DOC. 30 FOUNDATION OF GENERAL RELATIVITY 185
required general field equations of gravitation in mixed form.
Working back from these, we have in place of (47)
TL + T^aTya = - tf(TM" - ig^T!),.
J "9 = 1
It must be admitted that this introduction of the energy-
tensor of matter is not justified by the relativity postulate
alone. For this reason we have here deduced it from the
requirement that the energy of the gravitational field shall
act gravitatively in the same way as any other kind of energy.
But the strongest reason for the choice of these equations
lies in their consequence, that the equations of conservation
of momentum and energy, corresponding exactly to equations
(49) and (49a), hold good for the components of the total
energy. This will be shown in § 17.
§ 17. The Laws of Conservation in the General Case
Equation (52) may readily be transformed so that the
second term on the right-hand side vanishes. Contract (52)
with respect to the indices u and a, and after multiplying the
resulting equation by 1/2Sau, subtract it from equation (52).
^irßKß - = - + TZ). (52a)
On this equation we perform the operation d/dav. We have
~bxj)x, 7)xJ)x, ^Xft 7>Xx.
The first and third terms of the round brackets yield con-
tributions which cancel one another, as may be seen by
interchanging, in the contribution of the third term, the
summation indices a and a on the one hand, and ß and \
on the other. The second term may be re-modelled by (31),
so that we have
'bXn'dx, (g'ßKß) = i gaß fi.. (54)a2
The second term on the left-hand side of (52a) yields in the