Part 13
With the help of the extension of the four-vector, we can easily define “extension” of a co-variant tensor of any rank. This is a generalisation of the extension of the four-vector. We confine ourselves to the case of the extension of the tensors of the 2nd rank for which the law of formation can be clearly seen.
As already remarked every co-variant tensor of the 2nd rank can be represented as a sum of the tensors of the type A_{μ} B_{ν}.
It would therefore be sufficient to deduce the expression of extension, for one such special tensor. According to (26) we have the expressions
$$ \frac{\partial A_{\mu}}{\partial x_{\sigma}} - \begin{Bmatrix}\sigma\mu\\\tau\end{Bmatrix} A_{\tau} $$
$$ \frac{\partial B_{\nu}}{\partial x_{\sigma}} - \begin{Bmatrix}\sigma\mu\\\tau\end{Bmatrix} B_{\tau} $$
are tensors. Through outer multiplication of the first with B_{ν} and the 2nd with A_{μ} we get tensors of the third rank. Their addition gives the tensor of the third rank
(27)
$$ A_{\mu\nu\sigma} = \frac{\partial A_{\mu\nu}}{\partial x_{\sigma}} - \begin{Bmatrix}\sigma\mu\\\tau\end{Bmatrix} A_{\tau\nu} - \begin{Bmatrix}\sigma\nu\\\tau\end{Bmatrix} A_{\mu\tau} $$
where A_{μν} is put = A_{μ} B_{ν}. The right hand side of (27) is linear and homogeneous with reference to A_{μν}, and its first differential co-efficient, so that this law of formation leads to a tensor not only in the case of a tensor of the type A_{μ} B_{ν} but also in the case of a summation for all such tensors, _i.e._, in the case of any co-variant tensor of the second rank. We call A_{μνσ} the extension of the tensor A_{μν}. It is clear that (26) and (24) are only special cases of (27) (extension of the tensors of the first and zero rank). In general we can get all special laws of formation of tensors from (27) combined with tensor multiplication.
Some special cases of Particular Importance.
_A few auxiliary lemmas concerning the fundamental tensor._ We shall first deduce some of the lemmas much used afterwards. According to the law of differentiation of determinants, we have
(28) _dg_ = _g_^{μν} _g dg__{μν} = -_g__{μν} _gdg_^{μν}.
The last form follows from the first when we remember that
_g__{μν} _g_^{μ′ν} = δ^{μ′}_{μ} , and therefore _g__{μν}_g_^{μν} = 4,
consequently _g__{μν}_dg_^{μν} + _g_^{μν} _dg__{μν} = 0.
From (28), it follows that
(29)
$$ \frac{1}{\sqrt{-g}} \frac{\partial \sqrt{-g}}{\partial x_{\sigma}} = \frac{1}{2} \frac{\log (-g)}{\partial x_{\sigma}} = \frac{1}{2} g^{\mu\nu} \frac{\partial g_{\mu\nu}}{\partial x_{\sigma}} = - \frac{1}{2} g_{\mu\nu} \frac{\partial g^{\mu\nu}}{\partial x_{\sigma}} $$
Again, since _g__{μν} _g_^{νσ} = δ^{ν}_{μ} , we have, by differentiation,
$$ g_{\mu\sigma} dg^{\nu\sigma} = -g^{\nu\sigma} dg_{\mu\sigma} $$
or
$$ g_{\mu\sigma} \frac{\partial g^{\nu\sigma}}{\partial x_{\lambda}} = - g^{\nu\sigma} \frac{\partial g_{\mu\sigma}}{\partial x_{\lambda}} $$
By mixed multiplication with _g_^{στ} and _g__{νλ} respectively we obtain (changing the mode of writing the indices).
(31) _dg_^{μν} = -_g_^{μα} _g_^{νβ} _dg__{αβ}
∂_g_^{μν}/∂_x__{σ} = -_g_^{μα} _g_^{νβ} _dg__{αβ}
and
(32) _dg__{μν} = -_g__{μα} _g__{νβ} _dg_^{αβ}
∂_g__{μν}/∂_x__{σ} = -_g__{μα} _g__{νβ} ∂_g_^{αβ}/∂_x__{σ}.
The expression (31) allows a transformation which we shall often use; according to (21)
(33)
$$ \frac{\partial g_{\alpha\beta}}{\partial x_{\sigma}} = \begin{bmatrix}\alpha & & \sigma\ & \beta &\end{bmatrix} + \begin{bmatrix}\beta & & \sigma\ \alpha&\end{bmatrix} $$
If we substitute this in the second of the formula (31), we get, remembering (23),
(34)
$$ \frac{\partial g^{\mu\nu}}{\partial x_{\sigma}} = - ( g^{\mu\tau} \begin{Bmatrix}\tau & & \sigma\ \nu&\end{Bmatrix} + g^{\nu\tau} \begin{Bmatrix}\tau & & \sigma\ \mu&\end{Bmatrix} ) $$
By substituting the right-hand side of (34) in (29), we get
(29a)
$$ \frac{1}{\sqrt{-g}} \frac{\partial \sqrt{-g}}{\partial x_{\sigma}} = \begin{Bmatrix}\mu \sigma\\\mu\end{Bmatrix} $$
_Divergence of the contravariant four-vector._
Let us multiply (26) with the contravariant fundamental tensor _g_^{μν} (inner multiplication), then by a transformation of the first member, the right-hand side takes the form
(A)
$$ \frac{\partial}{\partial x_{\nu}} (g^{\mu\nu} A_{\mu}) - A_{\mu} \frac{\partial g^{\mu\nu}}{\partial x_{\nu}} - \frac{1}{2} g^{\tau\alpha} (\frac{\partial g_{\mu\alpha}}{\partial x_{\nu}} + \frac{\partial g_{ u\alpha}}{\partial x_{\mu}} - \frac{\partial g_{\mu\nu}}{\partial x_{\alpha}}) g^{\mu\nu} A_{\tau} $$
According to (31) and (29), the last member can take the form
(B)
$$ \frac{1}{2} \frac{\partial g^{\tau\nu}}{\partial x_{\nu}} A_{\tau} + \frac{1}{2} \frac{\partial g^{\mu\tau}}{\partial x_{\mu}} A_{\tau} + \frac{1}{\sqrt{-g}} \frac{\partial \sqrt{-g}}{\partial x_{\alpha}} g^{\mu\alpha} A_{\tau} $$
Both the first members of the expression (B), and the second member of the expression (A) cancel each other, since the naming of the summation-indices is immaterial. The last member of (B) can then be united with first of (A). If we put
_g_^{μν} A_{μ} = A^{ν},
where A^{ν} as well as A_{μ} are vectors which can be arbitrarily chosen, we obtain finally
$$ \Phi = \frac{1}{\sqrt{-g}} \frac{\partial}{\partial x_{\nu}} (\sqrt{-g} A^{\nu}) $$
This scalar is the _Divergence_ of the contravariant four-vector A^{ν}.
_Rotation of the (covariant) four-vector._
The second member in (26) is symmetrical in the indices μ, and ν. Hence A_{μν} - A_{νμ} is an antisymmetrical tensor built up in a very simple manner. We obtain
∂A_{μ} ∂A_{ν} (36) B_{μν} = -------------- - ------------ ∂_x__{ν} ∂_{_x_μ}
_Antisymmetrical Extension of a Six-vector._
If we apply the operation (27) on an antisymmetrical tensor of the second rank A_{μ{ν²}} and form all the equations arising from the cyclic interchange of the indices μ, ν, σ, and add all them, we obtain a tensor of the third rank
(37) B_{μνσ} = A_{μνσ} + A_{νσμ} + A_{σμν}
∂A_{μν} ∂A_{νσ} ∂A_{σμ} = ------------ + ------------- + ------------ ∂_x__{σ} ∂_x__{μ} ∂_x__{ν}
from which it is easy to see that the tensor is antisymmetrical.
_Divergence of the Six-vector._
If (27) is multiplied by _g_^{μα} _g_^{νβ} (mixed multiplication), then a tensor is obtained. The first member of the right hand side of (27) can be written in the form
$$ \frac{\partial}{\partial x_{\sigma}} (g^{\mu\alpha} g^{\nu\beta} A_{\mu\nu}) - g^{\mu\alpha} \frac{\partial g^{\nu\beta}}{\partial x_{\sigma}} A_{\mu\nu} - g^{\nu\beta} \frac{\partial g^{\mu\alpha}}{\partial x_{\sigma}} A_{\mu\nu} $$
If we replace _g_^{μα} _g_^{νβ} A_{μνσ} by A_{σ}^{αβ}, _g_^{μα} _g_^{νβ} A_{μν} by A^{αβ} and replace in the transformed first member
∂_g_^{νβ}/∂_x__{σ} and ∂_g_^{μα}/∂_x__{σ}
with the help of (34), then from the right-hand side of (27) there arises an expression with seven terms, of which four cancel. There remains
(38) $$ A^{\alpha\beta}_{\sigma} = \frac{\partial A^{\alpha\beta}}{\partial x_{\sigma}} + \begin{Bmatrix}\sigma & & \kappa\ \alpha end{Bmatrix} A^{\kappa\beta} + \begin{Bmatrix}\sigma & & \kappa\ \beta&\end{Bmatrix} A^{\alpha\kappa} $$
This is the expression for the extension of a contravariant tensor of the second rank; extensions can also be formed for corresponding contravariant tensors of higher and lower ranks.
We remark that in the same way, we can also form the extension of a mixed tensor A_{μ}^{α}
(39) $$ A^{\alpha}_{\mu\sigma} = \frac{\partial A^{\alpha}_{\mu}}{\partial x_{\sigma}} - \begin{Bmatrix}\sigma & & \mu\ \tau&\end{Bmatrix} A^{\alpha}_{\tau} + \begin{Bmatrix}\sigma & & \tau\ \alpha&\end{Bmatrix} A^{\tau}_{\mu} $$
By the reduction of (38) with reference to the indices β and σ(inner multiplication with δ_{β}^{σ}), we get a contravariant four-vector
$$ A^{\alpha} = \frac{\partial A^{\alpha\beta}}{\partial x_{\beta}} + \begin{Bmatrix}\beta & & \kappa\ \beta&\end{Bmatrix} A^{\alpha\kappa} + \begin{Bmatrix}\beta & & \kappa\ \alpha&\end{Bmatrix} A^{\kappa\beta} $$
On the account of the symmetry of
$$ \begin{Bmatrix}\beta & &\kappa\ \alpha&\end{Bmatrix} $$
with reference to the indices β and κ, the third member of the right hand side vanishes when A^{αβ} is an antisymmetrical tensor, which we assume here; the second member can be transformed according to (29a); we therefore get
(40) $$ A^{\alpha} = \frac{1}{\sqrt{-g}} \frac{\partial(\sqrt{-g} A^{\alpha\beta})}{\partial x_{\beta}} $$
This is the expression of the divergence of a contravariant six-vector.
_Divergence of the mixed tensor of the second rank._
Let us form the reduction of (39) with reference to the indices α and σ, we obtain remembering (29a)
(41) $$ \sqrt{-g} A_{\mu} = \frac{\partial(\sqrt{-g} A^{\sigma}_{\mu})}{\partial x_{\sigma}} - \begin{Bmatrix}\sigma & & \mu\ \tau&\end{Bmatrix} \sqrt{-g} A^{\sigma}_{\tau} $$
If we introduce into the last term the contravariant tensor A^{ρσ} = _g_^{ρτ} A^{σ}_{τ}, it takes the form
$$ - \begin{bmatrix}\sigma & & \mu\ \rho&\end{bmatrix} \sqrt{-g} A^{\rho\sigma} $$
If further A^{ρσ} or is symmetrical it is reduced to
$$ - \frac{1}{2} \sqrt{-g} \frac{\partial g_{\rho\sigma}}{\partial x_{\mu}} A^{\rho\sigma} $$
If instead of A^{ρσ}, we introduce in a similar way the symmetrical co-variant tensor A_{ρσ} = _g__{ρα} _g__{σβ} A^{αβ}, then owing to (31) the last member can take the form
$$ \frac{1}{2} \sqrt{-g} \frac{\partial g_{\rho\sigma}}{\partial x_{\mu}} A_{\rho\sigma} $$
In the symmetrical case treated, (41) can be replaced by either of the forms
(41a)
$$ \sqrt{-g} A{\mu} = \frac{\partial (\sqrt{-g} A^{\sigma}_{\mu})}{\partial x_{\sigma}} - \frac{1}{2} \frac{\partial g_{\rho\sigma}}{\partial x_{\mu}} \sqrt{-g} A^{\rho\sigma} $$
or
(41b)
$$ \sqrt{-g} A{\mu} = \frac{\partial (\sqrt{-g} A^{\sigma}_{\mu})}{\partial x_{\sigma}} + \frac{1}{2} \frac{\partial g_{\rho\sigma}}{\partial x_{\mu}} \sqrt{-g} A_{\rho\sigma} $$
which we shall have to make use of afterwards.
§12. The Riemann-Christoffel Tensor.
We now seek only those tensors, which can be obtained from the fundamental tensor _g_^{μν} by differentiation alone. It is found easily. We put in (27) instead of any tensor A^{μν} the fundamental tensor _g_^{μν} and get from it a new tensor, namely the extension of the fundamental tensor. We can easily convince ourselves that this vanishes identically. We prove it in the following way; we substitute in (27)
$$ A_{\mu\nu} = \frac{\partial A_{\mu}}{\partial x_{\nu}} - \begin{Bmatrix}\mu & & \nu\ \rho&\end{Bmatrix} A_{\rho} $$
_i.e._, the extension of a four-vector.
Thus we get (by slightly changing the indices) the tensor of the third rank
$$ A_{\mu\sigma\tau} = \frac{\partial^2 A_{\mu}}{\partial x_{\sigma} \partial x_{\tau}} - \begin{Bmatrix}\mu & & \sigma\ \rho&\end{Bmatrix} \frac{\partial A_{\rho}}{\partial x_{\tau}} - \begin{Bmatrix}\mu & & \tau\ \rho&\end{Bmatrix} \frac{\partial A_{\rho}}{\partial x_{\sigma}} - \begin{Bmatrix}\sigma & & \tau\ \rho&\end{Bmatrix} \frac{\partial A_{\mu}}{\partial x_{\rho}} + \begin{bmatrix} - \frac{\partial}{\partial x_{\tau}} \begin{Bmatrix}\mu&&\sigma\ \rho&\end{Bmatrix} + \begin{Bmatrix}\mu&&\tau\ \alpha\end{Bmatrix} \begin{Bmatrix}\alpha&&\sigma\ \rho&\end{Bmatrix} + \begin{Bmatrix}\sigma&&\tau\ \alpha\end{Bmatrix} \begin{Bmatrix}\alpha&&\mu\ \rho&\end{Bmatrix} \end{bmatrix} A_{\rho} $$
We use these expressions for the formation of the tensor A_{μστ} - A_{μτσ}. Thereby the following terms in A_{μστ} cancel the corresponding terms in A_{μτσ}; the first member, the fourth member, as well as the member corresponding to the last term within the square bracket. These are all symmetrical in σ, and τ. The same is true for the sum of the second and third members. We thus get
(43)
$$ A_{\mu\sigma\tau} - A_{\mu\tau\sigma} = B^{\rho}_{\mu\sigma\tau} A_{\rho} $$
$$ B^{\rho}_{\mu\sigma\tau} = - \frac{\partial}{\partial x_{\tau}} \begin{Bmatrix}\mu & & \sigma\ \rho&\end{Bmatrix} + \frac{\partial}{\partial x_{\sigma}} \begin{Bmatrix}\mu & & \tau\ \rho&\end{Bmatrix} - \begin{Bmatrix}\mu & & \sigma\ \alpha&\end{Bmatrix} \begin{Bmatrix}\alpha & & \tau\ \rho&\end{Bmatrix} + \begin{Bmatrix}\mu & & \tau\ \alpha&\end{Bmatrix} \begin{Bmatrix}\alpha & & \sigma\ \rho&\end{Bmatrix} $$
The essential thing in this result is that on the right hand side of (42) we have only A_{ρ}, but not its differential co-efficients. From the tensor-character of A_{μστ} - A_{μτσ}, and from the fact that A_{ρ} is an arbitrary four vector, it follows, on account of the result of §7, that B^{ρ}_{μστ} is a tensor (Riemann-Christoffel Tensor).
The mathematical significance of this tensor is as follows; when the continuum is so shaped, that there is a co-ordinate system for which _g__{μν}_’s_ are constants, B^{ρ}_{μστ} all vanish.
If we choose instead of the original co-ordinate system any new one, so would the _g__{μν}’s referred to this last system be no longer constants. The tensor character of B^{ρ}_{μστ} shows us, however, that these components vanish collectively also in any other chosen system of reference. The vanishing of the Riemann Tensor is thus a necessary condition that for some choice of the axis-system _g__{μν}’s can be taken as constants. In our problem it corresponds to the case when by a suitable choice of the co-ordinate system, the special relativity theory holds throughout any finite region. By the reduction of (43) with reference to indices to τ and ρ, we get the covariant tensor of the second rank
(44)
$$ B_{\mu\nu} = R_{\mu\nu} + S_{\mu\nu} $$
$$ R_{\mu\nu} = - \frac{\partial}{\partial x_{\alpha}} \begin{Bmatrix}\mu & & \nu\ \alpha&\end{Bmatrix} + \begin{Bmatrix}\mu & & \alpha\ \beta&\end{Bmatrix} \begin{Bmatrix}\nu & & \beta\ \alpha&\end{Bmatrix} $$
$$ S_{\mu\nu} = \frac{\partial \log \sqrt{-g}}{\partial x_{\mu} \partial x_{\nu}} - \begin{Bmatrix}\mu & & \nu\ \alpha&\end{Bmatrix} \frac{\partial \log \sqrt{-g}}{\partial x_{\alpha}} $$
_Remarks upon the choice of co-ordinates._—It has already been remarked in §8, with reference to the equation (18a), that the co-ordinates can with advantage be so chosen that √(-_g_) = 1. A glance at the equations got in the last two paragraphs shows that, through such a choice, the law of formation of the tensors suffers a significant simplification. It is specially true for the tensor B_{μν}, which plays a fundamental rôle in the theory. By this simplification, S_{μν} vanishes of itself so that tensor B_{μν} reduces to R_{μν}.
I shall give in the following pages all relations in the simplified form, with the above-named specialisation of the co-ordinates. It is then very easy to go back to the general covariant equations, if it appears desirable in any special case.
C. THE THEORY OF THE GRAVITATION-FIELD
§13. Equation of motion of a material point in a gravitation-field. Expression for the field-components of gravitation.
A freely moving body not acted on by external forces moves, according to the special relativity theory, along a straight line and uniformly. This also holds for the generalised relativity theory for any part of the four-dimensional region, in which the co-ordinates K_{0} can be, and are, so chosen that _g__{μν}’s have special constant values of the expression (4).
Let us discuss this motion from the stand-point of any arbitrary co-ordinate-system K₁; it moves with reference to K₁ (as explained in §2) in a gravitational field. The laws of motion with reference to K₁ follow easily from the following consideration. With reference to K₀, the law of motion is a four-dimensional straight line and thus a geodesic. As a geodetic-line is defined independently of the system of co-ordinates, it would also be the law of motion for the motion of the material-point with reference to K₁. If we put
(45) $$ \Gamma^{\tau}_{\mu\nu} = - \begin{Bmatrix}\mu & & \nu\ \tau&\end{Bmatrix} $$
we get the motion of the point with reference to K₁, given by
(46) $$ \frac{d^2 x_{\tau}}{ds^2} = \Gamma^{\tau}_{\mu\nu} \frac{dx_{\mu}}{ds} \frac{dx_{\nu}}{ds} $$
We now make the very simple assumption that this general covariant system of equations defines also the motion of the point in the gravitational field, when there exists no reference-system K₀, with reference to which the special relativity theory holds throughout a finite region. The assumption seems to us to be all the more legitimate, as (46) contains only the first differentials of _g__{μν}, among which there is no relation in the special case when K₀ exists.
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.
§15. Hamiltonian Function for the Gravitation-field. Laws of Impulse and Energy.
In order to show that the field equations correspond to the laws of impulse and energy, it is most convenient to write it in the following Hamiltonian form:—
(47a)
δ∫ H_d_τ = 0
H = _g_^{μν} γ^{α}_{μβ} γ^{β}_{να}
√(-_g_) = 1
Here the variations vanish at the limits of the finite four-dimensional integration-space considered.
It is first necessary to show that the form (47a) is equivalent to equations (47). For this purpose, let us consider H as a function of _g_^{μν} and _g_^{μν}_{σ} (= ∂_g_^{μν}/∂_x__{σ})
We have at first
δH = Γ^{α}_{μβ} Γ^{β}_{να} δ_g_^{μν} + 2_g_^{μν} Γ^{α}_{μβ} δΓ^{β}_{να}
= - Γ^{α}_{μβ} Γ^{β}_{να} δ_g_^{μν} + 2Γ^{α}_{μβ} δ(_g_^{μν}Γ^{β}_{να}).
But
$$ \delta(g^{\mu\nu} \Gamma^{\beta}_{\nu\alpha}) = - \frac{1}{2} \delta \begin{bmatrix}g^{\mu\nu} & g^{\beta\lambda}\end{bmatrix} (\frac{\partial g_{\nu\lambda}}{\partial x_{\alpha}} + \frac{\partial g_{\alpha\lambda}}{\partial x_{\nu}} - \frac{\partial g_{\alpha\nu}}{\partial x_{\lambda}}) $$
The terms arising out of the two last terms within the round bracket are of different signs, and change into one another by the interchange of the indices μ and β. They cancel each other in the expression for δH, when they are multiplied by Γ_{μβ}^{α}, which is symmetrical with respect to μ and β, so that only the first member of the bracket remains for our consideration. Remembering (31), we thus have:—
δH = -Γ_{μβ}^{α} Γ_{να}^{β} δ_g_^{μν} + Γ_{μβ}^{α} δ_g__{α}^{μβ}
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).
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. 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_{μ}^σ.
We thus get instead of (51), the tensor-equation
(52) $$ \frac{\partial}{\partial x_{\alpha}} (g^{\sigmaeta} \Gamma^{lpha}_{\mu\beta}) = - \kappa [(t^{\sigma}_{\mu} + T^{\sigma}_{\mu}) - \frac{1}{2} \delta^{\sigma}_{\mu} (t + T)] $$ $$ \sqrt{-g} = 1 $$
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) $$ \frac{\partial \Gamma^{lpha}_{\mu u}}{\partial x_{\alpha}} + \Gamma^{lpha}_{\mu\beta} \Gamma^{eta}_{\nu\alpha} = - \kappa (T_{\mu\nu} - \frac{1}{2} g_{\mu\nu} T) $$
$$ \sqrt{-g} = 1 $$
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.
The second term can be transformed according to (31). So that we get,
(54) ∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}γ_{μβ}^α) = ½ ∂³_g_^{αβ}/∂_x__{σ}∂_x__{β}∂_x__{μ}
The second member of the expression on the left-hand side of (52a) leads first to
- ½ ∂²/∂_x__{α}∂_x__{μ} (_g_^{λβ}Γ_{λβ}^α) or
to 1/4 ∂²/∂_x__{α}∂_x__{μ} [_g_^{λβ}_g_^{αδ}( ∂_g__{δλ}/∂_x__{β} + ∂_g__{δβ}/∂_x__{λ} - ∂_g__{λβ}/∂_x__{δ})].