Newtonian limit: slow motion in weak fields

Newtonian gravitation

The Newton's law of gravitation states that two bodies with masses m and M located at a relative distance r attract each other with the force
F=G mM
r2
 ,
(1)
where G ≈ 6.67×10−11Nm2kg−2 is the gravitational constant measured by Cavendish.
In other words a distribution of masses with density μ creates a gravitational potential φ([(r)\vec]) which satisfies the Poisson equation
∆φ = 4πGμ,
(2)
and a test body m in the gravitational potential φ is affected by a force
→
F
 
 =−m
→
∇
 
 φ.
(3)
The last equation can be cast into a variational form with the action
S=⌠
⌡ 		dt
 −mc2+ 1
2
 mv2−mφ
= −mc⌠
⌡ 		dt
 c− v2
2c
 + φ
c

 .
(4)
Comparing with S=−mc∫ds we get (in the limit c→∞)
ds2=
 1+ 2φ
c2

 c2dt2−d
→
r
 
 2
 
 .
(5)
In other words in the Newtonian limit the metric tensor can be approximated1 by gab=ηab+hab (where ηab is the Minkowski metric tensor and hab is a small correction) where the g00 component is
g00=1+ 2φ
c2
 .
(6)

Newtonian limit of general relativity

In this limit all fields are weak and all velocities are small. Only the 00 component of the energy-momentum tensor is non-vanishing, T00=μ where μ is the mass density of the matter. Therefore we shall only consider the 00 component of the Einstein's equation, R00=κ(T00−[1/2]g00T).
In the Newtonian limit g00=1 + 2φ, Γα00=−φ,α, R00=−φ,α,α=∆φ, where α = 1,2,3. The Einstein equation thus turns into the the Poisson's equation ∆φ = [1/2]κμ which is equivalent to the Newtonian theory if we put κ = [(8πG)/(c4)].

Gravitational waves.

In a weak gravitational field the space-time is almost flat and the metric tensor gab is equal to the flat metric ηab plus a small correction hab, gab = ηab+hab. The Riemann tensor to the lowest order in hab is
Rabcd = 1
2
 (had,bc + hbc,ad − hac,bd − hbd,ac).
(7)
If we choose coordinates such that (hab − [1/2]hδab),b=0, the Ricci tensor is simply
Rab = − 1
2
 hab,c,c
(8)
and the vacuum Einstein's equations turn into the ordinary wave equations

∂2
∂t2
 − ∆
 hab = 0.
(9)
The intensity of gravitational radiation by a system of slowly moving bodies is determined by its quadrupole moment Dαβ
− dE
dt
 = G
45c5
 (D"′αβ)2
(10)

Exercises

  1. Calculate the energy-momentum tensor Tab for a particle of mass m with the action S=−m∫ds. Hint: calculate the variation of the action with respect to δgab and represent it in the form δS = −1/2∫Tabδgabds.
  2. In the Minkowski space consider a scalar field ϕ with action
    S = ⌠
    ⌡     		dΩ
    		 − 1
    2
    		 ϕ,aϕ,a− 1
    2
    		 m2ϕ2
    and calculate its "translation-invariance" energy-momentum tensor,
    Tab= ∂L
    ∂ϕ,a
    		 ϕ,b−Lδab  .
    Rewrite the action in a generally covariant form and calculate its "metric-derivative" energy-momentum tensor,
    1
    2
    √
     
    −g
     
    		 Tab =
    δ(
    √
     
    −g
     
    		 L)
    δgab
    		   .

Footnotes:

1 where we have neglected the terms gαβ,  αβ = 1,2,3 since their contribution to ds2 is not multiplied by c2 and is thus neglidible compared to the contribution from g00.

File translated from TEX by TTH, version 3.77.
On 6 Oct 2007, 00:05.