On the Field of a Stationary Charged Spherical Source

Transcription

1 Volume PRORESS IN PHYSICS Aril, 009 On the Field of a Stationary Charged Sherical Source Nikias Stavroulakis Solomou 35, 533 Chalandri, reece nikias.stavroulakis@yahoo.fr The equations of gravitation related to the field of a sherical charged source imly the existence of an interdeendence between gravitation and electricity [5]. The resent aer deals with the joint action of gravitation and electricity in the case of a stationary charged sherical source. Let m and " be resectively the mass and the charge of the source, and let k be the gravitational constant. Then the equations of gravitation need secific discussion according as j"j < m k (source weakly charged) or j"j = m k or j"j > m k (source strongly charged). In any case the curvature radius of the shere bounding the matter ossesses a strictly ositive greatest lower hound, so that the source is necessarily an extended object. Pointwise sources do not exist. In articular, charged black holes do not exist. Introduction We recall that the field of an isotroic stationary sherical charged source is defined by solutions of the Einstein equations related to the stationary (4)-invariant metric ds = f () dt + f () (xdx) 4 (l ()) dx + (l ()) (l ()) (xdx) 3 5 (.) ( = kxk = x + x + x 3, l(0) = l (0)). The functions of one variable f(), f (), l (), l() are suosed to be C with resect to = kxk on the half-line [0 +[ (or, ossibly, on the entire real line ] +[), but since the norm kxk is not differentiable at the origin with resect to the coordinates x, x, x 3, these functions are not either. So, in general, the origin will aear as a singularity without hysical 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. A function of the norm kxk, say f(kxk), will be called smooth function of the norm, if: a). f(kxk) is C on R 3 f(0 0 0)g with resect to the coordinates x, x, x 3. b). Every derivative of f(kxk) with resect to the coordinates x, x, x 3 at the oints x R 3 f(0 0 0)g tends to a definite value as x! (0 0 0). Remark. In [3], [4] a smooth function of the norm is considered as a function C on R. However this last characterisation neglects the fact that the derivatives of the function are not directly defined at the origin. The roof of the following theorem aears in [3]. Theorem. f(kxk) is a smooth function of the norm if and only if the function of one variable f(u) is C on [0 [ and its right derivatives of odd order at u = 0 vanish. This being said, a significant simlification of the roblem results from the introduction of the radial geodesic distance = Z 0 l(u) du = () ((0) = 0) which makes sense in the case of stationary fields. Since () is a strictly increasing C function tending to + as! +, the inverse function = () is also a C strictly increasing function of tending to + as! +. So to the distance there corresonds a transformation of sace coordinates: with inverse y i = x i = () x i (i = 3) x i = () y i = () y i (i = 3): As shown in [4], these transformations involve smooth functions of the norm and since dx = by setting xdx = 3X i= dx i = 3X i= x i dx i = 0 (ydy) 0 4 (ydy) + 0 dy F () = f(()) F () = f (()) ()0 () L () = l (()) () 66 Nikias Stavroulakis. On the Field of a Stationary Charged Sherical Source

2 Aril, 009 PRORESS IN PHYSICS Volume and taking into account that L() = l(()) 0 () = we get the transformed metric: ds = (F dt + F (ydy)) L dy + L (ydy) : (.) Then = kyk and the curvature radius of the sheres = = const, is given by the function = () = L (): Moreover, instead of h = f, we have now the function H = H() = F (): This being said, we recall [5] that, with resect to (.), the field outside the charged sherical source is defined by the equations s dg d = l fl = c dg d g + g g = l g + g ( = km c, = k c j"j, g g + > 0, where k is the gravitational constant, m and " being resectively the mass and the charge of the source). The function h = f does not aear in these equations. Every function h = f satisfying the required conditions of differentiability and such that jhj l is allowable. We obtain a simler system of equations if we refer to the metric (.). Then r d d = F = c d d = c r + (.3) + = + (.4) jhj : So our roblem reduces essentially to the definition of the curvature radius () by means of the equation (.4) the study of which deends on the sign of the difference = k c 4 " km : A concise aroach to this roblem aeared first in the aer []. Source weakly charged ( < or j"j < m k) + = ( ) + vanishes for = = and = +. Moreover + < 0 if < < + and + > 0 if < or > + +. Since negative values of are not allowed and since the solution must be toologically connected, we have to consider two cases according as or 0 < + < +: The first case gives an unhysical solution, because cannot be bounded outside the source. So, it remains to solve the equation (.4) when describes the half-line + +. The value + is the greatest lower bound of the values of and is not reachable hysically, because F vanishes, and hence the metric degenerates for this value. However the value + must be taken into account for the definition of the mathematical solution. So, on account of (.4) the function () is defined as an imlicit function by the equation 0 + Z + or, after integration, udu u u + = ( 0 = const) + ln + + = (.) with > +. We see that the solution involves a new constant 0 which is not defined classically. To given mass and charge there corresond many ossible values of 0 deending robably on the size of the source as well as on its revious history, namely on its dynamical states receding the considered stationary one. From the mathematical oint of view, the determination of 0 necessitates an initial condition, for instance the value of the curvature radius of the shere bounding the matter. Let us denote by E() the left hand side of (.). The function E() is a strictly increasing function of such that E()! + as! +. Consequently (.) ossesses a unique strictly increasing solution () tending to + as! + The equation (.) allows to obtain two significant relations: Nikias Stavroulakis. On the Field of a Stationary Charged Sherical Source 67

3 Volume PRORESS IN PHYSICS Aril, 009 a) Since () = E() = = 0 + ln = = 0 + ln q + +! + as! + it follows that ()! + as! +. b) Since r () = E() = ln + ln + + q + +! as! + it follows that ()! as! +. Moreover from (.), it follows that the greatest lower bound + of the values of () is obtained for = 0. The characteristics of the solution deend on the sign of 0. Suose first that 0 < 0. Since function () is strictly increasing, we have (0) > +, which is hysically imossible, because the hysical solution () vanishes for = 0. Consequently there exists a strictly ositive value (the radius of the shere bounding the matter) such that the solution is valid only for. So, there exists no vacuum solution inside the ball kxk <. In other words, the ball kxk < lies inside the matter. Suose secondly that 0 = 0. Then (0) = + > 0 which contradicts also the roerties of the globally defined hysical solution. Consequently there exists a strictly ositive value (the radius of the shere bounding the matter) such that the solution is valid for. Suose thirdly that 0 > 0. Since the derivative 0 ( 0 ) = ( 0 ) = + q (( 0 )) ( 0 ) + ( 0 ) Fig. : rah of in the case where <. vanishes, so that F ( 0 ) = c 0 ( 0 ) = 0. The vanishing of F ( 0 ) imlies the degeneracy of the sacetime metric for = 0 and since degenerate metrics have no hysical meaning, there exists a value > 0 such that the metric is hysically valid for. There exists no vacuum solution for 0. The ball kxk 0 lies inside the matter. From the receding considerations it follows, in articular, that, whatever the case may be, a weakly charged source cannot be reduced to a oint. 3 Source with = (or j"j = m k) Since =, we have + = ( ), so that the equation (.4) is written as d d = d d = j j : Consider first the case where <. Then or whence a ln d = d = (a 0 = const) : If!, then! +, thus introducing a shere with infinite radius and finite measure. This solution is unhysical. It remains to examine the case where >. Then whence a ln + d = d = (a 0 = const) : To the infinity of values of a 0 there corresond an infinity of solutions which results from one of them, for instance from 68 Nikias Stavroulakis. On the Field of a Stationary Charged Sherical Source

4 Aril, 009 PRORESS IN PHYSICS Volume Fig. : rah of in the case where =. the solution obtained for a 0 = 0, by means of translations arallel to -axis. For each value of a 0, we have! + as!. The value is the unreachable greatest lower bound of the values of the corresonding solution () which is mathematically defined on the entire real line. If > 0 is the radius of the shere bounding the matter, only the restriction of the solution to the half-line [ +[ is hysically valid. In order to define the solution, we need the value of the corresonding constant a 0, the determination of which necessitates an initial condition, for instance the value ( ). In any case, the values of () for 0 are unhysical. Finally we remark that ()! + and! as! +: () 4 Source strongly charged ( > or j"j > m k) Since + = ( ) +, we have + > 0 for every value of. Regarding the function () = + = + we have ()! + as! 0 and ()! as! +: On the other hand the derivative vanishes for and moreover 0 () = = = " mc 0 () < 0 for < 0 () > 0 for > : It follows that the function () is strictly decreasing on the interval ] 0 [, strictly increasing on the half-line [ +[, so that m k = = " is the minimum of (). The behaviour of the solution on the half-line [ +[ is quite different from that on the interval ] 0 [. Several arguments suggest that only the restriction of the solution to the half-line [ +[ is hysically valid. a) Let 0 be the radius of the sherical source. In order to rove that the restriction of the solution to ] 0 [ is unhysical, we have only to rove that ( 0 ). We argue by contradiction assuming that ( 0 ) <. Since () is unbounded, there exists a value > 0 such that ( ) =. On the other hand, since () satisfies the equation (.4), namely r d d = + = () the function F = c d d = c () is strictly decreasing on the interval ] 0 [, and strictly increasing on the half-line [ +[. Such a behaviour of the imortant function F, which is involved in the law of roagation of light, is unexlained. We cannot indicate a cause comelling the function F first to decrease and then to increase outside the sherical source. The solution cannot be valid hysically in both intervals ] 0 [ and [ +[, and since the great values of are necessarily involved in the solution, it follows that only the half-line [ +[ must be taken into account. The assumtion that ( 0 ) < is to be rejected. b) The non-euclidean (or, more recisely, non-seudo-euclidean) roerties of the sacetime metric are induced by the matter, and this is why they become more and more aarent in the neighbourhood of the sherical source. On the contrary, when (or ) increases the sacetime metric tends rogressively to a seudo-euclidean form. This situation is exressed by the solution itself. In order to see this, we choose a ositive value b and integrate the equation (.4) in the half-line [b +[, b 0 + Z b udu u u + = (b 0 = const) and then writing down the exlicit exression resulting from the integration, we find, as reviously, that ()! + and! as! +: () Nikias Stavroulakis. On the Field of a Stationary Charged Sherical Source 69

5 Volume PRORESS IN PHYSICS Aril, 009 But, since L () = ()! and F = c (())! c as! +, the metric (.) tends effectively to a seudo- Euclidean form as! +. Now, if decreases, the non- Euclidean roerties become more and more aarent, so that q the minimum c of F, obtained for =, is related to the strongest non-euclidean character of the metric. For values of less than, the behaviour of the mathematical solution becomes unhysical. In fact, the metric loses rogressively its non-euclidean roerties, and, in articular, for =, we have 4 = + 4 = hence F = c and d d =. On account of = L, the last condition imlies = d d = L + dl d and since we have to do hysically with very small values of (in the neighbourhood of the origin), we conclude that L : = c, the metric is almost seudo-eucli- and since F dean, a henomenon inadmissible hysically in the neighbourhood of the source. So we are led to reject the restriction of the mathematical solution to the interval [ [. For values less than, the function F () increases raidly and tends to + as decreases, so that the restriction of the mathematical solution to the interval ] 0 [ is also hysically inadmissible. It follows that the restriction of the solution to the entire interval ] 0 [ is unhysical. c) Another argument suorting the above assertion is given in []. Let be the radius of the sherical source and assume that ( ) >. A radiation emitted radially from the shere bounding the matter is redshifted, and its redshift at the oints of a shere kxk = with > is given by the formula s Z( ) = + F (()) F (( )) = + (()) (( )) : Suose fixed and let us examine the variation of Z( ) considered as function of. If (or ( )) decreases, Z( ) increases and tends to its maximum, obtained for ( ) =, max Z( ) = + s (()) : Fig. 3: rah of in the case where >. If ( ) takes values less than, the henomenon is inverted: The redshift first decreases and then vanishes for a unique value ( ) ] [ with (( )) = (()). If ( ) decreases further, instead of a redshift, we have a blueshift. This situation seems quite unhysical, inasmuch as the vanishing of the redshift deends on the osition of the observer. In order to observe constantly a redshift, the condition ( ) is necessary. From the receding considerations we conclude that the value = " mc is the greatest lower bound of the curvature radius () outside the sherical strongly charged source. In articular, the curvature radius of the shere bounding the matter is " mc, so that a strongly charged source cannot be reduced to a oint. Our study does not exclude the case where the solution () attains its greatest lower bound, namely the case where the curvature radius of the shere bounding the matter is exactly equal to " mc. So, in order to take into account all ossible cases, the equation (.4) must be integrated as follows a 0 + Z = udu u u + = (a 0 = const) : If a 0 0, there exists a value > 0 such that the solution is valid only for. If a 0 > 0, the solution is valid for a 0, only if the shere bounding the matter has the curvature radius. Otherwise there exists a value > a 0 such that the solution is valid for. The exression mc " is also known in classical electrodynamics, but in the resent situation it aears on the basis Con- j"j m k = of new rinciles and with a different signification. sider, for instance, the case of the electron. Then = :00, so that the electron is strongly charged, and, from the oint of view of the classical electrodynamics, is a sherical object with radius " mc = :750 3 cm. Regarding the resent theory, we can only assert that, if 70 Nikias Stavroulakis. On the Field of a Stationary Charged Sherical Source

6 Aril, 009 PRORESS IN PHYSICS Volume the electron is a stationary sherical object, then it is a non- Euclidean ball such that the value :750 3 cm is the greatest lower bound of the ossible values of the curvature radius of the shere bounding it. The radius of the electron cannot be deduced from the resent theory. The roton is also strongly charged with j"j m k =: 0 8. The corresonding value " mc = :50 6 cm is less than that related to the electron by a factor of the order 0 3. So, if the roton is assumed to be sherical and stationary, it is not reasonable to accet that this value reresents its radius. This last is not definable by the resent theory. Submitted on January 3, 009 / Acceted on February 05, 009 References. Stavroulakis N. Paramètres cachés dans les otentiels des chams statiques. Annales Fond. Louis de Broglie, 98, v. 6(4), Stavroulakis N. Particules et articules test en relativité générale. Annales Fond. Louis de Broglie, 99, v. 6(), Stavroulakis N. Vérité scientifique et trous noires (deuxième artie) Symétries relatives au groue des rotations. Annales Fond. Louis de Broglie, 000, v. 5(), Stavroulakis N. Non-Euclidean geometry and gravitation. Progress in Physics, 006, v., Stavroulakis N. ravitation and electricity. Progress in Physics, 008, v., Nikias Stavroulakis. On the Field of a Stationary Charged Sherical Source 7