723

1198714(1 + 119890120593 sin120593 + 1198902 (1

2minus1

6cos 2120593))

+Λ1198714

31198723(1 minus

3

2119890120593 sin120593) minus 1198721198762

1198714(1 +

1

2119890120593 sin120593)

minus211987621198723

1198716(1 +

3

2119890120593 sin120593 + 31198902 (1

2minus1

6cos 2120593))

(14)

in the following conditions

31198721205832 ≪ 120583Λ

312058331198712≪ 120583

1198762

1198712≪ 1 211987621205832 ≪ 1 (15)

where 120583(120593) = (1198721198712)(1 + 119890 cos120593) is the analytical ellipticalsolution which has already been found in Newtonrsquos gravityand 119890 is the orbital eccentricity which has been considered asa small constant Therefore (14) can be further reduced afterneglecting the high order terms

120583 (120593)

=119872

11987121 + 119890 [cos120593 + (3119872

2

1198712minusΛ1198716

21198724minus1198762

21198712)120593 sin120593]

(16)

where the conditions 311987221198712 ≪ 1 and Λ119871631198724 ≪ 1 havealso been assumed Note that the above equation could befurther simplified as

120583 (120593) =119872

1198712[1 + 119890 cos (120593 minus 120576120593)] (17)

where we have set

120576 = (31198722

1198712minusΛ1198716

21198724minus1198762

21198712) (18)

For the perihelion of orbit 119903(120593) it satisfies cos(120593 minus 120576120593) = 1and hence 120593 = 2120587 + 2120587120576 Therefore the precession angle ofperihelion is

Δ120593 = 2120587120576 = 2120587(31198722

1198712minusΛ1198716

21198724minus1198762

21198712) (19)

Note that here 119871 is the angular momentum per unit restmass of the particle along the time-like geodesic Obviouslyboth the cosmological constant and the electric charge canaffect the precession angle of perihelion When both thecosmological constant and electric charge disappear in (19)the result recovers the standard general relativity result withSchwarzschild solution Δ120593 = 612058711987221198712 asymp 6120587119872119886 where 119886is the semimajor axis of the ellipse and 119886 asymp 1198712119872 when theeccentricity 119890 is small

4 Advances in High Energy Physics

Next we will discuss the deflection of light in theReissner-Nordstorm-AdS spacetime In this case the corre-sponding geodesic is the null geodesic that is 120581 = 0 Similarto the procedure dealt with in the perihelion precession weneed to find out the approximation solution of the orbit ofdeflection of light The master equation is

(119889120583

119889120593)2

+ 1205832 =1198642

1198712+ 21198721205833 minus

Λ

3minus 11987621205834 (20)

Note that the cosmological constant term is just a constantterm like11986421198712Therefore the cosmological constant will notaffect the deflection angle of light In fact (20) can be furthersimplified as

1198892120583

1198891205932+ 120583 = 31198721205832 minus 211987621205833 (21)

which shows more clearly that the cosmological constantdoes not affect the deflection angle Note that the approxi-mation solution of (21) is

120583 (120593) =1

119897sin120593 + 119872

1198972(1 minus cos120593)2

minus21198762

1198973(minus

3

8120593 cos120593 + 1

32sin 3120593)

(22)

where 119897 is a constant 120583(120593) = (1119897) sin120593 is in fact a straightline expressed in polar coordinates (120583 120593) and we have usedthe condition120583(0) = 0Therefore the deflection angle of light120573 can be obtained from the equation

120583 (120587 + 120573) = 0 (23)

After using the approximation

sin (120587 + 120573) asymp minus120573 cos (120587 + 120573) asymp minus1 (24)

the angle is

120573 =4119872

119897minus31205871198762

41198972 (25)

where the first term is just the well-known deflection angleof light in Schwarzschild spacetime while the second term isfrom the effect of the electric charge

32 Clifton-Barrow Solution In this subsection we willinvestigate the perihelion precession and deflection of lightin the Clifton-Barrow solution in 119891(119877) gravity In a general119891(119877) theory the uniqueness theorem of the sphericallysymmetric space becomes invalid 119877 may be not zero evenfor vacuum solutions Considering the 4-dimensional actionin the following form

119878 =1

16120587119866(int

M

1198894119909radicminus det (119892)119877119889+1

+int120597M

1198893119909radicminus det (ℎ)2119861) (26)

where 119889 is a constant 119861 represents the correspondingboundary term for the 119891(119877) term 119892 is the 4-dimensionalmetric ℎ denotes the induced metric on the boundary thecorresponding field equation reads

(1 + 119889) 119877119889119877120583] minus

1

2119877119889+1119892

120583] minus (1 + 119889) nabla120583nabla]119877119889

+ 119892120583] (1 + 119889)r119877119889 = 0

(27)

It is easy to see that the Schwarzschild metric is a solutionfor the above equation since Schwarzschild metric has avanishing Ricci scalar119877 In addition the119891(119877) gravity permitsa nontrivial spherically symmetric solution other than theSchwarzschildmetric and the called Clifton-Barrow solutionalso solves the field equation [8]

1198891199042 = minus119891 (119903) 1198891199052 +

1198891199032

ℎ (119903)+ 1199032 (1198891205792 + sin21205791198891206012) (28)

where

119891 (119903) = 1199032119889((1+2119889)(1minus119889)) +119888

119903(1minus4119889)(1minus119889)

ℎ (119903) =(1 minus 119889)2

(1 minus 2119889 + 41198892) (1 minus 2119889 (1 + 119889))

times (1 +119888

119903(1minus2119889+41198892)(1minus119889)

)

(29)

Here 119888 is a constant which reduces to the Schwarzschildmassparameter 119888 = minus2119872 in Einstein gravity

Therefore one can directly insert (29) into the masterequation (10) to discuss its perihelion precession and deflec-tion of light Note that after inserting the two functions aboveinto (10) it will be found that the equations are difficult tobe solved directly In order to explicitly show the effects onperihelion precession and deflection of light from the 119891(119877)gravity here we just consider the case that the constant 119889 isvery small since 119889 = 0 is just the Einstein gravity In this casethe master equation (10) turns out to be

1198892120583

1198891205932+ 120583 =

120581119872

1198712+ 31198721205832

+ [120581119872

1198712+1198642

11987121

120583minus 2120583 + 51198721205832

+(120581119872

1198712+ 31198721205832) ln 1

120583] 119889

(30)

where we have kept the linear terms of 119889 and neglected thehigher order terms

For the perihelion precession that is 120581 = 1 the aboveequation is still a little complicated which can be furthersimplified in the larger radius case that is 120583 = 1119903 sim 0 Thusthe above equation is

1198892120583

1198891205932+ 120583 =

120581119872

1198712+ 31198721205832 +

1198642

1198712119889

120583 (31)

Advances in High Energy Physics 5

and the approximate solution can be easily obtained

120583 (120593) =119872

1198712(1 + 119890 cos120593)

+31198723

1198714(1 + 119890120593 sin120593 + 1198902 (1

2minus1

6cos 2120593))

+1198642

119872(1 minus

1

2119890120593 sin120593)119889

(32)

which can be further simplified

120583 (120593) =119872

11987121 + 119890 [cos120593 + (3119872

2

1198712minus11986421198712

21198722119889)120593 sin120593]

(33)

Therefore the angle of perihelion precession is

Δ120593 = 2120587120576 = 2120587(31198722

1198712minus11986421198712

21198722119889) (34)

From the above simple discussion we can easily find thatthere can be indeed an effect from the 119891(119877) gravity on theperihelion precession

For the deflection of light that is 120581 = 0 the masterequation becomes

1198892120583

1198891205932+ 120583 = 31198721205832 + (

1198642

11987121

120583minus 2120583 + 51198721205832 + 31198721205832 ln 1

120583)119889

(35)

which can also be simplified in the larger radius case that is120583 = 1119903 sim 0 as

1198892120583

1198891205932+ 120583 = 31198721205832 +

1198642

1198712119889

120583 (36)

If we solve this equation in the exact same way used beforewe obtain

120583 (120593) =1

119897sin120593

+119872

1198972(1 minus cos120593)2 + 1198642119897

1198712(minus120593 cos120593 + sin120593 ln sin120593) 119889

(37)

which can be further simplified near 120593 = 120587

120583 (120593) =1

119897sin120593 + 119872

1198972(1 minus cos120593)2 + 1198642119897

1198712(minus120593 cos120593) 119889

(38)

Therefore the angle can be obtained

120573 =4119872

119897+1198642119897 (4119872 + 120587119897)

1198712119889 (39)

4 Conclusion and Discussion

In this paper the perihelion precession and deflection of lighthave been investigated in the 4-dimensional general spher-ically symmetric spacetime Since the perihelion precessionand deflection of light can be treated respectively as thetime-like and null geodesic in spacetime themaster equationof perihelion precession and deflection of light is obtainedMoreover the Reissner-Nordstorm-AdS solution in Einsteingravity and Clifton-Barrow solution in 119891(119877) gravity aretaken as the two examples for the application of this masterequation and the effects from the cosmological constantelectric charge and119891(119877) gravity on the perihelion precessionand deflection of light are investigated We find that theelectric charge can affect both the perihelion precession andthe deflection of light while the cosmological constant canonly affect the perihelion precession which are consistentwith the results in the previous works In addition afterconsidering the case with small 119889 we easily find that the119891(119877)gravity can also affect both the perihelion precession and thedeflection of light Several discussions related to our resultsare in order

(i) During investigating the effects from the electriccharge cosmological constant and 119891(119877) gravity on the peri-helion precession and deflection of light we have assumed theapproximate conditions like (15) and small 119889 Therefore theresults under other approximate conditions are interestingopen issues

(ii) The perihelion precession and deflection of lightcan contain the information of the spacetime that is thecosmological constant electric charge and small 119889 Partic-ularly if we just consider them in the solar system sincethe angle of perihelion precession and deflection of lightin the solar system can be detected by the experimentsthus the information like cosmological constant and electriccharge can be extracted from the angles by the experimentsTherefore we may also extract the information of darkmatter such as its distribution of energy density 120588(119903) throughdetecting the perihelion precession and deflection of lightby the experiments if we can first find the static solutioncontaining the dark matter which will be further studied inthe future work

(iii) A simple constraint on the cosmological constantAlthough the cosmological constant does not affect thedeflection of light it can affect the perihelion precessionFrom our results if the cosmological constant indeed existsin our universe it can also give effects in the solar systemTherefore we can give a simple constraint on the cosmologi-cal constant from the well-known phenomenonmdashperihelionprecession of Mercury in the solar system The experimentdata of anomalous precession angle is (4256 plusmn 094)10158401015840 percentury [26] Therefore the theoretical result should be in(4256 plusmn 094)10158401015840 Considering the simple case 119876 = 0 in (19)and recovering the constants 119866 and 119888 (19) becomes

Δ120593 = 2120587(311987221198662

11987121198882minus

Λ1198716

211987241198663) (40)

6 Advances in High Energy Physics

Since the theoretical result of the first term is 430310158401015840 percentury thus the possible contribution from cosmologicalconstant 120587Λ119871611987241198663 must be less than 14110158401015840 per centurywhere we just consider the positive cosmological constantwhich can accelerate our universe After inserting the con-stants the mass of the sun 119872 = 1989 times 1030 kg and theangular momentum of unit mass of Mercury 119871 = 272 times

1016m2sminus1 we can constrain the cosmological constant Λ lt

589times10minus11 kgm3 which is consistent with observation datain our universe Λ = 19 times 10minus25 kgm3

(iv) The subtlety in the deflection of light in the solarsystem Note that there is a gap between the experiment data(161 plusmn 04)10158401015840 and theoretical result 17510158401015840 in Schwarzschildspacetime [24 25] From the result in (25) one may considerthat this difference may be from the sunrsquos possible electriccharge effect However we will give a simple proof in thefollowing that it is not true After recovering the constantsthe angle of starlight deflection is

120573 =4119872119866

1198882119897minus31205871198762119866

411989721198884 (41)

where we have used the Gauss unit of119876 that is kg12m32sminus1fromwhichwe canfind that the charge could indeedmake theangle smaller and hence make the gap smaller If the chargecould make the angle smaller 110158401015840 that is (31205871198762119866411989721198884) times180120587times60times60 = 1 we can obtain the sunwith possible charge119876 = 11 times 1028 kg12m32sminus1 However note that the sunrsquoscharge can also affect the perihelion precession of Mercuryin (19) which can be recovered

Δ120593 = 2120587(311987221198662

11987121198882minus

1198762119866

211987121198882) (42)

where we have neglected the effect of cosmological constantsince the observation data in our universe Λ = 19 times10minus25 kgm3 is very smallTherefore after the simple calcula-tion we can obtain that the sunrsquos charge effect on the angle ofperihelion precession ofMercury will be120587119876211986611987121198882 = 0038which is larger than the first term 61205871198722119866211987121198882 = 497times10minus7This is obviously opposite to the experiment observationThus the gap between the experiment data and theoreticalresult for the starlight deflection in solar system could notcompletely come from the sunrsquos electric charge and a moresuitable explanation of this difference will still be an openissue

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (NSFC) under Grant no 11105004 andShanghai Key Laboratory of Particle Physics and Cosmologyunder Grant no 11DZ2260700 and partially by Grants from

NSFC (nos 10821504 10975168 and 11035008) and theMinistry of Science andTechnology ofChina underGrant no2010CB833004 Professor Hongsheng Zhang is supported bythe Program for Professor of Special Appointment (EasternScholar) at Shanghai Institutions of Higher Learning andNational Natural Science Foundation of China under Grantnos 11075106 and 11275128

References

[1] S Weinberg Gravitation and Cosmology Principles and Appli-cations of the General Theory of Relativity John Wiley amp SonsNew York NY USA 1972

[2] C W Misner K S Thorne and J A Wheeler Gravitation WH Freeman and Co San Francisco Calif USA 1973

[3] R M Wald General Relativity University of Chicago PressChicago Ill USA 1984

[4] S G Turyshev U E Israelsson M Shao et al ldquoSpace-basedresearch in fundamental physics and quantum technologiesrdquoInternational Journal of Modern Physics D vol 16 no 12 pp1879ndash1925 2008

[5] S G Turyshev ldquoExperimental tests of general relativityrdquoAnnualReview of Nuclear and Particle Science vol 58 pp 207ndash2482008

[6] T P Sotiriou and V Faraoni ldquo119891(119877) theories of gravityrdquo Reviewsof Modern Physics vol 82 no 1 pp 451ndash497 2010

[7] R G Cai L M Cao Y P Hu and N Ohta ldquoGeneralizedMisner-Sharp energy in f (R) gravityrdquo Physical Review D vol80 Article ID 104016 2009

[8] T Clifton and J D Barrow ldquoThe power of general relativityrdquoPhysical Review D Third Series vol 72 no 10 Article ID103005 2005 Erratum in Physical Review D Third Series vol90 Article ID 029902 2014

[9] H Zhang Y Hu and X Z Li ldquoMisner-Sharp mass in n-dimensional f(R) gravityrdquo Physical Review D vol 90 Article ID024062 2014

[10] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 1989

[11] P J Peebles and B Ratra ldquoThe cosmological constant and darkenergyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash6062003

[12] A Albrecht and C Skordis ldquoPhenomenology of a realisticaccelerating universe using only Planck-scale physicsrdquo PhysicalReview Letters vol 84 p 2076 2000

[13] M Li X D Li S Wang and Y Wang ldquoDark energyrdquo Commu-nications in Theoretical Physics vol 56 no 3 p 525 2011

[14] V Kagramanova J Kunz and C Lammerzahl ldquoSolar systemeffects in Schwarzschild-de Sitter space-timerdquo Physics Letters Bvol 634 pp 465ndash470 2006

[15] P Jetzer and M Sereno ldquoTwo-body problem with the cosmo-logical constant and observational constraintsrdquo Physical ReviewD vol 73 no 4 Article ID 044015 6 pages 2006

[16] L Iorio ldquoSolar System motions and the cosmological constanta new approachrdquo Advances in Astronomy vol 2012 Article ID268647 9 pages 2012

[17] L Iorio ldquoConstraining the angular momentum of the sun withplanetary orbital motions and general relativityrdquo Solar Physicsvol 281 no 2 pp 815ndash826 2012

[18] L Iorio ldquoConstraining the electric charges of some astro-nomical bodies in Reissner-Nordstrom spacetimes and generic

Advances in High Energy Physics 7

119903minus2-type power-law potentials from orbital motionsrdquo GeneralRelativity and Gravitation vol 44 no 7 pp 1753ndash1767 2012

[19] M Sereno and P Jetzer ldquoDark matter versus modificationsof the gravitational inverse-square law results from planetarymotion in the Solar systemrdquo Monthly Notices of the RoyalAstronomical Society vol 371 no 2 pp 626ndash632 2006

[20] M Sereno ldquoWeak field limit of Reissner-Nordstrom black holelensingrdquo Physical Review D vol 69 no 2 Article ID 023002 4pages 2004

[21] M Sereno and P Jetzer ldquoSolar and stellar system tests of thecosmological constantrdquo Physical Review D vol 73 Article ID063004 2006

[22] M Sereno ldquoInfluence of the cosmological constant on gravi-tational lensing in small systemsrdquo Physical Review D vol 77Article ID 043004 2008

[23] M Sereno ldquoRole of Λ in the cosmological lens equationrdquoPhysical Review Letters vol 102 no 2 Article ID 021301 2009

[24] F W Dyson A S Eddington and C Davidson ldquoA determi-nation of the deflection of light by the sunrsquos gravitational fieldfrom observations made at the total eclipse of May 29 1919rdquoPhilosophical Transactions of the Royal Society A vol 220 no571ndash581 pp 291ndash333 1920

[25] D Kennefick ldquoTesting relativity from the 1919 eclipsemdasha ques-tion of biasrdquo Physics Today vol 62 no 3 pp 37ndash42 2009

[26] G M Clemence ldquoThe relativity effect in planetary motionsrdquoReviews of Modern Physics vol 19 p 361 1947

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

4 Advances in High Energy Physics

Next we will discuss the deflection of light in theReissner-Nordstorm-AdS spacetime In this case the corre-sponding geodesic is the null geodesic that is 120581 = 0 Similarto the procedure dealt with in the perihelion precession weneed to find out the approximation solution of the orbit ofdeflection of light The master equation is

(119889120583

119889120593)2

+ 1205832 =1198642

1198712+ 21198721205833 minus

Λ

3minus 11987621205834 (20)

Note that the cosmological constant term is just a constantterm like11986421198712Therefore the cosmological constant will notaffect the deflection angle of light In fact (20) can be furthersimplified as

1198892120583

1198891205932+ 120583 = 31198721205832 minus 211987621205833 (21)

which shows more clearly that the cosmological constantdoes not affect the deflection angle Note that the approxi-mation solution of (21) is

120583 (120593) =1

119897sin120593 + 119872

1198972(1 minus cos120593)2

minus21198762

1198973(minus

3

8120593 cos120593 + 1

32sin 3120593)

(22)

where 119897 is a constant 120583(120593) = (1119897) sin120593 is in fact a straightline expressed in polar coordinates (120583 120593) and we have usedthe condition120583(0) = 0Therefore the deflection angle of light120573 can be obtained from the equation

120583 (120587 + 120573) = 0 (23)

After using the approximation

sin (120587 + 120573) asymp minus120573 cos (120587 + 120573) asymp minus1 (24)

the angle is

120573 =4119872

119897minus31205871198762

41198972 (25)

where the first term is just the well-known deflection angleof light in Schwarzschild spacetime while the second term isfrom the effect of the electric charge

32 Clifton-Barrow Solution In this subsection we willinvestigate the perihelion precession and deflection of lightin the Clifton-Barrow solution in 119891(119877) gravity In a general119891(119877) theory the uniqueness theorem of the sphericallysymmetric space becomes invalid 119877 may be not zero evenfor vacuum solutions Considering the 4-dimensional actionin the following form

119878 =1

16120587119866(int

M

1198894119909radicminus det (119892)119877119889+1

+int120597M

1198893119909radicminus det (ℎ)2119861) (26)

where 119889 is a constant 119861 represents the correspondingboundary term for the 119891(119877) term 119892 is the 4-dimensionalmetric ℎ denotes the induced metric on the boundary thecorresponding field equation reads

(1 + 119889) 119877119889119877120583] minus

1

2119877119889+1119892

120583] minus (1 + 119889) nabla120583nabla]119877119889

+ 119892120583] (1 + 119889)r119877119889 = 0

(27)

It is easy to see that the Schwarzschild metric is a solutionfor the above equation since Schwarzschild metric has avanishing Ricci scalar119877 In addition the119891(119877) gravity permitsa nontrivial spherically symmetric solution other than theSchwarzschildmetric and the called Clifton-Barrow solutionalso solves the field equation [8]

1198891199042 = minus119891 (119903) 1198891199052 +

1198891199032

ℎ (119903)+ 1199032 (1198891205792 + sin21205791198891206012) (28)

where

119891 (119903) = 1199032119889((1+2119889)(1minus119889)) +119888

119903(1minus4119889)(1minus119889)

ℎ (119903) =(1 minus 119889)2

(1 minus 2119889 + 41198892) (1 minus 2119889 (1 + 119889))

times (1 +119888

119903(1minus2119889+41198892)(1minus119889)

)

(29)

Here 119888 is a constant which reduces to the Schwarzschildmassparameter 119888 = minus2119872 in Einstein gravity

Therefore one can directly insert (29) into the masterequation (10) to discuss its perihelion precession and deflec-tion of light Note that after inserting the two functions aboveinto (10) it will be found that the equations are difficult tobe solved directly In order to explicitly show the effects onperihelion precession and deflection of light from the 119891(119877)gravity here we just consider the case that the constant 119889 isvery small since 119889 = 0 is just the Einstein gravity In this casethe master equation (10) turns out to be

1198892120583

1198891205932+ 120583 =

120581119872

1198712+ 31198721205832

+ [120581119872

1198712+1198642

11987121

120583minus 2120583 + 51198721205832

+(120581119872

1198712+ 31198721205832) ln 1

120583] 119889

(30)

where we have kept the linear terms of 119889 and neglected thehigher order terms

For the perihelion precession that is 120581 = 1 the aboveequation is still a little complicated which can be furthersimplified in the larger radius case that is 120583 = 1119903 sim 0 Thusthe above equation is

1198892120583

1198891205932+ 120583 =

120581119872

1198712+ 31198721205832 +

1198642

1198712119889

120583 (31)

Advances in High Energy Physics 5

and the approximate solution can be easily obtained

120583 (120593) =119872

1198712(1 + 119890 cos120593)

+31198723

1198714(1 + 119890120593 sin120593 + 1198902 (1

2minus1

6cos 2120593))

+1198642

119872(1 minus

1

2119890120593 sin120593)119889

(32)

which can be further simplified

120583 (120593) =119872

11987121 + 119890 [cos120593 + (3119872

2

1198712minus11986421198712

21198722119889)120593 sin120593]

(33)

Therefore the angle of perihelion precession is

Δ120593 = 2120587120576 = 2120587(31198722

1198712minus11986421198712

21198722119889) (34)

From the above simple discussion we can easily find thatthere can be indeed an effect from the 119891(119877) gravity on theperihelion precession

For the deflection of light that is 120581 = 0 the masterequation becomes

1198892120583

1198891205932+ 120583 = 31198721205832 + (

1198642

11987121

120583minus 2120583 + 51198721205832 + 31198721205832 ln 1

120583)119889

(35)

which can also be simplified in the larger radius case that is120583 = 1119903 sim 0 as

1198892120583

1198891205932+ 120583 = 31198721205832 +

1198642

1198712119889

120583 (36)

If we solve this equation in the exact same way used beforewe obtain

120583 (120593) =1

119897sin120593

+119872

1198972(1 minus cos120593)2 + 1198642119897

1198712(minus120593 cos120593 + sin120593 ln sin120593) 119889

(37)

which can be further simplified near 120593 = 120587

120583 (120593) =1

119897sin120593 + 119872

1198972(1 minus cos120593)2 + 1198642119897

1198712(minus120593 cos120593) 119889

(38)

Therefore the angle can be obtained

120573 =4119872

119897+1198642119897 (4119872 + 120587119897)

1198712119889 (39)

4 Conclusion and Discussion

In this paper the perihelion precession and deflection of lighthave been investigated in the 4-dimensional general spher-ically symmetric spacetime Since the perihelion precessionand deflection of light can be treated respectively as thetime-like and null geodesic in spacetime themaster equationof perihelion precession and deflection of light is obtainedMoreover the Reissner-Nordstorm-AdS solution in Einsteingravity and Clifton-Barrow solution in 119891(119877) gravity aretaken as the two examples for the application of this masterequation and the effects from the cosmological constantelectric charge and119891(119877) gravity on the perihelion precessionand deflection of light are investigated We find that theelectric charge can affect both the perihelion precession andthe deflection of light while the cosmological constant canonly affect the perihelion precession which are consistentwith the results in the previous works In addition afterconsidering the case with small 119889 we easily find that the119891(119877)gravity can also affect both the perihelion precession and thedeflection of light Several discussions related to our resultsare in order

(i) During investigating the effects from the electriccharge cosmological constant and 119891(119877) gravity on the peri-helion precession and deflection of light we have assumed theapproximate conditions like (15) and small 119889 Therefore theresults under other approximate conditions are interestingopen issues

(ii) The perihelion precession and deflection of lightcan contain the information of the spacetime that is thecosmological constant electric charge and small 119889 Partic-ularly if we just consider them in the solar system sincethe angle of perihelion precession and deflection of lightin the solar system can be detected by the experimentsthus the information like cosmological constant and electriccharge can be extracted from the angles by the experimentsTherefore we may also extract the information of darkmatter such as its distribution of energy density 120588(119903) throughdetecting the perihelion precession and deflection of lightby the experiments if we can first find the static solutioncontaining the dark matter which will be further studied inthe future work

(iii) A simple constraint on the cosmological constantAlthough the cosmological constant does not affect thedeflection of light it can affect the perihelion precessionFrom our results if the cosmological constant indeed existsin our universe it can also give effects in the solar systemTherefore we can give a simple constraint on the cosmologi-cal constant from the well-known phenomenonmdashperihelionprecession of Mercury in the solar system The experimentdata of anomalous precession angle is (4256 plusmn 094)10158401015840 percentury [26] Therefore the theoretical result should be in(4256 plusmn 094)10158401015840 Considering the simple case 119876 = 0 in (19)and recovering the constants 119866 and 119888 (19) becomes

Δ120593 = 2120587(311987221198662

11987121198882minus

Λ1198716

211987241198663) (40)

6 Advances in High Energy Physics

Since the theoretical result of the first term is 430310158401015840 percentury thus the possible contribution from cosmologicalconstant 120587Λ119871611987241198663 must be less than 14110158401015840 per centurywhere we just consider the positive cosmological constantwhich can accelerate our universe After inserting the con-stants the mass of the sun 119872 = 1989 times 1030 kg and theangular momentum of unit mass of Mercury 119871 = 272 times

1016m2sminus1 we can constrain the cosmological constant Λ lt

589times10minus11 kgm3 which is consistent with observation datain our universe Λ = 19 times 10minus25 kgm3

(iv) The subtlety in the deflection of light in the solarsystem Note that there is a gap between the experiment data(161 plusmn 04)10158401015840 and theoretical result 17510158401015840 in Schwarzschildspacetime [24 25] From the result in (25) one may considerthat this difference may be from the sunrsquos possible electriccharge effect However we will give a simple proof in thefollowing that it is not true After recovering the constantsthe angle of starlight deflection is

120573 =4119872119866

1198882119897minus31205871198762119866

411989721198884 (41)

where we have used the Gauss unit of119876 that is kg12m32sminus1fromwhichwe canfind that the charge could indeedmake theangle smaller and hence make the gap smaller If the chargecould make the angle smaller 110158401015840 that is (31205871198762119866411989721198884) times180120587times60times60 = 1 we can obtain the sunwith possible charge119876 = 11 times 1028 kg12m32sminus1 However note that the sunrsquoscharge can also affect the perihelion precession of Mercuryin (19) which can be recovered

Δ120593 = 2120587(311987221198662

11987121198882minus

1198762119866

211987121198882) (42)

where we have neglected the effect of cosmological constantsince the observation data in our universe Λ = 19 times10minus25 kgm3 is very smallTherefore after the simple calcula-tion we can obtain that the sunrsquos charge effect on the angle ofperihelion precession ofMercury will be120587119876211986611987121198882 = 0038which is larger than the first term 61205871198722119866211987121198882 = 497times10minus7This is obviously opposite to the experiment observationThus the gap between the experiment data and theoreticalresult for the starlight deflection in solar system could notcompletely come from the sunrsquos electric charge and a moresuitable explanation of this difference will still be an openissue

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (NSFC) under Grant no 11105004 andShanghai Key Laboratory of Particle Physics and Cosmologyunder Grant no 11DZ2260700 and partially by Grants from

NSFC (nos 10821504 10975168 and 11035008) and theMinistry of Science andTechnology ofChina underGrant no2010CB833004 Professor Hongsheng Zhang is supported bythe Program for Professor of Special Appointment (EasternScholar) at Shanghai Institutions of Higher Learning andNational Natural Science Foundation of China under Grantnos 11075106 and 11275128

References

[1] S Weinberg Gravitation and Cosmology Principles and Appli-cations of the General Theory of Relativity John Wiley amp SonsNew York NY USA 1972

[2] C W Misner K S Thorne and J A Wheeler Gravitation WH Freeman and Co San Francisco Calif USA 1973

[3] R M Wald General Relativity University of Chicago PressChicago Ill USA 1984

[4] S G Turyshev U E Israelsson M Shao et al ldquoSpace-basedresearch in fundamental physics and quantum technologiesrdquoInternational Journal of Modern Physics D vol 16 no 12 pp1879ndash1925 2008

[5] S G Turyshev ldquoExperimental tests of general relativityrdquoAnnualReview of Nuclear and Particle Science vol 58 pp 207ndash2482008

[6] T P Sotiriou and V Faraoni ldquo119891(119877) theories of gravityrdquo Reviewsof Modern Physics vol 82 no 1 pp 451ndash497 2010

[7] R G Cai L M Cao Y P Hu and N Ohta ldquoGeneralizedMisner-Sharp energy in f (R) gravityrdquo Physical Review D vol80 Article ID 104016 2009

[8] T Clifton and J D Barrow ldquoThe power of general relativityrdquoPhysical Review D Third Series vol 72 no 10 Article ID103005 2005 Erratum in Physical Review D Third Series vol90 Article ID 029902 2014

[9] H Zhang Y Hu and X Z Li ldquoMisner-Sharp mass in n-dimensional f(R) gravityrdquo Physical Review D vol 90 Article ID024062 2014

[10] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 1989

[11] P J Peebles and B Ratra ldquoThe cosmological constant and darkenergyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash6062003

[12] A Albrecht and C Skordis ldquoPhenomenology of a realisticaccelerating universe using only Planck-scale physicsrdquo PhysicalReview Letters vol 84 p 2076 2000

[13] M Li X D Li S Wang and Y Wang ldquoDark energyrdquo Commu-nications in Theoretical Physics vol 56 no 3 p 525 2011

[14] V Kagramanova J Kunz and C Lammerzahl ldquoSolar systemeffects in Schwarzschild-de Sitter space-timerdquo Physics Letters Bvol 634 pp 465ndash470 2006

[15] P Jetzer and M Sereno ldquoTwo-body problem with the cosmo-logical constant and observational constraintsrdquo Physical ReviewD vol 73 no 4 Article ID 044015 6 pages 2006

[16] L Iorio ldquoSolar System motions and the cosmological constanta new approachrdquo Advances in Astronomy vol 2012 Article ID268647 9 pages 2012

[17] L Iorio ldquoConstraining the angular momentum of the sun withplanetary orbital motions and general relativityrdquo Solar Physicsvol 281 no 2 pp 815ndash826 2012

[18] L Iorio ldquoConstraining the electric charges of some astro-nomical bodies in Reissner-Nordstrom spacetimes and generic

Advances in High Energy Physics 7

119903minus2-type power-law potentials from orbital motionsrdquo GeneralRelativity and Gravitation vol 44 no 7 pp 1753ndash1767 2012

[19] M Sereno and P Jetzer ldquoDark matter versus modificationsof the gravitational inverse-square law results from planetarymotion in the Solar systemrdquo Monthly Notices of the RoyalAstronomical Society vol 371 no 2 pp 626ndash632 2006

[20] M Sereno ldquoWeak field limit of Reissner-Nordstrom black holelensingrdquo Physical Review D vol 69 no 2 Article ID 023002 4pages 2004

[21] M Sereno and P Jetzer ldquoSolar and stellar system tests of thecosmological constantrdquo Physical Review D vol 73 Article ID063004 2006

[22] M Sereno ldquoInfluence of the cosmological constant on gravi-tational lensing in small systemsrdquo Physical Review D vol 77Article ID 043004 2008

[23] M Sereno ldquoRole of Λ in the cosmological lens equationrdquoPhysical Review Letters vol 102 no 2 Article ID 021301 2009

[24] F W Dyson A S Eddington and C Davidson ldquoA determi-nation of the deflection of light by the sunrsquos gravitational fieldfrom observations made at the total eclipse of May 29 1919rdquoPhilosophical Transactions of the Royal Society A vol 220 no571ndash581 pp 291ndash333 1920

[25] D Kennefick ldquoTesting relativity from the 1919 eclipsemdasha ques-tion of biasrdquo Physics Today vol 62 no 3 pp 37ndash42 2009

[26] G M Clemence ldquoThe relativity effect in planetary motionsrdquoReviews of Modern Physics vol 19 p 361 1947

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 5

and the approximate solution can be easily obtained

120583 (120593) =119872

1198712(1 + 119890 cos120593)

+31198723

1198714(1 + 119890120593 sin120593 + 1198902 (1

2minus1

6cos 2120593))

+1198642

119872(1 minus

1

2119890120593 sin120593)119889

(32)

which can be further simplified

120583 (120593) =119872

11987121 + 119890 [cos120593 + (3119872

2

1198712minus11986421198712

21198722119889)120593 sin120593]

(33)

Therefore the angle of perihelion precession is

Δ120593 = 2120587120576 = 2120587(31198722

1198712minus11986421198712

21198722119889) (34)

From the above simple discussion we can easily find thatthere can be indeed an effect from the 119891(119877) gravity on theperihelion precession

For the deflection of light that is 120581 = 0 the masterequation becomes

1198892120583

1198891205932+ 120583 = 31198721205832 + (

1198642

11987121

120583minus 2120583 + 51198721205832 + 31198721205832 ln 1

120583)119889

(35)

which can also be simplified in the larger radius case that is120583 = 1119903 sim 0 as

1198892120583

1198891205932+ 120583 = 31198721205832 +

1198642

1198712119889

120583 (36)

If we solve this equation in the exact same way used beforewe obtain

120583 (120593) =1

119897sin120593

+119872

1198972(1 minus cos120593)2 + 1198642119897

1198712(minus120593 cos120593 + sin120593 ln sin120593) 119889

(37)

which can be further simplified near 120593 = 120587

120583 (120593) =1

119897sin120593 + 119872

1198972(1 minus cos120593)2 + 1198642119897

1198712(minus120593 cos120593) 119889

(38)

Therefore the angle can be obtained

120573 =4119872

119897+1198642119897 (4119872 + 120587119897)

1198712119889 (39)

4 Conclusion and Discussion

In this paper the perihelion precession and deflection of lighthave been investigated in the 4-dimensional general spher-ically symmetric spacetime Since the perihelion precessionand deflection of light can be treated respectively as thetime-like and null geodesic in spacetime themaster equationof perihelion precession and deflection of light is obtainedMoreover the Reissner-Nordstorm-AdS solution in Einsteingravity and Clifton-Barrow solution in 119891(119877) gravity aretaken as the two examples for the application of this masterequation and the effects from the cosmological constantelectric charge and119891(119877) gravity on the perihelion precessionand deflection of light are investigated We find that theelectric charge can affect both the perihelion precession andthe deflection of light while the cosmological constant canonly affect the perihelion precession which are consistentwith the results in the previous works In addition afterconsidering the case with small 119889 we easily find that the119891(119877)gravity can also affect both the perihelion precession and thedeflection of light Several discussions related to our resultsare in order

(i) During investigating the effects from the electriccharge cosmological constant and 119891(119877) gravity on the peri-helion precession and deflection of light we have assumed theapproximate conditions like (15) and small 119889 Therefore theresults under other approximate conditions are interestingopen issues

(ii) The perihelion precession and deflection of lightcan contain the information of the spacetime that is thecosmological constant electric charge and small 119889 Partic-ularly if we just consider them in the solar system sincethe angle of perihelion precession and deflection of lightin the solar system can be detected by the experimentsthus the information like cosmological constant and electriccharge can be extracted from the angles by the experimentsTherefore we may also extract the information of darkmatter such as its distribution of energy density 120588(119903) throughdetecting the perihelion precession and deflection of lightby the experiments if we can first find the static solutioncontaining the dark matter which will be further studied inthe future work

(iii) A simple constraint on the cosmological constantAlthough the cosmological constant does not affect thedeflection of light it can affect the perihelion precessionFrom our results if the cosmological constant indeed existsin our universe it can also give effects in the solar systemTherefore we can give a simple constraint on the cosmologi-cal constant from the well-known phenomenonmdashperihelionprecession of Mercury in the solar system The experimentdata of anomalous precession angle is (4256 plusmn 094)10158401015840 percentury [26] Therefore the theoretical result should be in(4256 plusmn 094)10158401015840 Considering the simple case 119876 = 0 in (19)and recovering the constants 119866 and 119888 (19) becomes

Δ120593 = 2120587(311987221198662

11987121198882minus

Λ1198716

211987241198663) (40)

6 Advances in High Energy Physics

Since the theoretical result of the first term is 430310158401015840 percentury thus the possible contribution from cosmologicalconstant 120587Λ119871611987241198663 must be less than 14110158401015840 per centurywhere we just consider the positive cosmological constantwhich can accelerate our universe After inserting the con-stants the mass of the sun 119872 = 1989 times 1030 kg and theangular momentum of unit mass of Mercury 119871 = 272 times

1016m2sminus1 we can constrain the cosmological constant Λ lt

589times10minus11 kgm3 which is consistent with observation datain our universe Λ = 19 times 10minus25 kgm3

(iv) The subtlety in the deflection of light in the solarsystem Note that there is a gap between the experiment data(161 plusmn 04)10158401015840 and theoretical result 17510158401015840 in Schwarzschildspacetime [24 25] From the result in (25) one may considerthat this difference may be from the sunrsquos possible electriccharge effect However we will give a simple proof in thefollowing that it is not true After recovering the constantsthe angle of starlight deflection is

120573 =4119872119866

1198882119897minus31205871198762119866

411989721198884 (41)

where we have used the Gauss unit of119876 that is kg12m32sminus1fromwhichwe canfind that the charge could indeedmake theangle smaller and hence make the gap smaller If the chargecould make the angle smaller 110158401015840 that is (31205871198762119866411989721198884) times180120587times60times60 = 1 we can obtain the sunwith possible charge119876 = 11 times 1028 kg12m32sminus1 However note that the sunrsquoscharge can also affect the perihelion precession of Mercuryin (19) which can be recovered

Δ120593 = 2120587(311987221198662

11987121198882minus

1198762119866

211987121198882) (42)

where we have neglected the effect of cosmological constantsince the observation data in our universe Λ = 19 times10minus25 kgm3 is very smallTherefore after the simple calcula-tion we can obtain that the sunrsquos charge effect on the angle ofperihelion precession ofMercury will be120587119876211986611987121198882 = 0038which is larger than the first term 61205871198722119866211987121198882 = 497times10minus7This is obviously opposite to the experiment observationThus the gap between the experiment data and theoreticalresult for the starlight deflection in solar system could notcompletely come from the sunrsquos electric charge and a moresuitable explanation of this difference will still be an openissue

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (NSFC) under Grant no 11105004 andShanghai Key Laboratory of Particle Physics and Cosmologyunder Grant no 11DZ2260700 and partially by Grants from

NSFC (nos 10821504 10975168 and 11035008) and theMinistry of Science andTechnology ofChina underGrant no2010CB833004 Professor Hongsheng Zhang is supported bythe Program for Professor of Special Appointment (EasternScholar) at Shanghai Institutions of Higher Learning andNational Natural Science Foundation of China under Grantnos 11075106 and 11275128

References

[1] S Weinberg Gravitation and Cosmology Principles and Appli-cations of the General Theory of Relativity John Wiley amp SonsNew York NY USA 1972

[2] C W Misner K S Thorne and J A Wheeler Gravitation WH Freeman and Co San Francisco Calif USA 1973

[3] R M Wald General Relativity University of Chicago PressChicago Ill USA 1984

[4] S G Turyshev U E Israelsson M Shao et al ldquoSpace-basedresearch in fundamental physics and quantum technologiesrdquoInternational Journal of Modern Physics D vol 16 no 12 pp1879ndash1925 2008

[5] S G Turyshev ldquoExperimental tests of general relativityrdquoAnnualReview of Nuclear and Particle Science vol 58 pp 207ndash2482008

[6] T P Sotiriou and V Faraoni ldquo119891(119877) theories of gravityrdquo Reviewsof Modern Physics vol 82 no 1 pp 451ndash497 2010

[7] R G Cai L M Cao Y P Hu and N Ohta ldquoGeneralizedMisner-Sharp energy in f (R) gravityrdquo Physical Review D vol80 Article ID 104016 2009

[8] T Clifton and J D Barrow ldquoThe power of general relativityrdquoPhysical Review D Third Series vol 72 no 10 Article ID103005 2005 Erratum in Physical Review D Third Series vol90 Article ID 029902 2014

[9] H Zhang Y Hu and X Z Li ldquoMisner-Sharp mass in n-dimensional f(R) gravityrdquo Physical Review D vol 90 Article ID024062 2014

[10] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 1989

[11] P J Peebles and B Ratra ldquoThe cosmological constant and darkenergyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash6062003

[12] A Albrecht and C Skordis ldquoPhenomenology of a realisticaccelerating universe using only Planck-scale physicsrdquo PhysicalReview Letters vol 84 p 2076 2000

[13] M Li X D Li S Wang and Y Wang ldquoDark energyrdquo Commu-nications in Theoretical Physics vol 56 no 3 p 525 2011

[14] V Kagramanova J Kunz and C Lammerzahl ldquoSolar systemeffects in Schwarzschild-de Sitter space-timerdquo Physics Letters Bvol 634 pp 465ndash470 2006

[15] P Jetzer and M Sereno ldquoTwo-body problem with the cosmo-logical constant and observational constraintsrdquo Physical ReviewD vol 73 no 4 Article ID 044015 6 pages 2006

[16] L Iorio ldquoSolar System motions and the cosmological constanta new approachrdquo Advances in Astronomy vol 2012 Article ID268647 9 pages 2012

[17] L Iorio ldquoConstraining the angular momentum of the sun withplanetary orbital motions and general relativityrdquo Solar Physicsvol 281 no 2 pp 815ndash826 2012

[18] L Iorio ldquoConstraining the electric charges of some astro-nomical bodies in Reissner-Nordstrom spacetimes and generic

Advances in High Energy Physics 7

119903minus2-type power-law potentials from orbital motionsrdquo GeneralRelativity and Gravitation vol 44 no 7 pp 1753ndash1767 2012

[19] M Sereno and P Jetzer ldquoDark matter versus modificationsof the gravitational inverse-square law results from planetarymotion in the Solar systemrdquo Monthly Notices of the RoyalAstronomical Society vol 371 no 2 pp 626ndash632 2006

[20] M Sereno ldquoWeak field limit of Reissner-Nordstrom black holelensingrdquo Physical Review D vol 69 no 2 Article ID 023002 4pages 2004

[21] M Sereno and P Jetzer ldquoSolar and stellar system tests of thecosmological constantrdquo Physical Review D vol 73 Article ID063004 2006

[22] M Sereno ldquoInfluence of the cosmological constant on gravi-tational lensing in small systemsrdquo Physical Review D vol 77Article ID 043004 2008

[23] M Sereno ldquoRole of Λ in the cosmological lens equationrdquoPhysical Review Letters vol 102 no 2 Article ID 021301 2009

[24] F W Dyson A S Eddington and C Davidson ldquoA determi-nation of the deflection of light by the sunrsquos gravitational fieldfrom observations made at the total eclipse of May 29 1919rdquoPhilosophical Transactions of the Royal Society A vol 220 no571ndash581 pp 291ndash333 1920

[25] D Kennefick ldquoTesting relativity from the 1919 eclipsemdasha ques-tion of biasrdquo Physics Today vol 62 no 3 pp 37ndash42 2009

[26] G M Clemence ldquoThe relativity effect in planetary motionsrdquoReviews of Modern Physics vol 19 p 361 1947

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Advances in High Energy Physics

Since the theoretical result of the first term is 430310158401015840 percentury thus the possible contribution from cosmologicalconstant 120587Λ119871611987241198663 must be less than 14110158401015840 per centurywhere we just consider the positive cosmological constantwhich can accelerate our universe After inserting the con-stants the mass of the sun 119872 = 1989 times 1030 kg and theangular momentum of unit mass of Mercury 119871 = 272 times

1016m2sminus1 we can constrain the cosmological constant Λ lt

589times10minus11 kgm3 which is consistent with observation datain our universe Λ = 19 times 10minus25 kgm3

(iv) The subtlety in the deflection of light in the solarsystem Note that there is a gap between the experiment data(161 plusmn 04)10158401015840 and theoretical result 17510158401015840 in Schwarzschildspacetime [24 25] From the result in (25) one may considerthat this difference may be from the sunrsquos possible electriccharge effect However we will give a simple proof in thefollowing that it is not true After recovering the constantsthe angle of starlight deflection is

120573 =4119872119866

1198882119897minus31205871198762119866

411989721198884 (41)

where we have used the Gauss unit of119876 that is kg12m32sminus1fromwhichwe canfind that the charge could indeedmake theangle smaller and hence make the gap smaller If the chargecould make the angle smaller 110158401015840 that is (31205871198762119866411989721198884) times180120587times60times60 = 1 we can obtain the sunwith possible charge119876 = 11 times 1028 kg12m32sminus1 However note that the sunrsquoscharge can also affect the perihelion precession of Mercuryin (19) which can be recovered

Δ120593 = 2120587(311987221198662

11987121198882minus

1198762119866

211987121198882) (42)

where we have neglected the effect of cosmological constantsince the observation data in our universe Λ = 19 times10minus25 kgm3 is very smallTherefore after the simple calcula-tion we can obtain that the sunrsquos charge effect on the angle ofperihelion precession ofMercury will be120587119876211986611987121198882 = 0038which is larger than the first term 61205871198722119866211987121198882 = 497times10minus7This is obviously opposite to the experiment observationThus the gap between the experiment data and theoreticalresult for the starlight deflection in solar system could notcompletely come from the sunrsquos electric charge and a moresuitable explanation of this difference will still be an openissue

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (NSFC) under Grant no 11105004 andShanghai Key Laboratory of Particle Physics and Cosmologyunder Grant no 11DZ2260700 and partially by Grants from

NSFC (nos 10821504 10975168 and 11035008) and theMinistry of Science andTechnology ofChina underGrant no2010CB833004 Professor Hongsheng Zhang is supported bythe Program for Professor of Special Appointment (EasternScholar) at Shanghai Institutions of Higher Learning andNational Natural Science Foundation of China under Grantnos 11075106 and 11275128

References

[1] S Weinberg Gravitation and Cosmology Principles and Appli-cations of the General Theory of Relativity John Wiley amp SonsNew York NY USA 1972

[2] C W Misner K S Thorne and J A Wheeler Gravitation WH Freeman and Co San Francisco Calif USA 1973

[3] R M Wald General Relativity University of Chicago PressChicago Ill USA 1984

[4] S G Turyshev U E Israelsson M Shao et al ldquoSpace-basedresearch in fundamental physics and quantum technologiesrdquoInternational Journal of Modern Physics D vol 16 no 12 pp1879ndash1925 2008

[5] S G Turyshev ldquoExperimental tests of general relativityrdquoAnnualReview of Nuclear and Particle Science vol 58 pp 207ndash2482008

[6] T P Sotiriou and V Faraoni ldquo119891(119877) theories of gravityrdquo Reviewsof Modern Physics vol 82 no 1 pp 451ndash497 2010

[7] R G Cai L M Cao Y P Hu and N Ohta ldquoGeneralizedMisner-Sharp energy in f (R) gravityrdquo Physical Review D vol80 Article ID 104016 2009

[8] T Clifton and J D Barrow ldquoThe power of general relativityrdquoPhysical Review D Third Series vol 72 no 10 Article ID103005 2005 Erratum in Physical Review D Third Series vol90 Article ID 029902 2014

[9] H Zhang Y Hu and X Z Li ldquoMisner-Sharp mass in n-dimensional f(R) gravityrdquo Physical Review D vol 90 Article ID024062 2014

[10] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 1989

[11] P J Peebles and B Ratra ldquoThe cosmological constant and darkenergyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash6062003

[12] A Albrecht and C Skordis ldquoPhenomenology of a realisticaccelerating universe using only Planck-scale physicsrdquo PhysicalReview Letters vol 84 p 2076 2000

[13] M Li X D Li S Wang and Y Wang ldquoDark energyrdquo Commu-nications in Theoretical Physics vol 56 no 3 p 525 2011

[14] V Kagramanova J Kunz and C Lammerzahl ldquoSolar systemeffects in Schwarzschild-de Sitter space-timerdquo Physics Letters Bvol 634 pp 465ndash470 2006

[15] P Jetzer and M Sereno ldquoTwo-body problem with the cosmo-logical constant and observational constraintsrdquo Physical ReviewD vol 73 no 4 Article ID 044015 6 pages 2006

[16] L Iorio ldquoSolar System motions and the cosmological constanta new approachrdquo Advances in Astronomy vol 2012 Article ID268647 9 pages 2012

[17] L Iorio ldquoConstraining the angular momentum of the sun withplanetary orbital motions and general relativityrdquo Solar Physicsvol 281 no 2 pp 815ndash826 2012

[18] L Iorio ldquoConstraining the electric charges of some astro-nomical bodies in Reissner-Nordstrom spacetimes and generic

Advances in High Energy Physics 7

119903minus2-type power-law potentials from orbital motionsrdquo GeneralRelativity and Gravitation vol 44 no 7 pp 1753ndash1767 2012

[19] M Sereno and P Jetzer ldquoDark matter versus modificationsof the gravitational inverse-square law results from planetarymotion in the Solar systemrdquo Monthly Notices of the RoyalAstronomical Society vol 371 no 2 pp 626ndash632 2006

[20] M Sereno ldquoWeak field limit of Reissner-Nordstrom black holelensingrdquo Physical Review D vol 69 no 2 Article ID 023002 4pages 2004

[21] M Sereno and P Jetzer ldquoSolar and stellar system tests of thecosmological constantrdquo Physical Review D vol 73 Article ID063004 2006

[22] M Sereno ldquoInfluence of the cosmological constant on gravi-tational lensing in small systemsrdquo Physical Review D vol 77Article ID 043004 2008

[23] M Sereno ldquoRole of Λ in the cosmological lens equationrdquoPhysical Review Letters vol 102 no 2 Article ID 021301 2009

[24] F W Dyson A S Eddington and C Davidson ldquoA determi-nation of the deflection of light by the sunrsquos gravitational fieldfrom observations made at the total eclipse of May 29 1919rdquoPhilosophical Transactions of the Royal Society A vol 220 no571ndash581 pp 291ndash333 1920

[25] D Kennefick ldquoTesting relativity from the 1919 eclipsemdasha ques-tion of biasrdquo Physics Today vol 62 no 3 pp 37ndash42 2009

[26] G M Clemence ldquoThe relativity effect in planetary motionsrdquoReviews of Modern Physics vol 19 p 361 1947

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 7

119903minus2-type power-law potentials from orbital motionsrdquo GeneralRelativity and Gravitation vol 44 no 7 pp 1753ndash1767 2012

[19] M Sereno and P Jetzer ldquoDark matter versus modificationsof the gravitational inverse-square law results from planetarymotion in the Solar systemrdquo Monthly Notices of the RoyalAstronomical Society vol 371 no 2 pp 626ndash632 2006

[20] M Sereno ldquoWeak field limit of Reissner-Nordstrom black holelensingrdquo Physical Review D vol 69 no 2 Article ID 023002 4pages 2004

[21] M Sereno and P Jetzer ldquoSolar and stellar system tests of thecosmological constantrdquo Physical Review D vol 73 Article ID063004 2006

[22] M Sereno ldquoInfluence of the cosmological constant on gravi-tational lensing in small systemsrdquo Physical Review D vol 77Article ID 043004 2008

[23] M Sereno ldquoRole of Λ in the cosmological lens equationrdquoPhysical Review Letters vol 102 no 2 Article ID 021301 2009

[24] F W Dyson A S Eddington and C Davidson ldquoA determi-nation of the deflection of light by the sunrsquos gravitational fieldfrom observations made at the total eclipse of May 29 1919rdquoPhilosophical Transactions of the Royal Society A vol 220 no571ndash581 pp 291ndash333 1920

[25] D Kennefick ldquoTesting relativity from the 1919 eclipsemdasha ques-tion of biasrdquo Physics Today vol 62 no 3 pp 37ndash42 2009

[26] G M Clemence ldquoThe relativity effect in planetary motionsrdquoReviews of Modern Physics vol 19 p 361 1947

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of