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220 RELATIVITY, THE GENERAL THEORY

Einstein uses Eqs. 12.21 and 12.22 to discuss the properties of the energy and

momentum of a matter distribution with mass m (m being 'a characteristic con-

stant independent of the gravitational potential'). In particular he derives the

expression

for the energy-momentum tensor of pressureless matter, where p 0 = ml Va and

VQ = d£ is the rest-volume element of the material distribution. His next advance

is made with the help of Grossmann's Eq. 12.15. He conjectures that the energy-

momentum conservation laws must be of the generally covariant form

in which the second term expresses the action of the gravitational field on matter.

The geodesic equation of motion

[dr = (—g^dx'dx")^^2 is the proper time] for a particle with nonvanishing mass
is not found in EG (Einstein first derived this equation in 1914). It is important
to note this absence, since the two authors experienced some difficulty in recog-
nizing the connection between their work and the Newtonian limit. For later pur-
poses, it is helpful to recall how this limit is found for the equation of motion (Eq.
12.28) [W8]: (1) neglect die/dr relative to dt/dr (slow motion); (2) put dgjdt
= 0 (stationarity); (3) write

and retain only first-order terms in hm (weak-field approximation). Then one
obtains the Newtonian equation

where = — h^/2 is the Newtonian potential, so that

Nevertheless, though the discussion of the motion of matter was not complete,
all was going well so far, and the same continued to be true for electrodynamics.
Indeed, EG contains the correct generally covariant form of the Maxwell
equations: