IV

In order to demonstrate that the assumption of the group G_{_c_} for the

physical laws does not possibly lead to any contradiction, it is

unnecessary to undertake a revision of the whole of physics on the basis

of the assumptions underlying this group. The revision has already been

successfully made in the case of “Thermodynamics and Radiation,”[30] for

“Electromagnetic phenomena”,[31] and finally for “Mechanics with the

maintenance of the idea of mass.”

For this last mentioned province of physics, the question may be asked:

if there is a force with the components X, Y, Z (in the direction of the

space-axes) at a world-point (_x_, _y_, _z_, _t_), where the

velocity-vector is ([._x_], [._y_], [._z_], [._t_]), then how are we to

regard this force when the system of reference is changed in any

possible manner? Now it is known that there are certain well-tested

theorems about the ponderomotive force in electromagnetic fields, where

the group G_{_c_} is undoubtedly permissible. These theorems lead us to

the following simple rule; _if the system of reference be changed in any

way, then the supposed force is to be put as a force in the new

space-coordinates in such a manner, that the corresponding vector with

the components_

[._t_]X, [._t_]Y, [._t_]Z, [._t_]T,

_where_ T = 1/_c²_ ([._x_]/[._t_] X + [._y_]/[._t_] Y +

[._z_]/[._t_] Z) = 1/_c²_

(_the rate of

which work is done at the world-point_), _remains unaltered_.

This vector is always normal to the velocity-vector at P. Such a

force-vector, representing a force at P, may be called a _moving

force-vector at_ P.

Now the world-line passing through P will be described by a substantial

point with the constant _mechanical mass m_. Let us call _m-times_ the

velocity-vector at P as the _impulse-vector_, and _m-times_ the

acceleration-vector at P as the _force-vector of motion_, at P.

According to these definitions, the following law tells us how the

motion of a point-mass takes place under any moving force-vector[32]:

_The force-vector of motion is equal to the moving force-vector._

This enunciation comprises four equations for the components in the four

directions, of which the fourth can be deduced from the first three,

because both of the above-mentioned vectors are perpendicular to the

velocity-vector. From the definition of T, we see that the fourth simply

expresses the “Energy-law.” Accordingly _c²_-_times the component of the

impulse-vector in the direction of the t-axis is_ to be defined as _the

kinetic-energy_ of the point-mass. The expression for this is

_mc²_ _dt_/_d_τ = _mc²_ /√(1 - _v²_/_c²_)

_i.e._, if we deduct from this the additive constant _mc²_, we obtain

the expression ½ _mv²_ of Newtonian-mechanics up to magnitudes of _the

order of_ 1/_c²_. Hence it appears that _the energy_ depends _upon the

system of reference_. But since the _t_-axis can be laid in the

direction of any time-like axis, therefore the energy-law comprises, for

any possible system of reference, the whole system of equations of

motion. This fact retains its significance even in the limiting case c =

∞, for the axiomatic construction of Newtonian mechanics, as has already

been pointed out by T. R. Schütz.[33]

From the very beginning, we can establish the ratio between the units of

time and space in such a manner, that the velocity of light becomes

unity. If we now write √-1 _t_ = _l_, in the place of _l_, then the

differential expression

_d_τ² = -(_dx²_ + _dy²_ + _dz²_ + _dl²_),

becomes symmetrical in (_x_, _y_, _r_, _l_); this symmetry then enters

into each law, which does not contradict the _world-postulate_. We can

clothe the “essential nature of this postulate in the mystical, but

mathematically significant formula

3·10⁵ _km_ = √-1 Sec.

V

The advantages arising from the formulation of the world-postulate are

illustrated by nothing so strikingly as by the expressions which tell us

about the reactions exerted by a point-charge moving in any manner

according to the Maxwell-Lorentz theory.

Let us conceive of the world-line of such an electron with the charge

(_e_), and let us introduce upon it the “Proper-time” τ reckoned from

any possible initial point. In order to obtain the field caused by the

electron at any world-point P₁ let us construct the fore-cone belonging

to P₁ (_vide_ fig. 4). Clearly this cuts the unlimited world-line of the

electron at a single point P, because these directions are all time-like

vectors. At P, let us draw the tangent to the world-line, and let us

draw from P₁ the normal to this tangent. Let _r_ be the measure of P₁Q.

According to the definition of a fore-cone, _r_/_e_ is to be reckoned as

the measure of PQ. Now at the world-point P₁, the vector-potential of

the field excited by _e_ is represented by the vector in direction PQ,

having the magnitude _e_/_cr_, in its three space components along the

_x_-, _y_-, _z_-axes; the scalar-potential is represented by the

component along the _t_-axis. This is the elementary law found out by A.

Lienard, and E. Wiechert.[34]

If the field caused by the electron be described in the above-mentioned

way, then it will appear that the division of the field into electric

and magnetic forces is a relative one, and depends upon the time-axis

assumed; the two forces considered together bears some analogy to the

force-screw in mechanics; the analogy is, however, imperfect.

I shall now describe _the ponderomotive force which is exerted by one

moving electron upon another moving electron_. Let us suppose that the

world-line of a second point-electron passes through the world-point P₁.

Let us determine P, Q, _r_ as before, construct the middle-point M of

the hyperbola of curvature at P, and finally the normal MN upon a line

through P which is parallel to QP₁. With P as the initial point, we

shall establish a system of reference in the following way: the _t_-axis

will be laid along PQ, the _x_-axis in the direction of QP₁. The

_y_-axis in the direction of MN, then the _z_-axis is automatically

determined, as it is normal to the _x_-, _y_-, _z_-axes. Let [:_x_],

[:_y_], [:_z_], [:_t_] be the acceleration-vector at P, [._x_]₁, [._y_]₁

[._z_]₁, [._t_]₁ be the velocity-vector at P₁. Then the force-vector

exerted by the first election _e_, (moving in any possible manner) upon

the second election _e_, (likewise moving in any possible manner) at P₁

is represented by

-_e e₁_([._t₁_] - [._x₁_]/_c_)F,

_For the components F_{x}, F_{y}, F_{z}, F_{t} of the vector F the

following three relations hold_:—

_c_F_{_t_} - F_{_x_} = 1/_r²_, F_{_y_} = [:_y_]/(_c²__r_), F_{_z_} =

0,

_and fourthly this vector F is normal to the velocity-vector_ P₁, _and

through this circumstance alone, its dependence on this last

velocity-vector arises_.

If we compare with this expression the previous formulæ[35] giving the

elementary law about the ponderomotive action of moving electric charges

upon each other, then we cannot but admit, that the relations which

occur here reveal the inner essence of full simplicity first in four

dimensions; but in three dimensions, they have very complicated

projections.

In the mechanics reformed according to the world-postulate, the

disharmonies which have disturbed the relations between Newtonian

mechanics and modern electrodynamics automatically disappear. I shall

now consider the position of the Newtonian law of attraction to this

postulate. I will assume that two point-masses _m_ and _m₁_ describe

their world-lines; a moving force-vector is exercised by _m_ upon _m₁_,

and the expression is just the same as in the case of the electron, only

we have to write +_mm₁_ instead -_ee₁_. We shall consider only the

special case in which the acceleration-vector of _m_ is always zero:

then _t_ may be introduced in such a manner that _m_ may be regarded as

fixed, the motion of _m_ is now subjected to the moving-force vector of

_m_ alone. If we now modify this given vector by writing

-([^.]1/√(1-(_v_²/_c²_)) instead of [._t_] ([._t_] = 1 up to magnitudes

of the order (1[^.]/_c²_)), then it appears that Kepler’s laws hold good

for the position (_x₁_, _y₁_, _z₁_), of _m₁_ at any time, only in place

of the time _t₁_, we have to write the proper time τ₁ of _m₁_. On the

basis of this simple remark, it can be seen that the proposed law of

attraction in combination with new mechanics is not less suited for the

explanation of astronomical phenomena than the Newtonian law of

attraction in combination with Newtonian mechanics.

Also the fundamental equations for electro-magnetic processes in moving

bodies are in accordance with the world-postulate. I shall also show on

a later occasion that the deduction of these equations, as taught by

Lorentz, are by no means to be given up.

The fact that the world-postulate holds without exception is, I believe,

the true essence of an electromagnetic picture of the world; the idea

first occurred to Lorentz, its essence was first picked out by Einstein,

and is now gradually fully manifest. In course of time, the mathematical

consequences will be gradually deduced, and enough suggestions will be

forthcoming for the experimental verification of the postulate; in this

way even those, who find it uncongenial, or even painful to give up the

old, time-honoured concepts, will be reconciled to the new ideas of time

and space,—in the prospect that they will lead to pre-established

harmony between pure mathematics and physics.

Footnote 30:

Planck, Zur Dynamik bewegter systeme, Ann. d. physik, Bd. 26, 1908, p.

1.

Footnote 31:

H. Minkowski; the passage refers to paper (2) of the present edition.

Footnote 32:

Minkowski—Mechanics, appendix, page 65 of paper (2). Planck—Verh. d.

D. P. G. Vol. 4, 1906, p. 136.

Footnote 33:

Schütz, Gött. Nachr. 1897, p. 110.

Footnote 34:

Lienard, L’Eclairage électrique T. 16, 1896, p. 53. Wiechert, Ann. d.

Physik, Vol. 4.

Footnote 35:

K. Schwarzschild. Gött-Nachr. 1903. H. A. Lorentz, Enzyklopädie der

Math. Wissenschaften V. Art 14, p. 199.

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

(1) _ds²_ = - _d_X₁² - _d_X₂² - _d_X₃² + _d_X₄²

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 (_d_X₁,

_d_X₂, _d_X₃, _d_X₄) 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, _d_X’s would then be represented by

definite linear homogeneous expressions of the form

(2) _d_X_{ν} = σ_{σ}_a__{νσ}_dx__{σ}