Consider the spatially flat ( ) FRW metric in comoving Cartesian coordinates

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Differential geometry of curved spacetime · Einstein's field equations

The Geometry of Gravity, Ch FRW cosmology as an exact solution

Problem

Consider the spatially flat (

K=0

) FRW metric in comoving Cartesian coordinates,

ds^2 = -c^2dt^2 + a(t)^2\left(dx^2+dy^2+dz^2\right),

filled with a comoving perfect fluid of energy density

\rho(t)

and pressure

p(t)

: four-velocity

u^\mu=(1,0,0,0)

and

T^{\mu\nu}=(\rho+p)u^\mu u^\nu/c^2+pg^{\mu\nu}

. ((a)) Using the geodesic Lagrangian

\mathcal{L}=-c^2\dot t^2+a^2(\dot x^2+\dot y^2+\dot z^2)

and its Euler-Lagrange equations, show that the only nonzero Christoffel symbols are

\Gamma^0{}_{ij}=(a\dot a/c^2)\delta_{ij}

and

\Gamma^i{}_{0j}=\Gamma^i{}_{j0}=H\delta^i_j

, with

H\equiv \dot a/a

. ((b)) Hence show

R_{00}=-3\ddot a/a

and

R_{ij}=c^{-2}(a\ddot a+2\dot a^2)\delta_{ij}

, and that

R=\dfrac{6}{c^2}\left(\dfrac{\ddot a}{a}+H^2\right)

. ((c)) Assemble

G_{00}

and

G_{ij}

, and equate them to

(8\pi G/c^4)T_{00}

and

(8\pi G/c^4)T_{ij}

to obtain the two Friedmann equations for this universe. ((d)) State which of the two equations shows that pressure contributes to gravity, and use it to say why radiation (

p=\rho/3

) decelerates the expansion twice as strongly as dust (

p=0

) at the same energy density.

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