P R I N C E T O N L E C T U R E S A B S T R A C T S 5 3 5
In Euclidean geometry if we translate a vector along a closed curve, we must return to
the same vector, but this is not so in non-Euclidean geometry. The difference between the
two vectors is a vector and is given by: (39) where is a tensor of
the second order and is an expression involving the second derivatives of the ’s
We thus come upon the well known Riemann-Tensor. If this tensor vanishes, it
means that through any translation about a small closed curve, a vector will be transformed
into itself. We can then construct a Euclidean co-ordinate system, and there will be no grav-
itational field. Newton characterized a gravitational field by a single potential function ,
satisfying the Laplace equation involving the second derivatives of . What are the
analogous equations which the quantities defining a gravitational field must satisfy under
the relativity theory? We must obtain from R a tensor of the second order involving the sec-
ond derivatives of the g’s linearly; whose vanishing is the condition sought. The only way
of obtaining a tensor of the second order from R is by the equation:
(40)
which must be the tensor in question.
(41) gives ten differential equations for a gravitational field, supposing no
matter to be present. How do these equations look when matter is present? In the Newtonian
theory, we then have: (42)
where is the density of matter. We must expect analogous equations here. The analogue
of the density of matter is the density of energy, which has the tensor, not the scalar char-
acte[r.] The conservation equation for the impulse of the energy in the electrostatic field,
according to Maxwell, is:
(43)
Unless this is satisfied, the conservation of energy will not be valid. We take the volume
integral of this, noticing that the quantity vanishes on the boundary.
(44)
In the static field, the time derivative of a space integral vanishes, giving for
the four conservation laws:
(45)
(43) only has significance when no gravitation is present. It must then be possible to find in
the general theory, an energy tensor whose divergence is zero when there is no gravitation.
We get:
(46)
a R f a = f
R g
Riklm
ab
Rim gklRiklm =
Rim 0=
[p. 8]
4k –=
x
T
x1
T
1
x2
T
2
x3
T
3
x4
T
4
+ + + 0 = =
x1
T
1
x2
T
2
x3
T
3
x4
T
4
+ + +
dV
1 2 3 4 =
dx4
d
T
4
dV
0=
T
x
-------- -
– 0 =