On self-gravitation

Volume 64A, number 1 PHYSICS LETTERS 28 November 1977

ON SELF-GRAVITATION

H.-J. TREDER Zentra!institut fiir Astrophysik, AdW, Potsdam-Babeisberg, DDR

and

W. YOURGRAU Foundations of Physics, University of Denver, USA

Received 28 September 1977

Gravitational self-interaction permits a purely gravitational origin of inertia in the sense of Einstein’s bridge-models of elementary particles. However, these particles will be inconsistent and collapse by implosions.

Abraham, Lorentz et al. formulated for Maxwell’s electrodynamics the problem of self-interaction of the electron as cause of the particle’s inert mass m inter- preted as field mass. The Maxwell-Minkowski energy momentum tensor

&‘Tik = $&kFrsFrS - FirFkr (1)

with the electromagnetic field tensor Fik = Ak i - Ai k

is a variational derivative of the Maxwell-Larmor- Lagrange density

l* = q s”“@F,, iFnk = (p&/21; *.

According to Hilbert [I],

Tik = -‘+&l/2 G.

(24

h&T (2b)

For the case of the electrostatic field of a point charge e the vector potential Ai is equal to the Coulomb potential

Ai = phi4 = (e/r)hi4. (3)

The inertial mass m produced by this electrostatic field through self-induction amounts to*

* The commas as indices denote the partial derivative: sip = p,i. The Latin indices run from 1 to 4, and Greek indices from 1 to 3. Throughout this paper we use the symbols by

Einstein and Pauli [ 21.

mc2 = s (-g)1/2T44 d3x

V

=-& S(fg44P,pP,,gp” - ~,,~,,g~v) d3x (4) V

11-2 dJx = _ I l* dsx =I& V 2 yo

(p, v = 1,2,3 and 611” is the Kronecker symbol). However, the domain of integration V cannot cover

the total space x4 = 0, since eq. (4) diverges for r * 0.

Rather we have to exclude a neighborhood of the Coulomb singularity r = 0 by a cut-off radius ro. We

further determine this effective particle-radius SO that m is according to eq. (4) the experimental rest-mass of

the electron:

rQ = e2/2mc2. (5)

This is the electrodynamic theory of the particle mass [3-S].

The electron is, however, not self-consistent: mc2

= e2/2r,, > 0 is simply a Coulomb repulsion of the electronic charge e owing to self-interaction. The electron “explodes”, if it is not held together by an additional pressure from without, i.e., the PoincarC pressure. Indeed, in this situation the electron does not possess a real system at rest, for at such a system we would have

J (-g)1/2Tik d3x = 0. i, k # 4. (6a)

25


But

s (_@T

P V d3x =fillV 3 ~&s#/~T,~ d3x.

(6b)

/J,v=1,2,3.

Next to the metric energy-momentum tensor Tik

we define the “canonical” tensor by

For the electrostatic field we have

e U=TV P P ’ TF4 = tlp4 = 0:

yet

04 4 =L* = -T44,

Therefore,

inc2 = (--J#/~T~~ d3x s

= &g)1/2044 d3x +&g)‘/20p d3x

with the charge density u = e6(xV) [6].

(8a)

@b)

(8~)

Already in 19 19 Einstein dealt with the analogous

par-title problem in regard to the gravitational field. The gravitational field-mass of a particle can be repre-

sented as the result of the gravitational self-interaction of the particle and identified with the inertial particle mass nz*. But in CRT the energy-momentum tensor Tik of the particle is the very source of the gravitational field in conformity with the Einstein equations

Einstein’s problem leads also to the rather complicated

issue of eliminating the phenomenological matter tensor Tik from the mathematical particle model [7, 81 .

The equivalent of electrostatic Coulomb repulsion is the Newtonian gravitational attraction which deter- mines also the attraction of the particle mass m by it- self, as it were. The self-energy a Gm2/r is negative, and hereby follows rigorously the “implosion” of the particles, their collapse, owing to the self-attraction.

In CRT we have, instead of the Newtonian poten-

26

tial -Gm/r, the Schwarzschild metric. Expressed in iso-

tropic coordinates, it can be written as

46 FJv dxp dxV + [+$](dx”)?

This metric is the external spherically-symmetric field

of a mass-sphere, in particular, of a mass-point. The metric contains already the gravitational mass rn of the particle.

For an arbitrary static gravitational potential

gik,4 =o, g,4=0, v= I,?,3 (9)

the above-mentioned particle mass m is furnished by

the surface integrals

-4nn1 = s (-g)1/2g44r44vrz dS LJ f.i)no (10)

-1 -- $ (-g)1/2g44g44,V~zV df = ~-d”2R44 d3x 2

).a00 I’

= $3~ j--g)19T$ ~~ ;Tii) d3x.

Applied to pure matter

Tik = puiuk, ukuk = I

Eq. (10) signifies

(11)

4nm = -+ 4 (-d1’2g44g44,vn* df y=em

= 4r~&g)~l~T~” d3x.

The metrical energy-momentum tensor is, according to Hilbert and Lorentz, always equal to zero. For, the Einstein-Lagrange density

~ = (-g)‘i2glk(rakbriba ~ r~‘abbrjka)

and consequently

6d: = (-g)‘12(Rik - &kR); 6g’k

hence, by Einstein’s field equations, [ 1 .7-

(_g)-t/2 6d:+ 87rTjk = 0. ‘gik

(13)

(14a)

91

(14b)


However, there exists the non-covariant, non-symmetric “canonical energy-momentum-complex” of Einstein:

tik =A ( 6$ - --?L-- ymn,i

a !I mn ,k ) (15)

where the tensor density !lwrn = (-g)1/2g”“. With the aid of eq. (15) we can apply Einstein’s conservation law

(-g)I/27+ t t .k) = 0 1 I ,k ’ (16)

Further, we write for a static spherically-symmetric field

* 4= /J t4p =,o, t44 =J-e,

6 * [ UL!L~

167~ fi 48n ’ (17)

On the strength of eq. (16) the inertial particle mass

m* is equal to the gravitational mass m as the principle of equivalence requires, and thus we get

m* = m = _f

((_g)l/Q- 44 + t44) d3x. (18) V

[9, IO]. From eq. (18) one gains the insight that for the

static gravitational field, g, the integral over the density of the gravitational energy td4 - taken over the total space, V, from r = 0 to r * m - does not furnish any contribution to the particle’s inertia at all:

7-m r=l-

s tq4 d3x =&-r d: d3x = 0. (19)

r=O r=O

In the case of the spherically-symmetric field we now write

r-m

.f t,/‘d3x=0. (20)

r=O

Consequently,

r-m

m= s (g)1/2T44 d3x (2la) r=O

and for matter - eq. (11) with i, k f 4 - one obtains

i-300 r=.=

[ ((-g)‘j2T; + t/‘) d3x = 1 tik d3x =O. (21b)

r=O r=O

That is: The point-particle in CRT is self-consistent

and its self-gravitation has no effect upon the particle dynamics at all. In this respect are CRT and electrodynamics completely different. We conclude that Einstein’s particle problem, i.e., reduction of the inertial particle mass to its self-gravitation, seems thus un- solvable. However, there exists also in CRT the possi- bility of a cut-off for an “effective particle radius” through changing the topology of the domain of integration, V.

Einstein’s energy-momentum complex (-g)1/2Tjk + tik may be represented as the divergence of a superpotential Uikl = -Uilk. To treat this superpotential we now invoke - on the strength of Einstein’s “strong” principle of equivalence - the Ansatz of von Freud and Pauli:

(-g)1/2T.k t t ,k = I I j& vi”‘,l (22)

with

2U$ =(-g)1’2[$bg,b,k(6mn$c’ - &‘gkn)

- (&,kgkngmr - fr,kglkgmr) (23)

-I- (&,,Ng’k,k - &,‘gnk,J)]

[8, 111. By reason of eq. (23) we get once more the inertial

particle mass as a surface integral:

m*=m= s((-g)l/27’44 t t44) d3x = $ U44vnv df.

V r-0 (244

Furthermore, for i, k # 4 we write

s ((-g)‘i2Tik + tik) d3X = $ UikVnv df= 0. (24b)

ram

We suppose then that the whole particle is concen- trated within the Schwarzschild surface such that

g=g44 =Oor=m/2. (25)

On this surface we have

5 U4%rV df. (26a) r=m/2

Hence the volume-integral over the domain with g44 a 0 (r > m/2) outside the Schwarzschild surface (25)

27

Volume 64A. number 1 PHYSICS LETTERS 28 November 1977

~ (i.e., “excision” of the region g44 < 0) - gives the

result

m = s t4 4 d3x

r=m/2 (26b)

= $ U44vnv df -- $ U44vnV df.

r=+m r=m/2

The total inertial particle mass is therefore contained in the region g44 > 0 as the result of its self-gravitation, i.e. outside the Schwarzschild surface. This statement is the basis for Einstein’s “bridge-model” of elementary particles [ 121 . Einstein postulated for the inertial mass m* of a spherically-symmetric particle that

r*m

m* = s

t44 d3x (27a)

r=rg

and specified thereby the cut-off radius ru. The de- mand that - according to the principle of equivalence - the inertial mass m* (defined by (27a)) is equivalent

to the gravitational mass (determining the Schwarzschild metric) decides once for all: The effec-

tive particle radius r. from eq. (27) is the Schwarzschild

radius:

m* = m jr0 = m/2. (27b)

This particle model is the gravitational analogy to

Abraham’s electrons with the effective electron radius

r o = e2/2m. Neither those electrons nor the particles of Einstein and Rosen are self-consistent. The reason

is:

r-m

s t vd3X=- cc .f

Ubvhnh df

r=m/2 r=m/2

r-00

=- :“py- 14 4 d3x =$6,Ym

r=m/*

r=)m

= s

((-g)“*T,” + t/) d3x.

r=mf*

(28)

These non-vanishing integrals signify that the particles in the Einstein-Rosen model do not form a static system. In fact, they implode, because in this model t44 > 0 (for g44 > 0) is now the canonical energy density; also t44 = .C/lbn. The particle collapse can only be prevented by an additional internal pressure. It should

be emphasized that such a pressure does not appear in Einstein’s gravitational equations. In contrast, a point- like matter tensor (11) by its very nature guarantees the consistency of the particle.

This paper was begun at the Zentralinstitut fur Astrophysik, AdW, Potsdam-Babelsberg, DDR; con-

tinued by W.Y. at the Max-Planck-lnstitut fur Physik und Astrophysik, Munchen, BRD; and completed at the International Centre for Theoretical Physics, Trieste, Italy.

References

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141

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[61 I71 ISI

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On self-gravitation