In general relativity and differential geometry, the Bel–Robinson tensor is a tensor defined in the abstract index notation by:
T
a
b
c
d
=
C
a
e
c
f
C
b
e
d
f
+
1
4
ϵ
a
e
h
i
ϵ
b
e
j
k
C
h
i
c
f
C
j
k
d
f
Alternatively,
T
a
b
c
d
=
C
a
e
c
f
C
b
e
d
f
−
3
2
g
a
[
b
C
j
k
]
c
f
C
j
k
d
f
where
C
a
b
c
d
is the Weyl tensor. It was introduced by Lluís Bel in 1959. The Bel–Robinson tensor is constructed from the Weyl tensor in a manner analogous to the way the electromagnetic stress–energy tensor is built from the electromagnetic tensor. Like the electromagnetic stress–energy tensor, the Bel–Robinson tensor is totally symmetric and traceless:
T
a
b
c
d
=
T
(
a
b
c
d
)
T
a
a
c
d
=
0
In general relativity, there is no unique definition of the local energy of the gravitational field. The Bel–Robinson tensor is a possible definition for local energy, since it can be shown that whenever the Ricci tensor vanishes (i.e. in vacuum), the Bel–Robinson tensor is divergence-free:
∇
a
T
a
b
c
d
=
0
