PHYS5523 Theory of Relativity
Test 2: The Gravitational Field of a Single Body
November 2, 2011
General Instructions:
This is a three hour, open book test with three
problems. Complete all three problems.
Problem 1: A Radially Infalling Observer.
a) Demonstrate that a radially infalling observer always measures the speed
of radially moving photons (whether they are traveling inwards or outward) to
be
c
relative to his own velocity.
Accomplish this by finding the velocity of
both the photons and the observer himself in the reference frame of a radially
infalling observer as given by the GullstrandPainleve coordinates.
The Schwarschild metric in GullstrandPainleve coordinates is
ds
2
=

(1

2
M
r
)
dt
′
2
+ 2
radicalbigg
2
M
r
dt
′
dr
+
dr
2
+
r
2
dθ
2
+
r
2
sin
2
θdφ
2
(1)
where
t
′
is the GullstrandPainleve coordinate time, which is the same as the
proper time for an observer freelyfalling from infinity, and
r
,
θ
and
φ
are the
Schwarzschild coordinates of the same name.
b) Now consider the viewpoint of a radially infalling observer (observer 1)
compared to that of an observer stationary at infinity in the Schwarzschild
radial coordinate (observer 2). Each of them is observing photons emitted by
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 Fall '11
 Kennefick
 Black Holes, Theory Of Relativity, General Relativity, Black hole

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