The most important fact that we draw from experience as to the distribution of matter is that the relative velocities of the stars are very small as compared with the velocity of light. So I think that for the present we may base our reasoning upon the following approximative assumption. There is a system of reference relatively to which matter may be looked upon as being permanently at rest. With respect to this system, therefore, the contravariant energy-tensor Tμv of matter is, by reason of (5), of the simple form
The scalar ρ of the (mean) density of distribution may be a priori a function of the space co-ordinates. But if we assume the universe to be spatially finite, we are prompted to the hypothesis that ρ is to be independent of locality. On this hypothesis we base the following considerations.
As concerns the gravitational field, it follows from the equation of motion of the material point
that a material point in a static gravitational field can remain at rest only when g44 is independent of locality. Since, further, we presuppose independence of the time co-ordinate x4 for all magnitudes, we may demand for the required solution that, for all xv,
Further, as always with static problems, we shall have to set
It remains now to determine those components of the gravitational potential which define the purely spatial-geometrical relations of our continuum (g11, g12, . . . g33). From our assumption as to the uniformity of distribution of the masses generating the field, it follows that the curvature of the required space must be constant. With this distribution of mass, therefore, the required finite continuum of the x1, x2, x3, with constant x4, will be a spherical space.
We arrive at such a space, for example, in the following way. We start from a Euclidean space of four dimensions, ξ1, ξ2, ξ3, ξ4, with a linear element dσ; let, therefore,
In this space we consider the hyper-surface
where R denotes a constant. The points of this hyper-surface form a three-dimensional continuum, a spherical space of radius of curvature R.
The four-dimensional Euclidean space with which we started serves only for a convenient definition of our hyper-surface. Only those points of the hyper-surface are of interest to us which have metrical properties in agreement with those of physical space with a uniform distribution of matter. For the description of this three-dimensional continuum we may employ the co-ordinates ξ1, ξ2, ξ3 (the projection upon the hyper-plane ξ4 = 0) since, by reason of (10), ξ4 can be expressed in terms of ξ4, ξ2, ξ3. Eliminating ξ4 from (9), we obtain for the linear element of the spherical space the expression
where δμv = 1, if μ = v; δμv = 0, if μ ≠ v, and The co-ordinates chosen are convenient when it is a question of examining the environment of one of the two points ξ1 = ξ2 = ξ3 = 0.
Now the linear element of the required four-dimensional spacetime universe is also given us. For the potential gμv, both indices of which differ from 4, we have to set
which equation, in combination with (7) and (8), perfectly defines the behaviour of measuring-rods, clocks, and light-rays.
§ 4. ON AN ADDITIONAL TERM FOR THE FIELD EQUATIONS OF GRAVITATION
My proposed field equations of gravitation for any chosen system of co-ordinates run as follows:—
The system of equations (13) is by no means satisfied when we insert for the gμv the values given in (7), (8), and (12), and for the (contravariant) energy-tensor of matter the values indicated in (6). It will be shown in the next paragraph how this calculation may conveniently be made. So that, if it were certain that the field equations (13) which I have hitherto employed were the only ones compatible with the postulate of general relativity, we should probably have to conclude that the theory of relativity does not admit the hypothesis of a spatially finite universe.
However, the system of equations (14) allows a readily suggested extension which is compatible with the relativity postulate, and is perfectly analogous to the extension of Poisson’s equation given by equation (2). For on the left-hand side of field equation (13) we may add the fundamental tensor gμv, multiplied by a universal constant, –λ, at present unknown, without destroying the general covariance. In place of field equation (13) we write
This field equation, with λ sufficiently small, is in any case also compatible with the facts of experience derived from the solar system. It also satisfies laws of conservation of momentum and energy, because we arrive at (13a) in place of (13) by introducing into Hamilton’s principle, instead of the scalar of Riemann’s tensor, this scalar increased by a universal constant; and Hamilton’s principle, of course, guarantees the validity of laws of conservation. It will be shown in § 5 that field equation (13a) is compatible with our conjectures on field and matter.
§ 5. CALCULATION AND RESULT
Since all points of our continuum are on an equal footing, it is sufficient to carry through the calculation for one point, e.g. for one of the two points with the co-ordinates
Then for the gμv in (13a) we have to insert the values
wherever they appear differentiated only once or not at all. We thus obtain in the first place
From this we readily discover, taking (7), (8), and (13) into account, that all equations (13a) are satisfied if the two relations
or
are fulfilled.
Thus the newly introduced universal constant λ defines both the mean density of distribution ρ which can remain in equilibrium and also the radius R and the volume 2π2R3 of spherical space. The total mass M of the universe, according to our view, is finite, and is in fact
Thus the theoretical view of the actual universe, if it is in correspondence with our reasoning, is the following. The curvature of space is variable in time and place, according to the distribution of matter, but we may roughly approximate to it by means of a spherical space. At any rate, this view is logically consistent, and from the standpoint of the general theory of relativity lies nearest at hand; whether, from the standpoint of present astronomical knowledge, it is tenable, will not here be discussed. In order to arrive at this consistent view, we admittedly had to introduce an extension of the field equations of gravitation which is not justified by our actual knowledge of gravitation. It is to be emphasized, however, that a positive curvature of space is given by our results, even if the supplementary term is not introduced. That term is necessary only for the purpose of making possible a quasi-static distribution of matter, as required by the fact of the small velocities of the stars.
* p is the mean density of matter, calculated for a region which is large as compared with the distance between neighbouring fixed stars, but small in comparison with the dimensions of the whole stellar system.
* de Sitter, Akad. van Wetensch, te Amsterdam, 8 Nov., 1916.
DO GRAVITATIONAL FIELDS
PLAY AN ESSENTIAL PART
IN THE STRUCTURE OF
THE ELEMENTARY PARTICLES
OF MATTER?
BY
A. EINSTEIN
Translated from “Spielen Gravitationsfelder im Aufber der materiellen Elementarteilchen eine wesentliche Rolle?” Sitzungsberichte der Preussischen Akad. d. Wissenschaften, 1919.
NEITHER the Newtonian nor the relativistic theory of gravitation has so far led to any advance in the theory of the constitution of matter. In view of this fact it will be shown in the following pages that there are reasons for thinking that the elementary formations which go to make up the atom are held together by gravitational forces.
§ 1. DEFECTS OF THE PRESENT VIEW
Great pains have been taken to elaborate a theory which will account for the equilibrium of the electricity constituting the electron. G. Mie, in particular, has devoted deep researches to this question. His theory, which has found considerable support among theoretical physicists, is based mainly on the introduction into the energy-tensor of supplementary terms depending on the components of the electro-dynamic potential, in addition to the energy terms of the Maxwell-Lorentz theory. These new terms, which in outside space are unimportant, are neve
rtheless effective in the interior of the electrons in maintaining equilibrium against the electric forces of repulsion. In spite of the beauty of the formal structure of this theory, as erected by Mie, Hilbert, and Weyl, its physical results have hitherto been unsatisfactory. On the one hand the multiplicity of possibilities is discouraging, and on the other hand those additional terms have not as yet allowed themselves to be framed in such a simple form that the solution could be satisfactory.
So far the general theory of relativity has made no change in this state of the question. If we for the moment disregard the additional cosmological term, the field equations take the form
where Gμv denotes the contracted Riemann tensor of curvature, G the scalar of curvature formed by repeated contraction, and Tμv the energy-tensor of “matter.” The assumption that the Tμv do not depend on the derivatives of the gμv is in keeping with the historical development of these equations. For these quantities are, of course, the energy-components in the sense of the special theory of relativity, in which variable gμv do not occur. The second term on the left-hand side of the equation is so chosen that the divergence of the left-hand side of (1) vanishes identically, so that taking the divergence of (1), we obtain the equation
which in the limiting case of the special theory of relativity gives the complete equations of conservation
Therein lies the physical foundation for the second term of the left-hand side of (1). It is by no means settled a priori that a limiting transition of this kind has any possible meaning. For if gravitational fields do play an essential part in the structure of the particles of matter, the transition to the limiting case of constant gμv would, for them, lose its justification, for indeed, with constant gμv there could not be any particles of matter. So if we wish to contemplate the possibility that gravitation may take part in the structure of the fields which constitute the corpuscles, we cannot regard equation (1) as confirmed.
Placing in (1) the Maxwell-Lorentz energy-components of the electromagnetic field ϕμv,
we obtain for (2), by taking the divergence, and after some reduction,*
where, for brevity, we have set
In the calculation we have employed the second of Maxwell’s systems of equations
We see from (4) that the current-density must everywhere vanish. Therefore, by equation (1), we cannot arrive at a theory of the electron by restricting ourselves to the electro-magnetic components of the Maxwell-Lorentz theory, as has long been known. Thus if we hold to (1) we are driven on to the path of Mie’s theory.†
Not only the problem of matter, but the cosmological problem as well, leads to doubt as to equation (1). As I have shown in the previous paper, the general theory of relativity requires that the universe be spatially finite. But this view of the universe necessitated an extension of equations (1), with the introduction of a new universal constant λ, standing in a fixed relation to the total mass of the universe (or, respectively, to the equilibrium density of matter). This is gravely detrimental to the formal beauty of the theory.
§ 2. THE FIELD EQUATIONS FREED OF SCALARS
The difficulties set forth above are removed by setting in place of field equations (1) the field equations
where Tμv denotes the energy-tensor of the electromagnetic field given by (3).
The formal justification for the factor – in the second term of this equation lies in its causing the scalar of the left-hand side,
to vanish identically, as the scalar gvμTμv of the right-hand side does by reason of (3). If we had reasoned on the basis of equations (1) instead of (1a), we should, on the contrary, have obtained the condition G = 0, which would have to hold good everywhere for the gμv, independently of the electric field. It is clear that the system of equations [(1a), (3)] is a consequence of the system [(1), (3)], but not conversely.
We might at first sight feel doubtful whether (1a) together with (6) sufficiently define the entire field. In a generally relativistic theory we need n – 4 differential equations, independent of one another, for the definition of n independent variables, since in the solution, on account of the liberty of choice of the co-ordinates, four quite arbitrary functions of all co-ordinates must naturally occur. Thus to define the sixteen independent quantities gμv and ϕμv we require twelve equations, all independent of one another. But as it happens, nine of the equations (1a), and three of the equations (6) are independent of one another.
Forming the divergence of (1a), and taking into account that the divergence of Gμv – gμv vanishes, we obtain
From this we recognize first of all that the scalar of curvature G in the four-dimensional domains in which the density of electricity vanishes, is constant. If we assume that all these parts of space are connected, and therefore that the density of electricity differs from zero only in separate “world-threads,” then the scalar of curvature, everywhere outside these world-threads, possesses a constant value G0. But equation (4a) also allows an important conclusion as to the behaviour of G within the domains having a density of electricity other than zero. If, as is customary, we regard electricity as a moving density of charge, by setting
we obtain from (4a) by inner multiplication by Jσ, on account of the antisymmetry of ϕμv, the relation
Thus the scalar of curvature is constant on every world-line of the motion of electricity. Equation (4a) can be interpreted in a graphic manner by the statement: The scalar of curvature plays the part of a negative pressure which, outside of the electric corpuscles, has a constant value G0. In the interior of every corpuscle there subsists a negative pressure (positive G – G0) the fall of which maintains the electro-dynamic force in equilibrium. The minimum of pressure, or, respectively, the maximum of the scalar of curvature, does not change with time in the interior of the corpuscle.
We now write the field equations (1a) in the form
On the other hand, we transform the equations supplied with the cosmological term as already given
Subtracting the scalar equation multiplied by , we next obtain
Now in regions where only electrical and gravitational fields are present, the right-hand side of this equation vanishes. For such regions we obtain, by forming the scalar,
In such regions, therefore, the scalar of curvature is constant, so that λ may be replaced by G0. Thus we may write the earlier field equation (1) in the form
Comparing (9) with (10), we see that there is no difference between the new field equations and the earlier ones, except that instead of Tμv as tensor of “gravitating mass” there now occurs which is independent of the scalar of curvature. But the new formulation has this great advantage, that the quantity λ appears in the fundamental equations as a constant of integration, and no longer as a universal constant peculiar to the fundamental law.
§ 3. ON THE COSMOLOGICAL QUESTION
The last result already permits the surmise that with our new formulation the universe may be regarded as spatially finite, without any necessity for an additional hypothesis. As in the preceding paper I shall again show that with a uniform distribution of matter, a spherical world is compatible with the equations.
In the first place we set
Then if Pik and P are, respectively, the curvature tensor of the second rank and the curvature scalar in three-dimensional space, we have
It therefore follows for our case that
We pursue our reflexions, from this point on, in two ways. Firstly, with the support of equation (1a). Here Tμv denotes the energy-tensor of the electro-magnetic field, arising from the electrical particles constituting matter. For this field we have everywhere
The individual are quantities which vary rapidly with position; but for our purpose we no doubt may replace them by their mean values.
We therefore have to choose
and therefore
In consideration of what has been shown hitherto, we obtain in place of (1a)
The scalar of equation (13) agrees with (14). It is on this account that our fundamental equa
tions permit the idea of a spherical univers. For from (13) and (14) follows
and it is known* that this system is satisfied by a (three-dimensional) spherical universe.
But we may also base our reflexions on the equations (9). On the right-hand side of (9) stand those terms which, from the phenomenological point of view, are to be replaced by the energy-tensor of matter; that is, they are to be replaced by
where ρ denotes the mean density of matter assumed to be at rest. We thus obtain the equations
From the scalar of equation (16) and from (17) we obtain
and consequently from (16)
which equation, with the exception of the expression for the co-efficient, agrees with (15). By comparison we obtain
This equation signifies that of the energy constituting matter three-quarters is to be ascribed to the electromagnetic field, and one-quarter to the gravitational field.
A Stubbornly Persistent Illusion Page 11
