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Astrophysics ASTR3415
Homework Assignment No.2
(Due 10/24/05, 5pm)
1. In a particular coordinate system, the components of a (2,0) tensor T are symmetric (meaning that
T ij  T ji for all i, j). Show that the components of T must be symmetric in any coordinate
system.
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2. An orthogonal metric is one for which g   0 and g   0 unless    .
As you saw in the lectures, the general expression for the Christoffel symbols in terms of the
metric and its partial derivatives is


 12 g  g ,  g ,   g ,  ,
where the repeated upper and lower index  indicates summation over  and commas denote
partial differentiation. By eliminating the terms in the above expression which are zero, show that
for an orthogonal metric


 0 if  ,  , are all different


  12 g  g,


   12 g  g ,
  12 g  g ,
(note: the repeated usage of  and  in the above expressions does not denote the Einstein
summation convention. If you’re unsure about the notation here, come and ask me).
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3. The invariant interval, expressed in polar coordinates x  r cos  , y  r sin  , for 2-D Euclidean
space is given by
ds 2  dr 2  r 2 d 2
Use the results of Q.2 to show that
rrr  0 , r   r and rr  0
rr  0 ,   0 and r 
1
r
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

 

4. Consider the vector field V with Cartesian components V x ,V y  x 2  3 y, y 2  3x .
a. Using the transformation law for a (1,0) tensor and the results of Q.7 of Homework

Assignment 1, show that the components V r ,V 
 of the vector field with respect to the
 
usual polar coordinate basis er , e  may be written as


V r  r 2 cos 3   sin 3   6r cos  sin 


V   3 cos 2   sin 2   r cos  sin  sin   cos  
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b. Using the formula for the covariant derivative V i ; j  V i , j   ijkV k , verify that the

covariant derivative of V has the following components with respect to the Cartesian
basis
V x ; x  2 x, V x ; y  3, V y ; x  3, V y ; y  2 y
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c. The covariant derivative of a (1,0) tensor transforms as a (1,1) tensor.
Using the
transformation law for the components of the covariant derivative
V ;j
i
x  i x l k
 k
V ;l
x x  j
show that

V  ;r  2 cos  sin  sin   cos    3r cos 2   sin 2 

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d. Now, using the results given in Q.3 and Q.4a, apply the explicit formula for the covariant
derivative (as given in Q.4b only this time in a polar coordinate system) and verify that
you get the same result for V  ;r as you did in Q.4c. (This will also be the case for all the
other components of the covariant derivative).
[6]
5. In relativity we frequently measure time in units of length – e.g. we replace 1 second by the
distance (in metres) travelled by light in this time. This conversion is equivalent to setting
c  1.
a. Given that in SI units
G  6.67  10 11 N m 2 kg 2 and
1 N  1 kg ms 2 , show that
G  7.41 10 28 m kg 1
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b. Hence show that setting G  1 is equivalent to measuring mass in units of length, such
that
1 m  1.34  10 27 kg .
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c. The Schwarzschild radius of a star of mass M is given by
RS 
or (setting
2GM
c2
c  1 and G  1) RS  2M . Using the conversion factor from Q.5b,
calculate the Schwarzschild radius of:
i. the Earth (mass  6  10 24 kg );
ii. the Sun (mass  2  10 30 kg );
iii. a supermassive black hole (mass = 10 7 solar masses)
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Total: [50]