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

J¹ = i_x ... J⁴ = 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.