A2 Introduction to G.R. – Model Answers
3.19
The Harvard experiment is a famous example of gravitational redshift. Since the
gamma ray photons travelled upwards a distance h , where h is much smaller than
the radius of the Earth, we can use the approximate expression gh / c 2 for the change
in gravitational potential energy of (i.e. no need to integrate). Hence the gravitational
redshift can be approximated by


3.20

gh
9.98  22.6

 2.5 10 15
2
c
9 1016
This is a generalisation of the treatment in 3.19, where we do need to integrate over
distance. We need to know the value of the redshift of photons released from the
surface of Sirius B, and which escape to infinity. For small distances, we know



gh
c2
Hence integrating, we have

d

 
0

GMdr
GM
 2 ,
2 2
c r
c r0
r0
 ln(  / 0 )   
where  0 ,   , r0 , M denote respectively, the proper frequency at the surface, the frequency at
a very large distance, the radius of Sirius B, and the mass of the dwarf star. Noting that the
right hand side is small, we can expand the left hand side to get
ln(  / 0 )  ln( 1   / 0 )   / 0  GM /( c 2 r0 )
  / 0 
(6.67  1011 )( 2.1 10 30 )
 2.8  10 4.
(9  1016 )(5.5  10 6 )
(Since this formula was derived in the notes it was acceptable simply to use it, but some
people derived it).
N.B. There was a typographical error in the Tutorial Handbook: the radius of Sirius B should
have been given as 5.5  10 6 m; the quoted figure of 5.5  108 m is too big for a white dwarf.
Also, the mass of Sirius B was not given in the question, but was taken to be approximately
equal to the Sun’s mass.
General comment: Be careful about round-off error. G.R. effects are usually VERY
small, which means that you need to carry a lot of significant figures through your
calculation to ensure that they don’t disappear altogether! I was fairly lenient with this,
since Q.3.20 involved some approximation (the mass of Sirius B) anyhow, but there were
a range of answers differing at the second significant figure due to round-off error.