Page:The principle of relativity (1920).djvu/216

 Therefore in the place of the equations (47), we obtain

(51) { [part]/[part]x_{α} (g^{σβ} Γ^α_{μβ}) = -κ(t^σ_{μ} - 1/2 δ^σ_{μ} t)

{ [sqrt](-g) = 1.

§16. General formulation of the field-equation of Gravitation.

The field-equations established in the preceding paragraph for spaces free from matter is to be compared with the equation [nabla]^2φ = 0 of the Newtonian theory. We have now to find the equations which will correspond to Poisson's Equation [nabla]^2φ = 4πκρ ([p1**]ρ signifies the density of matter).

The special relativity theory has led to the conception that the inertial mass (Träge Masse) is no other than energy. It can also be fully expressed mathematically by a symmetrical tensor of the second rank, the energy-tensor. We have therefore to introduce in our generalised theory energy-tensor Τ^α_{σ} associated with matter, which like the energy components t^α_{σ} of the gravitation-field (equations 49, and 50) have a mixed character but which however can be connected with symmetrical covariant tensors. The equation (51) teaches us how to introduce the energy-tensor (corresponding to the density of Poisson's equation) in the field equations of gravitation. If we consider a complete system (for example the Solar-system) its total mass, as also its total gravitating action, will depend on the total energy of the system, ponderable as well as gravitational.