A cloud of non-interacting dust particles (pressureless fluid, ) has stress-energy tensor

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Differential geometry of curved spacetime · Differential geometry of curved spacetime

The Geometry of Gravity, Ch Matter in curved spacetime: minimal coupling and relativistic hydrodynamics

Problem

A cloud of non-interacting dust particles (pressureless fluid,

p=0

) has stress-energy tensor

T^{\mu\nu}=\rho u^\mu u^\nu

, with

\rho

the rest-frame energy density and

u^\mu

the four-velocity of a fluid element, normalised by

u_\mu u^\mu=-1

. ((a)) Starting from

\nabla_\mu T^{\mu\nu}=0

, contract with

u_\nu

and use

u_\mu u^\mu=-1

to obtain the continuity equation

\nabla_\mu(\rho u^\mu)=0

. Substitute this back into

\nabla_\mu T^{\mu\nu}=0

to show that each dust element satisfies

u^\mu\nabla_\mu u^\nu=0

, the geodesic equation. ((b)) A weak, static, spherically symmetric gravitational field has the metric

ds^2=-\left(1+\dfrac{2\Phi}{c^2}\right)c^2\,dt^2+dx^2+dy^2+dz^2

, with

|\Phi/c^2|\ll1

and

\Phi=\Phi(x,y,z)

time-independent (the Newtonian-limit metric). Using the definition of the Christoffel symbols, show that to leading order in

\Phi/c^2

\Gamma^i_{\ 00}=\partial_i\Phi

. ((c)) A dust particle is instantaneously at rest in these coordinates, so

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

with

u^0=dt/d\tau\approx1

for

v\ll c

. Write the spatial component of the geodesic equation for this particle and show it reduces to

\dfrac{d^2x^i}{dt^2}=-\partial_i\Phi

. ((d)) Take

\Phi(r)=-GM/r

, the potential of a point mass

M

, and apply the result of part (c) at the surface of a body with

M=1.99\times10^{30}\ \mathrm{kg}

and radius

R=6.96\times10^{8}\ \mathrm{m}

(the Sun's mass and radius). Find the magnitude of the coordinate acceleration

|d^2x^i/dt^2|

of a dust particle instantaneously at rest at that radius. Use

G=6.674\times10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}

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