Mathematics of Physics and Engineering

Einstein's Field Equations 105

called the Minkowski space, after the German mathematician HERMANN
MINKOWSKI (1864-1909), who introduced relativistic space-time in 1907 as
a part of a new theory of electrodynamics.
Mathematics takes (2.4.17) to a whole new level of abstraction by writ-
ing

(ds)^2 = {dx, dy, dz, dt) g (dx, dy, dz, dt)T', (2.4.20)

where (dx,dy,dz,dt) is the row vector, g is the metric tensor, that is,
a 4 x 4 symmetric matrix with non-zero determinant, and (dx, dy, dz, dt)T
is the column vector, Formula (2.4.20) defines a metric in M^4 , and ds is
called the line element corresponding to the metric g. As a result, the
same vector space K^4 can lead to different metric spaces, corresponding to
different choices of the matrix g. While this construction is possible in any
number of dimensions, and is the subject of pseudo-Riemannian geometry
(the standard Riemannian geometry deals with positive-definite matrices
g), we will only consider the four-dimensional case. The matrix g can, in
principle, be different at different points (x, y, z, t). A metric corresponding
to a constant matrix g is called flat; in particular, the metric (2.4.18)
used in special relativity is often called the flat Minkowski metric. In
general relativity, the metric is not flat, and the matrix g is different
at different points, that is, each element fljj of g is a function of (x,y, z,t).
We will discuss general relativity in Section 2.4.3.
If the matrix g is the same at all points, then, after a suitable change
of basis, we can assume that g is diagonal, see Exercise 8.1.5 on page 454.
For a non-constant matrix, a simultaneous reduction to a diagonal form at
all points at once might not be possible. A non-constant matrix g that is
diagonal everywhere still gives rise to a non-flat metric.
EXERCISE 2.4.4. c (a) What matrix corresponds to the usual Euclidean
metric? Hint: the identity matrix, (b) Write the components of the matrix g
corresponding to the metric (2.4-18). Hint: the matrix is diagonal.

2.4.3 Einstein's Field Equations and General Relativity

As we discussed on page 95, Einstein adopted the hypothesis that physical

laws should be stated in a form that is invariant under coordinate trans-

formations between frames. In his theory of special relativity, the frames

move with constant relative velocity, and there is only one possible coordi-

nate transformation: the Lorentz transformation. In his theory of general