A.2. The Levi-Civita Tensor Density
The Levi-Civita symbol    is defined as

 
1

  1
0

for
e v e n p e r m u t a ti0o, n1 ,o2f , 3
  ,  ,  ,   o d d p e r m u t a t i o 0n , o1 f, 2 , 3

otherwise

in every 4-D coordinate system.
Consider now the Levi-Civita tensor ˆ   which
x  .
equals to    in one coordinate system
x


In another coordinate system
   ' x   , we have
ˆ  ' ' ' '    '  '  '  ' ˆ       '  '  '  '    
(A.12)
 det    '    ' ' ' '

  ' ' ' ' 
1
  '  '  '  '    
'
det   
(A.14)
Thus, the Levi-Civita symbol is a tensor density of weight 1. The metric
determinant g can be written as
g  det  g       g 0 g 1g 2 g 3 
Thus, g is a scalar density of weight 2.
1     
 
g  g  g  g 
4!
(A.15)
[ A tensor density of weight n is an object
that transforms like a tensor but with an extra factor
 det  
n
]
For convenience,
we define the covariant symbol by        . Note that using the metric tensor
to lower indices gives
g g g g        g   
(A.16)
Since the left-hand side is a tensor density of weight 1,    must be a tensor
density of weight +1.
This can also be seen from
  '  '  '  '       ' ' ' ' det   ' 

  ' ' ' ' 
1
  '  '   '  '   

det   ' 
(A.17)
 det   '    '  '  '  '   
Finally, it is easy to see that
1
g
   and
g    transform like tensors.