The Particle Problem The General: JUI Ie35
The Particle Problem The General: JUI Ie35
The Particle Problem The General: JUI Ie35
The writers investigate the possibility of an atomistic found. The combined system of gravitational and electro-
theory of matter and electricity which, while excluding magnetic equations are treated similarly and lead to a
singularities of the field, makes use of no other variables similar interpretation. The most natural elementary
than the g&of the general relativity theory and the pof charged particle is found to be one of zero mass. The many-
the Maxwell theory. By the consideration of a simple particle system is expected to be represented by a regular
example they are led to modify slightly the gravitational solution of the field equations corresponding to a space of
equations which then admit regular solutions for the static two identical sheets joined by many bridges. In this case,
spherically symmetric case. These solutions involve the because of the absence of singularities, the field equations
mathematical representation of physical space by a space determine both the field and the motion of the particles.
of two identical sheets, a particle being represented by a The many-particle problem, v;hich would decide the value
"bridge" connecting these sheets. One is able to under- of the theory, has not yet been treated.
stand why no neutral particles of negative mass are to be
N spite of its great success in various fields, the One would be inclined to answer this question
-- present theoretical physics is still far from in the negative in view of the fact that the
being able to provide a unified foundation on Schwarzschild solution for the spherically sym-
which the theoretical treatment of all phenomena metric static gravitational field and Reissner s
could be based. We have a general relativistic extension of this solution to the case when 'an
theory of macroscopic phenomena, which how- electrostatic field is also present each have a
ever has hitherto been unable to account for the singularity. Furthermore the last of the Maxwell
atomic structure of matter and for quantum equations, which expresses the vanishing of the
effects, and we have a quantum theory, which is divergence of the (contravariant) electrical field
able to account satisfactorily for a large number density, appears to exclude in general the exis-
of atomic and quantum phenomena but which by tence of charge densities, hence also of electrical
its very nature is unsuited to the principle of particles.
relativity. Under these circumstances it does not For these reasons writers have occasionally
seem superfluous to raise the question as to what noted the possibility that material particles might
extent the method of general relativity provides be considered as singularities of the field. This
the possibility of accounting for atomic phenom- point of view, however, we cannot accept at all.
ena. It is to such a possibility that we wish to For a singularity brings so much arbitrariness
call attention in the present paper in spite of the into the theory that it actually nullifies its laws.
fact that we are not yet able to decide whether A pretty confirmation of this was imparted in a
this theory can account for quantum phenomena. letter to one of the authors by L. Silbersfein. As
The publication of this theoretical method is is well known, Levi-Civita and Weyl have given
nevertheless justified, in our opinion, because it a general method for finding axially symmetric
provides a clear procedure, characterized by a static solutions of the gravitational equations.
minimum of assumptions, the carrying out of By this method one can readily obtain a solution
which has no other diAiculties to overcome than which, except for two point singularities lying on
those of a mathematical nature. the axis of symmetry, is everywhere regular and.
The question with which we are concerned can is Euclidean at infinity. Hence if one admitted
be put as follows: Is an atomistic theory of matter singularities as representing particles one would
and electri'city conceivable which, while exclud- have here a case of two particles not accelerated
ing singularities in the field, makes use of no by their gravitational interaction, which would
other field variables than those of the gravita- certainly be excluded physically. Every field
tional field (g) and those of the electromagnetic theory, in our opinion, must therefore adhere to
field in the sense of Max&veil (vector poten- the fundamental principle that singularities of
tials, p)? the field are to be excluded.
73
A. E I NSTE I N AN D N. ROS EN
In the following we shall show that it is possible that in g'RI, there is no longer any denominator.
&
The main value of the considerations we are thoughts. If besides the pure gravitational field
presenting consists in that they point the way to a other field variables are also present, the field
satisfactory treatment of gravitational mechan- equations of gravitation are
ics. One of the imperfections of the original rela-
~ik g gi k~ Tiki'
tivistic theory of gravitation was that as a field
theory it was not complete; it introduced the where T;k is the "material" energy tensor, i.e. ,
independent postulate that the law of motion of a that part of the mathematical expression of the
particle is given by the equation of the geodesic. ' energy which does not depend exclusively on the
A complete field theory knows only fields and not g. In the case of the phenomenological represen-
the concepts of particle and motion. For these
tation of matter if it is to be considered as
must not exist independently of the field but are "dust-like, " that is, without pressure one takes
to be treated as part of it. On the basis of the 7'" = p(dx'/ds) (dx" /ds),
description of a particle without singularity one
has the possibility of a logically more satisfactory where p is the density-scalar, dx'/ds the velocity-
treatment of the combined problem: The problem vector of the matter. It is to be noted that T44 is
of the field and that of motion coincide. accordingly a positive quantity.
If several particles are present, this case corre- In general the additional field-variables satisfy
sponds to finding a solution without singularities such differential equations that, in consequence
of the modified Eqs. (3a), the solution represent- of them, the divergence T, k. g
vanishes. As
ing a space with two congruent sheets connected the divergence of the left side of (4) vanishes
"
by several discrete "bridges. Every such solu- identically, this means that among all the field
tion is at the same time a solution of the field equations those four identities exist which are
problem and of the motion problem. needed for. their compatibility. Through this con-
In this case it will not be possible to describe dition, in certain cases, the structure of Tik, not
the whole field by means of a single coordinate however its sign, is determined. It appears
system without introducing singularities. The natural to choose this sign in such a way that
simplest procedure appears to be to choose co- the component 7'4' (in the limit of the special
ordinate systems in the following way: relativity theory) is always positive.
(1) One coordinate system to describe one of The Maxwell electromagnetic field, as is well
the congruent sheets. With respect to this system known, is represented by the antisymmetric
the field will appear to be singular at every bridge. field tensor p~(= Brp/Bx" Brp/Bx &), which satis-
(2) One coordinate system for every bridge, fies the field equations
to provide a description of the field at the bridge
and in the neighborhood of the latter, which is
free from singularities. These equations have the well-known conse-
Between the coordinates of the sheet system quence that the divergence of the tensor
and those of each bridge system there must exist
T'k gg 'k(P PP Pia Pk
outside of the hypersurfaces g=0, a regular co-
ordinate transformation with nonvanishing de- vanishes. The sign has been so chosen that T4 is
terminant. positive for the case of the special relativity
theory. If one puts this Tik into the gravitational
f3. CQMEINED FIEI.D. ELEcTRIcITY Eqs. (4), then the latter together with (6) and
The simplest method of fitting electricity into (7) form a theory .of gravitation and electricity.
the conceptual framework of the general theory
It so happens that we are forced to put the
negative of the above into the gravitational equa-
of relativity is based on the following train of
tions if it is to be possible to obtain static spher-
' To be sure, this weakness was formally avoided in the ically symmetric solutions of the equations, free
original theory of relativity by the introduction of the
energy tensor into the field equations. It was clear, how- from singularities, which could represent elec-
ever, from the very beginning that this was'only a pro- trical particles. Making this change of sign one
visory completion of the theory in the sense of a phe-
non&enological in t:erpreta tion. finds as the required solution
VAIe TI CLE PROI3LEM I N RELATIVI T Y
y4= c/r, (4. SUMMARY AND GENERAL REMARKS
ds =
1
2m/r c'/2r"-
df
(8)
If one solves the equations of the general
theory of relativity for the static spherically
symmetric case, with or without an electrostatic
2m field, one finds that singularities occur in the
r' (d
O'-+-sin'Od p')+ 1 d&'. solutions. If one modifies the equations in an
2r'/
~ ~