Mathematics of Physics and Engineering

Einstein's Field Equations 109

eral relativity is that the physical laws are independent of a particular frame,
inertial or not, in space-time, that is, the equations are invariant under gen-
eral coordinate transformations from one frame to another; what makes spe-
cial relativity special is the restriction of this invariance to inertial frames.
When combined with differential geometry and tensor calculus, whose con-
structions do not depend on a particular coordinate system and are valid
in every space, Euclidean or not, this invariance assumption motivates the
formulation of equations (2.4.22). Before formulating the equations in 1915,
Einstein studied Riemannian geometry and also learned tensor calculus di-
rectly from one of the creators of the theory, the Italian mathematician
TULLIO LEVI-CIVITA (1873-1941). We give a brief and elementary sum-
mary of tensors on page 457 in Appendix.

In classical Newtonian theory, the gravitational field is described by the
partial differential equation <pxx + (pyy + tpzz = 4irGp, where <p is the poten-
tial of the field and p is the density of the gravitating mass; see Exercise 3.3.8
on page 168 below. Einstein started with a relation, -Ayfe] = 4irGTij[g],
with g instead of <p and the stress-energy tensor instead of the usual mass
density, and assumed that the tensor Aij depends linearly on the second-
order partial derivatives of 0. In tensor analysis, it is proved that such a
tensor A^ [g] must have the form

Ai:j [g\ = a Ri:j [g] + (3 R[g]gij + 7 0y

for some real numbers a, 0, 7, with R^ and R denned in (2.4.26) and
(2.4.28), respectively. Physical considerations, such as dimension analysis
and conservation of energy and momentum, result in the values a = —2/3 =
c^4 /2, 7 = 0. For more details, see the book Theory of Relativity by W. Pauli,
1981.
To complete the general discussion of (2.4.22), let us mention some other
related terms and names:

• The collection of Rij[g], i,j = 1,...,4, is called the Ricci curvature
  tensor (also Ricci curvature or Ricci tensor), after the Italian mathemati-
  cian GREGORIO RICCI-CURB ASTRO (1853-1925), whose most famous pub-
  lication The Absolute Differential Calculus, written together with his former
  student T. Levi-Civita, was published with his name truncated to Ricci.


• The number R[g] is called the scalar curvature or Ricci scalar.


• Each rL is called a Christoffel symbol, after the German mathe-
  matician ELWIN BRUNO CHRISTOPFEL (1829-1900). Because of the way
  the numbers nfc change when the coordinate system is changed, the col-
  lection of Christoffel symbols is not a tensor.