Chapter VIII: Appendix: Mechanics and the Relativity-Postulate (3)

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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,

$$ g_{\mu\sigma} dg^{\nu\sigma} = -g^{\nu\sigma} dg_{\mu\sigma} $$

$$ g_{\mu\sigma} \frac{\partial g^{\nu\sigma}}{\partial x_{\lambda}} = - g^{\nu\sigma} \frac{\partial g_{\mu\sigma}}{\partial x_{\lambda}} $$

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)

$$ \sqrt{-g} A{\mu} = \frac{\partial (\sqrt{-g} A^{\sigma}_{\mu})}{\partial x_{\sigma}} - \frac{1}{2} \frac{\partial g_{\rho\sigma}}{\partial x_{\mu}} \sqrt{-g} A^{\rho\sigma} $$

(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

{ ∂_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})].

The expression arising out of the last member within the round bracket vanishes according to (29) on account of the choice of axes. The two others can be taken together and give us on account of (31), the expression

-½ ∂³_g_^{αβ}/∂_x__{α}∂_x__{β}∂_x__{μ}

So that remembering (54) we have

(55) ∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}Γ_{μβ}^α
- ½ δ_{μ}^σ _g_^{λβ} Γ_{λβ}^α) = 0.

identically.

From (55) and (52a) it follows that

(56) ∂/∂_x__{σ} (_t__{μ}^σ + T_{μ}^σ) = 0

From the field equations of gravitation, it also follows that the conservation-laws of impulse and energy are satisfied. We see it most simply following the same reasoning which lead to equations (49a); only instead of the energy-components of the gravitational-field, we are to introduce the total energy-components of matter and gravitational field.

§18. The Impulse-energy law for matter as a consequence of the
field-equations.

If we multiply (53) with ∂_g_^{μν}/∂_x__{σ}, we get in a way similar to §15, remembering that

_g__{μν} ∂_g_^{μν}/∂_x__{σ} vanishes,

the equations ∂_t__{σ}^α/∂_x__{α} - ½ ∂_g_^{μν}/∂_x__{σ} T_{μν} = 0

or remembering (56)

(57) ∂T_{σ}^α/∂_x__{α} + ½ ∂_g_^{μν}/∂_x__{σ} T_{μν} = 0

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.

D. THE “MATERIAL” PHENOMENA.

The Mathematical auxiliaries developed under ‘B’ at once enables us to generalise, according to the generalised theory of relativity, the physical laws of matter (Hydrodynamics, Maxwell’s Electro-dynamics) as they lie already formulated according to the special-relativity-theory. The generalised Relativity Principle leads us to no further limitation of possibilities; but it enables us to know exactly the influence of gravitation on all processes without the introduction of any new hypothesis.

It is owing to this, that as regards the physical nature of matter (in a narrow sense) no definite necessary assumptions are to be introduced. The question may lie open whether the theories of the electro-magnetic field and the gravitational-field together, will form a sufficient basis for the theory of matter. The general relativity postulate can teach us no new principle. But by building up the theory it must be shown whether electro-magnetism and gravitation together can achieve what the former alone did not succeed in doing.

§19. Euler’s equations for frictionless adiabatic liquid.

Let _p_ and ρ, be two scalars, of which the first denotes the pressure and the last the density of the fluid; between them there is a relation. Let the contravariant symmetrical tensor

T^{αβ} = -_g_^{αβ} _p_ + ρ _dx__{α}/_ds_ _dx__{β}/_ds_ (58)

be the contra-variant energy-tensor of the liquid. To it also belongs the covariant tensor

(58a) T_{μν} = -_g__{μν} _p_ + _g__{μα} _dx__{α}/_ds_ _g__{μβ}
_dx__{β}/_ds_ ρ

as well as the mixed tensor

(58b) T^α_{σ} = -δ^α_{σ} _p_ + _g__{σβ} _dx__{β}/_ds_ _dx__{α}/_ds_
ρ.

If we put the right-hand side of (58b) in (57a) we get the general hydrodynamical equations of Euler according to the generalised relativity theory. This in principle completely solves the problem of motion; for the four equations (57a) together with the given equation between _p_ and ρ, and the equation

_g__{αβ} _dx__α/_ds_ _dx__{β}/_ds_ = 1,

are sufficient, with the given values of _g__{αβ}, for finding out the six unknowns

_p_, ρ, _dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_ _dx₄_/_ds_.

If _g__{μν}’s are unknown we have also to take the equations (53). There are now 11 equations for finding out 10 functions _g_, so that the number is more than sufficient. Now it is be noticed that the equation (57a) is already contained in (53), so that the latter only represents (7) independent equations. This indefiniteness is due to the wide freedom in the choice of co-ordinates, so that mathematically the problem is indefinite in the sense that three of the space-functions can be arbitrarily chosen.

§20. Maxwell’s Electro-Magnetic field-equations.

Let φ_{ν} be the components of a covariant four-vector, the electro-magnetic potential; from it let us form according to (36) the components F_{ρσ} of the covariant six-vector of the electro-magnetic field according to the system of equations

(59) F_{ρσ} = ∂φ_{ρ}/∂_x__{σ} - ∂φ_{σ}/∂_x__{ρ}.

From (59), it follows that the system of equations

(60) ∂F_{ρσ}/∂_x__{τ} + ∂F_{στ}/∂_x__{ρ} + ∂F_{τρ}/∂_x__{σ} = 0

is satisfied of which the left-hand side, according to (37), is an anti-symmetrical tensor of the third kind. This system (60) contains essentially four equations, which can be thus written:—

{ ∂F₂₃/∂_x₄_ + ∂F₃₄/∂_x₂_ ∂F₄₂/∂_x₃_ = 0
{
{ ∂F₃₄/∂_x₁_ + ∂F₄₁/∂_x₃_ ∂F₁₃/∂_x₄_ = 0
(60a) {
{ ∂F₄₁/∂_x₂_ + ∂F₁₂/∂_x₄_ ∂F₂₄/∂_x₁_ = 0
{
{ ∂F₁₂/∂_x₃_ + ∂F₂₃/∂_x₁_ ∂F₃₁/∂_x₂_ = 0.

This system of equations corresponds to the second system of equations of Maxwell. We see it at once if we put

{ F₂₃ = H_{_x_} F₁₄ = E_{_x_}
{
(61) { F₃₁ = H_{_y_} F₂₄ = E_{_y_}
{
{ F₁₂ = H_{_z_} F₃₄ = E_{_z_}

Instead of (60a) we can therefore write according to the usual notation of three-dimensional vector-analysis:—

{ ∂H/∂_t_ + rot E = 0
(60b) {
{ div H = 0.

The first Maxwellian system is obtained by a generalisation of the form given by Minkowski.

We introduce the contra-variant six-vector F_{αβ} by the equation

(62) F^{μν} = _g_^{μα} _g_^{νβ} F_{αβ},

and also a contra-variant four-vector J^μ, which is the electrical current-density in vacuum. Then remembering (40) we can establish the system of equations, which remains invariant for any substitution with determinant 1 (according to our choice of co-ordinates).

(63) ∂F^{μν}/∂_x__{ν} = J^μ

If we put

{ F²³ = H′_{_x_} F¹⁴ = -E′_{_x_}
{
(64) { F³¹ = H′_{_y_} F²⁴ = -E′_{_y_}
{
{ F¹² = H′_{_z_} F³⁴ = -E′_{_z_}

which quantities become equal to H_{_x_} ... E_{_x_} in the case of the special relativity theory, and besides

J^1 = _i__{_x_} ... J^4 = ρ

we get instead of (63)

{ rot H′ - ∂E′/∂_t_ = _i_
(63a) {
{ div E′ = ρ

The equations (60), (62) and (63) give thus a generalisation of Maxwell’s field-equations in vacuum, which remains true in our chosen system of co-ordinates.

_The energy-components of the electro-magnetic field._

Let us form the inner-product

(65) K_{σ} = F_{σμ} J^μ.

According to (61) its components can be written down in the three-dimensional notation.

{ K₁ = ρE_{_x_} + [_i_, H]_{x}

(65a) { — — —

{ K₄ = — (_i_, E).

K_{σ} is a covariant four-vector whose components are equal to the negative impulse and energy which are transferred to the electro-magnetic field per unit of time, and per unit of volume, by the electrical masses. If the electrical masses be free, that is, under the influence of the electro-magnetic field only, then the covariant four-vector K_{σ} will vanish.

In order to get the energy components T_{σ}^ν of the electro-magnetic field, we require only to give to the equation K_{σ} = 0, the form of the equation (57).

From (63) and (65) we get first,

K_{σ} = F_{σμ} ∂F_{μν}/∂_x__{ν}

= ∂/∂_x__{ν} (F_{σμ} F^{μν}) - F^{μν} ∂F_{σμ}/∂_x__{ν}.

On account of (60) the second member on the right-hand side admits of the transformation—

F^{μν} ∂F_{σμ}/∂_x__{ν} = -½ F^{μν} ∂F_{μν}/∂_x__{σ}

= -½ _g_^{μα} _g_^{νβ} F_{αβ} ∂F_{μν}/∂_x__{σ}.

Owing to symmetry, this expression can also be written in the form

= -1/4 [_g_^{μα} _g_^{νβ} F_{αβ} ∂F_{μν}/∂_x__{σ}

+ _g_^{μα} _g_^{νβ} ∂F_{αβ}/∂_x__{σ} F_{μν}],

which can also be put in the form

- 1/4 ∂/∂_x__{σ} (_g_^{μα} _g_^{νβ} F_{αβ} F_{μν})

+ 1/4 F_{αβ} F_{μν} ∂/∂_x__{σ} (_g_^{μα} _g_^{νβ}).

The first of these terms can be written shortly as

- 1/4 ∂/∂_x__{σ} (F^{μν} F_{μν}),

and the second after differentiation can be transformed in the form

- ½ F^{μτ} F_{μν} _g_^{νρ} ∂_g__{στ}/∂_x__{σ}.

If we take all the three terms together, we get the relation

(66) K_{σ} = ∂τ_{σ}^ν/∂_x__{ν} - ½ _g_^{τμ} ∂_g__{μν}/∂_x__{σ}
τ_{τ}^ν

where

(66a) τ_{σ}^ν = -F_{σα} F^{να} + 1/4 δ_{σ}^ν F_{αβ} F^{αβ}.

On account of (30) the equation (66) becomes equivalent to (57) and (57a) when K_{σ} vanishes. Thus τ_{σ}^ν’s are the energy-components of the electro-magnetic field. With the help of (61) and (64) we can easily show that the energy-components of the electro-magnetic field, in the case of the special relativity theory, give rise to the well-known Maxwell-Poynting expressions.

We have now deduced the most general laws which the gravitation-field and matter satisfy when we use a co-ordinate system for which √(-_g_) = 1. Thereby we achieve an important simplification in all our formulas and calculations, without renouncing the conditions of general covariance, as we have obtained the equations through a specialisation of the co-ordinate system from the general covariant-equations. Still the question is not without formal interest, whether, when the energy-components of the gravitation-field and matter is defined in a generalised manner without any specialisation of co-ordinates, the laws of conservation have the form of the equation (56), and the field-equations of gravitation hold in the form (52) or (52a); such that on the left-hand side, we have a divergence in the usual sense, and on the right-hand side, the sum of the energy-components of matter and gravitation. I have found out that this is indeed the case. But I am of opinion that the communication of my rather comprehensive work on this subject will not pay, for nothing essentially new comes out of it.

E. §21. Newton’s theory as a first approximation.

We have already mentioned several times that the special relativity theory is to be looked upon as a special case of the general, in which _g__{μν}’s have constant values (4). This signifies, according to what has been said before, a total neglect of the influence of gravitation. We get one important approximation if we consider the case when _g__{μν}’s differ from (4) only by small magnitudes (compared to 1) where we can neglect small quantities of the second and higher orders (first aspect of the approximation.)

Further it should be assumed that within the space-time region considered, _g__{μν}’s at infinite distances (using the word infinite in a spatial sense) can, by a suitable choice of co-ordinates, tend to the limiting values (4); _i.e._, we consider only those gravitational fields which can be regarded as produced by masses distributed over finite regions.

We can assume that this approximation should lead to Newton’s theory. For it however, it is necessary to treat the fundamental equations from another point of view. Let us consider the motion of a particle according to the equation (46). In the case of the special relativity theory, the components

_dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_,

can take any values. This signifies that any velocity

_v_ = √((_dx₁_/_dx₄_)² + (_dx₂_/_dx₄_)² + (_dx₃_/_dx₄_)²)

can appear which is less than the velocity of light in vacuum (_v_ < 1). If we finally limit ourselves to the consideration of the case when _v_ is small compared to the velocity of light, it signifies that the components

_dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_,

can be treated as small quantities, whereas _dx₄_/_ds_ is equal to 1, up to the second-order magnitudes (the second point of view for approximation).

Now we see that, according to the first view of approximation, the magnitudes γ_{μν}^τ’s are all small quantities of at least the first order. A glance at (46) will also show, that in this equation according to the second view of approximation, we are only to take into account those terms for which μ = ν = 4.

By limiting ourselves only to terms of the lowest order we get instead of (46), first, the equations:—

_d²__x__{τ}/_dt²_ = Γ₄₄^τ, where _ds_ = _dx₄_ = _dt_,

or by limiting ourselves only to those terms which according to the first stand-point are approximations of the first order,

§17. The laws of conservation in the general case.

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 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})].

-½ ∂³_g_^{αβ}/∂_x__{α}∂_x__{β}∂_x__{μ}

So that remembering (54) we have

(55) ∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}Γ_{μβ}^α
- ½ δ_{μ}^σ _g_^{λβ} Γ_{λβ}^α) = 0.

identically.

From (55) and (52a) it follows that

(56) ∂/∂_x__{σ} (_t__{μ}^σ + T_{μ}^σ) = 0

§18. The Impulse-energy law for matter as a consequence of the
field-equations.

If we multiply (53) with ∂_g_^{μν}/∂_x__{σ}, we get in a way similar to §15, remembering that

_g__{μν} ∂_g_^{μν}/∂_x__{σ} vanishes,

the equations ∂_t__{σ}^α/∂_x__{α} - ½ ∂_g_^{μν}/∂_x__{σ} T_{μν} = 0

or remembering (56)

(57) ∂T_{σ}^α/∂_x__{α} + ½ ∂_g_^{μν}/∂_x__{σ} T_{μν} = 0

(57a) ∂T_{σ}^α/∂_x__{α} = -Γ_{σβ}^α T_{α}^β.

D. THE “MATERIAL” PHENOMENA.

§19. Euler’s equations for frictionless adiabatic liquid.

Let _p_ and ρ, be two scalars, of which the first denotes the pressure and the last the density of the fluid; between them there is a relation. Let the contravariant symmetrical tensor

T^{αβ} = -_g_^{αβ} _p_ + ρ _dx__{α}/_ds_ _dx__{β}/_ds_ (58)

be the contra-variant energy-tensor of the liquid. To it also belongs the covariant tensor

(58a) T_{μν} = -_g__{μν} _p_ + _g__{μα} _dx__{α}/_ds_ _g__{μβ}
_dx__{β}/_ds_ ρ

as well as the mixed tensor

(58b) T^α_{σ} = -δ^α_{σ} _p_ + _g__{σβ} _dx__{β}/_ds_ _dx__{α}/_ds_
ρ.

_g__{αβ} _dx__α/_ds_ _dx__{β}/_ds_ = 1,

are sufficient, with the given values of _g__{αβ}, for finding out the six unknowns

_p_, ρ, _dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_ _dx₄_/_ds_.

§20. Maxwell’s Electro-Magnetic field-equations.

(59) F_{ρσ} = ∂φ_{ρ}/∂_x__{σ} - ∂φ_{σ}/∂_x__{ρ}.

From (59), it follows that the system of equations

(60) ∂F_{ρσ}/∂_x__{τ} + ∂F_{στ}/∂_x__{ρ} + ∂F_{τρ}/∂_x__{σ} = 0

is satisfied of which the left-hand side, according to (37), is an anti-symmetrical tensor of the third kind. This system (60) contains essentially four equations, which can be thus written:—

This system of equations corresponds to the second system of equations of Maxwell. We see it at once if we put

{ F₂₃ = H_{_x_} F₁₄ = E_{_x_}
{
(61) { F₃₁ = H_{_y_} F₂₄ = E_{_y_}
{
{ F₁₂ = H_{_z_} F₃₄ = E_{_z_}

Instead of (60a) we can therefore write according to the usual notation of three-dimensional vector-analysis:—

{ ∂H/∂_t_ + rot E = 0
(60b) {
{ div H = 0.

The first Maxwellian system is obtained by a generalisation of the form given by Minkowski.

We introduce the contra-variant six-vector F_{αβ} by the equation

(62) F^{μν} = _g_^{μα} _g_^{νβ} F_{αβ},

(63) ∂F^{μν}/∂_x__{ν} = J^μ

If we put

{ F²³ = H′_{_x_} F¹⁴ = -E′_{_x_}
{
(64) { F³¹ = H′_{_y_} F²⁴ = -E′_{_y_}
{
{ F¹² = H′_{_z_} F³⁴ = -E′_{_z_}

which quantities become equal to H_{_x_} ... E_{_x_} in the case of the special relativity theory, and besides

J^1 = _i__{_x_} ... J^4 = ρ

we get instead of (63)

{ rot H′ - ∂E′/∂_t_ = _i_
(63a) {
{ div E′ = ρ

The equations (60), (62) and (63) give thus a generalisation of Maxwell’s field-equations in vacuum, which remains true in our chosen system of co-ordinates.

_The energy-components of the electro-magnetic field._

Let us form the inner-product

(65) K_{σ} = F_{σμ} J^μ.

According to (61) its components can be written down in the three-dimensional notation.

{ K₁ = ρE_{_x_} + [_i_, H]_{x}

(65a) { — — —

{ K₄ = — (_i_, E).

In order to get the energy components T_{σ}^ν of the electro-magnetic field, we require only to give to the equation K_{σ} = 0, the form of the equation (57).

From (63) and (65) we get first,

K_{σ} = F_{σμ} ∂F_{μν}/∂_x__{ν}

= ∂/∂_x__{ν} (F_{σμ} F^{μν}) - F^{μν} ∂F_{σμ}/∂_x__{ν}.

On account of (60) the second member on the right-hand side admits of the transformation—

F^{μν} ∂F_{σμ}/∂_x__{ν} = -½ F^{μν} ∂F_{μν}/∂_x__{σ}

= -½ _g_^{μα} _g_^{νβ} F_{αβ} ∂F_{μν}/∂_x__{σ}.

Owing to symmetry, this expression can also be written in the form

= -1/4 [_g_^{μα} _g_^{νβ} F_{αβ} ∂F_{μν}/∂_x__{σ}

+ _g_^{μα} _g_^{νβ} ∂F_{αβ}/∂_x__{σ} F_{μν}],

which can also be put in the form

- 1/4 ∂/∂_x__{σ} (_g_^{μα} _g_^{νβ} F_{αβ} F_{μν})

+ 1/4 F_{αβ} F_{μν} ∂/∂_x__{σ} (_g_^{μα} _g_^{νβ}).

The first of these terms can be written shortly as

- 1/4 ∂/∂_x__{σ} (F^{μν} F_{μν}),

and the second after differentiation can be transformed in the form

- ½ F^{μτ} F_{μν} _g_^{νρ} ∂_g__{στ}/∂_x__{σ}.

If we take all the three terms together, we get the relation

(66) K_{σ} = ∂τ_{σ}^ν/∂_x__{ν} - ½ _g_^{τμ} ∂_g__{μν}/∂_x__{σ}
τ_{τ}^ν

where

(66a) τ_{σ}^ν = -F_{σα} F^{να} + 1/4 δ_{σ}^ν F_{αβ} F^{αβ}.

E. §21. Newton’s theory as a first approximation.

_dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_,

can take any values. This signifies that any velocity

_v_ = √((_dx₁_/_dx₄_)² + (_dx₂_/_dx₄_)² + (_dx₃_/_dx₄_)²)

_dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_,

can be treated as small quantities, whereas _dx₄_/_ds_ is equal to 1, up to the second-order magnitudes (the second point of view for approximation).

By limiting ourselves only to terms of the lowest order we get instead of (46), first, the equations:—

_d²__x__{τ}/_dt²_ = Γ₄₄^τ, where _ds_ = _dx₄_ = _dt_,

or by limiting ourselves only to those terms which according to the first stand-point are approximations of the first order,

$$ \frac{d^2 x_{\tau}}{dt^2} = \begin{bmatrix}44\\\tau\end{bmatrix} $$ (\tau = 1, 2, 3)

$$ \frac{d^2 x_{4}}{dt^2} = - \begin{bmatrix}4^4\\4\end{bmatrix] $$

If we further assume that the gravitation-field is quasi-static, _i.e._, it is limited only to the case when the matter producing the gravitation-field is moving slowly (relative to the velocity of light) we can neglect the differentiations of the positional co-ordinates on the right-hand side with respect to time, so that we get

(67) _d²__x__{τ}/_dt²_ = -½ ∂_g₄₄_/∂_x__{τ} (τ, = 1, 2, 3)

This is the equation of motion of a material point according to Newton’s theory, where _g_₄₄/₂ plays the part of gravitational potential. The remarkable thing in the result is that in the first-approximation of motion of the material point, only the component _g₄₄_ of the fundamental tensor appears.

Let us now turn to the field-equation (53). In this case, we have to remember that the energy-tensor of matter is exclusively defined in a narrow sense by the density ρ of matter, _i.e._, by the second member on the right-hand side of 58 [(58a, or 58b)]. If we make the necessary approximations, then all component vanish except

τ₄₄ = ρ = τ.

On the left-hand side of (53) the second term is an infinitesimal of the second order, so that the first leads to the following terms in the approximation, which are rather interesting for us:

$$ \frac{\partial}{\partial x_{1}} \begin{bmatrix}\mu\nu\\1\end{bmatrix} + \frac{\partial}{\partial x_{2}} \begin{bmatrix}\mu\nu\\2\end{bmatrix} + \frac{\partial}{\partial x_{3}} \begin{bmatrix}\mu\nu\\3\end{bmatrix} + \frac{\partial}{\partial x_{4}} \begin{bmatrix}\mu\nu\\4\end{bmatrix} $$

By neglecting all differentiations with regard to time, this leads, when μ = ν =4, to the expression

$$ - \frac{1}{2} ( \frac{\partial^2 g_{44}}{\partial x^2_{1}} + \frac{\partial^2 g_{44}}{\partial x^2_{2}} + \frac{\partial^2 g_{44}}{\partial x^2_{3}} ) = - \frac{1}{2} V^2 g_{44} $$

The last of the equations (53) thus leads to

(68) ▽² _g₄₄_ = κρ.

The equations (67) and (68) together, are equivalent to Newton’s law of gravitation.

For the gravitation-potential we get from (67) and (68) the exp.

(68a.) -κ/(8π) ∫ ρ_d_τ/_r_

whereas the Newtonian theory for the chosen unit of time gives

-K/_c²_ ∫ρ_d_τ/_r_,

where K denotes usually the gravitation-constant. 6.7 x 10⁻⁸; equating them we get

(69) κ = 8πK/_c²_ = 1.87 x 10⁻²⁷.

§22. Behaviour of measuring rods and clocks in a statical
gravitation-field. Curvature of light-rays. Perihelion-motion of the
paths of the Planets.

In order to obtain Newton’s theory as a first approximation we had to calculate only _g₄₄_, out of the 10 components _g__{μν} of the gravitation-potential, for that is the only component which comes in the first approximate equations of motion of a material point in a gravitational field.

We see however, that the other components of _g__{μν} should also differ from the values given in (4) as required by the condition _g_ = -1.

For a heavy particle at the origin of co-ordinates and generating the gravitational field, we get as a first approximation the symmetrical solution of the equation:—

{ _g__{ρσ} = -δ_{ρσ} - α(_x__{ρ} _x__{σ})/_r³_ (ρ and σ 1, 2,
3)
{
(70) { _g__{ρ4} = _g__{4ρ} = 0 (ρ 1, 2, 3)
{
{ _g₄₄_ = 1 - α/_r_.

δ_{ρσ} is 1 or 0, according as ρ = σ or not and _r_ is the quantity

+√(_x₁²_ + _x₂²_ + _x₃²_).

On account of (68a) we have

(70a) α = κM/4π

where M denotes the mass generating the field. It is easy to verify that this solution satisfies approximately the field-equation outside the mass M.

Let us now investigate the influences which the field of mass M will have upon the metrical properties of the field. Between the lengths and times measured locally on the one hand, and the differences in co-ordinates _dx__{ν} on the other, we have the relation

_ds²_ = _g__{μν} _dx__{μ} _dx__{ν}.

For a unit measuring rod, for example, placed parallel to the _x_ axis, we have to put

_ds²_ = -1, _dx₂_ = _dx₃_ = _dx₄_ = 0

then -1 = _g_₁₁_dx₁²_.

If the unit measuring rod lies on the _x_ axis, the first of the equations (70) gives

_g₁₁_ = -(1 + α/_r_).

From both these relations it follows as a first approximation that

(71) _dx_ = 1 - α/2_r_.

The unit measuring rod appears, when referred to the co-ordinate-system, shortened by the calculated magnitude through the presence of the gravitational field, when we place it radially in the field.

Similarly we can get its co-ordinate-length in a tangential position, if we put for example

_ds²_ = -1, _dx₁_ = _dx₃_ = _dx₄_ = 0, _x₁_ = _r_, _x₂_ = _x₃_ = 0

we then get

(71a) -1 = _g₂₂_ _dx₂²_ = -_dx₂²_.

The gravitational field has no influence upon the length of the rod, when we put it tangentially in the field.

Thus Euclidean geometry does not hold in the gravitational field even in the first approximation, if we conceive that one and the same rod independent of its position and its orientation can serve as the measure of the same extension. But a glance at (70a) and (69) shows that the expected difference is much too small to be noticeable in the measurement of earth’s surface.

We would further investigate the rate of going of a unit-clock which is placed in a statical gravitational field. Here we have for a period of the clock

_ds_ = 1, _dx₁_ = _dx₂_ _dx₃_ = 0;

then we have

1 = _g₄₄__dx₄²_

_dx₄_ = 1/√(_g_₄₄) = 1/√(1 + (_g_₄₄ - 1)) = 1 - (_g_₄₄ - 1)/2

or _dx₄_ = 1 + _k_/8π ∫ ρ_d_τ/_r_.

Therefore the clock goes slowly what it is placed in the neighbourhood of ponderable masses. It follows from this that the spectral lines in the light coming to us from the surfaces of big stars should appear shifted towards the red end of the spectrum.

Let us further investigate the path of light-rays in a statical gravitational field. According to the special relativity theory, the velocity of light is given by the equation

-_dx₁²_ - _dx₂²_ - _dx₃²_ + _dx₄²_ = 0;

thus also according to the generalised relativity theory it is given by the equation

(73) _ds²_ = _g__{μν} _dx__{μ} _dx__{ν} = 0.

If the direction, _i.e._, the ratio _dx₁_ : _dx₂_ : _dx₃_ is given, the equation (73) gives the magnitudes

_dx₁_/_dx₄_, _dx₂_/_dx₄_, _dx₃_/_dx₄_,

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The Principle of RelativityChapter VIII: Appendix: Mechanics and the Relativity-Postulate (3)

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