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FIELD THEORIES OF GRAVITATION: THE FIRST FIFTY YEARS 233

whence

Note further that Eqs. 13.7 and 13.8 yield

in which m has disappeared. Equations 13.10 and 13.2 form the basis of Nord-

stroms first theory, in which he identified p with the 'rest mass density' [N2].

I shall leave aside further details of this theory, which left much to be desired, and

turn at once to his 'second theory', which he proposed in 1913 [N3]. Though it

was not to survive, it deserves to be remembered as the first logically consistent

relativistic field theory of gravitation ever formulated.

The main idea (which Nordstrom owed to von Laue and Einstein) is that the

only possible source for his scalar gravitational field is

the trace of the energy momentum tensor T^1 " (rim is, as usual, the Minkowski

metric). All the physical conclusions of the theory are due to Nordstrom himself.

I shall not follow his derivations, however, but instead describe the simple trick,

reported by Einstein at the Vienna meeting [Ell], which leads rapidly to the

desired result.

In Eq. 13.10, put $ = c^2 In $. Then

This equation can be derived from the variational principle

Once one has a variational principle, one can derive the equation for the energy
momentum tensor of a particle with rest mass m and rest volume V (p = m/V),
where the particle is treated as a continuum distributed over the rest volume V:

and for its divergence