The Principle of Relativity

by Albert Einstein; H. Minkowski

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 K₀ 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)"

we get the motion of the point with reference to K₁, given by

"(46)"

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)"

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∫ Hdτ = 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 δg^μν + 2g^μν Γ^α_μβ δΓ^β_να

= - Γ^α_μβ Γ^β_να δg^μν + 2Γ^α_μβ δ(g^μνΓ^β_να).

But

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 δg^μν + Γ_μβ^α δg_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) { -2κt_σ^α = g_σ^μν ∂H/∂g_α^μν - δ_σ^α H.

Owing to the relations (48), the equations (47) and (34),

(50) κt_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_σ⁴ dV} = ∫(t_σ¹ α₁

+ t_σ² α₂ + t_σ³ α₃)dS

where α₁, α₂, α₂ are the direction-cosines of the inward-drawn normal to the surface-element dS 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_nμ

- 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)"

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)"

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.