Unied Field Theory of Gravitation and Electricity

Albert Einstein
translation by A. Unzicker and T. Case

Session Report of the Prussian Academy of Sciences, pp. 414-419 July 25th, 1925

Among the theoretical physicists working in the eld of the general theory of relativity there should be a consensus about the consubstantiality of the gravitational and electromagnetic eld. However, I was not able to succeed in nding a convincing formulation of this connection so far. Even in my article published in these session reports (XVII, p. 137, 1923) which is entirely based on the foundations of Eddington, I was of the opinion that it does not reect the true solution of this problem. After searching ceaselessly in the past two years I think I have now found the true solution. I am going to communicate it in the following. The applied method can be characterized as follows. First, I looked for the formally most simple expression for the law of gravitation in the absence of an electromagnetic eld, and then the most natural generalization of this law. This theory appeared to contain Maxwells theory in rst approximation. In the following I shall outline the scheme of the general theory ( 1) and then show in which sense this contains the law of the pure gravitational eld ( 2) and Maxwells theory ( 3).

1. The general theory

We consider a 4-dimensional continuum with an ane connection, i.e. a -eld which denes in nitesimal vector shifts according to the relation dA = A dx . (1)

We do not assume symmetry of the with respect to the indices and . From these quantities we can derive the Riemannian tensors
R. =

+ + + x x

and

+ + + (2) x x in a well-known manner. Independently from this ane connection we introduce a contravariant tensor density g , whose symmetry properties we leave undetermined as well. From both quantities we obtain the scalar density H = g R (3)
R = R. =

and postulate that all the variations of the integral J = with respect to the vanish.

H dx1 dx2 dx3 dx4

and as independent (i.e. not to be varied at the boundaries) variables

The variation with respect to the

g yields the 16 equations

R = 0, (4) at rst the 64 equations (5)

the variation with respect to the

g g + g + g + g g = 0. x x

We are going to begin with some considerations that allow us to replace the eqns. (5) by simpler ones. If we contract the l.h.s. of (5) by and or and , we obtain the equations 3 g + g + g ( ) = 0. x (6) (7)

g g = 0. x x If we further introduce the quantities g which are the normalized subdeterminants of the thus fulll the equations
g g = g g = .

and

and if we now multiply (5) by g , after pulling up one index the result may be written as follows: g lg g + + ( ) + + g = 0, (8) 2g x x while denotes the determinant of g . The equations (6) and (8) we write in the form 1 g lg g f = 3 g ( ) = x + g = g + , (9) x whereby f stands for a certain tensor density. It is easy to prove that the system (5) is equivalent to the system g + g + g g + f = 0 (10) x in conjunction with (7). By pulling down the upper indices we obtain the relations g g = = g g, g wherebyg is a covariant tensor g + g + g + g + g = 0, (10a) x whereby is a covariant vector. This system, together with the two systems given above, g g =0 (7) x x and 0 = R = + + , (4) x x are the result of the variational principle in the most simple form. Looking at this result, it is remarkable that the vector occurs besides the tensor (g ) and the quantities . To obtain consistency with the known laws of gravitation end electricity, we have to interpret the symmetric part of g as metric tensor and the skew-symmetric part as electromagnetic eld, and we have to assume the vanishing of , which will be done in the following. For a later analysis (e.g. the problem of the electron), we will have to keep in mind that the Hamiltonian principle does not indicate a vanishing . Setting to zero leads to an overdetermination of the eld, since we have 16 + 64 + 4 algebraically independent dierential equations for 16 + 64 variables. 2

 2. The pure gravitational eld as special case

Let the g be symmetric. The equations (7) are fullled identically. By changing to in (10a) and subtraction we obtain in easily understandable notation , + , , , = 0. (11)

If is called the skew-symmetric part of with respect to the last two indices, (11) takes the form , + , = 0 or , = , . (11a) This symmetry property of the rst two indices contradicts the antisymmetry of the last ones, as we learn from the series of equations , = , = , = , = , = , . This, in conjunction with (11a), compels the vanishing of all . Therefore, the are symmetric in the last two indices as in Riemannian geometry. The equations (10a) can be resolved in a well-known manner, and one obtains g g 1 g g + . (12) 2 x x x Equation (12), together with (4) is the well-known law of gravitation. Had we presumed the symmetry of the g at the beginning, we would have arrived at (12) and (4) directly. This seems to be the most simple and coherent derivation of the gravitational equations for the vacuum to me. Therefore it should be seen as a natural attempt to encompass the law of electromagnetism by generalizing these considerations rightly. Had we not assumed the vanishing of the , we would have been unable to derive the known law of the gravitational eld in the above manner by assuming the symmetry of the g . Had we assumed the symmetry of both the g and the instead, the vanishing of would have been a consequence of (9) or (10a) and (7); we would have obtained the law of the pure gravitational eld as well.

3. Relations to Maxwells theory

If there is an electromagnetic eld, that means the g or the g do contain a skew-symmetric part, we cannot solve the eqns. (10a) any more with respect to the , which signicantly complicates the clearness of the whole system. We succeed in resolving the problem however, if we restrict ourselves to the rst approximation. We shall do this and once again postulate the vanishing of . Thus we start with the ansatz g = + + , (13) whereby the should be symmetric, and the skew-symmetric, both should be innitely small in rst order. We neglect quantities of second and higher orders. Then the are innitely small in rst order as well. Under these circumstances the system (10a) takes the more simple form + g + + = 0. x (10b)

After applying two cyclic permutations of the indices , and two further equations appear. Then, out of the three equations we may calculate the in a similar manner as in the symmetric case. One obtains g g 1 g = + . (14) 2 x x x 3

Eqn. (4) is reduced to the rst and third term. If we put the expression from (14) therein, one obtains 2 g 2 g 2 g 2 g + + = 0. (15) x2 x x x x x x Before further consideration of (15), we develop the series from equation (7). Firstly, out of (13) follows that the approximation we are interested in yields

g = + ,
Regarding this, (7) transforms to = 0. x Now we put the expressions given by (13) into (15) and obtain with respect to (17) 2 2 2 2 + + =0 x2 x x x x x x 2 = 0. x2

(16)

(17)

(18)

(19)

The expressions (18), which may be simplied as usual by proper choice of coordinates, are the same as in the absence of an electromagnetic eld. In the same manner, the equations (17) and (19) for the electromagnetic eld do not contain the quantities which refer to the gravitational eld. Thus both elds are - in accordance with experience - independent in rst approximation. The equations (17), (19) are nearly equivalent to Maxwells equations of empty space. (17) is one Maxwellian system. The expressions + + , x x x which1 according to Maxwell should vanish, do not vanish necessarily due to (17) and (19), but their divergences of the form + + x x x x however do. Thus (17) and (19) are substantially identical to Maxwells equations of empty space. Concerning the attribution of to the electric and magnetic vectors (a and h) I would like to make a comment that claims validity independently from the theory presented here. According to classical mechanics that uses central forces to every sequence of motion V there is an inverse motion V , that passes the same congurations by taking an inverse succession. This inverse motion V is formally obtained from V by substituting x =x y =y z =z t = t in the latter one. We observe a similar behavior, according to the general theory of relativity, in the case of a pure gravitational eld. To achieve the solution V out of V , one has to substitute t = t into all eld functions and to change the sign of the eld components g14 , g24 , g34 and the energy components T14 ,
1

This appears to be a misprint. The rst term should be squared.

T24 , T34 . This is basically the same procedure as applying the above transformation to the primary motion V . The change of signs in g14 , g24 , g34 and in T14 , T24 , T34 is an intrinsic consequence of the transformation law for tensors. This generation of the inverse motion by transformation of the time coordinate (t = t) should be regarded as a general law that claims validity for electromagnetic processes as well. There, an inversion of the process changes the sign of the magnetic components, but not those of the electric ones. Therefore one should have to assign the components 23 , 31 , 12 to the electric eld and 14 , 24 , 34 to the magnetic eld. We have to give up the inverse assignment which was in use as yet. It was preferred so far, since it seems more comfortable to express the density of a current by a vector rather than by a skew-symmetric tensor of third rank. Thus in the theory outlined here, (7) respectively (17) is the expression for the law of magnetoelectric induction. In accordance, at the r.h.s. of the equation there is no term that could be interpreted as density of the electric current. The next issue is, if the theory developed here renders the existence of singularity-free, centrally symmetric electric masses comprehensible. I started to tackle this problem together with Mr. J. Grommer, who was at my disposal ceaselessly for all calculations while analyzing the general theory of relativity in the last years. At this point I would like to express my best thanks to him and to the International educational board which has rendered possible the continuing collaboration with Mr. Grommer.