Page:The principle of relativity (1920).djvu/217

 This can be expressed, by putting in (51), in place of energy-components t_{μ}^σ of gravitation-field alone the sum of the energy-components of matter and gravitation, i.e.,

t_{μ}^σ + T? & below]_{μ}^σ.

We thus get instead of (51), the tensor-equation

(52) { [part]/[part]x_{α}(g^{σβ}Γ_{μβ}^α) = -κ[(t_{μ}^σ + T_{μ}^σ) - 1/2 δ_{μ}^σ(t + T)]

{ [sqrt](-g) = 1

where T = T_{μ}^μ (Laue's Scalar). These are the general field-equations of gravitation in the mixed form. In place of (47), we get by working backwards the system

(53) { [part]Γ_{μν}^α/[part]x_{α} + Γ_{μβ}^αΓ_{να}^β = -κ(Τ_{μν} - 1/2 g_{μν}T)

{ [sqrt](-g) = 1.

It must be admitted, that this introduction of the energy-tensor of matter cannot be justified by means of the Relativity-Postulate alone; for we have in the foregoing analysis deduced it from the condition that the energy of the gravitation-field should exert gravitating action in the same way as every other kind of energy. The strongest ground for the choice of the above equation however lies in this, that they lead, as their consequences, to equations expressing the conservation of the components of total energy (the impulses and the energy) which exactly correspond to the equations (49) and (49a). This shall be shown afterwards.