Mathematics of Physics and Engineering

Einstein's Field Equations 107

^•[fl]-^[fl]fl« = ^T«[fl], i,j = l,..A, (2.4.22)

where Rij, R, and T^- are non-linear partial differential operators acting
on the metric tensor Q.
Our most immediate goal now is to understand the meaning of every
component of (2.4.22).

• The metric tensor a = (gij(x), i,j = 1,...,4) is a symmetric 4x4
  matrix defined at the points x = (a;^1 , a;^2 , a;^3 , a;^4 ) of relativistic space-time.
  This matrix has a nonzero determinant and continuous second-order partial
  derivatives of all components %%j(x) for most x (say, all but finitely many).
  This matrix defines a metric on space-time by defining the line element
  ar every point x according to the formula
  4
  (ds)^2 (x)= ]T gij(x)dxidxj; (2.4.23)

compare this with (2.4.20) on page 105. Summations of this kind will
appear frequently in our discussion. To minimize the amount of writing,
we will use Einstein's summation convention, assuming summation in
an expression over an index from 1 to 4 if the index appears in the expression
twice. With this convention, (2.4.23) becomes

(ds)^2 (x) = Qi:j{x) dxi dxj. (2.4.24)

EXERCISE 2.4.5.c The Kronecker symbol S is defined by

l=J' (2.4.25)

Verify that, according to Einstein's summation convention, 5\ = 4.

• The operators R^ in (2.4.22) are denned as follows:
  f)Vk (T) dT^(x)
  KM*) = -^

J

• —^F

i
+rS(*)r^(a')-Cfc(x)r?i(x)> (2.4.26)
where

rS(-)-j.~(«)(^ + ^

i

-^

4

) P«7)

and gmn(x), m,n = 1,...,4, are the components of the inverse matrix
g~^1 (x), that is, gmn(x)g„j(x) = 6™ for every x, with 6™ defined in