Opsta relativnost

1 0 0

, , 23. 2015.

w w w.gravity.ipb.ac.rs/gr100

100

100 171031 - , . 1979. 36 36 : 18 , , loop- . .

: :

2

1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 : , . . . . . . . . 19

3 O . . . . . . . . . . . 31

4 - ? . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

100 , 23. 2015. 3

; .

100 , , , . , , . , , 100 , , . 100 , 23. . Sci-ence - 100 , Classical and Quantum Gravity- Milestones in General Relativity. , , , 1915. 13 :

(1919) (1929) - (1952) , (1959) - (1963) (1963) (1965) (1965) (1965) (1970-74) (1974) AdS/CFT (1998) (2005).

, , . 1905. , 2005. - . a, .

, : ?1 , :

1 ,

4

( ), , . , . , , , , : , . , 1879. 1913. , , . , , , 1926. . , . , 1905. 1915, . , , . , . - . , 1960. ; , . , , .

1905. , , . , , , 2. , ( ) . , ( : 1916) , , , , . , .

: , .

100 , 23. 2015. 5

, : , , . : , . , . , . , : . , . ; . , 1954: , . : , , . : , , : .

, , , , ? , . , , , . , . , . , , , . . () , , . , , . , . , , , 100 , . : .

6

. , , , : @..

100 2015.1:715

, , ,

. , (1915/1916) , . , . , . ( ).

7

, , .

, Naturales quaestiones

. , , . (4. ...), . , , :

?

, . , . , .

. . (15641642). , . : Dialogo (1632) Discorsi (1638). [Discorsi].

, 100 200 , 200 .

. . ().

. ( ), .

, .

. , , . (1642-1726), . , . , . , . , . . . Philosophiae Naturalis Principia Mathematica (1687). , . , ( ) .

8

. (15641642)

. (16421726)

, . , , . (18791955) ().

.

. , . , , , . , . , . .

. () .

(1907). . , :

, . , . ?

, :

. .

100 , 23. 2015. 9

, .

. . , , , -. - .

, . 1854, . (18261866) , . . (17771855); . , , (18021860) (17921856).

. (18261866)

. (18791955)

., , . 1913, . , . 20. , .

. , . . = 4, , . , , :

R 12gR| {z }

=8G

c4T| {z }

:

10

1915. , , 1916. 1915. . (18621943) . ; 20- 1915. , , . , 25. , . , . :

. .

. , . . : , , , . : - , . : ( , 1981). : (1915/16), ( ); - : + , ( ), . ( ), . : , , (95% ).

. , . (18851955) , . , , . , :

() (P;Q) (x; x + x) s2 = gxx , g ;

() (P;Q) (P 0; Q0), , (s0)2 = s2.

(x; x+ x) ds2 = gx

x , ; () (). , () , . , .

100 , 23. 2015. 11

(x), .

, (x) , .

, , B(x). / 5.

( ). . - , , . , , B(x) , .

, 1929. , [ U(1)]. , .

. (1929), (1954) [ SU(2)] () . . , . , (1956) L(1; 3), , . , . , , - ; . (1932, ) . (19261999) 1960-.

. , (1961) . , , L(1; 3) T (4).

12

, AL AT . , AT , . . (), - . ? , , ; , - , .

T. (1932, )

. (19261999)

.

(i) , , (ii) , ,

.

, . . (1918) , , . , . , . (18691951), , (1922), . , , - , .

? , . 1960, . , , . Gravity Probe A (1976) Gravity ProbeB (2004). . r = 1 cm, L = 4:6gr cm2/sec. , (1980) :

(); ( L), 850 !

100 , 23. 2015. 13

.

. , , . , , (2008) , 1998, . , . , . , (JP = 0+) , . , . .

, Naturales quaestiones

1. B. Homann, Relativity and its roots (Freeman, New York, 1983).2. A. Einstein, ber das Relativittsprinzip und die aus demselben gezogenen Folgerungen, Jarb. der Radioaktiv-

itt u. Elektronik, 4 (1907) 411462;A. Einstein, Erklrung der Perihelbewegung der Merkur aus der Relativittstheorie, Sitzungsber. preuss. Akad.Wiss. 47, 2 (1915) 831839;A. Einstein, Die grundlage der allgemeinen Relativittstheorie, Ann. Phys. 49 (1916) 769822;I. Todorov, Einstein and Hilbert: the creation of general relativity, arXiv:physics/0504179;B. Riemann, On the Hypotheses which lie at the Bases of Geometry, translated from German by William King-don Cliord in Nature (1873),http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/WKCGeom.html

3. H. Weyl, Space-Time-Matter (Dover Publications, New York, 1952), translated from the 4th edition (1922) ofRaum-Zeit-Materie (1918) by H. L. Brose;H. Weyl, Elektron und Gravitation I, Zeitschri f. Physik, 56 (1929) 330352;L. ORaifeartaigh,e Dawning of Gauge eory (Princeton Univ. Press, Princeton, 1997).

4. T. Kibble, Lorentz invariance and the gravitational field, J. Math. Phys. 2 (1961) 212221;D. W. Sciama, On the analogy between charge and spin in general relativity, in: Recent Developments in GeneralRelativity, Festschri for Infeld (Pergamon Press, Oxford, 1962) 415439;E. Cartan, On a generalization of the notion of Riemann curvature and spaces with torsion (in French), ComptesRendus Acad. Sci. (Paris) 174 (1922) 593595;K.-F. Shie, J. M. Nester, and H.-J. Yo, Torsion cosmology and the accelerating universe, Phys. Rev. D 78 (2008)023522;M. Blagojevi and F. W. Hehl (eds.), Gauge eories of Gravitation, A Reader with Commentaries (ImperialCollege Press, London, 2013).

14

:

, . . .

A (1908): P , , (1 + /c2) E E/c2.

(1913): ( ) , , () 1/10000 . , , . , .

= T ;

g [] , . . .

(1914): 1913. .

(1915): . . , ; . , . , , , , . , . ,

R = T ;

100 , 23. 2015. 15

1. , .

@2g

@x@x= 0:

, . ( 4. 1915. 11. ) , . , . , . T , T+t T . T = 0 . [ ] , .

G = T ;

[ G ]. ( 11. , 18. ) [ ] . , , , 45 , .

.

Gim = (Tim 12gimT )

[ Gim ]..... ... , - . ( 25. , 2. ) , . - , . , , .

16

, . , [ ] .

(1915): I: L, :

g ;@g@x

;@2g@x@x

;

A;@A@x

;

ZL(g)1/2d4x

14 g ; A: II ( ). L [] x L, . g , L

L = R+ Lelec;

R ( ).( 20. , 1915.)

(1916): . . . , . . . .

G

24

X;

@3J@t3

:"

(1917): , ; . ; . : 14 g ; A , .

100 , 23. 2015. 17

:, , : @..

100 2015.2:1930

, ,

(). a, , , .

19

. ( ) , . , - .

, , . , : .

. ( IV ) :

; . .

, (, ).

. XVI , , . . (1643-1727) .

(1564-1642). 1 . s t2. . ?

. II . .

2, mi, II a , mg ,

1 II . . , .

2 () .

20 : ,

. II :

~F = mi~a ; (1.1)

, , :

~F = mg~g ; (1.2)

je ~g . :

~a =mgmi

~g ; (1.3)

mg/mi .

pmg/mi.

, . , . XIX 109.

(1848-1919)

(1879-1955) , .

. 1665. . , . , , III . , , .

100 , 23. 2015. 21

1684. , . . , .

. XIX . .

35 . 43 . ( ) . - . , . . , , XIX .

.

. . , , . . .

22 : ,

1907. , . , , . .

(1879-1955)

( , . ). . , , . . 1911. . . , 1913. -., - 3 . XVIII , XX .

, .

. n (x1; x2; : : : xn) (x1; x2; : : : xn) (x1 + dx1; x2 + dx2; : : : xn + dxn) :

ds2 = g(x1; x2; : : : ; xn)dxdx ; (2.1)

3 - .

100 , 23. 2015. 23

g 4. :

x0 = x0(x1; x2; : : : ; xn) ; = 1; 2; : : : ; n : (2.2)

, . (2.2). '(x1; x2; : : : ; xn) (2.2) , :

'0(x01; x02; : : : ; x0n) = '(x1; x2; : : : ; xn) : (2.3)

ds2.n n .

(A1; A2; : : : ; An) (2.2) :

A0 =@x0

@xA ; (2.4)

, . (A1; A2; : : : ; An) :

A0 =@x

@x0A : (2.5)

, . . , 5

:

A0 =@x0

@x@x0

@xA : (2.6)

:

gg = ; (2.7)

.

. :A = gA

: (2.8)

. ( ) @A := @A@x . . -.

4 , . .

5 (A ) (A ).

24 : ,

, -. . 6. A. :

A = Adx ; (2.9)

. , . =

. :

rA = dA A rAdx : (2.10)

, (2.9) (2.10):

rA = @A + A : (2.11)

, . rg = 0. :

du

ds= 0 ; u =

dx

ds: (2.12)

:

du

ds+ u

u = 0 : (2.13)

(2.13) , . . muu . . . ,

6 .

100 , 23. 2015. 25

. :

A =1

2RAf

; (2.14)

je f ,

R = @ @ + ; (2.15)

. R R :

R := gR ; R := g

R : (2.16)

:

R 12g =

8G

c4T ; (2.17)

je T ( ), G , c .

. : 7.

( ) R = 0

8 , .

rT = 0. . , .

. 1916. . -

7 .8 . :

R = 0.

26 : ,

1915.

:

ds2 =1 rg

r

c2dt2 dr

2

1 rgr

r2 d2 + sin2 '2 ; (2.18) je rg = 2Gm/c2 m. 3 km, 0; 9 cm. , , . (2.18) . - , , = /2.

. , :

=6Gm

c2a(1 e2) ; (2.19)

m , a , a e . (2.19) 43 . .

. :

=2rg

=4Gm

c2: (2.20)

1; 75 . . 29. 1919, , ,

100 , 23. 2015. 27

. ( ) 1,6 1,98 . .

29. 1919.

.

. . . , .

: , 9. . , , . . . . .

. . . ( ) - :

ds2 = c2dt2 a2(t) d2 + sin2 (d2 + sin2 d'2) : (2.21)9 .

28 : ,

:

a 9a0c

2

2

1/3t2/3 : (2.22)

, . . . , . , . . :

v = H` ; (2.23)

` , H . v 50 75 km/s .

. , . , .

. , , . .

( ). ., . .

, . , - . , , , . . , .

. . - , . . 1930, , . 10,

10

100 , 23. 2015. 29

- , .

, ,- , . .

.

30 : ,

O : @...

100 2015.3:3141

, , , ,

, . . - . 1919. , . , . . , 1960.. , . - , , , , . - .

31

, .

() . - , .

:

. [1]. , - . , . , , .

[2]:

. ( - , ) (a , ).

. . .

. , .

, - .

, , , .

. . - ds2 g(x)

ds2 = g(x)dxdx :

g(x) 4 4 , -, dx - .

() ( , ) .

:

32 O

: , , g =9; 8m/s2. . , . . , .

:

() , ( , ).

() 1 .

() - .

: : :

XVII . 1885. 1908. . mg mi A B:

= 2

mAimAg

mBimBg

mAimAg

+mBimBg

:

, ( 4000 ) < 109. , , .

. 2008. : Et-Wash , . =(1:0 1:4) 1013.

. 1013 1018. - , , . 2016. MICROSCOPE (Micro-Satellite trane Compense pour lObservation duPrincipe dEquivalence) (CNES). 230 . MICROSCOPE ,

1 a , . .

100 , 23. 2015. 33

MICROSCOPE.

. , . 1015. , , 1018.

. , . lP = ( -hG/c3)1/2 1033 cm: -h ,G c . lP . , ( , ). ( , ) , E p. E = pc, . , . , , 1013.

- . , , . . . , . , . U 1 + U/c2 . . .

. . 10%

34 O

104. .

. -. - -. . , .

. 1915., . , . , , 43 . . . . 4300 ! . . , . , 400 .

. . 1919. . : . , . , , 1915. 100:75.

, 29. 1919. : . - 100:60000:31, 100:98 000:12. 6. 1919. . , . . , - . , 7. :

100 , 23. 2015. 35

. . , [4].

, . 15 :

, , .

. -, . - 1973. , , . 100:66000:19. . . 1929. - 30 . , . ( ), . . . , . , . 1922. . 1924. , . . . .

. , , . - , . .

. , , . . , .

36 O

. 0, h

g

= 0

1 gh

c2

;

, / = gh/c2. 10 / 1015., 1959. 1965. . (- ) -. - 57Fe . - 57Fe, 22,5 . -, . - . , - . . v = 7:5 107m/s . -. . 1%, . - . , . .

. 1965. . . , . , , . , . 3 . , , . , , . . .

- . - . , . .

100 , 23. 2015. 37

1971. . , . 0, = 0 ( 1).

(GPS). , , 1 km . GPS , - . 2010. exp = (230 20) ns, th = (246 3) ns.

. 2005. . . 2010. 27Al+ 1015Hz 1017Hz. . 10m/s , 3,3 . .

. . - - . , 30% , . , , ( ). 0; 01%.

1 - = 0. (ns).

184 18 96 10 144 14 179 18 40 23 275 21 59 10 273 7

38 O

.

. (), . , ,

. , , . , . j : , . , . 1964. . , 2002. - 0; 0021%.

. 1967. . - . , , . , . . , , . , , .

- . . , .

1974. (PSR B1913+16). , . . 70 , . - , . , . . , , , , , , ,

100 , 23. 2015. 39

, . 1%. 0; 1%. .

. . . , = GM/Rc2 1. = 7 1010, 105, 0:2 1. , .

. GM/R3c2 -, , l. GM/R3c2 l2 . , .

, - . - . . - (Sgr ) .

() : , . , - . , , .

- , Sgr , je 40 , 20 , 10 . .

: 30 . Sgr A* 4 4.6 .

: - S2 Sgr A 17 , 3.7 SgrA. .

: Sgr A. , :

40 O

. , .

. : , . SgrA .

Sgr A . 18 , - . ( UCLA ) 2018. S-2 . , 16 , 0:2. S-2 2019.

. , . . , , . . 4.

50 , . , . 150 .

1. S. G. Turyshev, Experimental Tests of General Relativity, Ann. Rev. Nucl. Part. Sci. 58 (2008) 207

[arXiv:0806.1731 [gr-qc]].2. C. M. Will, e Confrontation between General Relativity and Experiment, Living Rev. Rel. 17 (2014) 4

[arXiv:1403.7377 [gr-qc]].3. E. Berti, E. Barausse, V. Cardoso, L. Gualtieri, P. Pani, U. Sperhake, L. C. Stein and N. Wex et al., Testing

General Relativity with Present and Future Astrophysical Observations, [arXiv:1501.07274 [gr-qc]].4. C. M. Will, e 1919 measurement of the deflection of light, 2015 Class. Quantum Grav. 32 124001,

[arXiv:1409.7812 [physics.hist-ph]].

100 , 23. 2015. 41

.

42 O

- : @..

100 2015.4:4350

, ,

() . . , . ( ) ( .). (LIGO ).

43

. , , . . . . . t ~r ( ) u(~r; t)

@2

@x2+

@2

@y2+

@2

@z2 1

v2@2

@t2

u(~r; t) = 0 : (1)

v ( ). u(~r; t) ( ) ( ).

19. , , . ~E ~B . , () . - . , c = 3 108m/s. (1888) . . . . , - - , , , . (1905) ().

20. . , , - ( ). - ()1915. .

. . . ( 1 2)

~F = G m1m2j~r2 ~r1j3 (~r2 ~r1) ; (2)

G = 6:67 1011Nm2/kg2 . , , .

44 - ?

. 20. . , . - . . ( ).

. . . , ( ).

. - . . ,

R 12gR =

8G

c4T ; [ = 0; 1; 2; 3] (3)

, , , . , , - / ( ).

- -. -. , . , . ( ) , . .

. . ( ) .

. (3) (. )

g = g(0) + h ; (4)

h () g(0) , ( )

@2

@x2+

@2

@y2+

@2

@z2 1

c2@2

@t2

h = 0 ; (5)

100 , 23. 2015. 45

c . g(0) 1. , h . . . . , .

, . . (. ). , , . g(x)

ds2 = 3=03=0g(x)dx

dx : (6)

A B, . . .

. . 1/r. . 2014. . , . , . 1993. ( ).

. . . , .

.. . - , . . . , . (

1, . -. . .

46 - ?

AURIGA

MiniGRAIL

, ALLEGRO). .

MiniGRAIL . (), 1150 20 mK. . 2-4 kHz, . AURIGA INFN- . 3 mK.

100 , 23. 2015. 47

LISA

.. . - . .

. LIGO (Laser Interferometer Gravitational Wave Observatory).LIGO : ( ) ( ). 2-4 . 1 . () . , LIGO . (VIRGO, GEO 600, TAMA 300) .

. . . , .

, (LISA). 5 . , .

.. (107105Hz). () INFN- (). (). 1 .

48 - ?

VIRGO

. . 10 GHz.

. . . . . (

100 , 23. 2015. 49

), , . 1/r , . .

1. C. M. Misner, K. S. orne, J. A. Wheeler, Gravitation, W. H. Freeman and Co., San Francisco, 1973.2. L.D. Landau, E.M. Lifshitz,e Classical eory of Fields, Pergsmon Press, 1971.3. http://en.wikipedia.org/wiki/Gravitational-wave observatory.

50 - ?

, : @.

100 2015.5:5156

, , ,

, . , , . , . 95 . ( ) ( ). , .

51

. , , . - -. .

1915. ., , . ( ) (1). , 1929. , , .

R 12gR+ g =

8G

c4T : (1)

u R , g -, R , G , c T .

1929. , . . , . .

,

. , , . .

52

, . , , . . , 1948. . (76 , 24 1 ) . . . , , : , . .

, , . , 13 . , , . , , 1940. , . 2.7 50 . 1950. . , 1965. , , , - . .

,

: , ,

100 , 23. 2015. 53

. , , . ( ) . , , - .

. , . .

, . ( ), . , . ( ).

. . , , . , . , , . , . , , , .

. , :

1) ,2)

,3) 13.7 (

),4) , . ,

. ,

. . . , -, . ,

54

( ). .

.

. , 1998. , . ( ) . , . , . , . , ( ), .

. . . . , (1), , , . 124 , . . ,

100 , 23. 2015. 55

.

. . , .

. , . , (5 , 23 72 ). . , , . 200 .

56

: : @..

100 2015.6:5766

, - , , ,

, , . , , . . , ; - - , . . , - ,, . - , .

57

, , 2014. , . , 2014. . (Christopher Nolan), , (Inseption) , (Jonathan Nolan). , . , , , , . - () , .

: Science Fiction,

, ., (Kiporne) (Caltech). , , . . [1, 2], [3], . .

1. (): , . , -, . . , [4]. : , .

:

- , - (Schwarzschild) . , . , (Kerr) , . , , , . . - , . () . , , . , .

, . () .

, .

58 :

, , , , . , . ( ) , . - 3. 2 , , - . , (). - . - , .

. ( ). ( ) , 1 [5]. , . , . -.

-? ,, ? .

2. - . . ( ). () -. . : , . - . - . , , - : , 1.

: ,-

, , () . , A

1 : , , , , .

100 , 23. 2015. 59

1- :

(t; x; y; z) A0 (t+ dt; x+ dx; y + dy; z + dz)

ds2 = dt2 + dx2 + dy2 + dz2: (1)

, . , . A A0 1. , , A B 1. , , , A C 1.

, (1) dt2 [s2] dx2, dy2 dz2 [m2]. : 1, c = 1. , . 2500000 . c = 1, v < 1.

. , , . - ( ) 45 . , 45 2, 1. 45 , .

. , . A 2 A. . A. () A . , A . A. A,

2 x = t x =vt, v < 1.

60 :

2- : ,

A . , A -, A. , . . 2, A B, C D. , C A B, A B . D A C .

,,

. , , -, .

3. : , (Karl Schwarzschild) . - M . -, - , . , , .

ds2 = 1 2MG

r

dt2 + 1

1 2MGr

dr2 + r2(d2 + sin d'2): (2)

M , - (, ). , (2) - . , r , ' , 3.

. r ! 1, , M , (2) (1). - . .

2MG . ,

r = 2MG, ( ) .

100 , 23. 2015. 61

3 .

. r = 2MG . , r = 2MG , , . , [4], 3.

? r =0 - ( ), r = 0 . , . , . , , r = 2MG.

3.1. . r, r = 2MG, 4. r ; , . r = 2MG , r = 2MG . ? A , A . P Q . . Q r = 2MG4. ( )

3 .

ds2 = dx2 + dy2 + dz2

= dr2 + r2(d2 + sin d'2)

r = 0, ' = 0 ' = 2 = 0 = .

4 Q : , ' , r . ,

62 :

P r . P . , , P , , . , , P . Q r = 2MG, P . Q r = 2MG , P , P ( r = 2MG ), . 4.

4- -: ,

. P , Q . , , . 5. Q ( ) P ( ), . , P Q . . , . , , .

( -).5 , .

, , .

100 , 23. 2015. 63

3.2. ? (Cooper) (Tars) ?

, . (2) . , t . 6.

, t v.

ds2 = (1 2MGr

)dv2 + 2dvdr + r2(d2 + sin d'2): (3)

5. . Q, , , P . , . , , P . , r > 2MG. , r = 0. Q , r, r. , . , , , r, .

, . , . , . , . .

3.3. , , - . , . .

6 Q ? , ., , . . . . , .

64 :

5- -:

. , , , . . , . Q, . , , . . , , () .

4. , , , -? . - ( ) , , . , () (); . , .

. . : . , , . . .

, . (Madeleine LEngle), . , . , .

100 , 23. 2015. 65

1. O. James, E. Tunzelmann, P. Franklin and K. S. orne, Gravitational Lensing by Spinning Black Holes in

Astrophysics, and in the Movie Interstellar, Class. Quant. Grav. 32 (2015) 6, 065001.2. O. James, E. Tunzelmann, P. Franklin and K. S. orne, Visualizing Interstellars Wormhole,

arXiv:1502.03809[gr-qc].3. K. S. orne, e Science of Interstellar, W. W. Norton & Company, New York, London (2014).4. C. W. Misner, K. S. orne and J. A. Wheeler, Gravitation, W. H. Freeman, San Francisco (1973).5. M. S. Morris and K. S.orne, Wormholes in space-time and their use for interstellar travel: A tool for teach-

ing general relativity, Am. J. Phys. 56 (1988) 395-412.

66 :

: @..

100 2015.7:6772

,

. . , . .

67

1. (). . -

R 12gR+

c2g=

8G

c4T ; (1)

G . T -. . : . A . aa - , - . .

, . - m .

ds2 = 1 2Gm

rc2

dt2 +

1

1 2Gmrc2

dr2 + r2d: (2)

. r = 2Gm r = 0. . , r = 2Gm . . . r = 2Gm , . -. , . , . . (2)

RR =

48m2

r6: (3)

r = 0 -, . -. -. t . , r < 2Gm. , . . . , , . ,ms,

rh =2Gm

c2= 3

m

mSkm :

3km, a 1cm. .

. . : , . .

68

2. , je . . . , . , , . . . . . .

, , . , . . , . .. , . . -h!/c mc , . , . . , , . .

, . , . . . , . . , , . , , . , . , , .

? , . . . . , . .

. : -h, , c ,G. , ,

lp =

r-hGc3

= 1; 62 1033cm;

mp =

r-hcG

= 2:18 105g;

100 , 23. 2015. 69

tp =

r-hGc5

= 5:39 1044s;

Ep =

r-hc5G

= 1:22 1019GeV:

.

3. , . , . 1976. , . . . . , ,. .

-.

(x) =

Zd3k

a(k)u(k) + ay(k)u(k)

;

u(k) , a(k) i ay(k) . :

[a(k); ay(q)] = kq; (4)[a(k); a(q)] = 0; (5)

[ay(k); ay(q)] = 0: (6)

, j0; xi a(k) j0; xi = 0. . x , y ,

(y) =

Zd3k

b(k)v(k) + by(k)v(k)

:

[b(k); by(q)] = kq; (7)[b(k); b(q)] = 0; (8)

[by(k); by(q)] = 0: (9)

j0; yi , . j0; xi 6= j0; yi. , . , . b(k) a(k) :

b(k) =

Zd3q

kqa(q) + kqa

y(q);

70

kq kq . a(k) j0; xi ,h0; xj ay(k)a(k) j0; xi = 0

, b(k)

h0; xj by(k)b(k) j0; xi = 1e-h!kT 1

:

.

T =1

8m= 6 108ms

mK:

, , . . , . . .

4. . . - . , . . .

. , . , . . 2, . , .. . , , . . - .

, - . . . , . , .

- . ,

[x; y ] = i :

.

100 , 23. 2015. 71

loop , , .- , . . - .

5. , . , . .

1. S. Caroll, Spacetime and Geometry, An Introduction to General Relativity, AW (2003)2. S. Kiefer, Quantum Gravity, OUP (2012)3. K. Becker, M. Becker and J. Schwarz, Stringeory and M-theory, CUP (2007)

72

knjiga-final.pdf00-Maja01-Blagojevic02-Cvetkovic03-Latas04-Bojan05-Dejan06-Marija07-voja

Documents

Opsta relativnost