Outline of the LecturesOutline of the Lectures

Lecture 1: The Einstein Equivalence PrincipleLecture 2: Post-Newtonian Limit of GRLecture 3: The Parametrized Post-Newtonian FrameworkLecture 4: Tests of the PPN Parameters

Outline of the LecturesOutline of the Lectures

Lecture 1: The Einstein Equivalence PrincipleLecture 2: Post-Newtonian Limit of GRLecture 3: The Parametrized Post-Newtonian Framework

Parametrizing the PN metric Conservation laws Equations of motion - photons Equations of motion - massive bodies Equations of motion - gyroscopes Locally measured gravitation constant G The Strong Equivalence Principle

Lecture 4: Tests of the PPN Parameters

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g00 = −1+ 2U − 2U 2 + ˙ ̇ X

+ 4Φ1 + 4Φ2 + 2Φ3 + 6Φ4

g0i = −4Vi

gij = δij (1+ 2U)

The PN metric in GRThe PN metric in GR

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g i j = gij − 2λ 2(δ ijU −U ij )

g 0 i

= goi + (λ1 + λ 2)(Vi −W i)

g 0 0

= g00 + 2λ1(A + B − Φ1) + 2λ 2(U 2 + ΦW − Φ2)

Effect of a PN gauge changeEffect of a PN gauge change

Parametrizing the post-Newtonian Parametrizing the post-Newtonian metricmetric

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g00 = −1+ 2U − 2βU 2 + (1+ α 2) ˙ ̇ X − 2ξΦW

+(2γ + 2 + α 3)Φ1 + 2(3γ − 2β +1+ ς 2 + ξ )Φ2

+2(1+ ς 3)Φ3 + 6(γ + ς 4 )Φ4 + (ς1 − 2ξ )B

g0i = −4γ +1

2+

α 1

8

⎛

⎝ ⎜

⎞

⎠ ⎟Vi

gij = δ ij (1+ 2γU)

Effect of a BoostEffect of a Boost

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g 0 0

(ξ ,τ ) = g00(ξ ,τ ) − (α 1 −α 2 −α 3)w2U −α 2wiw jU ij

+ (2α 3 −α 1)w jV j

g 0 i

(ξ ,τ ) = g0i(ξ ,τ ) − 12 (α 1 − 2α 2)w iU −α 2w

jU ij

g i j (ξ ,τ ) = gij (ξ ,τ )

The parametrized post-Newtonian The parametrized post-Newtonian (PPN) framework(PPN) framework

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g00 = −1+ 2U − 2βU 2 + (1+ α 2) ˙ ̇ X − 2ξΦW

+(2γ + 2 + α 3)Φ1 + 2(3γ − 2β +1+ ς 2 + ξ )Φ2

+2(1+ ς 3)Φ3 + 6(γ + ς 4 )Φ4 + (ς1 − 2ξ )B

−(α 1 −α 2 −α 3)w2U −α 2wiw jU ij + (2α 3 −α 1)w

jV j

g0i = −4γ +1

2+

α 1

8

⎛

⎝ ⎜

⎞

⎠ ⎟Vi − 1

2 (α 1 − 2α 2)w iU −α 2wjU ij

gij = δij (1+ 2γU)

ParameterWhat it measures, relative

to general relativityValue in

GR

Value in scalar

tensor theory

Value in semi-conservative

theories

How much space curvature produced by unit mass?

1(1+)/(2+)

How “nonlinear’’ is gravity? 1 1 +

Preferred-location effects? 0 0

Preferred-frame effects?

0 0

0 0

0 0 0

Is momentum conserved?

0 0 0

0 0 0

0 0 0

0 0 0

PPN Parameters and their SignificancePPN Parameters and their Significance

PPN n-bodyPPN n-bodyequation of equation of motionmotion

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(aa )Newt =(ma )P

(ma )I

∇(mb )A

rab

⎛

⎝ ⎜

⎞

⎠ ⎟

b≠a

∑

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(ma )P

(ma )I

=1− (4β − γ − 3− 103 ξ −α 1 + 2

3 α 2 − 23 ζ 1 − 1

3 ζ 2)Ωa

ma

(mb )A

(mb )I

=1− (4β − γ − 3− 103 ξ − 1

2 α 3 − 13 ζ 1 − 2ζ 2)

Ωb

mb

+ ζ 3

Eb

mb

− ( 32 α 3 + ζ 1 − 3ζ 4 )

Pb

mb

Ω = − 12

ρ ′ ρ

| x − ′ x |∫∫ d3x d3 ′ x , E = ρΠ∫ d3x, P = p∫ d3x

Newtonian” part of the n-body accelerationNewtonian” part of the n-body acceleration

PPN n-bodyPPN n-bodyequation of equation of motionmotion

The problem of motionThe problem of motion

Geodesic motion 1916 - Droste, De Sitter - n-body equations of motion 1918 - Lense & Thirring - motion in field of spinning body 1937 - Levi-Civita - center-of-mass acceleration 1938 - Eddington & Clark - no acceleration 1937 - EIH paper & Robertson application 1960s - Fock & Chandrasekhar - PN approximation 1967 - the Nordtvedt effect& the PPN framework 1974 - numerical relativity - BH head-on collision 1974 - discovery of PSR 1913+16

The Strong Equivalence Principle (SEP)The Strong Equivalence Principle (SEP)

All bodies fall with the same accelerationWeak Equivalence Principle (WEP)

In a local freely falling frame, all physics is independent of frame’s velocity

Local Lorentz Invariance (LLI)In a local freely falling frame, all physics is independent of frame’s location

Local Position Invariance (LPI)

W€

gμν → η μν

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gμν →η μν

φ → φ0

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gμν → η μν

K μ → (K 0,0,0,0)

→ (γK 0,γK 0W ,0,0)