256 RELATIVITY, THE GENERAL THEORYfor gy, as mentioned above, applied this time to compute unbound orbits.* The
discussion of the momentous consequences of this result will be reserved for Chap-
ter 16.- November the Twenty-Fifth \E1\:
The work is done. The conservation laws are satisfied: yg = 1 is no equation of
principle but rather an important guide to the choice of convenient coordinate
systems. The identity Eq. 14.9, thought earlier to have major physical implica-
tions, is replaced by a triviality. The calculations of the week before remain
unaffected:
Any physical theory that obeys special relativity can be incorporated into the
general theory of relativity; the general theory does not provide any criterion
for the admissibility of that physical theory.... Finally the general theory of
relativity is closed as a logical structure.[El]
Note that Eq. 14.15 is equivalent to R" - g^R/2 = -K.T".
In Section 12d, I mentioned that Einstein did not know the Bianchi identities
[W8]
when he did his work with Grossmann. He still did not know them on November
25 and therefore did not realize that the energy-momentum conservation laws
follow automatically from Eqs. 14.15 and 14.16. Instead, he used these conser-
vation laws as a constraint on the theory! I paraphrase his argument. Start from
Eq. 14.15 but with the coefficient % replaced by a number a to be determined.
Differentiate Eq. 14.15 covariantly and use Eq. 14.17. Next take the trace of Eq.
14.15, then differentiate. One finds that (R" + a(\ - 4a)-y7?):, = 0 (use
gi** = 0)- Choose coordinates such that \fg = 1. See if there is a solution for a.
One finds a = & Einstein's choice of coordinates is of course admissible, but it
is an unnecessary restriction that prevented him from discovering Eq. 14.16 as a
generally covariant relation. We shall see in Section 15c how the Bianchi identities
finally entered physics.
Einstein's brief belief in Eq. 14.9 may have been a useful mistake, since he had
discovered that funny equation by the same compatibility method. In the case of
Eq. 14.8, the relations are r = — /cTand r£ = 0. The term on the left-hand side
in Eq. 14.9 arose because in the November 4 paper Einstein had redefined his
*Einstein inserted those gf, into gf,dx"dx' = 0 and then applied Huyghens" principle.