On the Use of the Energy-Momentum Principle in General Relativity

The primary purpose of this article is to obtain from the general relativity form of the energy-momentum principle certain new consequences which are needed for later work that the author has in mind. In addition, it is the intention to give at the same time a somewhat comprehensive and coherent treatment of the principle and its consequences, which it is hoped will increase the confidence and facility of physicists in the use of this important part of the general theory of relativity. In carrying out the investigation, it has seemed desirable for English readers, to take Eddington's "Mathematical Theory of Relativity" as a starting point, and this has incidentally led to a new form of deduction for certain consequences of the energy-momentum principle that were already known.

After presenting the energy-momentum principle in the form discovered by Einstein and showing its application to the case of the conservation of energy in an isolated system, an important expression is derived which gives the total densities of energy and momentum in the form of a divergence. This expression is equivalent to one previously obtained by Einstein but on account of the starting point adopted is derived and expressed in terms of the quantities ${\mathfrak{g}}^{\mu \nu }$ and ${\mathfrak{g}}_{\alpha }^{\mu \nu }$ instead of the $g^{\mu \nu }$ and $g_{\alpha }^{\mu \nu }$. Following this, the limiting values at large distances from an isolated material system are obtained for the quantities ${\mathfrak{g}}^{\alpha \beta }\frac{\partial \mathfrak{L}}{\partial {\mathfrak{g}}_{\gamma }^{\alpha \beta }}$ and ${\mathfrak{g}}^{\alpha 4}\frac{\partial \mathfrak{L}}{\partial {\mathfrak{g}}_{\gamma }^{\alpha 4}}$. These values, which have considerable use, have not previously received explicit expression. This is followed by a deduction from our present starting point of Einstein's famous relation $U=m$ between the energy and gravitational producing mass of an isolated system. An important expression is then obtained which gives the energy of a quasi-static isolated system in the form of an integral which has to be extended only over the portion of space actually occupied by matter or radiation. This expression has not previously received a satisfactory derivation. The result is used to obtain an expression for the energy of a spherical distribution of a perfect fluid, and it is then shown that this expression, in the case of a sphere of ordinary material, approaches in a sufficiently weak field to the classical expression for energy including the potential gravitational energy. This result is not only intrinsically useful, but also shows for a particular case that a higher order of approximation to the general relativity value for total energy is obtained by including the classical gravitational energy than by going at once to flat space-time as is often done. Finally, a general consideration is given to the problem of determining the conditions imposed on those changes from one static state to another which could occur in a non-isolated system forming part of a larger static system, without changing the distribution of matter and radiation outside the boundary and without contravening the energy-momentum principle as applied to the system as a whole.