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Lecture 5Classical and Modern Tests of

General Relativity

Ho Jung Paik

University of Maryland

February 8, 2007

Physics 798G Spring 2007

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From Newton to Einstein 1

• Galileo’s Universality of Free Fall: All bodies fall with the same acceleration regardless of their mass or internal composition.

⇒ Newton’s Weak Equivalence Principle: mP (passive grav) = mI

Stationary

Free Fall

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From Newton to Einstein 2

• Einstein Equivalence Principle: In a freely falling frame, all the laws of physics behave as if gravity is absent.

(EEP ⇒ WEP + LLI + LPI)

⇒ Metric theories of gravity(curvature of spacetime)

• Strong Equivalence Principle: WEP is valid for self-gravitating bodies as well as test bodies. (GR may uniquely embody SEP.)

Special Relativity ⇒ General Relativity

−

=

=

θ

η

η

µν

νµµν

2sin2000020000100001

,2

rr

dxdxds

=

=

−

−

−−

θ

µν

νµµν

2sin20000200

000

000

,

1

221

221

2

rr

g

dxdxgds

rc

GM

rc

GM

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General Relativity 1

• Law of gravity: Newton: ⇒ Einstein:

• “Field” equations: Maxwell: ⇒ Einstein: (φ ≈ 0, v « c = 1)

• Equation of motion: Newton: ⇒ Einstein:

( )( ) fluid)(perfect sor energy ten-stress :

,

)(curvatureensor Einstein t :)(

,,,,21

21

µννµµν

βδαγαδβγαγβδβγαδαβγδβµαναβ

µν

αβαβ

µνµνµν

ρ pguupTggggRRgR

RggRG

++=

−+−==

−=

ρφ Gπ42 −=∇ µνµν TcGG 4

π8=

mF

dtxd=2

2

equation geodesic

02

2

=Γ+τττ

βαµαβ

µ

ddx

ddx

dxd

.cπ41 ,0

,01 ,π4

JEBB

BEE

=∂∂

−×∇=⋅∇

=∂∂

+×∇=⋅∇

tc

tcρ

( ) sconnection metric :,,,21

βαγαγβγβααβγ ggg +−=Γ

.16 ,0

,0 ,π4

vE

BB

BEE

πρ

ρ

−≈∂∂

−×∇≈⋅∇

≈∂∂

+×∇≈⋅∇

t

tg

gg

ggg

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General Relativity 2

• Weak field, low velocity limit: ⇒

• Wave equation: EM wave: Gravitational wave:

• Expanding universe ⇒ Big Bang cosmology

• Generalized field equation:

λ: cosmological constant (~10−29g/cm3 : “dark energy”?)

µνµν TcGG 4

π8= ρφ Gπ42 −=∇

µνµνµνµν η hghtc

+==

∂−∇ ,011

222 ,0),(11

222 =

∂−∇ BE

tc

µνµνµν λ TcGgG 4

π8=−

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Classical Tests of General Relativity 1

1. Perihelion shift of MercuryThe deviation of Mercury orbit around the sun from its Kepler orbit due to a correction to the 1/r2 law.

Einstein: < 0.1Oblateness of the sun?

undetectedAnother planet (Vulcan)?

43.1 ± 0.5Additional shift

43.0General Relativity

532Perturbation from other planets

Perihelion shift (arcsec/century)

Cause

τφ

)1(π6

22 eaM

cG

dtd

−=

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Classical Tests of General Relativity 2

2. Deflection of lightWhen light passes near the sun, its path is slightly bent.

Einstein: , twice the Newtonian effect

⇒ Gravitational lensingbM

cG S2

4=∆φ

1.760 ± 0.016Fomalont and Sramek(1976)

1.98 ± 0.16 (Sobral)1.16 ± 0.40 (Principe)

Eddington (1919)

1.750General Relativity

Deflection (arcsec)Prediction/Experiment

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Classical Tests of General Relativity 3

3. Gravitational red shiftClocks in a gravitational potential well observed from a stationary clock at a distant point appear to tick slower.

Einstein:

(Einstein Equivalence Principle)

22 cgh

cU

=∆

=∆νν

4 × 10−10 for h = 10,000 km

5 × 10−15 for h = 22.6 mRed shift

7 × 10−5Vessot (1980)“Gravity Probe A”

10−1

10−2

Pound-Rebka (1960)Pound-Snider (1964)

Agreement with GRExperiment

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Modern Tests of General Relativity 1

1. Weak Equivalence Principle and the Inverse-Square Law

For a generalized potential: ( )cmr

mmGrVb

r h=+−= − λα λ ,e1)( /21

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Modern Tests of General Relativity 2

2. Local Lorentz Invariance and Local Position Invariance

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Modern Tests of General Relativity 3

3. Gravitational time delay (Shapiro effect)As the radar and radio beams pass close to the Sun, a delay in the transit time is measured. This delay is caused by the gravitational force of the Sun.

Gravitational time delay:

10−1Shapiro (1968)Radar to Mars

10−3Shapiro et al. (1970’s)Radio from Viking

Agreement with GR

Experiment

+

=∆ 14ln4

23 brr

cGMt ME

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Modern Tests of General Relativity 4

4. Strong Equivalence Principle

Free fall is independent of gravitational self energy GM2/r.

⇒ Uniquely GR

(The Nordtvedt effect)

< 10−3Binary pulsar (2004)strong field

< 1.6 × 10−3Lunar laser ranging (1972- )weak field

Limit on |η| Experiment

22 ,'1

rcGMEEE

mm

selfselfselfi

g −=++= ηη

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Modern Tests of General Relativity 5

5. Binary pulsar timing

GR predicts that such a system will radiate energy in gravitational waves, causing the stars to slowly spiral towards each other.

Decay of the orbit period:

45

33

512)(1

acMGe

dtd ητ

τ−=

−2.40(9) × 10−12Taylor et al. (1975- )−2.403(2) × 10−12General Relativitydτ /dtPrediction/Experiment