Master Thesis

With this metric, the velocity perturbation in the rest frame results by the normalization condition g µνu µ u µ =−1 to:
u µ = 1�
i
1−AY , vY
a
�
, u µ=a � 1−AY ,(v−B)Y i�
(5.11)
The choice of coordinates in general relativity is free. Thus the physics should be independent of a displacement by a
small vector X= T∂ 0+ L i ∂ i. This yields the gauge transformation:
˜ h=h+ a −2 LX ¯g , (5.12)
where L X is the Lie derivative and h is the scalar + vector + tensor perturbation of the metric.
Comparing ˜ h and h, one can read off the behavior of the scalar, vector and tensor amplitudes under the gauge transformation:
à = A+ ˙a
a T+ ˙T (5.13)
˜B = B−˙L−kT (5.14)
˜B (V) = B (V) − ˙L (V)
(5.15)
˜H L = HL+ ˙a k
T+ L
a 3
(5.16)
˜H T = HT− kL (5.17)
˜H (V) = H (V) − kL (V)
˜H (T) = H (T)
(5.18)
(5.19)
It turns out, that the vector perturbations B (V) and H (V) are damped (see [27] and [28]). So they can be neglected today.
The obviously gauge invariant metric perturbation H (T) will characterize gravitational waves. Though it is not coupled
to the rest of the perturbation variables it will not be treated in the following.
Common known gauges are:
Longitudinalgauge: This gauge is defined by H (long)
T = 0 and B (long) = 0. The remaining perturbation variables A=Ψand
HL=Φ are like the background quantities diagonal entries of the metric and have the meaning of a gravitational
potential and a constant curvature. For this reason it is also named Conformal Newtonian gauge.
Synchronous gauge: The synchronous gauge is defined by: A s= 0, B s= 0 .
One introduces the gauge invariant Bardeen potentials, which are defined as:
5.1.2 Perturbationsof theenergymomentum tensor
In the rest frame the general form of the energy momentum tensor
can be rewritten to an eigenvector problem by multiplying u ν :
Requiring a density and pressure perturbation:
Ψ = A− ˙a
a k−1 σ− k −1 ˙σ (5.20)
Φ = H L+ 1
3 H T− ˙a
a k−1 σ , (5.21)
withσ=k −1 ˙H T− B . (5.22)
T µ
ν =(ρ+p)uµ uν+ pg µ
ν , (5.23)
T µ
ν uν =−ρu ν
(5.24)
ρ= ¯ρ(1+δY) , p=¯p(1+π LY) , (5.25)
i(f ull)
and an anisotropic, traceless (Π j = 0), space-like pressure perturbation (named shear):
i(f ull)
Π j =ΠY i
j +Π(V) Y (V)i
j
+Π(T) Y (T)i
j
(5.26)
30 5.1 Perturbations of thebackground quantities