18. Experimental tests of gravitational theory
1
18. EXPERIMENTAL TESTS OF
GRAVITATIONAL THEORY
Revised November 2009 by T. Damour (IHES, Bures-sur-Yvette, France).
Einstein’s General Relativity, the current “standard” theory of gravitation, describes
gravity as a universal deformation of the Minkowski metric:
g
μν
(
x
λ
) =
η
μν
+
h
μν
(
x
λ
)
,
where
η
μν
= diag(
−
1
,
+1
,
+1
,
+1)
.
(18
.
1)
Alternatively, it can be defined as the unique, consistent, local theory of a massless
spin-2 field
h
μν
, whose source must then be the total, conserved energy-momentum
tensor [1].
General Relativity is classically defined by two postulates. One postulate
states that the Lagrangian density describing the propagation and self-interaction of the
gravitational field is
L
Ein
[
g
μν
] =
c
4
16
πG
N
√
gg
μν
R
μν
(
g
)
,
(18
.
2)
R
μν
(
g
) =
∂
α
Γ
α
μν
−
∂
ν
Γ
α
μα
+ Γ
β
αβ
Γ
α
μν
−
Γ
β
αν
Γ
α
μβ
,
(18
.
3)
Γ
λ
μν
=
1
2
g
λσ
(
∂
μ
g
νσ
+
∂
ν
g
μσ
−
∂
σ
g
μν
)
,
(18
.
4)
where
G
N
is Newton’s constant,
g
=
−
det(
g
μν
), and
g
μν
is the matrix inverse of
g
μν
. A
second postulate states that
g
μν
couples universally, and minimally, to all the fields of
the Standard Model by replacing everywhere the Minkowski metric
η
μν
. Schematically
(suppressing matrix indices and labels for the various gauge fields and fermions and for
the Higgs doublet),
L
SM
[
ψ, A
μ
, H, g
μν
] =
−
1
4
√
gg
μα
g
νβ
F
a
μν
F
a
αβ
−
√
g
ψ γ
μ
D
μ
ψ
−
1
2
√
gg
μν
D
μ
HD
ν
H
−
√
g V
(
H
)
−
λ
√
g
ψ Hψ ,
(18
.
5)
where
γ
μ
γ
ν
+
γ
ν
γ
μ
= 2
g
μν
, and where the covariant derivative
D
μ
contains, besides the
usual gauge field terms, a (spin-dependent) gravitational contribution Γ
μ
(
x
) [2].
From
the total action follow Einstein’s field equations,
R
μν
−
1
2
Rg
μν
=
8
πG
N
c
4
T
μν
.
(18
.
6)
Here
R
=
g
μν
R
μν
,
T
μν
=
g
μα
g
νβ
T
αβ
, and
T
μν
= (2
/
√
g
)
δ
L
SM
/δg
μν
is the (symmetric)
energy-momentum tensor of the Standard Model matter. The theory is invariant
K. Nakamura
et al.
, JPG
37
, 075021 (2010) (http://pdg.lbl.gov)
July 30, 2010
14:36