9780192806727.pdf

234 RELATIVITY, THE GENERAL THEORY

in which all reference to a particle with mass m has disappeared. Einstein pro-

posed to consider Eq. 13.16 to hold whatever material (and electromagnetic) sys-

tem generates T*". Put

This is Nordstrom's 'second' field equation. It follows from Eqs. 13.16 and 13.17

that

where

is the energy momentum tensor of the gravitational field. Thus the theory is
Lorentz invariant and also satisfies the conservation laws.
Now to the equivalence principle. Consider a totally static closed system. This
obeys (integrated over the system) jrj<£e = 0 (i = 1, 2, 3). Hence, JTtfjt = —
E/c^2 , where E is its total energy. The same relation is also true for a system in
statistical equilibrium provided E is considered as the time average, over the sys-
tem.* Go to the static weak-field limit, i^/c^2 = 1 + 4>/c^2 , where $ is the New-
tonian potential. Then Eq. 13.17 becomes

and we have the desired result that the gravitational mass is proportional to the
total energy of the system.** As Einstein put it later, in this theory the equivalence
principle is a statistical law [E12].
About one quarter of Einstein's Vienna report, 'On the Current Status of the
Gravitation Problem,' is devoted to Nordstrom's work.f He commented only
briefly on Abraham's contributions, noting that he considered it a requirement of
any future theory that special relativity be incorporated and that Abraham had
not done so. When in the subsequent discussion Mie remarked that Nordstrom's
theory was an outgrowth of Abraham's work, Einstein replied: psychologically
yes, logically no. The incorporation of the equivalence principle was another desi-

*See [L2]. The average is to be taken over a time that evens out pressure fluctuations.
**This is the weak equivalence principle in the sense of Dicke, who further showed that the Nord-
strom theory does not satisfy the strong equivalence principle, according to which in a nonrotating
free-falling laboratory the laws of physics are those of gravity-free space, assumed to be everywhere
the same [Dl].

-(-Einstein also used this occasion to withdraw an objection to the scalar theory which he had raised
in his paper with Grossmann [El]. For other comments on scalar gravitation, see [W2].