2l8 RELATIVITY, THE GENERAL THEORY
Of particular interest for EG is the covariant divergence of a second-rank tensor
T" rwm
* For dcfiniteness, «**** is defined in a locally cartesian system where 0 denotes the time direction, 123
the space directions.where
For a symmetric T"" we have:
a relation that Einstein used in the discussion of energy-momentum conservation.
As a further instance of covariant differentiation, the equation
should be mentioned [ W6]. This is one of the relations that threw Einstein off the
track for some time.
Grossmann devotes a special section to antisymmetric tensors. He notes that
Eq. 12.13 implies that
He also points out that ^^/Vg is a contravariant fourth-rank tensor derived
from the Levi-Civita symbol ^^ = + 1( —1) if a@y8 is an even (odd) permuta-
tion of 0123, zero otherwise.* As a result
is a tensor, the dual of Fyl.
Grossmann's concluding section starts as follows. 'The problem of the formu-
lation of the differential equations of a gravitation field draws attention to the
differential invariants. .. and ... covariants of ... ds^2 = g^dx^dx".' He then
presents to Einstein the major tensor of the future theory: the 'Ghristoffel four-
index symbol,' now better known as the Riemann-Christoffel tensor [W7]: