Part 3

How to fit electricity into General Relativity

If besides the pure gravitational field, other field variables are also present, the field equations of gravitation are=

Rik—%gikR= "Tile; (4) where T“, is the “material” energy tensor, i.e.,

that part of the mathematical expression of the energy which does not depend exclusively on the g”. In the case of the phenomenological representation of matter—if it is to be considered as “dust-like,” that is, without pressure—one takes

T=  p(dxi/ds)(dxk/ds), 

where p is the density-scalar, dxi/ds the velocity—vector of the matter. It is to be noted that T4“ is accordingly a positive quantity.

In general, the additional field-variables satisfy such differential equations that, in consequence of them, the divergence Tikmgkm vanishes.

As the divergence of the left side of (4) vanishes identically, this means that among all the field equations those 4 identities exist which are needed for their compatibility.

Through this condition, in certain cases, the structure of Tik, not however its sign, is determined. It appears natural to choose this sign in such a way that the component T44 (in the limit of the special relativity theory) is always positive. The Maxwell electromagnetic field, as is well known, is represented by the antisymmetric field tensor

go,,(=6<p,,/6x”—6go,/6x), 

which satisfies the field equations

g0; agw = 0' (6) 

These equations have the well-known consequence that the divergence of the tensor

Tik=igik¢awafima¢ka (7) vanishes. 

The sign has been so chosen that T44 is positive for the case of the special relativity theory. If one puts this Ti], into the gravitational Eqs. (4), then the latter together with (6) and (7) form a theory of gravitation and electricity.

This weakness was formally avoided in the original theory of relativity by the introduction of the energy tensor into the field equations. It was clear, however, from the very beginning that this was’only a provisory completion of the theory in the sense of a phenomenological interpretation.

                ¢1=¢2=e3=0,            <pI=c/r,                        
                    1                                                 
 ds'z = —»——s—<—                (er                                
           1—2’I’VL/T—62/27’2                               (8)    
                     ‘                    2m       6‘2             
       ~r2(d02+sm20d<p2) —l— (1 —— ——~—> (W.                       1'    Zr2                

Here m has obviously the significance of a gravitating mass, 6 that of an electrical charge.

It turns out that also in this case there is no difficulty in forming a solution without singularity corresponding to the solution just given.4

We find that the mass m is not determined by the electrical charge 6, but that e and m are independent constants of integration. It also turns out that for the removal of the singularity it is not necessary to take the ponderable mass m positive. In fact, as we shall show immediately, there exists a solution free from singularities for which the mass constant 1% vanishes. Because we believe that these massless solutions are the physically important ones we will consider here the case m=0.

The field equations without denominators are=

VM =  ion.  901nm    g2lewgw=  0,           (9)      

mass. One is therefore led, according to this theory, to consider as a

g2 (Riki Wager  rig/Mango) = 0.

where in the last equation the term in R has been omitted because it vanishes in consequence of (7), by which To,“ is zero.

If in Eq. (8) (with m=0) one replaces r by the variable it according to the equation it? = r2— 62/2 one obtains

¢1=s02=908=0. ¢4=6/(u2+2/2)h
d5? =  du2 (u2+e2/2)(d02+sin20d¢2)   J. (8a) + [2u2/(2u2le2) jdf).             

This solution is free from singularities for all finite points in the space of two sheets and the charge is again represented by a bridge between the sheets. It is the representation of an elementary electrical particle without mass.

4 If we had taken the usual sign for Tm, the solution would involve +e" instead of ——e‘2. It would then not be possible, by making a coordinate transformation, to obtain a solution free from singularities.

It so happens that we are forced to put the negative of the above into the gravitational equations if it is to be possible to obtain static spherically symmetric solutions of the equations, free from singularities, which could represent electrical particles. Making this change of sign one finds as the required solution

Part 4 Summary

If we solve the equations of General Relativity for the static spherically symmetric case, with or without an electrostatic field, one finds that singularities occur in the solutions.

If one modifies the equations in an unessential manner so as to make them free from denominators, regular solutions can be obtained, provided one treats the physical space as consisting of two congruent sheets. The neutral, as well as the electrical, particle is a portion of space connecting the two sheets (bridge). In the hypersurfaces of contact of the two sheets the deter minant of the g,” vanishes.

Only by investigations of these solutions will one be able to determine the extent to which the theory accounts for the facts. For now, we cannot even know whether regular solutions with more than one bridge exist at all.

We might expect that processes, in which several elementary particles take part, correspond to regular solutions of the field equations with several bridges between the two equivalent sheets corresponding to the physical space.

Our theory explains the atomistic character of matter and the fact that there exist no negative neutral masses. This introduces no new variables other than the g,,., and < p,,,.

This it can claim to be complete (or closed). Our theory does not contain the quantum phenomena. But it might.

Thus it might turn out that only such regular many-bridge solutions can exist for which the “charges” of the electrical bridges are numerically equal to one another and only two different “masses” occur for the mass bridges, and for which the stationary “motions” are subject to restrictions like those which we encounter in the quantum theory. A general relativistic theory of matter which contains no new hypothetical elements is possible.

The most natural electrical particle in the theory is one without gravitating mass. This means that the electron or proton is a two-bridge problem.

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