The Mathematical Theory of RelativityEddington, Arthur Stanley, Sir
Science
The Mathematical Theory of Relativity
Eddington, Arthur Stanley, Sir
Relativity (Physics)
equality or identity of two physical quantities is simpler than to ponder over
the behaviour of the quantity which is their difference---distinguished though
it may be by the important property of being incapable of existing!
According to the views reached in this chapter the law of gravitation
$G_{\mu\nu} = 0$ is not to be regarded as an expression for the natural texture of the
continuum, which can only be forcibly broken at points where some extraneous
agent (matter) is inserted. The differentiation of occupied and unoccupied
space arises from our particular outlook on the continuum, which, as explained
above, is such that the Hamiltonian derivatives of the principal invariant~$G$
stand out as active agents against the passive background. It is therefore
the regions in which these derivatives vanish which are regarded by us as
unoccupied; and the law $G_{\mu\nu} = 0$ merely expresses the discrimination made
by this process.
Among the minor points discussed, we have considered the speed of propagation
of gravitational influence. It is presumed that the speed is that
\PageSep{148}
of light, but this does not appear to have been established rigorously. Any
absolute influence must be measured by an invariant, particularly the invariant
$B_{\mu\nu\sigma}^{\rho} B_{\rho}^{\mu\nu\sigma}$. The propagation of this invariant does not seem to have
been investigated.
The ordinary potential energy of a weight raised to a height is not counted
\index{Potential energy}%
as energy in our theory and does not appear in our energy-tensor. It is found
superfluous because the property of our energy-tensor has been formulated
as a general law which from the absolute point of view is simpler than the
formal law of conservation. The potential energy and momentum~$\mf{t}_{\mu}^{\nu}$ needed
if the formal law of conservation is preserved is not a tensor, and must be
regarded as a mathematical fiction, not as representing any significant condition
of the world. The pseudo-energy-tensor~$\mf{t}_{\mu}^{\nu}$ can be created and destroyed
at will by changes of coordinates; and even in a world containing no attracting
matter (flat space-time) it does not necessarily vanish. It is therefore impossible
to regard it as of a nature homogeneous with the proper energy-tensor.
\PageSep{149}
\Chapter{V}{Curvature of Space and Time}
\index{Curvature!of 4-dimensional manifold}%
\Section{65.}{Curvature of a four-dimensional manifold}
In the general Riemannian geometry admitted in our theory the~$g_{\mu\nu}$ may
be any $10$~functions of the four coordinates~$x_{\mu}$.
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