Specifying Sources in General Relativity

It is demonstrated that the dynamical equations for a massless scalar field, incoherent matter, or incoherent radiation may be put into such forms that they may be solved without a metric. The equations reduce to the usual forms when four conditions on the form of the metric are imposed, which do not restrict its nature. The conservation laws ${\tilde{T}}_{\mu ;\nu }^{\nu }=0\mathrm{}$ are thereby satisfied before the integration of the Einstein equations is attempted, ensuring that solutions will exist. The equations of motion of a perfect fluid in isentropic flow are similarly reformulated nonmetrically; but since five conditions must be placed on the metric to reduce the theory to its usual form, there is one real restriction on the metrical geometry implied in this case.