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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] = 0
Eddington 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.
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