Part 2 The Schwarzschild Solution

Two Sheets of Spacetime

Schwarzschild found the spherically symmetric static solution of the gravitational equations

ds2 = - (1/1-2m/r)  dr2 - r2 (d 2 + sin2 d 2) + (1-2m/r)dt2       (5)
(r>2m, θ from 0 to 7r, «5 from 0 to r); 

the variables x1, x2, x3, x4 are here r, Φ, t.

The vanishing of the determinant of the guv for 0= 0 is unimportant, since the corresponding (spatial) direction is not preferred. On the other hand gm for r=2m becomes infinite and hence we have there a singularity.

If one introduces in place of r a new variable

according to the equation it?

= r  2m,                           

one obtains for ds2 the expression

(132 =  4(u2 + 2m)dn2

~ (tt2i- 2m)2(d62+sin20do2) +~   dt2.   (5a)   
ifl-iZm

These new guv are regular functions for all values of the variables. For n=0, however, £44 vanishes, hence also the determinant g. This does not prevent the field equations (3a), which have no denominators, from being satisfied for all values of the independent variables.

This solution of the (new) field equations is therefore free from singularities for all finite points.

The hypersurface u=0 (or in the original variables, 1’= 2m) plays here the same role as the hypersurfaee x1=0 in the previous example.

As it varies from — co to + 00, 7 varies from +00 to 2m and then again from 2m to +00. This solution (5a) in the space of 1’, 6, gs leads to the following conclusion.

The four-dimensional space is described mathematically by two congruent parts or “sheets,” corresponding to u>0 and u <0 joined by a hyperplane 7= 2m or u=0 in which g vanishes.2

We call such a connection between the two sheets a “bridge.” We see now in the given solution, free from singularities, the mathematical representation of an elementary particle (neutron or neutrino). Characteristic of the theory we are presenting is the description of space by means of two sheets.

A spatially finite bridge connects these sheets as an electrically-neutral elementary particle. This=

  • matches the field equations without introducing new field quantities to describe the density of matter
  • one is also able to understand the atomistic character of matter as well as the fact that there can be no particles of negative mass.

The latter is made clear by the following considerations. If we had started from a Schwarzschild solution with negative m, we should not have been able to make the solution regular by introducing a new variable 14 instead of 7’; that is to say, no “bridge” is possible that corresponds to a particle of negative mass.

If we consider once more the solution. (1) from the standpoint of the information we have acquired from the Schwarzschild solution, we see that there also the two congruent halves of the space for x1> 0 and x1<0 can be interpreted as

two sheets each corresponding to the same physical space. In this sense the example represents a gravitational field, independent of x2 and x3, which ends in a plane covered with mass and forming a boundary of the space.

In this example, as well as in the Schwarzschild case, a solution free from singularities at all finite points is made possible by the introduction of the modified gravitational Eqs. (3a). 2 Because of the symmetry about the hypersurface g =0, the Sign of g does not change at this hypersurface.

The main value of the considerations we are presenting consists in that they point the way to a satisfactory treatment of gravitational mechanics. One of the imperfections of the original relativistic theory of gravitation was that as a field theory it was not complete; it introduced the independent postulate that the law of motion of a particle is given by the equation of the geodesic.3

A complete field theory knows only fields and not the concepts of particle and motion.

For these must not exist independently of the field but are to be treated as part of it.

On the basis of the description of a particle without singularity one has the possibility of a logically more satisfactory treatment of the combined problem= The problem of the field and that of motion coincide.

If several particles are present, this case corresponds to finding a solution without singularities of the modified Eqs. (3a), the solution representing a space with two congruent sheets connected by several discrete “bridges.” Every such solution is at the same time a solution of the field problem and of the motion problem.

In this case it will not be possible to describe the whole field by means of a single coordinate system without introducing singularities. The simplest procedure appears to be to choose coordinate systems in the following way=

  1. One coordinate system to describe one of the congruent sheets. With respect to this system the field will appear to be singular at every bridge.

  2. One coordinate system for every bridge, to provide a description of the field at the bridge and in the neighborhood of the latter, which is free from singularities.

Between the coordinates of the sheet system and those of each bridge system there must exist outside of the hypersurfaces g=0, a regular coordinate transformation with nonvanishing determinant.

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