The Principle of Relativity · Albert Einstein
C. THE THEORY OF THE GRAVITATION-FIELD
If γ_{μν}^{τ}’s vanish, the point moves uniformly and in a straight line; these magnitudes therefore determine the deviation from uniformity. They are the components of the gravitational field. §14. The Field-equation of Gravitation in the absence of matter. In the following, we differentiate gravitation-field from matter in the sense that everything besides the gravitation-field will be signified as matter; therefore the term includes not only matter in the usual sense, but also the electro-dynamic field. Our next problem is to seek the field-equations of gravitation in the absence of matter. For this we apply the same method as employed in the foregoing paragraph for the deduction of the equations of motion for material points. A special case in which the field-equations sought-for are evidently satisfied is that of the special relativity theory in which _g__{μν}’s have certain constant values. This would be the case in a certain finite region with reference to a definite co-ordinate system K₀. With reference to this system, all the components B^{ρ}_{μστ} of the Riemann’s Tensor [equation 43] vanish. These vanish then also in the region considered, with reference to every other co-ordinate system. The equations of the gravitation-field free from matter must thus be in every case satisfied when all B^{ρ}_{μστ} vanish. But this condition is clearly one which goes too far. For it is clear that the gravitation-field generated by a material point in its own neighbourhood can never be transformed _away_ by any choice of axes, _i.e._, it cannot be transformed to a case of constant _g__{μν}’s.
Therefore it is clear that, for a gravitational field free from matter, it is desirable that the symmetrical tensors B_{μν} deduced from the tensors B^{ρ}_{μστ} should vanish. We thus get 10 equations for 10 quantities _g__{μν} which are fulfilled in the special case when B^{ρ}_{μστ}’s all vanish. Remembering (44) we see that in absence of matter the field-equations come out as follows; (when referred to the special co-ordinate-system chosen.) (47) $$ \frac{\partial \Gamma^{\alpha}_{\mu\nu}}{\partial x_{\alpha}} + \Gamma^{\alpha}_{\mu\beta} \Gamma^{\beta}_{\mu\alpha} = 0 $$ $$ \sqrt{-g} = 1 $$ $$ \Gamma^{\alpha}_{\mu\nu} = - \begin{Bmatrix}\mu & & \nu\ \alpha&\end{Bmatrix} $$ It can also be shown that the choice of these equations is connected with a minimum of arbitrariness. For besides B_{μν}, there is no tensor of the second rank, which can be built out of _g__{μν}’s and their derivatives no higher than the second, and which is also linear in them. It will be shown that the equations arising in a purely mathematical way out of the conditions of the general relativity, together with equations (46), give us the Newtonian law of attraction as a first approximation, and lead in the second approximation to the explanation of the perihelion-motion of mercury discovered by Leverrier (the residual effect which could not be accounted for by the consideration of all sorts of disturbing factors). My view is that these are convincing proofs of the physical correctness of my theory.
Therefore (48) ∂H/∂_g_^{μν} = -Γ_{μβ}^{α} Γ_{να}^{β} ∂H/∂_g__{σ}^{μν} = Γ_{μν}^{σ} If we now carry out the variations in (47a), we obtain the system of equations (47b) ∂/∂_x__{α} ( ∂H/∂_g__{α}^{μν} ) - ∂H/∂_g_^{μν} = 0, which, owing to the relations (48), coincide with (47), as was required to be proved. If (47b) is multiplied by _g__{σ}^{μν}, since ∂_g__{σ}^{μν}/∂_x__{α} = ∂_g__{α}^{μν}/∂_x__{σ} and consequently _g__{σ}^{μν} ∂/∂_x__{α} (∂H/∂_g__{α}^{μν}) = ∂/∂_x__{α} (_g__{σ}^{μν} ∂H/∂_g__{α}^{μν}) - ∂H/∂_g__{α}^{μν} ∂_g__{α}^{μν}/∂_x__{σ} we obtain the equation ∂/∂_x__{α} (_g__{σ}^{μν} ∂H/∂_g__{α}^{μν}) - ∂H/∂_x__{σ} = 0 or { ∂_t__{σ}^α/∂_x__{α} = 0 (49) { -2κ_t__{σ}^{α} = _g__{σ}^{μν} ∂H/∂_g__{α}^{μν} - δ_{σ}^{α} H. Owing to the relations (48), the equations (47) and (34), (50) κ_t__{σ}^{α} = ½ δ_{σ}^{α} _g_^{μν} Γ_{μβ}^{α} Γ_{να}^{β} - _g_^{μν} Γ_{μβ}^{α} Γ_{νσ}^{β}. It is to be noticed that _t__{σ}^{α} is not a tensor, so that the equation (49) holds only for systems for which √-_g_ = 1. This equation expresses the laws of conservation of impulse and energy in a gravitation-field. In fact, the integration of this equation over a three-dimensional volume V leads to the four equations (49a) _d_/_dx₄_ {∫_t__{σ}^4 _d_V} = ∫(_t__{σ}^1 α₁ + _t__{σ}² α₂ + _t__{σ}³ α₃)_d_S where α₁, α₂, α₂ are the direction-cosines of the inward-drawn normal to the surface-element _d_S in the Euclidean Sense. We recognise in this the usual expression for the laws of conservation. We denote the magnitudes _t_^α_{σ} as the energy-components of the gravitation-field.
I will now put the equation (47) in a third form which will be very serviceable for a quick realisation of our object. By multiplying the field-equations (47) with _g_^{νσ}, these are obtained in the mixed forms. If we remember that _g_^{νσ} ∂Γ^α_{μν}/∂_x__{α} = ∂/∂_x__{α} (_g_^{νσ} Γ^α_{μν}) - ∂_g_^{νσ}/∂_x__{α} Γ^α_{μν}, which owing to (34) is equal to ∂/∂_x__{α} (._g_^{νσ} Γ^α_{μν}) - _g_^{νβ} Γ^σ_{αβ} Γγ^α_{μν} - _g_^{σβ} Γ^ν_{βα} Γ^α_{μν}, or slightly altering the notation, equal to ∂/∂_x__{α} (_g_^{σβ} Γ^α_{μβ}) - _g_^{mn} Γ^σ_{mβ} Γ^β_{_n_μ} - _g_^{νσ} Γ^α_{μβ} Γ^β_{να}. The third member of this expression cancels with the second member of the field-equations (47). In place of the second term of this expression, we can, on account of the relations (50), put κ (_t_^σ_{μ} - ½ δ^σ_{μ} _t_), where _t_ = _t_^α_{α} Therefore in the place of the equations (47), we obtain (51) { ∂/∂_x__{α} (_g_^{σβ} Γ^α_{μβ}) = -κ(_t_^σ_{μ} - ½ δ^σ_{μ} _t_) { √(-_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 ▽²φ = 0 of the Newtonian theory. We have now to find the equations which will correspond to Poisson’s Equation ▽²φ = 4πκρ (ρ signifies the density of matter).
(53) $$ \frac{\partial \Gamma^{lpha}_{\mu u}}{\partial x_{\alpha}} + \Gamma^{lpha}_{\mu\beta} \Gamma^{eta}_{\nu\alpha} = - \kappa (T_{\mu\nu} - \frac{1}{2} g_{\mu\nu} T) $$ $$ \sqrt{-g} = 1 $$ 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. §17. The laws of conservation in the general case. The equations (52) can be easily so transformed that the second member on the right-hand side vanishes. We reduce (52) with reference to the indices μ and σ and subtract the equation so obtained after multiplication with ½ δ_{μ}^σ from (52). We obtain, (52a) ∂/∂_x__{α}(_g_^{σβ} Γ_{μβ}^α - ½ δ_{μ}^σ _g_^{λβ} Γ_{λβ}^α) = -κ(_t__{μ}^σ + T_{μ}^σ) we operate on it by ∂/∂_x__{σ}. Now, ∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}Γ_{μβ}^α) = -½ ∂²/∂_x__{α}∂_x__{σ} [_g_^{σβ} _g_^{αλ}(∂_g__{μλ}/∂_x__{β} + ∂_g__{βλ}/∂_x__{μ} - ∂_g__{μβ}/∂_x__{λ})]. The first and the third member of the round bracket lead to expressions which cancel one another, as can be easily seen by interchanging the summation-indices α, and σ, on the one hand, and β and λ, on the other.
A comparison with (41b) shows that these equations for the above choice of co-ordinates (√(-_g_) = 1) asserts nothing but the vanishing of the divergence of the tensor of the energy-components of matter. Physically the appearance of the second term on the left-hand side shows that for matter alone the law of conservation of impulse and energy cannot hold; or can only hold when _g_^{μν}’s are constants; _i.e._, when the field of gravitation vanishes. The second member is an expression for impulse and energy which the gravitation-field exerts per time and per volume upon matter. This comes out clearer when instead of (57) we write it in the form of (47). (57a) ∂T_{σ}^α/∂_x__{α} = -Γ_{σβ}^α T_{α}^β. The right-hand side expresses the interaction of the energy of the gravitational-field on matter. The field-equations of gravitation contain thus at the same time 4 conditions which are to be satisfied by all material phenomena. We get the equations of the material phenomena completely when the latter is characterised by four other differential equations independent of one another.
Ideas discussed in this chapter
Text from Project Gutenberg, public domain in the United States.