Part 1

The Wormhole Singularity

by Einstein and Rosen

We propose an atomistic theory of matter and electricity which uses only `gm` of General Relativity and Maxwell’s theory, while excluding singularities of the ﬁeld.

Through a simple example, these modify slightly the gravitational equations.

• This then allows regular solutions for the static spherically symmetric case.

These solutions involve the mathematical representation of physical space by a space of two identical sheets which has a particle acting as a “bridge” connecting these sheets.

• This allows us to understand why no neutral particles of negative mass are to be found.

The combined system of gravitational and electromagnetic equations are treated similarly and lead to a similar interpretation.

The most natural elementary charged particle is one of zero mass.

The many-particle system will then be represented by a regular solution of the ﬁeld equations of a space of 2 identical sheets joined by many bridges. In this case, because of the absence of singularities, the ﬁeld equations determine both the ﬁeld and the motion of the particles.

We do not go into the many-particle problem which would decide the value of the theory.

This is because theoretical physics is still far from providing a uniﬁed foundation of all phenomena.

General Relativity for macroscopic phenomena is unable to account for:

• the atomic structure of matter
• quantum effects

Quantum theory can explain many atomic and quantum phenomena, but, by its very nature is unsuitable for Relativity.

How can General relativity explain atomic phenomena?

We explain it here.

We think that our theoretical method is nevertheless justiﬁed because it provides a clear procedure with minimum of assumptions and can be done with math.
We think that an atomistic theory of matter and electricity is possible, one that only uses:

• gravitational ﬁeld (`guy`)
• the electromagnetic ﬁeld in the sense of Maxwell (vector potentials, `W`)

Our rivals will say that this is impossible because:

• the Schwarzschild solution for a black hole has a Newtonian singularity
• the last of the Maxwell equations*, excludes the existence of charge densities and electrical particles

*This expresses the vanishing of the divergence of the (con‘travariant) electrical ﬁeld density

This is why writers have occasionally noted the possibility that material particles might be considered as singularities of the ﬁeld.

We reject these because a singularity brings so much arbitrariness into General Relativity that it actually nulliﬁes its laws.

L. Silberstein wrote to Albert about this, as Levi—Civita and Weyl gave a way to find axially symmetric static solutions of the gravitational equations.

This gives us a solution which, except for 2 point singularities lying on the axis of symmetry, is everywhere regular and is Euclidean at inﬁnity.

Hence if Newtonian singularities are particles, then in this case, we would have 2 particles not accelerated by their gravitational interaction, which would certainly be excluded physically. We think that every ﬁeld theory must therefore adhere to the fundamental principle that Newtonian singularities of the ﬁeld are to be excluded. We explain how.

Part 1: A Special Kind Of Singularity And Its Removal

The ﬁrst step to General relativity is the “Principle of Equivalence”:

A reference system uniformly accelerated in a gravity-free space is ”at rest,” provided one interprets the condition of the space with respect to it as a homogeneous gravitational ﬁeld.

The latter is exactly described by the metric ﬁeld*

``````d32= ~dx12—dx22~—dx32+a2x12dx42.                 (1)
``````

*This metric ﬁeld does not represent the whole Minkowski space but only part of it. Thus, the transformation that converts d3? = —— dsi2—d522—d532+d542
into (1) is 51= 36] cosh ax4, 53= 963, . £2=x2s £4=x1 smh am.
It follows that only those points for which 5123= 5154? corre spond to points for which (1) is the metric.

The g“, of this ﬁeld satisfy in general the equations

``````Rim", = O,                     (2)
``````

and hence the equations

``````RH = Rmklm = 0.                        (3)
``````

The `guv` corresponding to (1) are regular for all ﬁnite points of space—time. Nevertheless one cannot assert that Eqs. (3) are satisﬁed by (1) for all ﬁnite values of `x1,...x4.`

This is due to the fact that the determinant `g` of the `guv` vanishes for `x1=0`.

The contravariant `guv` therefore become inﬁnite and the tensors `Rim` and `R..` take on the form `0/0`.

From the standpoint of Equation 3, the hyperplane `x1=0` then represents a singularity of the ﬁeld.

Can the ﬁeld law of gravitation (and later on the ﬁeld law of gravitation and electricity) be modiﬁed in a natural way without essential change so that the solution (1) would satisfy the ﬁeld equations for all ﬁnite points, i.e., also for x1=0?

W. Mayer says that one can make `Rim...` and `R1` into rational functions of the `guv` and their ﬁrst two derivatives by multiplying them by suitable powers of g.

It is easy to show that in `gQRk`: There is no longer any denominator.

If then we replace (3) by `Rkl*=g2Rkl=0y` (33—) this system of equations is satisﬁed by (1) at all ﬁnite points. This amounts to introducing in place of the g” the cofactors [gnu] of the g,” in g in order to avoid the occurrence of denominators.

One is therefore operating with tensor densities of a suitable weight instead of with tensors.

In this way, one succeeds in avoiding singularities of that special kind which is characterized by the vanishing of g.

The solution (1) naturally has no deeper physical signiﬁcance insofar as it extends into spatial inﬁnity.

It allows one to see however to what extent the regularization of the hypersurfaces `g=0` leads to a theoretical representation of matter, regarded from the standpoint of the original theory.

Thus, in the framework of the original theory one has the gravitational equations `Rik—%gtI= R= —Tzk,(4)` where `T“`, is the tensor of mass or energy density.

To interpret (l) in the framework of this theory we must approximate the line element by a slightly different one which avoids the singularity `g=0`.

Accordingly we introduce a small constant a and let `ds2 = —dx12 —alx22 ~dx32+ (a2x12+0)dx42i (1a)`

The smaller `a(>0)` is chosen, the nearer does this gravitational ﬁeld come to that of (1). If one calculates from this the (ﬁctitious) energy tensor Tm one obtains as nonvanishing components

``````T22 = T23 = aQ/la/(1+a2x12/a)2.
``````

We see then that the smaller one takes a the sigma, the more is the tensor concentrated in the neighborhood of the hypersurface `x1=0`.

From the standpoint of the original theory the solution (1) contains a singularity which corresponds to an energy or mass concentrated in the surface `x1=0`. From the standpoint of the modiﬁed theory, however, (1) is a solution of (3a), free from singularities, which describes the “ﬁeld-producing mass,” without requiring for this the introduction of any new ﬁeld quantities.

All equations of the absolute differential calculus can be written in a form free from denominators, whereby the tensors are replaced by tensor densities of suitable weight.

In the case of the solution (1) the whole ﬁeld consists of two equal halves, separated by the surface of symmetry x1=0, such that for the corresponding points (x1, x2, x3, x4) and (—x1, x2, x3, x4) the g,;, are equal.

As a result we ﬁnd that, although we are permitting the determinant g to take on the value 0 (for 961= 0), no change of sign of g and in general no change in the “inertial index” of the quadratic form (1) occurs. These features are of fundamental importanee from the point of View of the physical interpretation, and will be encountered again in the solutions to be considered later.

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