GR Notes for Wednesday, September 24th, 2014
Peisen Ma
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Tensor Densities
First let’s recall the Levi-Civita symbol which is defined as
˜µ1 µ2 ···µn = +1
(1)
if µ1 µ2 · · · µn is an even permutation of 01 · · · (n − 1);
˜µ1 µ2 ···µn = −1
(2)
if µ1 µ2 · · · µn is an odd permutation of 01 · · · (n − 1);
˜µ1 µ2 ···µn = 0
(3)
if otherwise.
Given any n × n matrix Mµµ0 , we have the following property
˜µ01 µ02 ···µ0n Det(Mµµ0 ) = ˜µ1 µ2 ···µn Mµµ01 Mµµ02 · · · Mµµ0nn
1
Now if we set
2
0
Mµµ0 = ∂xµ /∂xµ
we can get
(4)
(5)
0
˜µ01 µ02 ···µ0n = |
where we have used
∂xµ1 ∂xµ2
∂xµn
∂xµ
|˜
·
·
·
µ µ ···µ
0
0
∂xµ 1 2 n ∂xµ1 ∂xµ2
∂xµ0n
(6)
0
∂xµ
1
=
|
|
Det(Mµµ0 )
∂xµ
(7)
Thus we can see the Levi-Civita symbol transforms in a way close to the
tensor transformation law except for the determinant out front, so objects
transforming in this way are called Tensor Densities.
Define the determinant of metric gµν as
g = Det(gµν )
1
(8)
Now consider the following relation
gµ0 ν 0 = gµν
then we get
∂xµ ∂xν
∂xµ0 ∂xν 0
(9)
0
∂xµ
∂xµ
g = g| µ0 |2 = g| µ |−2
∂x
∂x
Therefore g is called Tensor Density and it’s NOT a scalar.
0
(10)
Since tensor density does not transform as a tensor does, we can define a
tensor as
µ1 µ2 ···µn = g 1/2 ˜µ1 µ2 ···µn
(11)
which will transform according to the tensor transformation law.
2
Causality
A causal curve is defined to be one which is timelike or null everywhere.
Given any subset S of a manifold M, we can define the causal future of
S, denoted J + (S), to be the set of points that can be reached from S by
following a future-directed causal curve.
Given any subset S of a manifold M, the chronological future I + (S) is
the set of points that can be reached by following a future-directed timelike
curve.
Causal past J − and chronological past I − are defined analogously.
A subset S of manifold M is called achronal if no two points in S are connected by a timelike curve.
Given a closed achronal set S, the future domain of dependence of S,
denoted D+ (S), is defined as the set of all points p such that every pastmoving inextendible causal curve through p must intersect, and we define
the boundary of D+ (S) to be the future Cauchy horizon H + (S). And
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likewise we can define D− (S) and H − (S).
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