Schwarzschild Metric and Event Horizons in General Relativity, Study notes of Physics

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PHY312 - lecture 15

Simon Catterall

Summary of lecture

Exterior spacetime to spherically symmetric, timeindependent, non-spinning gravitational source is givenby Schwarzschild metric

s 2

A

r

c 2

t 2

A

r

r 2

r 2

θ 2

with

A

r

r S r r S

GM/c 2

Schwarzschild radius.

Global coordinate system.

r

and

t

“far away” coordinate

and time. Visualize at fixed

t

by embedding in 3D space ->

surface. Profile is just

A

r

Event horizon

A

at event horizon. Time passes infinitely slowly

relative to far away observer. Photons emitted from event horizon are redshifted toinfinite wavelength and fail to escape to infinity.

r ∆ t

cA

0 as r

r S

For

r < r S

all matter including light must fall to the

singularity

r

. Time and space coordinates flip.

Thus event horizon divides spacetime into 2 causallydisconnected regions – stuff inside the event horizoncan never influence what happens outside.

What happens at

r

r

S

Notice that an observer close to

r S

will not notice

anything odd – velocity of a radially moving light beamwill be

c

just as per normal:

r shell ∆ t shell

c for light

In fact the metric in shell coordinates is flat! SR applies. He/she will see no change as the light ray crosses theevent horizon, no violent redshifting etc For a freely falling observer gravity does not even exist -so that there cannot be anything that happens at

r

r S

provided that tidal effects are small

PHY312 - lecture 15 – p. 5

FOR in GR

Only one physical motion through spacetime. Many FOR of reference can be used to view it eg shell,FFF, far away (global) frame. They agree

only

on invariant intervals – not things like

distance, times, velocities etc. Different systems may be better/worse for figuring outdifferent things eg presence of event horizon. Notice that shell and FFF are physical frames whereobservers can make local measurements. Global frame different. No real global observer - butdoes give useful picture of entire trajectory of testparticle in spacetime.

Radial stretching done right

Said that strictly should treat

r

t

etc as

infinitessimal. But so far taken them finite. OK provided

r

small compared to the characteristic

scale of the curvature

r S

Calculation of Sun stretching fine. But the solar massblack hole? Should really use an integral

ds

A

r

dr

Hence

s

r 2 r 1 dr (

r S /r

1 / 2

Can be done exactly (or numerically).