Lecture 23
Linear GR
Objectives:
• Linearised GR
Reading: Schutz 8; Hobson 17; Rindler 15
23.1
Approximating GR
The non-linearity of GR makes it difficult to solve in most situations. It is
useful to develop an approximate form of the field equations for the common
case of weak fields.
In weak fields we can assume that there are coordinates xα in which the
metric can be written
gαβ = ηαβ + hαβ ,
where |hαβ | ¿ 1. Using this the field equations
1
Rαβ − Rgαβ = kTαβ ,
2
can be approximated to first order in h.
e.g. the connection
1 αδ
g (gδγ,β + gβδ,γ − gβγ,δ ) ,
2
1 αδ
η (hδγ,β + hβδ,γ − hβγ,δ ) ,
=
2
Γα βγ =
is first-order in h, so the Riemann tensor boils down to
Rρ αβγ = Γρ αγ,β − Γρ αβ,γ .
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LECTURE 23. LINEAR GR
Eventually one finds:
h,αβ + 2hαβ − η γδ (hαγ,δβ + hδβ,αγ ) − (2h − hσρ ,σρ ) ηαβ = 2kTαβ ,
where h = η αβ hαβ and
2 = η σρ ∂σ ∂ρ = ∂σ ∂ σ =
1 ∂2
− ∇2 ,
2
2
c ∂t
is the D’Alembertian or wave operator.
23.2
Lorenz Gauge
The choice of hαβ is not unique; it depends on the underlying coordinates.
This can be used to simplify the linearised equations. For instance consider
the coordinate transform
x′α = xα + ²α ,
with ²α and its derivatives ¿ 1 (easier here not to put primes on indices;
hαβ is not a tensor). Then
∂x′γ ∂x′δ ′
g ,
∂xα ∂xβ γδ¡
¢ ′
,
= (δαγ + ²γ ,α ) δβδ + ²δ ,β gγδ
gαβ =
so
Thus
¢
¢¡
¡
ηαβ + hαβ = (δαγ + ²γ ,α ) δβδ + ²δ ,β ηγδ + h′γδ .
ηαβ + hαβ = ηαβ + ²δ ,β ηαδ + ²γ ,α ηγβ + h′αβ ,
and so
h′αβ = hαβ − ²α,β − ²β,α .
Very similar to gauge transformation of EM where the physics is invariant
to transforms of the 4-potential of the form
A′α = Aα + ψ,α ,
where ψ is some scalar field.
Choose ²α to simplify field equations. In particular choosing coordinates
such that
1
hαβ ,β = η αβ h,β ,
2
(“Lorenz gauge”), then the field equations reduce to
1
2hαβ − ηαβ 2h = 2kTαβ .
2
Do not try to
remember this!
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LECTURE 23. LINEAR GR
Further simplification comes from defining
1
h̄αβ = hαβ − hηαβ ,
2
(“trace reversal” since h̄ = −h). The Lorenz gauge becomes
h̄αβ,β = 0,
while the field equations reduce to
2h̄αβ = 2kTαβ ,
or in full:
µ
¶
1 ∂2
16πG
2
− ∇ h̄αβ = − 4 Tαβ .
2
2
c ∂t
c
There is still some remaining freedom: the same relations survive coordinate
transforms x′α = xα + ²α provided
2²α = 0.
23.3
Newtonian limit [not in lectures]
Consider a time-independent, weak-field. Setting k = −8πG/c4 , and 2 =
−∇2 , the field equations become
∇2 h̄αβ =
16πG αβ
T ,
c4
which has the form of Poisson’s equation. If all mass is stationary, then only
T 00 = ρc2 is significant so we have
∇2 h̄00 =
16πGρ
,
c2
and by analogy with
∇2 φ = 4πGρ,
we can immediately write
4φ
,
c2
where φ is the Newtonian potential. All other components = 0.
h̄00 =
From this we deduce h = −h̄ = −4φ/c2 , and since
1
hαβ = h̄αβ + hη αβ ,
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LECTURE 23. LINEAR GR
96
we find
2φ
,
c2
Finally, since g αβ = η αβ + hαβ , and lowering indices we find
µ
¶
µ
¶
¢
2φ
2φ ¡ 2
2
2
2
ds = c 1 + 2 dt − 1 − 2
dx + dy 2 + dz 2 .
c
c
h00 = h11 = h22 = h33 =
This approximate metric is useful for studying gravitational lensing around
anything more complex than a point mass, e.g. a star plus planets, or clusters
of galaxies.