ASSIGNMENTS Week 7 (F. Saueressig) Cosmology 14/15 (NWI

ASSIGNMENTS Week 7 (F. Saueressig)
Cosmology 14/15 (NWI-NM026C)
Prof. A. Achterberg, Dr. S. Larsen and Dr. F. Saueressig
In the lectures we derived Einstein’s equations
Gµν ≡ Rµν − 12 gµν R = 8πGTµν
(1)
where Gµν is called Einstein tensor,
Rαβ = ∂γ Γγ αβ − ∂β Γγ αγ + Γγ αβ Γδ γδ − Γγ αδ Γδ γβ
(2)
is the Ricci tensor, R = gαβ Rαβ denotes the Ricci scalar, and Tµν is the stress-energy tensor
capturing the matter content of spacetime. In this assignment, we will study the dynamics of
linear perturbations around flat space. This framework allows to construct the gravitational wave
solutions of general relativity (Exercise 4) and fixes the coupling constants between the curvature
and matter sector of Einstein’s equations (Exercise 5).
Exercise 1: Vacuum solutions of Einstein’s equations are Ricci flat
Show that in vacuum, Tµν = 0, Einstein’s equations (1) reduce to
Rµν = 0 ,
(3)
implying that in the absence of matter spacetime must be Ricci flat.
Exercise 2: Derivation of the linearized Einstein equations
In many physical settings, it is a good approximation to expand the curved spacetime metric gµν
around the flat Minkowski metric η = diag[1, −1, −1, −1], setting
gαβ (xα ) = ηαβ + hαβ (xα ) .
(4)
This expansion can be carried out systematically to any order in the perturbations hαβ . In this
exercise, we derive the linearized Einstein equations controlling the dynamics of hαβ by keeping
the first order terms in hαβ while terms which are quadratic in powers of hαβ are neglected. This
computation can be carried out along the following lines:
a) Substitute the ansatz (4) into the definition of the Christoffel symbol to obtain an explicit
formula for δΓµ αβ valid at first order in hαβ .
b) Based on your result from a) show that the linearized Ricci tensor δRαβ is given by
δRαβ =
1
2
(−✷hαβ + ∂α Vβ + ∂β Vα ) .
(5)
Here all indices are raised and lowered with the Minkowski space metric ηαβ and
Vα ≡ ∂γ hγ α − 12 ∂α hγ γ ,
✷ ≡ η µν ∂µ ∂ν .
(6)
c) Using the intermediate result (5), write down the linearized Einstein equations governing
the dynamics of hαβ . How do these equations simplify in vacuum?
Exercise 3: Gauge transformations
Under a change of coordinates x′α = x′α (xµ ) the metric gαβ (x) transforms according to
∂xγ ∂xδ
gγδ (x) .
∂x′α ∂x′β
Now consider the following “infinitesimal” change of coordinates
′
gαβ
(x′ ) =
x′α = xα + ξ α (xµ ) ≃ xα + ξ α (x′µ ) ,
(7)
(8)
where ξ α are four arbitrary functions which are taken to be “small” in the sense that terms
quadratic in ξ α and terms of the form hµν ξ α can be neglected.
a) Show that the transformation (8) induces
h′αβ = hαβ − ∂α ξβ − ∂β ξα .
(9)
b) Show that one can choose ξα in such a way that
Vα′ = ∂β h′ β α − 12 ∂α h′ γ γ = 0 .
(10)
This condition is called harmonic gauge, because it reduces the linearized Einstein’s equations to a set of (harmonic) wave-equations. Hints: derive a partial differential equation of
the form ✷ξα = fα . What is the precise relation between fα and hµν ? Solve the partial
differential, e.g., by applying Fourier transforms or Green function methods.
c) Adopting harmonic gauge still does not fix the freedom of choosing a coordinate system
completely. Show that there is still a residual gauge freedom, which allows to transform the
coordinates by an harmonic function
✷ξα (x) = 0 .
(11)
Thus starting from a perturbation h′αβ obeying the harmonic gauge condition and carrying
out a change of coordinates satisfying (11) results in a metric h′′αβ still satisfying Vα′′ = 0.
Exercise 4: Gravitational wave solutions (hand-in exercise)
The aim of this exercise is then to construct the gravitational wave solutions of general relativity.
Combining the results of exercises 2 and 3, given a coordinates system implementing harmonic
gauge, the linearized Einstein’s equations in vacuum are given by the wave equation
−✷ hαβ (x) = 0 .
(12)
In addition to satisfying the equations of motion, the solutions then have to obey the harmonic
gauge condition (10) and we can use the residual gauge freedom (11) to cast them into a canonical
form.
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a) Solve (12) using the Fourier transform of hαβ (x). What is the propagation speed of the
gravitational waves?
b) Consider the general solution found in part a). Show that the residual gauge freedom (11)
can be used to set h00 = 0 and h0i = 0.
c) Rotate your coordinate system so that the gravitational wave moves along the z-axis, i.e.,
consider the specific class of solutions with frequency ω whose three-momentum ~k = (0, 0, ω)
is aligned to the z-axis. Starting from the solution constructed in part b), work out the
constraints originating from the harmonic gauge condition (10). Explain why gravitational
waves are transverse.
d) Implementing the constraints from b) and c), write down the general solution for a gravitational wave propagating along the z-axis. Use your result to explain why gravitational
waves come with two polarizations.
Exercise 5: Fixing Einstein’s equations (optional, instructive for future theoreticians)
In the lecture, the static, weak-field metric has been given in form of the line-element (c = 1)
ds2 = 1 + 2Φ(xi ) dt2 − 1 − 2Φ(xi ) dx2 + dy 2 + dz 2 .
(13)
The potential Φ(xi ) depends on the spatial coordinates xi only and is obtained as a solution of
the Poisson equation
δij ∂i ∂j Φ(xi ) = 4πGρ
(14)
with ρ being the energy density of the non-relativistic matter source.
a) Compare the line-element (13) to the expansion (4) to find the relation between hαβ and Φ.
b) Recast the 00-component of the linearized Einstein equations coupled to matter in terms
of Φ. Show that the result agrees with the Poisson equation (14) if the coupling in front of
the stress-energy tensor is given by 8πG. Note that this computation is actually required
in order to fix the coupling between the space-time curvature and matter sectors entering
Einstein’s equations.
c) Check Einstein’s equation stated in Longairs book. This equation is not of the form (1),
while it indeed gives rise to the “correct” Friedman equations. Try to explain the origin of
this “discrepancy”.
Remark: the techniques used in this problem set can easily be adapted to the study of cosmological
perturbations. In this case, the flat Minkowski metric entering (4) is replaced by the FriedmanRobertson-Walker metric and hαβ describes fluctuations around the homogeneous and isotropic
cosmological solution.
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