Chapter 14 of 16 · 3972 words · ~20 min read

Part 14

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,

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})].

-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.