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:—

d²x_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^σβΓ_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^σβγ_μβ^α)

= ½ ∂³g^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^λβΓ_λβ^α) or

to 1/4 ∂²/∂x_α∂x_μ [g^λβg^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 ∂³g^αβ/∂x_α∂x_β∂x_μ

So that remembering (54) we have

(55) ∂²/∂x_α∂x_σ (g^σβΓ_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.