In his book [E, sect. 54] Sir Arthur Eddington derives Einstein's General Theory of Relativity using an argument similar to the one which will appear in a coming section of this paper. Let:
R[i][j][k][L] = C[j][L][i][!k] - C[j][k][i][!L] + C[j][k][m] g'[m][n] C[i][L][n] - C[j][L][m] g'[m][n] C[i][k][n] R[j][k] = g'[p][q] R[p][j][k][q] = g'[i][L] C[j][k][m] g'[m][n] C[i][L][n] - g'[i][L] C[j][L][m] g'[m][n] C[i][k][n] + g'[i][L] C[j][L][i][!k] - g'[i][L] C[j][k][i][!L] R = g'[r][s] R[r][s] = g'[r][s] g'[p][q] R[p][r][s][q]
Einstein's law of gravitation takes the following form:
R[i][k] - 1/2 g[i][k] (R - 2L) = -8pi N T[i][k]T[i][k] is the density of mass and momentum; actually g'[j][i] T[i][k] has the more familiar units. The factor of -8pi N (where N is Newton's gravitational constant) allows T to contain mass and momentum densities where the masses are in kilograms, or whatever mass units were used to determine N. L is the notorious ``cosmological constant''. Presently it is popular to speculate that it has a substantial nonzero value, although Einstein did not have experimental data to fit, and set it zero because that would produce an asymptotically flat space. Matter and momentum are supposed to be conserved:
g[t][u] T[i][t][/u] = 0Eddington demonstrates that the left side of Einstein's law of gravitation, when similarly differentiated and summed, comes out identically zero, and thus Einstein has a conserved source substance for his Ricci tensor formula, whether or not its relation to physical masses were known empirically. A number of points obtrude themselves. First, I find it aesthetically unpleasant that a range of values of the cosmological constant equally yield an Einstein tensor which is conserved. Second, a frequent complaint about Einstein's law of gravitation is that it does not fit into the convenient and soluble structure of Fredholm's equation, or in the usual phrasing, that you can't solve for the metric tensor given the mass and momentum tensor. Of course Green's function is closely related to the distance function, whose derivative is the metric tensor, and so it's an implicit job to determine the metric tensor by inverting Fredholm's equation using Green's function, but at least you can start from an estimated Green's function and do successive approximations, within reasonable limits of field intensity. But Einstein's law is simply not in a form to which Green's function applies. It would be valuable to transform the law into a possibly more complex form which can be processed by standard algorithms. Einstein's law is attractive for its economy of equations. All the structure of the space is contained within its distance function, a scalar function of two points. The distance function can be recovered from the metric tensor, which is its gradient versus each argument separately, subsequently restricted to the diagonal where both arguments are equal. As the distance function is symmetric end to end, the metric tensor has n(n+1)/2 independent components, where n is the dimension of the space, which are further restricted by Ricci's theorem: the covariant derivative of the metric tensor (in any direction) is zero. This is equivalent to the restriction that the second exterior derivative of a gradient is zero. The metric tensor is then differentiated and certain sums are performed, yielding a tensor of the same type, which is then set equal to the source term, which is a symmetric tensor having n(n+1)/2 independent components restricted by a conservation law. It would appear that there is a good balance between the number of gravitational law equations and additional restrictions, and the components in the metric tensor and its restrictions. This balance very much attracted Einstein and Eddington.