Spacetime and Geometry · Problem 1

Quantum Field Theory in Curved Spacetime — Sean Carroll

Physics — General Relativity · cosmology-extra-dimensions-friedmann

Problem

Consider an

(N + n + 1)

-dimensional spacetime with coordinates

\{t, x^I, y^i\}

, where

I

goes from 1 to

N

and

i

goes from 1 to

n

. Let the metric be

ds^2 = -dt^2 + a^2(t)\delta_{IJ}\,dx^I dx^J + b^2(t)\gamma_{ij}(y)\,dy^i dy^j,

where

\delta_{IJ}

is the usual Kronecker delta and

\gamma_{ij}(y)

is the metric on an

n

-dimensional maximally symmetric spatial manifold. Imagine that we normalize the metric

\gamma

such that the curvature parameter

k = \frac{R(\gamma)}{n(n-1)}

is either

+1

0

, or

-1

, where

R(\gamma)

is the Ricci scalar corresponding to the metric

\gamma_{ij}

. (a) Calculate the Ricci tensor for this metric. (b) Define an energy-momentum tensor in terms of an energy density

\rho

and pressure in the

x^I

and

y^i

directions,

p^{(N)}

and

p^{(n)}

\begin{aligned} T_{00} &= \rho \\ T_{IJ} &= a^2 p^{(N)} \delta_{IJ} \\ T_{ij} &= b^2 p^{(n)} \gamma_{ij}. \end{aligned}

Plug the metric and

T_{\mu\nu}

into Einstein's equations to derive Friedmann-like equations for

a

and

b

(three independent equations in all). (c) Derive equations for the energy density and the two pressures at a static solution where

\dot{a} = \dot{b} = \ddot{a} = \ddot{b} = 0

, in terms of

k

n

, and

N

. Use these to derive expressions for the equation-of-state parameters

w^{(N)} = p^{(N)}/\rho

and

w^{(n)} = p^{(n)}/\rho

, valid at the static solution.