The Generalised Principle of Relativity 89-163

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The Foundation of the Generalised Theory of Relativity
By A. Einstein.
From Annalen der Physik 4.49.1916.

The theory which is sketched in the following pages forms the most wide-going generalization conceivable of what is at present known as “the theory of Relativity;” this latter theory I differentiate from the former “Special Relativity theory,” and suppose it to be known. The generalization of the Relativity theory has been made much easier through the form given to the special Relativity theory by Minkowski, which mathematician was the first to recognize clearly the formal equivalence of the space like and time-like co-ordinates, and who made use of it in the building up of the theory. The mathematical apparatus useful for the general relativity theory, lay already complete in the “Absolute Differential Calculus,” which were based on the researches of Gauss, Riemann and Christoffel on the non-Euclidean manifold, and which have been shaped into a system by Ricci and Levi-civita, and already applied to the problems of theoretical physics. I have in part B of this communication developed in the simplest and clearest manner, all the supposed mathematical auxiliaries, not known to Physicists, which will be useful for our purpose, so that, a study of the mathematical literature is not necessary for an understanding of this paper. Finally in this place I thank my friend Grossmann, by whose help I was not only spared the study of the mathematical literature pertinent to this subject, but who also aided me in the researches on the field equations of gravitation.

A Principal considerations about the Postulate of Relativity.

§ 1. Remarks on the Special Relativity Theory.

The special relativity theory rests on the following postulate which also holds valid for the Galileo-Newtonian mechanics.

If a co-ordinate system K be so chosen that when referred to it, the physical laws hold in their simplest forms these laws would be also valid when referred to another system of co-ordinates K′ which is subjected to an uniform translational motion relative to K. We call this postulate “The Special Relativity Principle.” By the word special, it is signified that the principle is limited to the case, when K′ has uniform translatory motion with reference to K, but the equivalence of K and K′ does not extend to the case of non-uniform motion of K′ relative to K.

The Special Relativity Theory does not differ from the classical mechanics through the assumption of this postulate, but only through the postulate of the constancy of light-velocity in vacuum which, when combined with the special relativity postulate, gives in a well-known way, the relativity of synchronism as well as the Lorenz-transformation, with all the relations between moving rigid bodies and clocks.

The modification which the theory of space and time has undergone through the special relativity theory, is indeed a profound one, but a weightier point remains untouched. According to the special relativity theory, the theorems of geometry are to be looked upon as the laws about any possible relative positions of solid bodies at rest, and more generally the theorems of kinematics, as theorems which describe the relation between measurable bodies and clocks. Consider two material points of a solid body at rest; then according to these conceptions there corresponds to these points a wholly definite extent of length, independent of kind, position, orientation and time of the body.

Similarly let us consider two positions of the pointers of a clock which is at rest with reference to a co-ordinate system; then to these positions, there always corresponds, a time-interval of a definite length, independent of time and place. It would be soon shown that the general relativity theory can not hold fast to this simple physical significance of space and time.

§ 2. About the reasons which explain the extension of the relativity-postulate.

To the classical mechanics (no less than) to the special relativity theory, is attached an episteomological defect, which was perhaps first cleanly pointed out by E. Mach. We shall illustrate it by the following example; Let two fluid bodies of equal kind and magnitude swim freely in space at such a great distance from one another (and from all other masses) that only that sort of gravitational forces are to be taken into account which the parts of any of these bodies exert upon each other. The distance of the bodies from one another is invariable. The relative motion of the different parts of each body is not to occur. But each mass is seen to rotate by an observer at rest relative to the other mass round the connecting line of the masses with a constant angular velocity (definite relative motion for both the masses). Now let us think that the surfaces of both the bodies (S₁ and S₂) are measured with the help of measuring rods (relatively at rest); it is then found that the surface of S₁ is a sphere and the surface of the other is an ellipsoid of rotation. We now ask, why is this difference between the two bodies? An answer to this question can only then be regarded as satisfactory from the episteomological standpoint when the thing adduced as the cause is an observable fact of experience. The law of causality has the sense of a definite statement about the world of experience only when observable facts alone appear as causes and effects.

The Newtonian mechanics does not give to this question any satisfactory answer. For example, it says:—The laws of mechanics hold true for a space R₁ relative to which the body S₁ is at rest, not however for a space relative to which S₂ is at rest.

The Galiliean space, which is here introduced is however only a purely imaginary cause, not an observable thing. It is thus clear that the Newtonian mechanics does not, in the case treated here, actually fulfil the requirements of causality, but produces on the mind a fictitious complacency, in that it makes responsible a wholly imaginary cause R₁ for the different behaviours of the bodies S₁ and S₂ which are actually observable.

A satisfactory explanation to the question put forward above can only be thus given:—that the physical system composed of S₁ and S₂ shows for itself alone no conceivable cause to which the different behaviour of S₁ and S₂ can be attributed. The cause must thus lie outside the system. We are therefore led to the conception that the general laws of motion which determine specially the forms of S₁ and S₂ must be of such a kind, that the mechanical behaviour of S₁ and S₂ must be essentially conditioned by the distant masses, which we had not brought into the system considered. These distant masses, (and their relative motion as regards the bodies under consideration) are then to be looked upon as the seat of the principal observable causes for the different behaviours of the bodies under consideration. They take the place of the imaginary cause R₁. Among all the conceivable spaces R₁ and R₂ moving in any manner relative to one another, there is a priori, no one set which can be regarded as affording greater advantages, against which the objection which was already raised from the standpoint of the theory of knowledge cannot be again revived. The laws of physics must be so constituted that they should remain valid for any system of co-ordinates moving in any manner. We thus arrive at an extension of the relativity postulate.

Besides this momentous episteomological argument, there is also a well-known physical fact which speaks in favour of an extension of the relativity theory. Let there be a Galiliean co-ordinate system K relative to which (at least in the four-dimensional region considered) a mass at a sufficient distance from other masses move uniformly in a line. Let K′ be a second co-ordinate system which has a uniformly accelerated motion relative to K. Relative to K′ any mass at a sufficiently great distance experiences an accelerated motion such that its acceleration and the direction of acceleration is independent of its material composition and its physical conditions.

Can any observer, at rest relative to K′, then conclude that he is in an actually accelerated reference-system? This is to be answered in the negative; the above-named behaviour of the freely moving masses relative to K′ can be explained in as good a manner in the following way. The reference-system K′ has no acceleration. In the space-time region considered there is a gravitation-field which generates the accelerated motion relative to K′.

This conception is feasible, because to us the experience of the existence of a field of force (namely the gravitation field) has shown that it possesses the remarkable property of imparting the same acceleration to all bodies. The mechanical behaviour of the bodies relative to K′ is the same as experience would expect of them with reference to systems which we assume from habit as stationary; thus it explains why from the physical stand-point it can be assumed that the systems K and K′ can both with the same legitimacy be taken as at rest, that is, they will be equivalent as systems of reference for a description of physical phenomena.

From these discussions we see, that the working out of the general relativity theory must, at the same time, lead to a theory of gravitation; for we can “create” a gravitational field by a simple variation of the co-ordinate system. Also we see immediately that the principle of the constancy of light-velocity must be modified, for we recognise easily that the path of a ray of light with reference to K′ must be, in general, curved, when light travels with a definite and constant velocity in a straight line with reference to K.

§ 3. The time-space continuum. Requirements of the general Co-variance for the equations expressing the laws of Nature in general.

In the classical mechanics as well as in the special relativity theory, the co-ordinates of time and space have an immediate physical significance; when we say that any arbitrary point has x₁ as its X₁ co-ordinate, it signifies that the projection of the point-event on the X₁-axis ascertained by means of a solid rod according to the rules of Euclidean Geometry is reached when a definite measuring rod, the unit rod, can be carried x₁ times from the origin of co-ordinates along the X₁ axis. A point having x₄ = t₁ as the X₄ co-ordinate signifies that a unit clock which is adjusted to be at rest relative to the system of co-ordinates, and coinciding in its spatial position with the point-event and set according to some definite standard has gone over x₄ = t periods before the occurrence of the point-event.

This conception of time and space is continually present in the mind of the physicist, though often in an unconscious way, as is clearly recognised from the role which this conception has played in physical measurements. This conception must also appear to the reader to be lying at the basis of the second consideration of the last paragraph and imparting a sense to these conceptions. But we wish to show that we are to abandon it and in general to replace it by more general conceptions in order to be able to work out thoroughly the postulate of general relativity,—the case of special relativity appearing as a limiting case when there is no gravitation.

We introduce in a space, which is free from Gravitation-field, a Galiliean Co-ordinate System K (x, y, z, t) and also, another system K′ (x′ y′ z′ t′) rotating uniformly relative to K. The origin of both the systems as well as their z-axes might continue to coincide. We will show that for a space-time measurement in the system K′, the above established rules for the physical significance of time and space can not be maintained. On grounds of symmetry it is clear that a circle round the origin in the XY plane of K, can also be looked upon as a circle in the plane (X′, Y′) of K′. Let us now think of measuring the circumference and the diameter of these circles, with a unit measuring rod (infinitely small compared with the radius) and take the quotient of both the results of measurement. If this experiment be carried out with a measuring rod at rest relatively to the Galiliean system K we would get π, as the quotient. The result of measurement with a rod relatively at rest as regards K′ would be a number which is greater than π. This can be seen easily when we regard the whole measurement-process from the system K and remember that the rod placed on the periphery suffers a Lorenz-contraction, not however when the rod is placed along the radius. Euclidean Geometry therefore does not hold for the system K′; the above fixed conceptions of co-ordinates which assume the validity of Euclidean Geometry fail with regard to the system K′. We cannot similarly introduce in K′ a time corresponding to physical requirements, which will be shown by all similarly prepared clocks at rest relative to the system K′. In order to see this we suppose that two similarly made clocks are arranged one at the centre and one at the periphery of the circle, and considered from the stationary system K. According to the well-known results of the special relativity theory it follows—(as viewed from K)—that the clock placed at the periphery will go slower than the second one which is at rest. The observer at the common origin of co-ordinates who is able to see the clock at the periphery by means of light will see the clock at the periphery going slower than the clock beside him. Since he cannot allow the velocity of light to depend explicitly upon the time in the way under consideration he will interpret his observation by saying that the clock on the periphery actually goes slower than the clock at the origin. He cannot therefore do otherwise than define time in such a way that the rate of going of a clock depends on its position.

We therefore arrive at this result. In the general relativity theory time and space magnitudes cannot be so defined that the difference in spatial co-ordinates can be immediately measured by the unit-measuring rod, and time-like co-ordinate difference with the aid of a normal clock.

The means hitherto at our disposal, for placing our co-ordinate system in the time-space continuum, in a definite way, therefore completely fail and it appears that there is no other way which will enable us to fit the co-ordinate system to the four-dimensional world in such a way, that by it we can expect to get a specially simple formulation of the laws of Nature. So that nothing remains for us but to regard all conceivable co-ordinate systems as equally suitable for the description of natural phenomena. This amounts to the following law:—

That in general, Laws of Nature are expressed by means of equations which are valid for all co-ordinate systems, that is, which are covariant for all possible transformations. It is clear that a physics which satisfies this postulate will be unobjectionable from the standpoint of the general relativity postulate. Because among all substitutions there are, in every case, contained those, which correspond to all relative motions of the co-ordinate system (in three dimensions). This condition of general covariance which takes away the last remnants of physical objectivity from space and time, is a natural requirement, as seen from the following considerations. All our well-substantiated space-time propositions amount to the determination of space-time coincidences. If, for example, the event consisted in the motion of material points, then, for this last case, nothing else are really observable except the encounters between two or more of these material points. The results of our measurements are nothing else than well-proved theorems about such coincidences of material points, of our measuring rods with other material points, coincidences between the hands of a clock, dial-marks and point-events occurring at the same position and at the same time.

The introduction of a system of co-ordinates serves no other purpose than an easy description of totality of such coincidences. We fit to the world our space-time variables (x₁ x₂ x₃ x₄) such that to any and every point-event corresponds a system of values of (x₁ x₂ x₃ x₄). Two coincident point-events correspond to the same value of the variables (x₁ x₂ x₃ x₄); i.e., the coincidence is characterised by the equality of the co-ordinates. If we now introduce any four functions (x′₁ x′₂ x′₃ x′₄) as co-ordinates, so that there is an unique correspondence between them, the equality of all the four co-ordinates in the new system will still be the expression of the space-time coincidence of two material points. As the purpose of all physical laws is to allow us to remember such coincidences, there is a priori no reason present, to prefer a certain co-ordinate system to another; i.e., we get the condition of general covariance.

§ 4. Relation of four co-ordinates to spatial and time-like measurements.

Analytical expression for the Gravitation-field.

I am not trying in this communication to deduce the general Relativity-theory as the simplest logical system possible, with a minimum of axioms. But it is my chief aim to develop the theory in such a manner that the reader perceives the psychological naturalness of the way proposed, and the fundamental assumptions appear to be most reasonable according to the light of experience. In this sense, we shall now introduce the following supposition; that for an infinitely small four-dimensional region, the relativity theory is valid in the special sense when the axes are suitably chosen.

The nature of acceleration of an infinitely small (positional) co-ordinate system is hereby to be so chosen, that the gravitational field does not appear; this is possible for an infinitely small region. X₁, X₂, X₃ are the spatial co-ordinates; X₄ is the corresponding time-co-ordinate measured by some suitable measuring clock. These co-ordinates have, with a given orientation of the system, an immediate physical significance in the sense of the special relativity theory (when we take a rigid rod as our unit of measure). The expression

had then, according to the special relativity theory, a value which may be obtained by space-time measurement, and which is independent of the orientation of the local co-ordinate system. Let us take ds as the magnitude of the line-element belonging to two infinitely near points in the four-dimensional region. If ds² belonging to the element (dX₁, dX₂, dX₃, dX₄) be positive we call it with Minkowski, time-like, and in the contrary case space-like.

To the line-element considered, i.e., to both the infinitely near point-events belong also definite differentials dx₁, dx₂, dx₃, dx₄, of the four-dimensional co-ordinates of any chosen system of reference. If there be also a local system of the above kind given for the case under consideration, dX’s would then be represented by definite linear homogeneous expressions of the form

If we substitute the expression in (1) we get

We would afterwards see that the choice of such a system of co-ordinates for a finite region is in general not possible.

B
Mathematical Auxiliaries for Establishing the General Covariant Equations.

We have seen before that the general relativity-postulate leads to the condition that the system of equations for Physics, must be co-variants for any possible substitution of co-ordinates x₁, ... x₄; we have now to see how such general co-variant equations can be obtained. We shall now turn our attention to these purely mathematical propositions. It will be shown that in the solution, the invariant ds, given in equation (3) plays a fundamental rôle, which we, following Gauss’s Theory of Surfaces, style as the line-element.

The fundamental idea of the general co-variant theory is this:—With reference to any co-ordinate system, let certain things (tensors) be defined by a number of functions of co-ordinates which are called the components of the tensor. There are now certain rules according to which the components can be calculated in a new system of co-ordinates, when these are known for the original system, and when the transformation connecting the two systems is known. The things herefrom designated as “Tensors” have further the property that the transformation equation of their components are linear and homogeneous; so that all the components in the new system vanish if they are all zero in the original system. Thus a law of Nature can be formulated by putting all the components of a tensor equal to zero so that it is a general co-variant equation; thus while we seek the laws of formation of the tensors, we also reach the means of establishing general co-variant laws.

5. Contra-variant and co-variant Four-vector.

"(5)."

"(5a)."

the expressions

As in the above equation Bσ′ are independent of one another and perfectly arbitrary, it follows that the transformation law is:—

Remarks on the simplification of the mode of writing the expressions. A glance at the equations of this paragraph will show that the indices which appear twice within the sign of summation [for example ν in (5)] are those over which the summation is to be made and that only over the indices which appear twice. It is therefore possible, without loss of clearness, to leave off the summation sign; so that we introduce the rule: wherever the index in any term of an expression appears twice, it is to be summed over all of them except when it is not expressedly said to the contrary.

The difference between the co-variant and the contra-variant four-vector lies in the transformation laws [(7) and (5)]. Both the quantities are tensors according to the above general remarks; in it lies its significance. In accordance with Ricci and Levi-civita, the contravariants and co-variants are designated by the over and under indices.

§ 6. Tensors of the second and higher ranks.

Contravariant tensor:—If we now calculate all the 16 products Aμν of the components Aμ Bν, of two contravariant four-vectors

(8) Aμν = AμBν

nsformation relation (9), a contravariant tensor of the second rank. Not every such tensor can be built from two four-vectors, (according to 8). But it is easy to show that any 16 quantities Aμν, can be represented as the sum of AμBν of properly chosen four pairs of four-vectors. From it, we can prove in the simplest way all laws which hold true for the tensor of the second rank defined through (9), by proving it only for the special tensor of the type (8).

Contravariant Tensor of any rank:—It is clear that corresponding to (8) and (9), we can define contravariant tensors of the 3rd and higher ranks, with 4³, etc. components. Thus it is clear from (8) and (9) that in this sense, we can look upon contravariant four-vectors, as contravariant tensors of the first rank.

Co-variant tensor.

"(11)."

By means of these transformation laws, the co-variant tensor of the second rank is defined. All re-marks which we have already made concerning the contravariant tensors, hold also for co-variant tensors.

Remark:—

It is convenient to treat the scalar Invariant either as a contravariant or a co-variant tensor of zero rank.

Mixed tensor. We can also define a tensor of the second rank of the type

which is co-variant with reference to μ and contravariant with reference to ν. Its transformation law is

"(13)."

Naturally there are mixed tensors with any number of co-variant indices, and with any number of contra-variant indices. The co-variant and contra-variant tensors can be looked upon as special cases of mixed tensors.

Symmetrical tensors:—

A contravariant or a co-variant tensor of the second or higher rank is called symmetrical when any two components obtained by the mutual interchange of two indices are equal. The tensor Aμν or Aμν is symmetrical, when we have for any combination of indices

(14) Aμν = Aνμ

or

(14a) Aμν = Aνμ.

It must be proved that a symmetry so defined is a property independent of the system of reference. It follows in fact from (9) remembering (14)

Antisymmetrical tensor.

§ 7. Multiplication of Tensors.

Reduction in rank of a mixed Tensor.

The proof that the result of reduction retains a truly tensorial character, follows either from the representation of tensor according to the generalisation of (12) in combination with (6) or out of the generalisation of (13).

Inner and mixed multiplication of Tensors.

Proof:—According to the above assumption, for any substitution we have

From the inversion of (9) we have however

The co-variant fundamental tensor—In the invariant expression of the square of the linear element

The co-variant fundamental tensor.

According to the well-known law of Determinants

or according to (16) also in the form

Now according to the rules of multiplication, of the fore-going paragraph, the magnitudes

If we introduce it in our expression, we get

Determinant of the fundamental tensor.

According to the law of multiplication of determinants, we have

On the other hand we have

Invariant of volume.

From this by applying the law of multiplication twice, we obtain

or

"(A)."

On the other hand the law of transformation of the volume element

is according to the wellknown law of Jacobi.

"(B)."

by multiplication of the two last equations (A) and (B) we get

Should √(-g) vanish at any point of the four-dimensional continuum it would signify that to a finite co-ordinate volume at the place corresponds an infinitely small “natural volume.” This can never be the case; so that g can never change its sign; we would, according to the special relativity theory assume that g has a finite negative value. It is a hypothesis about the physical nature of the continuum considered, and also a pre-established rule for the choice of co-ordinates.

If however (-g) remains positive and finite, it is clear that the choice of co-ordinates can be so made that this quantity becomes equal to one. We would afterwards see that such a limitation of the choice of co-ordinates would produce a significant simplification in expressions for laws of nature.

In place of (18) it follows then simply that

dτ′ = d

from this it follows, remembering the law of Jacobi,

"(19)."

With this choice of co-ordinates, only substitutions with determinant 1 are allowable.

It would however be erroneous to think that this step signifies a partial renunciation of the general relativity postulate. We do not seek those laws of nature which are co-variants with regard to the transformations having the determinant 1, but we ask: what are the general co-variant laws of nature? First we get the law, and then we simplify its expression by a special choice of the system of reference.

Building up of new tensors with the help of the fundamental tensor.

Through inner, outer and mixed multiplications of a tensor with the fundamental tensor, tensors of other kinds and of other ranks can be formed.

Example:—

We would point out specially the following combinations:

(complement to the co-variant or contravariant tensors)

Similarly

§ 9. Equation of the geodetic line (or of point-motion).

Its equation is

From this equation, we can in a wellknown way deduce 4 total differential equations which define the geodetic line; this deduction is given here for the sake of completeness.

Let λ, be a function of the co-ordinates xν; this defines a series of surfaces which cut the geodetic line sought-for as well as all neighbouring lines from P₁ to P₂. We can suppose that all such curves are given when the value of its co-ordinates xν are given in terms of λ. The sign δ corresponds to a passage from a point of the geodetic curve sought-for to a point of the contiguous curve, both lying on the same surface λ.

Then (20) can be replaced by

But

So we get by the substitution of δω in (20a), remembering that

after partial integration,

are the equations of geodetic line; since along the geodetic line considered we have ds ≠ 0, we can choose the parameter λ, as the length of the arc measured along the geodetic line. Then w = 1, and we would get in place of (20c)

Or by merely changing the notation suitably,

"20d"

where we have put, following Christoffel,

"21"

Here we have put, following Christoffel,

§ 10. Formation of Tensors through Differentiation.

Relying on the equation of the geodetic line, we can now easily deduce laws according to which new tensors can be formed from given tensors by differentiation. For this purpose, we would first establish the general co-variant differential equations. We achieve this through a repeated application of the following simple law. If a certain curve be given in our continuum whose points are characterised by the arc-distances s, measured from a fixed point on the curve, and if further φ, be an invariant space function, then dφ/ds is also an invariant. The proof follows from the fact that dφ as well as ds, are both invariants

Since

so that

is also an invariant for all curves which go out from a point in the continuum, i.e., for any choice of the vector dxμ. From which follows immediately that

is a co-variant four-vector (gradient of φ).

From the interchangeability of the differentiation with regard to μ and ν, and also according to (23) and (21) we see that the bracket

is symmetrical with respect to μ and ν.

"25"

"26"

Now the right hand side of (25) multiplied by ψ is

Through addition follows the tensor character of

With the help of the extension of the four-vector, we can easily define “extension” of a co-variant tensor of any rank. This is a generalisation of the extension of the four-vector. We confine ourselves to the case of the extension of the tensors of the 2nd rank for which the law of formation can be clearly seen.

It would therefore be sufficient to deduce the expression of extension, for one such special tensor. According to (26) we have the expressions

are tensors. Through outer multiplication of the first with Bν and the 2nd with Aμ we get tensors of the third rank. Their addition gives the tensor of the third rank

"(27)"

Some special cases of Particular Importance.

A few auxiliary lemmas concerning the fundamental tensor. We shall first deduce some of the lemmas much used afterwards. According to the law of differentiation of determinants, we have

The last form follows from the first when we remember that

From (28), it follows that

"(29)"

and

The expression (31) allows a transformation which we shall often use; according to (21)

"(33)"

If we substitute this in the second of the formula (31), we get, remembering (23),

"(34)"

By substituting the right-hand side of (34) in (29), we get

"(29a)"

Divergence of the contravariant four-vector.

Let us multiply (26) with the contravariant fundamental tensor gμν (inner multiplication), then by a transformation of the first member, the right-hand side takes the form

"(A)"

According to (31) and (29), the last member can take the form

"(B)"

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

Rotation of the (covariant) four-vector.

Antisymmetrical Extension of a Six-vector.

from which it is easy to see that the tensor is antisymmetrical.

Divergence of the Six-vector.

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)"

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.

"(39)"

On the account of the symmetry of

"(40)"

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)"

In the symmetrical case treated, (41) can be replaced by either of the forms

"(41a)"

or

"(41b)"

which we shall have to make use of afterwards.

§12. The Riemann-Christoffel Tensor.

i.e., the extension of a four-vector.

Thus we get (by slightly changing the indices) the tensor of the third rank

"(43)"

"(44)"

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.

"(45)"

"(46)"

§14. The Field-equation of Gravitation in the absence of matter.

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 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:—

Here the variations vanish at the limits of the finite four-dimensional integration-space considered.

We have at first

But

If we now carry out the variations in (47a), we obtain the system of equations

which, owing to the relations (48), coincide with (47), as was required to be proved.

and consequently

we obtain the equation

or

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

which owing to (34) is equal to

or slightly altering the notation, equal to

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

Therefore in the place of the equations (47), we obtain

§16. General formulation of the field-equation of Gravitation.

We thus get instead of (51), the tensor-equation

"(52)"

"(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.

We obtain,

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,

The second member of the expression on the left-hand side of (52a) leads first to

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

So that remembering (54) we have

identically.

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

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.

or remembering (56)

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.

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.

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

as well as the mixed tensor

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

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

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

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

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

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

If we put

we get instead of (63)

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

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

Kσ 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 Kσ will vanish.

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

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

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

which can also be put in the form

The first of these terms can be written shortly as

and the second after differentiation can be transformed in the form

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

where

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 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 γμντ’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:—

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

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

This is the equation of motion of a material point according to Newton’s theory, where g₄₄/₂ 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:

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:

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) ∫ ρdτ/r

whereas the Newtonian theory for the chosen unit of time gives

-K/c² ∫ρdτ/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 gμν 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 gμν 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:—

δρσ 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

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 dxν on the other, we have the relation

ds² = gμν dxμ dxν.

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 = g₁₁dx₁².

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/2r.

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.

then we have

1 = g₄₄dx₄²

dx₄ = 1/√(g₄₄) = 1/√(1 + (g₄₄ - 1)) = 1 - (g₄₄ - 1)/2

or dx₄ = 1 + k/8π ∫ ρdτ/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 = gμν dxμ dxν = 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,

and with it the velocity,

√((dx₁/dx₄)² + (dx₂/dx₄)² + (dx₃/dx₄)²) = γ,

Let us find out the curvature which a light-ray suffers when it goes by a mass M at a distance Δ from it. If we use the co-ordinate system according to the above scheme, then the total bending B of light-rays (reckoned positive when it is concave to the origin) is given as a sufficient approximation by

where (73) and (70) gives

γ = √(-g₄₄/g₂₂) = 1 - α/2r (1 + x₂²/r²).

The calculation gives

B = 2α/Δ = KM/2πΔ.

A ray of light just grazing the sun would suffer a bending of 1·7″, whereas one coming by Jupiter would have a deviation of about ·02″.

If we calculate the gravitation-field to a greater order of approximation and with it the corresponding path of a material particle of a relatively small (infinitesimal) mass we get a deviation of the following kind from the Kepler-Newtonian Laws of Planetary motion. The Ellipse of Planetary motion suffers a slow rotation in the direction of motion, of amount

In this Formula ‘a’ signifies the semi-major axis, c, the velocity of light, measured in the usual way, e, the eccentricity, τ, the time of revolution in seconds.

The calculation gives for the planet Mercury, a rotation of path of amount 43″ per century, corresponding sufficiently to what has been found by astronomers (Leverrier). They found a residual perihelion motion of this planet of the given magnitude which can not be explained by the perturbation of the other planets.

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