On the field of a stationary charged spherical source.

The equations of gravitation related to the field of a spherical charged source imply the existence of an interdependence between gravitation and electricity . The present paper deals with the joint action of gravitation and electricity in the case of a stationary charged spherical source. Let m and [epsilon] be respectively the mass and the charge of the source, and let k be the gravitational constant. Then the equations of gravitation need specific discussion according as [absolute value of [epsilon]] < m ([square root of k]) (source weakly charged) or [absolute value of [epsilon]] = m ([square root of k]) or [absolute value of [epsilon]] > m ([square root of k]) (source strongly charged). In any case the curvature radius of the sphere bounding the matter possesses a strictly positive greatest lower hound, so that the source is necessarily an extended object. Pointwise sources do not exist. In particular, charged black holes do not exist.

1 Introduction

We recall that the field of an isotropic stationary spherical charged source is defined by solutions of the Einstein equations related to the stationary [THETA](4) -invariant metric

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.1)

([rho] = [parallel]x[parallel] = ([square root of [x.sup.2.sub.1] + [x.sup.2.sub.2] + [x.sup.2.sub.3]]), [l.sub.1](0) = [l.sub.1](0)). The functions of one variable are supposed to be C[infinity] with respect to [rho] = [parallel]x[parallel] on the half-line [0, +[infinity][(or, possibly, on the entire real line] - [infinity], +[infinity][), but since the norm [parallel]x[parallel] is not differentiable at the origin with respect to the coordinates [x.sub.1], [x.sub.2], [x.sub.3], these functions are not either. So, in general, the origin will appear as a singularity without physical meaning. In order to avoid the singularity, the considered functions must be smooth functions of the norm in the sense of the following definition.

Definition 1.1. A function of the norm [parallel]a[parallel], say f([parallel]x[parallel]), will be called smooth function of the norm, if:

a). f([parallel]x[parallel]) is C[infinity] on [R.sup.3] - {(0,0,0)} with respect to the coordinates [x.sub.1], [x.sub.2], [x.sub.3].

b). Every derivative of f([parallel]x[parallel]) with respect to the coordinates [x.sub.1], [x.sub.2], [x.sub.3] at the points x [member of] [R.sup.3] -{(0,0,0)} tends to a definite value as x [right arrow] (0,0,0).

Remark 1.1. In ,  a smooth function of the norm is considered as a function C[infinity] on R. However this last characterisation neglects the fact that the derivatives of the function are not directly defined at the origin.

The proof of the following theorem appears in .

Theorem 1.1. f([parallel]x[parallel]) is a smooth function of the norm if and only if the function of one variable f(u) is C[infinity] on [0, [infinity][ and its right derivatives of odd order at u = 0 vanish.

This being said, a significant simplification of the problem results from the introduction of the radial geodesic distance

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which makes sense in the case of stationary fields.

Since [beta]([rho]) is a strictly increasing C[infinity] function tending to +[infinity] as [rho][right arrow] +[infinity], the inverse function [rho] = [gamma]([delta]) is also a C[infinity] strictly increasing function of [delta] tending to +[infinity] as [delta] [right arrow] +[infinity]. So to the distance [delta] there corresponds a transformation of space coordinates:

[y.sub.i] = [delta]/[rho] [x.sub.i] = [beta]([rho])/[rho][x.sub.i], (i = 1,2,3),

with inverse

[x.sub.i] = [rho]/[beta]([rho]) [y.sub.i] = [gamma]([delta])/[delta][y.sub.i], (i = 1,2,3).

As shown in , these transformations involve smooth functions of the norm and since

xdx = [3.summation over (i=1)] [x.sub.i][dx.sub.i] = [gamma][gamma]'/[delta](ydy), [dx.sup.2] = [3.summation over (i=1)][dx.sup.2.sub.i] = ([[gamma]'.sup.2]/[[delta].sup.2] - [[gamma].sup.2]/[[delta].sup.4]) [(ydy).sup.2] + [[gamma]'.sup.2]/[[delta].sup.2] [dy.sup.2]

by setting

F([delta]) = f([gamma]([delta])), [F.sub.1]([delta]) = [f.sub.1]([gamma]([delta]))[gamma]([delta]) [gamma]'([delta])/[delta], [L.sub.1]([delta]) = [l.sub.1]([gamma]([delta]))[gamma]([delta])/[delta],

and taking into account that

L([delta]) = l([gamma]([delta]))[gamma]'([delta]) = 1

we get the transformed metric:

[ds.sup.2] = [(Fdt + [F.sub.1] (ydy)).sup.2] - ([L.sup.2.sub.1] [dy.sup.2] + 1 - [L.sup.2.sub.1]/[[delta].sup.2][(ydy).sup.2]). (1.2)

Then [delta] = [parallel]y[parallel] and the curvature radius of the spheres [delta] = = const, is given by the function

G = G([delta]) = [delta][L.sub.1]([delta]).

Moreover, instead of h = [rho][f.sub.1], we have now the function

H = H([delta]) = [delta][F.sub.1]([delta]).

This being said, we recall  that, with respect to (1.1), the field outside the charged spherical source is defined by the equations

fl = c dg/d[rho], dg/d[rho] = l([square root of 1 - 2[micro]/g + [v.sup.2]/[g.sup.2]]) = l ([square root of [g.sup.2] - 2[micro]g + [v.sup.2]/g]),

([micro] = km/[c.sup.2], v = ([square root of k])/[c.sup.2], [g.sup.2] - 2[micro]g + [v.sup.2] > 0, where k is the gravitational constant, to and e being respectively the mass and the charge of the source).

The function h = [rho][f.sub.1] does not appear in these equations. Every function h = [rho][f.sub.1] satisfying the required conditions of differentiability and such that [absolute value of h] [less than or equal to] l is allowable.

We obtain a simpler system of equations if we refer to the metric (1.2). Then

F = c dG/d[delta] = c([square root of 1 - 2[micro]/G + [v.sup.2]/[G.sup.2]]), (1.3)

dG/d[delta] = ([square root of 1 - 2[micro]/G + [v.sup.2]/[G.sup.2]]) = ([square root of [G.sup.2] - 2[micro]G + [v.sup.2]/G]), (1.4)

So our problem reduces essentially to the definition of the curvature radius G([delta]) by means of the equation (1.4) the study of which depends on the sign of the difference

[v.sup.2] - [[micro].sup.2] = k/[c.sup.4]([[epsilon].sup.2] - [km.sup.2]).

A concise approach to this problem appeared first in the paper .

2 Source weakly charged ([v.sup.2] < [[micro].sup.2] or [absolute value of [epsilon]] < m ([square root of k]))

[G.sup.2] - 2[micro]G + [v.sup.2] or [absolute value of [epsilon]] < m ([square root of k]) = [micro] - 2[micro]G + ([square root of [[micro].sup.2] - [v.sup.2]]) and G - [micro] + ([square root of [[micro].sup.2] - [v.sup.2]]). Moreover [G.sup.2] - - 2[micro]G + [v.sup.2] < 0 if [micro] - ([square root of [[micro].sup.2] - [v.sup.2]]) < G < [micro] + ([square root of [[micro].sup.2] - [v.sup.2]]) and [G.sup.2] - 2[micro]G + [v.sup.2] > 0 if G < [micro] - ([square root of [[micro].sup.2] - [v.sup.2]]) or G > [micro] + + ([square root of [[micro].sup.2] - [v.sup.2]]). Since negative values of G are not allowed and since the solution must be topologically connected, we have to consider two cases according as

0 < G [less than or equal to] [micro] - ([square root of [[micro].sup.2] - [v.sup.2]])

or

[micro] + ([square root of [[micro].sup.2] - [v.sup.2]]) [less than or equal to] G < +[infinity].

The first case gives an unphysical solution, because G cannot be bounded outside the source. So, it remains to solve the equation (1.4) when G describes the half-line [[micro] + ([square root of [[micro].sup.2] - [v.sup.2]]), +[infinity] [. The value [micro] + ([square root of [[micro].sup.2] - [v.sup.2]]) is the greatest lower bound of the values of G and is not reachable physically, because F vanishes, and hence the metric degenerates for this value. However the value [micro] + ([square root of [[micro].sup.2] - [v.sup.2]]) must be taken into account for the definition of the mathematical solution. So, on account of (1.4) the function G([delta]) is defined as an implicit function by the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

or, after integration,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

with G > [micro] + ([square root of [[micro].sup.2] - [v.sup.2]]).

We see that the solution involves a new constant [[delta].sub.0] which is not defined classically. To given mass and charge there correspond many possible values of [[delta].sub.0] depending probably on the size of the source as well as on its previous history, namely on its dynamical states preceding the considered stationary one. From the mathematical point of view, the determination of [[delta].sub.0] necessitates an initial condition, for instance the value of the curvature radius of the sphere bounding the matter.

Let us denote by E(G) the left hand side of (2.1). The function E(G) is a strictly increasing function of G such that E(G) [right arrow] + [infinity] as G [right arrow] +[infinity]. Consequently (2.1) possesses a unique strictly increasing solution G([delta]) tending to +[infinity] as [delta] [right arrow] + [infinity]

The equation (2.1) allows to obtain two significant relations:

a) Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

it follows that [delta] - G([delta]) [right arrow] + [infinity] as [delta] [right arrow] +[infinity].

b) Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

it follows that [delta]/G([delta]) 1 as [delta] [right arrow] +[infinity].

Moreover from (2.1), it follows that the greatest lower bound [micro] + [square root of [[micro].sup.2] - [v.sup.2] of the values of G([delta]) is obtained for [delta] = [[delta].sub.0]. The characteristics of the solution depend on the sign of [[delta].sub.0].

Suppose first that [[delta].sub.0] < 0. Since function G([delta]) is strictly increasing, we have G(0) > [micro] + ([square root of [[micro].sup.2] - [v.sup.2]]), which is physically impossible, because the physical solution G([delta]) vanishes for [delta] = 0. Consequently there exists a strictly positive value [[delta].sub.1] (the radius of the sphere bounding the matter) such that the solution is valid only for [delta] [greater than or equal to] [[delta].sub.1]. So, there exists no vacuum solution inside the ball [parallel]x[parallel] < [[delta].sub.1]. In other words, the ball [parallel]x[parallel] < [[delta].sub.1] lies inside the matter.||x|| < Si lies inside the matter.

Suppose secondly that [[delta].sub.0] = 0. Then

G(0) = [micro] + ([square root of [[micro].sup.2] - [v.sup.2]]) > 0,

which contradicts also the properties of the globally defined physical solution. Consequently there exists a strictly positive value [[delta].sub.1] (the radius of the sphere bounding the matter) such that the solution is valid for [delta] [greater than or equal to] [[delta].sub.1].

Suppose thirdly that [[delta].sub.0] > 0. Since

G([[delta].sub.0]) = [micro] + ([square root of [[micro].sup.2] - [v.sup.2]]),

the derivative

G'([[delta].sub.0]) = [square root of [(G([[delta].sub.0])).sup.2] - 2[micro]G([[delta].sub.0]) + [v.sup.2]/G([[delta].sub.0])

[FIGURE 1 OMITTED]

vanishes, so that F([[delta].sub.0]) = cG'([[delta].sub.0]) = 0. The vanishing of F([[delta].sub.0]) implies the degeneracy of the spacetime metric for [delta] = [[delta].sub.0] and since degenerate metrics have no physical meaning, there exists a value [[delta].sub.1] > [[delta].sub.0] such that the metric is physically valid for [delta] [greater than or equal to] [[delta].sub.1]. There exists no vacuum solution for [delta] [less than or equal to] [[delta].sub.0]. The ball [parallel]x[parallel] [less than or equal to] [[delta].sub.0] lies inside the matter.

From the preceding considerations it follows, in particular, that, whatever the case may be, a weakly charged source cannot be reduced to a point.

3 Source with [[micro].sup.2] = [v.sup.2] (or [absolute value of [epsilon]] = m ([square root of k]))

Since [[micro].sup.2] = [v.sup.2], we have [G.sup.2] - 2[micro]G + [v.sup.2] = [(G - y).sup.2], so that the equation (1.4) is written as

dG/d[delta] = [absolute value of G - [micro]]/G.

Consider first the case where G < [micro]. Then

dG/d[delta] = G - [micro]/G or (1 - [micro]/[micro] - G)dG = -d[delta],

whence

[a.sub.0] + G + [micro]ln (1 - G/[micro]) = -[delta], ([a.sub.0] = const).

If G [right arrow] [micro], then [delta] [right arrow] +[infinity], thus introducing a sphere with infinite radius and finite measure. This solution is unphysical. It remains to examine the case where G > [micro]. Then

(1 + [micro]/G - [micro])dG = d[delta],

whence

[a.sub.0] + G + [micro]In (G/[micro] - 1) = [delta], ([a.sub.0] = const).

To the infinity of values of [a.sub.0] there correspond an infinity of solutions which results from one of them, for instance from the solution obtained for [a.sub.0] = 0, by means of translations parallel to [delta]-axis.

[FIGURE 2 OMITTED]

For each value of [a.sub.0], we have [delta] [right arrow] + [infinity] as G [right arrow] [micro]. The value is the unreachable greatest lower bound of the values of the corresponding solution G([delta]) which is mathematically defined on the entire real line. If [[delta].sub.1] > 0 is the radius of the sphere bounding the matter, only the restriction of the solution to the half-line [[[delta].sub.1], +[infinity][ is physically valid. In order to define the solution, we need the value of the corresponding constant [a.sub.0], the determination of which necessitates an initial condition, for instance the value G([[delta].sub.1]). In any case, the values of G([delta]) for [delta] [less than or equal to] 0 are unphysical.

Finally we remark that

[delta] - G([delta]) [right arrow] +[infinity] and [delta]/G([delta]) [right arrow] 1 as [delta] [right arrow] +[infinity]

4 Source strongly charged ([v.sup.2] > [[micro].sup.2] or [absolute value of [epsilon]] > m ([square root of k]))

Since [G.sup.2] - 2[micro]G + [v.sup.2] = [(G - [micro]).sup.2] + [v.sup.2] - [[micro].sup.2], we have [G.sup.2] - 2[micro]G + [v.sup.2] for every value of G. Regarding the function

[PHI](G) = 1 - 2[micro]/G + [v.sup.2]/[G.sup.2] = [G.sup.2] - 2[micro]G + [v.sup.2]/[G.sup.2],

we have

[PHI](G) [right arrow] +[infinity] and [PHI](G)[right arrow] 1 as G [right arrow] +[infinity].

On the other hand the derivative

[PHI]'(G) = 2/[G.sup.2]([micro] - [v.sup.2]/G),

vanishes for

G = [v.sup.2]/[micro] = [[epsilon].sup.2]/[mc.sup.2]

and moreover

[PHI]'(G) < 0 for G < [v.sup.2]/[micro],

[PHI]'(G) < 0 for G < [v.sup.2]/[micro],

It follows that the function [PHI](G) is strictly decreasing on the interval ]0, [v.sup.2]/[micro][, strictly increasing on the half-line [[v.sup.2]/[micro], +[infinity] so that

[PHI]([v.sup.2]/[micro]) = 1 - [[micro].sup.2]/[v.sup.2] = 1 - [(m([square root of k])/[epsilon]).sup.2]

is the minimum of [PHI](G).

The behaviour of the solution on the half-line [[v.sup.2]/[micro], +[infinity][ is quite different from that on the interval ] 0, [v.sup.2]/[micro][. Several arguments suggest that only the restriction of the solution to the half-line [[v.sup.2]/[micro], +[infinity][ is physically valid. a) Let [[delta].sub.0] be the radius of the spherical source. In order to prove that the restriction of the solution to is]0, [v.sup.2][micro] unphysical, we have only to prove that G([[delta].sub.0]) [greater than or equal to] [v.sup.2]/[micro]. We argue by con tradiction assuming that G([[delta].sub.0]) < [v.sup.2]/[micro]. Since G([delta]) is unbounded, there exists a value [[delta].sub.1] > [[delta].sub.0] such that G([delta]) = [v.sup.2]/[micro]. On the other hand, since G([delta]) satisfies the equation (1.4), namely

dG/d[delta] = ([square root of 1 - 2[micro]/G + [v.sup.2]/[G.sup.2]]) = ([square root of [PHI](G)]),

the function

F = c dG/d[delta] = c ([square root of [PHI](G)])

is strictly decreasing on the interval ]0, [v.sup.2]/[micro][, and strictly increasing on the half-line [[v.sup.2]/[micro] +[infinity][. Such a behaviour of the important function F, which is involved in the law of propagation of light, is unexplained. We cannot indicate a cause compelling the function F first to decrease and then to increase outside the spherical source. The solution cannot be valid physically in both intervals and ]0, [v.sup.2]/[micro] and [[v.sup.2]/[micro], +[infinity][, since the great values of G are necessarily involved in the solution, it follows that only the half-line [[v.sup.2]/[micro], +[infinity][ must be taken into account. The assumption that G([[delta].sub.0]) < [v.sup.2]/[micro] is to be rejected.

b) The non-Euclidean (or, more precisely, non-pseudo-Euclidean) properties of the spacetime metric are induced by the matter, and this is why they become more and more apparent in the neighbourhood of the spherical source. On the contrary, when (5 (or G) increases the spacetime metric tends progressively to a pseudo-Euclidean form. This situation is expressed by the solution itself. In order to see this, we choose a positive value [b.sub.1] and integrate the equation (1.4) in the half-line [[b.sub.1], +[infinity] [,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and then writing down the explicit expression resulting from the integration, we find, as previously, that

[delta] - G([delta]) [right arrow] +[infinity] and [delta]/G([delta]) [right arrow] 1 as [delta] [right arrow] +[infinity]

But, since

[L.sub.1]([delta]) = G([delta])/[delta] [right arrow] and F = c ([square root of [PHI](G([delta]))]) [right arrow] c

as [delta] [right arrow] +[infinity], the metric (1.2) tends effectively to a pseudo-Euclidean form as [delta] [right arrow] +[infinity]. Now, if [delta] decreases, the non-Euclidean properties become more and more apparent, so that the minimum c[square root of 1 - [[micro].sup.2]/[v.sup.2] of F, obtained for G = [v.sup.2]/[micro], is related to the "strongest non-Euclidean character of the metric". For values of G less than [v.sup.2]/[micro], the behaviour of the mathematical solution becomes unphysical. In fact, the metric loses progressively its non-Euclidean properties, and, in particular, for G = [v.sup.2]/2[micro], we have

[PHI]([v.sup.2]/2[micro]) = 1 - 4[[micro].sup.2]/[v.sup.2] + 4[[micro].sup.2]/[v.sup.2] = 1,

hence F ([v.sup.2]/2[micro]) = c and dG/d[delta] = 1.

On account of G = [delta][L.sub.1], the last condition implies

1 = dG/d[delta] = [L.sub.1] + [delta] d[L.sub.1]/d[delta]

and since we have to do physically with very small values of [delta] (in the neighbourhood of the origin), we conclude that

[L.sub.1]([v.sup.2]/2[micro]) [approximately equal to] 1.

and since F ([v.sup.2]/2[micro]) = c, the metric is almost pseudo-Euclidean, a phenomenon inadmissible physically in the neighbourhood of the source. So we are led to reject the restriction of the mathematical solution to the interval [[v.sup.2]/2[micro], [v.sup.2]/[micro][. For values less than [v.sup.2]/2[micro], the function F(G) increases rapidly and tends to +[infinity] as decreases, so that the restriction of the mathematical solution to the interval]0, [v.sup.2]/2[micro][ is also physically inadmissible. It follows that the restriction of the solution to the entire interval]0, [v.sup.2]/[micro] is unphysical.

c) Another argument supporting the above assertion is given in .

Let [[delta].sub.1] be the radius of the spherical source and assume that G([[delta].sub.1]) > [v.sup.2]/[micro]. A radiation emitted radially from the sphere bounding the matter is redshifted, and its redshift at the points of a sphere [parallel]x[parallel] = [delta] with [delta] > [[delta].sub.1] is given by the formula

Z([delta], [[delta].sub.1]) = -1 + F(G([delta]))F(G([[delta].sub.1])) = -1 + [square root of [PHI](G([delta]))/[PHI](G([[delta].sub.1])).

Suppose [delta] fixed and let us examine the variation of Z ([delta], [[delta].sub.1]) considered as function of [[delta].sub.1]. If [[delta].sub.1] (or (G([[delta].sub.1])) decreases, Z([delta], [[delta].sub.1]) increases and tends to its maximum, obtained for G([[delta].sub.1]) = [v.sup.2]/[micro],

maxZ([delta], [[delta].sub.1]) = -1 + ([square root of [PHI](G([delta]))/1 - [[micro].sup.2]/[v.sup.2]]).

[FIGURE 3 OMITTED]

If G([[delta].sub.1]) takes values less than [v.sup.2]/[micro], the phenomenon is inverted: The redshift first decreases and then vanishes for a unique value G([[delta].sub.1]) [member of]] [v.sup.2]/2[micro], [v.sup.2]/[micro][ with [PHI](G([[delta].sub.1])) = [PHI](G([delta])). If G([[delta].sub.1]) decreases further, instead of a redshift, we have a blueshift. This situation seems quite unphysical, inasmuch as the vanishing of the redshift depends on the position of the observer. In order to observe constantly a redshift, the condition G([[delta].sub.1]) [greater than or equal to] [v.sup.2]/[micro] is necessary.

From the preceding considerations we conclude that the value

[v.sup.2]/[micro] = [[epsilon].sup.2]/[mc.sup.2]

is the greatest lower bound of the curvature radius G([delta]) outside the spherical strongly charged source. In particular, the curvature radius of the sphere bounding the matter is [greater than or equal to] [[epsilon].sup.2]/[mc.sup.2], so that a strongly charged source cannot be reduced to a point. Our study does not exclude the case where the solution G([delta]) attains its greatest lower bound, namely the case where the curvature radius of the sphere bounding the matter is exactly equal to [[epsilon].sup.2]/[mc.sup.2]. So, in order to take into account all possible cases, the equation (1.4) must be integrated as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If [a.sub.0] [less than or equal to] 0, there exists a value [[delta].sub.1] > 0 such that the solution is valid only for 5 .

If [a.sub.0] > 0, the solution is valid for [[delta].sub.1] [greater than or equal to] [a.sub.0], only if the sphere bounding the matter has the curvature radius [v.sup.2]/[micro]. Otherwise there exists a value [[delta].sub.1] > [a.sub.0] such that the solution is valid for [delta] [greater than or equal to] [[delta].sub.1].

The expression [[epsilon].sup.2]/[mc.sup.2] is also known in classical electrodynamics, but in the present situation it appears on the basis of new principles and with a different signification. Consider, for instance, the case of the electron. Then [absolute value of [epsilon]]/m ([square root of k]) = 2.02 x [l0.sup.21], so that the electron is strongly charged, and, from the point of view of the classical electrodynamics, is a spherical object with radius [[epsilon].sup.2]/[mc.sup.2] = 2.75 x [10.sup.-13] cm.

Regarding the present theory, we can only assert that, if the electron is a stationary spherical object, then it is a non-Euclidean ball such that the value 2.75 x [10.sup.-13] cm is the greatest lower bound of the possible values of the curvature radius of the sphere bounding it. The radius of the electron cannot be deduced from the present theory.

The proton is also strongly charged with [absolute value of [epsilon]]/m ([square root of k]) = l.l x [l0.sup.18].

The corresponding value [[epsilon].sup.2]/[mc.sup.2] = 1.5 x [l0.sup.-16] cm is less than that related to the electron by a factor of the order [10.sup.-3]. So, if the proton is assumed to be spherical and stationary, it is not reasonable to accept that this value represents its radius. This last is not definable by the present theory.

Submitted on January 31, 2009 / Accepted on February 05, 2009

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Nikias Stavroulakis

Solomou 35, 15233 Chalandri, Greece

E-mail: nikias.stavroulakis@yahoo.fr
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