Light deflection in Weyl gravity

GREX

27-29th October 2004

Sophie PIREAUX , UMR5562, Observatoire MIDI-PYRENEES

ObservatoireMidi-Pyrénnées

Based on articles:

Ø Light deflection in Weyl gravity: critical distances for photon paths. S. Pireaux, Classical and Quantum Gravity 21(2004) 1897-1913.

gr-qc/0403071

Ø Light deflection in Weyl gravity: constraints on the linear parameter.

S. Pireaux, Classical and Quantum Gravity 21 (2004) 4317-4333.

gr-qc/0408024

Plan

5 Constraints on linear parameter5.1 Solar system experiments: VLBI, CASSINI5.2 Beyond solar system experiments: microlenses, mirages

1 Introduction1.1 General relativity = final theory for gravitation1.2 Interesting features of Weyl theory1.3 Light deflection = good probe for Weyl theory …

2 Weyl theory2.1Weyl action2.2 Gravitational potential2.3 Mannheim-Kazanasparametrization2.4 Weak- versus the strong-field limit

3 Light deflection in Weyl theory3.1 Geodesic equation3.2 Critical radii for photons3.3 Conditions for light deflection

4 Amazing features of strong field regime4.1 Accumulation point4.2 Peculiar alignment configuration4.3 Observable regions of space

1. Introduction1.1 General relativity = final theory…Theoretical point of view:

- why Einstein-Hilbert action?- no quantum field theory perturbation- no conformal invariance- validity of Newtonian potential on very short/long distances?- …

Experimental point of view:- flat velocity distribution at galactic distances?-…

1.2 Interesting features of Weyl theory- conformal invariance- deviations Newtonian potential on long distances- could explain flat rotation curves without dark matter?- …

1.3 Light deflection = good probe for Weyl theory

2. The Weyl theory

2.1 Weyl action

αβγδ

αβγδWWgdxIW

4

ngravitatio � −=

���

��� +−−= � 24

31

2 RRRRRgdx αβ

αβ

αβγδ

αβγδ

Conformal transformations

… leave the action invariant

αβγδ

µ

αβγδ χ WxW )(2�

αβ

µ

αβ χ gxg )(2�

Weyl potential

Newtonian potential

( ) 22222

W

2

22332

2)(V cr

kcrcc

rr WW

WW

WWW −+−−−= γγβγββ

Gravitation equations

= Bach equations

= Einstein equations

0 2 =+≡ βααµβνµανβ

αβ

µν WWRB

0=µνR

Static spherically symmetric solution

Schwarzschild solution

νµ

µν dxdxgds =2

( )���

���

+−�

��

−�

��

⋅= 22222

-1

2

W22

2

W2 sin c

2V-1

c2V

-1 )( ϕϑϑχ µ ddrdrdtcx

If 0==

WWkγ

2c

MGN

W=β

2.2 Gravitational potential

( ) 22222

W

2

22332

2)(V cr

kcrcc

rr WW

WW

WWW −+−−−= γγβγββ

Newtonian term(weak field)

linear term(strong field)

Cosmological termconstant term

Breaks conformal transformationto asymptotically flat space.

-term negligible.WW

γβ

-term important only on cosmological distances.Does not contribute to photon motion.W

k

2.3 Mannheim-Kazanas parametrization

… to fit galactic rotation curves without dark matter 0>W

γ

( ) 11410 +++≈ mMgalaxyW

β

12610 −−+≈ mW

γ

… but based on assumption that cst)(2 =�

xχ

2.4 Weak- versus the strong-field limit

Weak field radius: Newtonian term dominates over the linear term

( )

W

WWWWWW

fieldweakr

γ

γβγβγβ

2

32

14131

2

−���

���

−+

±±+±<<<

( )

W

termWW

γγβ 1neglected

±≈−

mationparametrizKMandif

26 12

10+−=

+≈χ

Strong field radius: linear term dominates, Newtonian term neglected

… in this regime:

+ conditions on the radius to insure

W

W

fieldsr

γβ2

trong>>>

}

��

�

��

�

�

��

��

�

=⋅=⋅=⋅=⋅

≈

+

+

+

+

−=

cluster a

10for 102

10for 106

10for 102

galaxy a 10for 102

1522

1421

1321

1120

12

Sun

Sun

Sun

Sun

ationparametrizKMandif

MMm

MMm

MMm

MMm

χ

222

0W

2

2)(V cr

kcrr WW

W−+=

=

γβ

( )2222222222 sin (r)B (r)A ϕϑϑ ddrdrdtcds +−−=

>0>0

12 =χ

3. Light deflection in Weyl theoryA/ The geodesic equation: photons ( ) or massive particles ( )

( )���

��� +−+−==−

2243

3 32

2rJrrdr

dVF

WWW

geodesic

geodesic

�γββ

���

���+

3

23

rWWγβ

���

��� +−+

22

1JrW

�γ

���

��� −+

220

J

rk

W

�

“kinetic energy” “ total energy”

2

2

22

2

2

2

)(21

)(1J

E

c

rV

J

r

rd

drW =

���

��� +���

��� ++�

�

���

� χλ

�

0≡� 0>�

geodesicV = “geodesic potential”

diverges

ϕλ d

dr

rd

dr2

1≡where

function of the type of particle

• Photon geodesics are independent of unknown

light deflection is a good probe for Weyl gravity

)(2 �

xχ

• -term = null for photonsnon-null for massive particles:

positive (attractive) if negative (repulsive) if

0<W

k

Wk

0>W

k

• -term = positive (attractive) if negative (repulsive) if

0>W

γ0<

Wγ

WWγβ

• -term = if , negative (repulsive) for photons / relativistic particlespositive (attractive) for massive particles

… and vice-versa if

Wγ 0>

Wγ

0<W

γ

• Newtonian term = always positive (attractive)

3.2 Critical radii for photons

W

rγ2

min−≈

Wr β3

max≈

W

rγ3

inflection−≈

… is a particular scaleW

γ/1−

W

nullr

γ1−≈

let and …0≠W

γ0≠W

β

or

On large distance scales:linear term dominates

On short distance scales:Newtonian term dominates

Schwarzchild likebound orbits

3.3 Conditions for light deflection

r = 0

… unbound orbits,

… if convergent, if divergent0)(ˆ0

>rα 0)(ˆ0

<rα

…in the weak field regime

}

cluster a

10for 102

10for 108

10for 102galaxy a 10for 102

for 108

1522

1421

1321

1120

14

12

��

��

�

=⋅=⋅=⋅=⋅=⋅

≈

+

+

+

+

+

−=

Sun

Sun

Sun

Sun

SunationparametrizKMandif

MMm

MMm

MMmMMm

MMmχ

r > r000r < r0 00

r = r000

( )0

0

0 2

3322)(ˆ r

rr

WWW

WWW

fieldweakγπγβγββα −+−=

The light deflection angle

( )0

0

0 2

3322)(ˆ r

rr

WWW

WWW

fieldweakγπγβγββα −+−=

W

Wrγβ

20 0

≈

0>W

γCritical radius from the weak field deflection angle for

…in the strong field regime

The types of orbits allowed can be classified:

if : , hyperbolic (e >1)

if : , hyperbolic (e >1)

, parabolic (e =1)

, elliptic (e <1), circular case (e =0) for

0 r∀

nullrr =

0

nullrr <

0

nullrr >

0 min0rr =

0>W

γ

0<W

γ

( )initialr

rr

W

W

W

W

ϕϕγ

γγ

β

±±+−

−==

sin 2

1

/2

0

0

r

re

W

W

γγ

0 2+≡with , the excentricity

Open versus closed orbits in the strong regime

0>W

γ

0<W

γ

The light deflection angle

���

����

�

+−=

=

0

0

00 2arcsin2)(ˆ

r

rr

W

W

W γγαβ

… recover the weak field regime

00 /)(ˆ rr

Wfieldstrongweakγα −≈

4.1 Accumulation point:

4. Amazing features of strong field regime for a negative parameter

4.2 Peculiar alignment configuration:

4.3 Observable regions of space:

( )0

0

0 2

3322)(ˆ r

rr

WWW

WWW

fieldweakγπγβγββα −+−=

linear parameter estimateW

γ

5. Constraints on linear parameter

5.1 Solar system experiments: VLBI, CASSINI

PPN parameter estimateγ

( )0

0

12)(ˆ

r

GMr

fieldweak

γα +=0

0

0

4)(ˆ r

rr

W

W

fieldweakγβα −≅

2

c

MGSunN

W=β

extrapolate at solar limb

VLBI

[ ] -1-17-18 m 103.1 ,109.7 ⋅+⋅−∈W

γ

[ ] -1-18-18 m 103.1 ,107.1 ⋅+⋅−∈W

γextrapolate at solar lim

b

linear parameter estimateW

γ

00045.099983.0 ±=γ

0017.09996.0 ±=γ[Lebach et al. 1995]

[Shapiro et al. 2004]

PPN parameter estimateγVery Long Baseline

Interferometry

Medicina 25m

CASSINI mission Doppler Measurements

[ ] -1-19-20 m 107.2 ,102.1 ⋅+⋅−∈W

γextrapolate at solar lim

b

linear parameter estimateW

γ

[Bertoti et al. 2003]

PPN parameter estimateγ

( ) 5103.21.21 −×±−=−γ

5.2 Beyond solar system experiments: microlenses, mirages

Mirages exist

Abell 2218 (Hubble, NASA)

Constraints on a negative linear parameter

If , that separates0<W

γ E

nullr

: bound orbits light deflection not possible

: unbound orbits light deflection possible

nullrr >

0

nullrr <

0

0for 0 <<≈W

OL

null

OL

E D

r

D

r γϑ

0for m 107.1 3.0

1

1-31

0<�

��

< −

W

LE

W zh γ

ϑγ

Hubble empirical law (D, z)

arsecs ]0.55; 0.75]

~

Microlensing or lensing light curves

Lens equation (weak field limit):

0- n1

1-

2

W

2 =+ WIsI

ϑϑϑϑ���

with = angular radius of Weyl ring

OS

OLLS

W D

DD n

Wγ≡

EWϑϑ

Wn1

1+

≡

Wϑ

…but corrective factor small, maybe negligible (lens statistic required)??

• � critical radii function of : structure space-time (photons)

Summary of results

• They are physical or not according to the sign of Wγ

… light deflection always possible in ANY limit.0>W

γ

… which separatesbound/unbound orbits.0<W

γ E

nullr

light deflection always possible in THIS limit.nullfieldweakrr ≈

… which separates convergent/divergent light deflection;

it is also function of the deflector mass.

E

0 0r

Wγ

light deflection always divergent in THIS limit.0 0 rr

fieldstrong≈

0=W

γ … General Relativity

• Light deflection = good probe for Weyl theory:

0<W

γ-131 m 10−<W

γ for … from the existence of mirages

-119 m 10−<W

γ … from Solar System experiments (CASSINI)

0>W

γ0=W

γ 0<W

γ… BUT does not select between , or

Present analysis could be refined: amazing features?, different lens-mass models, mirage statistics …

… future missions: improve estimate of PPN improve estimate of

- GAIA [GAIA repport 2000] :

- LATOR [Turyshev et al 2004] :

- …

γW

γ-710 5~at ⋅γ-810 5~at ⋅γ

BIBLIOGRAPHY included in articles:

Ø Light deflection in Weyl gravity: critical distances for photon paths. S. Pireaux, Classical and Quantum Gravity 21(2004) 1897-1913.

gr-qc/0403071

Ø Light deflection in Weyl gravity: constraints on the linear parameter.

S. Pireaux, Classical and Quantum Gravity 21 (2004) 4317-4333.

gr-qc/0408024