466
V. Massart and M. B. Paranjape
the review article of Mashhoon [2] and that of Ruggiero and Tartaglia [3]. This work
is inspired by a previous work of M. Paranjape [4].
We begin with the full Einstein equations,
R μν −
1
2
g μν R = 8πGT μν .
(1)
We linearize the equations by expanding the metric around the Minkowski vacuum
with small perturbations : g μν = η μν + h μν , |h μν |λ1. We then compute the metric
connection (the Christoffel symbols), the Riemann tensor, the Ricci tensor, and the
curvature scalar in the approximation where we neglect terms of order h 2 and higher.
We make the following gauge choice,
∂
μ (h μν −
1
2
η μν h
λ
λ ) ≡ ∂
μ ¯
h μν = 0,
(2)
which fixes four of the degrees of freedom of the metric. This leads to the linearized
Einstein equations,
¯
h μν = −16πGT μν .
(3)
This is a wave equation with a source, it describes propagating gravitational waves
in the absence of sources.
Remarkably, Eq. (3) looks essentially identical to Maxwell’s equations for the
electromagnetic potential in Lorenz gauge with sources,
ν
= 4πj
ν .
This similarity is very useful since electromagnetism is well understood analytically
and intuitively, in particular we know the physical (retarded) solution,
A
ν
= 4π
D R (x − x
)J
ν (x
)d
4 x
,
as well as the general form of the corresponding electric and magnetic fields, where
D R (x − x ) is the retarded Green function of the d’Alembertian.
We will use the non-relativistic limit for the sources in the linearized Einstein
equations. The space–space part of the stress energy tensor on the right-hand side
of Eq. (3) can be neglected as typically T ij ∼ v i v j while T 0i ∼ v i and T 00 ∼ v 0 .
Then in the weak-field, non-relativistic limit, keeping terms to first order in v i , we
find for the gravitational analog of the electromagnetic gauge potential ¯
h 0ν (x),
¯
h 0ν (x) = −16πG
D R (x − x
)T
0ν (x
)d
4 x
.
(4)
V. Massart and M. B. Paranjape
the review article of Mashhoon [2] and that of Ruggiero and Tartaglia [3]. This work
is inspired by a previous work of M. Paranjape [4].
We begin with the full Einstein equations,
R μν −
1
2
g μν R = 8πGT μν .
(1)
We linearize the equations by expanding the metric around the Minkowski vacuum
with small perturbations : g μν = η μν + h μν , |h μν |λ1. We then compute the metric
connection (the Christoffel symbols), the Riemann tensor, the Ricci tensor, and the
curvature scalar in the approximation where we neglect terms of order h 2 and higher.
We make the following gauge choice,
∂
μ (h μν −
1
2
η μν h
λ
λ ) ≡ ∂
μ ¯
h μν = 0,
(2)
which fixes four of the degrees of freedom of the metric. This leads to the linearized
Einstein equations,
¯
h μν = −16πGT μν .
(3)
This is a wave equation with a source, it describes propagating gravitational waves
in the absence of sources.
Remarkably, Eq. (3) looks essentially identical to Maxwell’s equations for the
electromagnetic potential in Lorenz gauge with sources,
ν
= 4πj
ν .
This similarity is very useful since electromagnetism is well understood analytically
and intuitively, in particular we know the physical (retarded) solution,
A
ν
= 4π
D R (x − x
)J
ν (x
)d
4 x
,
as well as the general form of the corresponding electric and magnetic fields, where
D R (x − x ) is the retarded Green function of the d’Alembertian.
We will use the non-relativistic limit for the sources in the linearized Einstein
equations. The space–space part of the stress energy tensor on the right-hand side
of Eq. (3) can be neglected as typically T ij ∼ v i v j while T 0i ∼ v i and T 00 ∼ v 0 .
Then in the weak-field, non-relativistic limit, keeping terms to first order in v i , we
find for the gravitational analog of the electromagnetic gauge potential ¯
h 0ν (x),
¯
h 0ν (x) = −16πG
D R (x − x
)T
0ν (x
)d
4 x
.
(4)