GENERAL RELATIVITY
Problem sheet 2
1.
Show that in a two dimensional flat space, the condition for a path between two fixed
points to have minimum length, i.e.
  d  0,
is that the path should take the form of a straight line, i.e. y = mx+c.
dy 
d  dx  dy  1 
dx  Ldx
dx 
2
Hint: begin with the simple relation
2
2
where
dy 
L  1
dx 
2
You may assume that the variational condition given above, which may now be written
as
  Ldx  0 ,
with the end points of the path understood to be fixed, is equivalent to the analogue of
Lagrange's equation,


d   L  L

0
dx  dy  y
 dx 
where, in our case, L is clearly a function of dy/dx only. Calculate the appropriate
derivative of L and hence solve for the relation between y and x.
2.
Show that the partial derivative of the metric tensor may be expressed as

g ,   g  
  g 
Hint: start from the definition of the Christoffel symbol, and substitute into the RHS.
3.
Employing the tensor transformation law for the metric tensor g, show directly that
the transformation law for the Christoffel symbol is
1
x  x  x  
 x   2 x 


   x   x   x      x  x   x 

 
4.
The surface of a sphere of radius a is characterised by the metric
d
2
 a2d 2  a2 sin 2  d 2
where  and  are spherical polar coordinates. Taking x 1   , x 2   , evaluate all
Christoffel symbols in this coordinate system.
5.
For the surface of a sphere of radius a, Christoffel symbols required to be evaluated in
polar coordinates in question 4 of the second problem sheet. Use the results there to
show that
   12 (log g )

,
where g is the determinant of the matrix g. Evaluate also the components of the
curvature tensor and show that the curvature scalar R = -2/a2.
6.
Taking for granted the properties of the fully covariant form of the curvature tensor,
R    R ,
R    R  , R   R ,
show that in a two dimensional space only one component of the curvature tensor,
R1212 say, is independent.
7.
For the surface of a sphere of radius a, Christoffel symbols required to be evaluated in
polar coordinates in question 4 of the second problem sheet. Use the results there to
show that
   12 (log g )

,
where g is the determinant of the matrix g. Evaluate also the components of the
curvature tensor and show that the curvature scalar equals -2/a2.
8.
Contracting indices of the 1st and 2nd Bianchi identities to prove that the Einstein's
tensor is symmetric and divergence free.
2