D2 xk x dxv dxv

where

3xx 32%a 3x^3xV

is the Christoffel symbol of the second kind, also called the 'affine connection', or simply the 'connection coefficient'. Applying the same transformation to the line element ds - which must be invariant under a coordinate transformation, since it is a property of the underlying spacetime - we find that ds2 = gvv dxv dxv, where the metric tensor takes the form

_ 3\$a gvv = 3^ 3^nap' Within the framework of relativity, there are no forces acting on an object in free fall, and so the effect of gravity is determined by the affine connection. By itself, this is not particularly useful, since Eqn (12.8) requires knowledge of the local inertial coordinates at each point. However, it can be shown by direct

See, for example, Kline (1972), Chapter 37.

calculation that

= 1 ¿o (dgz, dg»° »v 2 g V9 x » + d x v d x o J

and, hence, the metric tensor can be considered as the gravitational potential.75 The final twists and turns in the creation of general relativity are evident from a series of four dramatic communications made by Einstein to the Prussian Academy during November 1915. The first was on 4 November and the second, a week later. In these, Einstein began the return toward general covariance -which had been his goal right from the start - but he was still held back by some of his previous misconceptions. It was Mercury that provided the key. A calculation of the perihelial motion based on the equations with which he was now working yielded 43" per century for the advance of Mercury. Overjoyed, he wrote that his theory explains ... quantitatively ... the secular rotation of the orbit of Mercury, discovered by Leverrier,... without the need of any special hypothesis.

In his communication of this result on 18 November, he also noted that his new theory gave a deflection of light which was twice his earlier prediction. There were still some problems, but the quantitative success with Mercury meant there was no doubt he was on the right track. In the course of his analysis, Einstein realized crucially that the form of the equations in the case of a weak, static field could be more general than he had supposed hitherto. With the stumbling block removed, Einstein soon grasped the true nature of the solution he was looking for and, on 25 November, communicated the final form for the field equations, concluding that '... the general theory of relativity is closed as a logical structure'. The change to the final set of equations did not affect the calculations of perihelion shift and light deflection of the week before.

75 See Weinberg (1972), p. 75. The relation between r»v and g»v is given in Grossman's part of the Entwurf paper. Note that g»v is the inverse of g»v, i.e. govg»o = 5», where 5» is the Kronecker delta.

76 See Pais (1982), Section 14c, Norton (1984), Mehra (1998).

Quoted from Pais (1982), p. 256. Remarkably, the same field equations were derived by Hilbert almost simultaneously, though he was in pursuit of a much grander objective - an axiomatic theory of the world. Einstein had spent a week in Gottingen earlier in 1915 lecturing on general relativity and had got to know Hilbert. During November, Einstein and Hilbert were in constant correspondence (postcards sent between Gottingen and Berlin would be delivered the following day) and there has been much discussion on Hilbert's influence on Einstein's derivation of his field equations. It would, however, appear unlikely that Hilbert influenced Einstein at all during this period, other than through his encouragement. For further details, see Earman and Glymour (1978a), Pais (1982), Section 14d.

In Newtonian theory, the gravitational potential @ satisfies Poisson's equation V2\$ = —4npG (see p. 339), where p is the mass density of the source of the gravitational field. In general relativity, gravity is determined from the tensor giv and so, if we want any chance of recovering the Newtonian formula in an appropriate limit, the role of the mass density must be played also by a second-order tensor. This is the energy-momentum stress tensor Tiv first introduced by Minkowski, which serves to define the distribution of energy and momentum throughout space-time. Where the energy-momentum tensor is zero, general relativity must reduce to special relativity and, hence, whatever is to appear in place of V2\$ must in such cases be zero. This provides a vital clue.

The most general tensor that can be constructed from the metric tensor and its first and second derivatives, linear in the second derivatives, is the RiemannChristoffel curvature tensor RXivk, defined by arx arx rX ____i pn pX _pn pX

To obtain a tensor of rank two, we can form the contraction

0 0