arXiv:gr-qc/0306088v12 13 Dec 2003 o h ag rnfrain nG a epu clarify us framework help wider can standing GR such long in of some transformations analysis gauge We the careful GR. for of a scheme that mathematical think used commonly the in ilatrsm ftewl known well the of some this course alter Of GR. will in transformations gauge transformations admissible of gauge a functions, a in used the of behavior (2), rprcoc ftequantities the of choice a proper in a GR in a gauge is the GR coor- that of fact known choice well a theory. the to up gauge reflects up problem This i.e., physical functions, dinates. given arbitrary a four of to solution the determine ∗ hc mrewe n xrse h DdAebr op- form d’Alembert the 4D in the erator expresses one when emerge which -al fi[email protected]fia.bg E-mail: esalcl h hneo h ag xn expressions fixing gauge the of change the call shall We codn otesadr etok 1 h xn of fixing the [1] textbooks standard the to According h isensequations: Einstein’s The broad without es.Ti a eesnilyepn h class the expand essentially we way This sense. Γ ¯ n rlmnr odtoso h analytical the on conditions preliminary any edo igemsiepitpril ihnneobr mas bare nonzero with particle point so massive only single but a distinguishable of physically two-parameter field a are exist which there (GR) equations and relativity center general the in at that singularity point with space-times metric h iepedHletsfr fShazcidslto,w M solution, Schwarzschild < of M form Hilbert’s widespread the eet h eutcnb ecie sacag fteNwo p Newton the of change a de as the described determine be and can so particle result mass point The point ph a a new defect. of of a defect description and correct mass field a the tational achieving for way coordinates this normal in Novel source. a as e one: fied aiod ihadfiiesnuaiya h lc ftematt the of 04.20.Dw place 04.20.Jb, the 04.20.Cv, at number(s): singularity PACS definite a with manifolds peial ymti ttcsaetmscetdb n po one by physic created the space-times of static characteristics symmetric invariant spherically give we addition In rvttoa il fMsiePitPril nGnrlRe General in Particle Point Massive of Field Gravitational µ = tlzn aiu agso h ailcodnt egv d a give we coordinate radial the of gauges various Utilizing .INTRODUCTION I. − p g 0 µν physical 1 osntsleteEnti qain ihamsiepitp point massive a with equations Einstein the solve not does , | ϕ G g holonomic ∇ G | ν µ g µ = µν = ∇ − ∂ ν κT h bu aa nentoa etefrTertclPhysi Theoretical for Centre International Salam Abdus The G λ = rbes fe l,i the in all, After problems. N ν µ M/ p g µν rm srpeetdby represented is frame ae orhe olvr,Sfi 14 Bulgaria. 1164, Sofia Boulevard, Bourchier James 5 | g | mathematical r g ∂ λν + µ taaCsir 1 41 ret,Italy. Trieste, 34014 11, Costiera Strada ∂ G ν aut fPyis oaUniversity, Sofia Physics, of Faculty N , eateto hoeia Physics, Theoretical of Department − M/c Γ ¯ µ 2 ∂ ν ln results . M M .P Fiziev P. P. 0 (1) (2) n orsodn oicto ftefour-interval. the of modification corresponding a and and n dto ftemngah[] 3,adterefer- base the the on and coordinates [3], of change [2], The monograph the therein.) ences of edition ond rpriso h ouin aigatasto oanother physical to transition sector. and a gauge making topological solution, the geometrical, of properties the both change n ieet4 suoReana pc-iemani- space-time pseudo-Riemannian folds 4D different fine cito sagueter,uigtepicplframe principal the using theory, gauge de- bundle a geometrical differential as modern scription its of framework solution. the the of properties global essential the the the While problem, fixes fixing sector. gauge first local this inessential in same an the gauge solves at latter (regular) and the lives solution fixing the time which choosing in in sector things: gauge coordinates different a space-time two of simultaneously choice means the- the GR gauge view from of because point an possible ”change ory is to a This as solution represented coordinates”. physical be of given can one, a different from essentially transitions that is clpoete.Fragvnslto hr xssaclass a exists there solution given a of For phys- and properties. sec- topological gauge ical geometrical, different the different to in owning belong tors, symmetry solutions Such same space. the base with solutions different real have the of properties the way adequate objects. an physical in reflect are to constructions mathematical the applications physical ihu hnigteeesnilpoete.I contrast, a In performing properties. by essential these changing without ngoerclsnetedffrn ouin nG de- GR in solutions different the sense geometrical In n a nesadbte hspclaiyo Rin GR of peculiarity this better understand can One ti elkonta nteguetere emay we theories gauge the in that known well is It regular ∗ lyadgoerclydffrn lse of classes different geometrically and ally M ihdpnsol nteKpeinmass Keplerian the on only depends hich aumotietesnuaiy eshow We singularity. the outside vacuum aiyo uhsltost h Einstein the to solutions such of family s n as hs pc-ie r analytic are space-times These mass. int eo hmdsrb h gravitational the describe them of me F rei R eas nrdc gravi- a introduce also We GR. in urce edneo h ouin nti mass this on solutions the of pendence (1 rparticle. er , 3) srpino ttcshrclysym- spherically static of escription ags hc le h omo h solution the of form the alter which gauges, s M otential sclcaso agsaeproposed, are gauges of class ysical { M g (1 µν 0 , npriua,w hwthat show we particular, In . 3) ( singular { ril’ teseeg tensor stress-energy article’s x g ) ϕ µν } N utepiti Rformalism GR in point subtle A . ( = x ) − } cs, ag rnfrainoecan one transformation gauge G Se o xml,tesec- the example, for (See, . N M/r lativity oamodi- a to 2
(1,3) (3) space gµν (x) induces automatically a nontrivial 3D space δµν , changing the topology of this part of M { } R { } rA r change of the frames on the fibre of frames. Singular the base space. As a result the quantity C d ( ), which coordinate transformations may produce a change of the is gauge invariant under regular gauge transformations, rA r H gauge sector of the solution, because they may change now changes its value to C d ( )+2πNα, where N is the topology of the frame bundle, adding new singu- the winding number of the cycle C around the axes OZ. lar points and(or) singular sub-manifolds, or removing Under such singular gaugeH transformation the solution of some of the existing ones. For example, such transforma- some initial physical problem will be transformed onto a tions can alter the fundamental group of the base space solution of a completely different problem. (1,3) gµν(x) , the holonomy group of the affine con- At present the role of singular gauge transformations M { } (1,3) nection on F gµν (x) , etc. Thus one sees that in the above physical theories is well understood. coordinate changesM in{ broad} sense are more than a pure In contrast, we still don’t have systematic study of the alteration of the labels of space-time points. classes of physically, geometrically and topologically dif- If one works in the framework of the theory of smooth ferent solutions in GR, created by singular gauge trans- real manifolds, ignoring the analytical properties of the formations, even in the simple case of static spherically solutions in the complex domain, one is generally allowed symmetric space-times with only one point singularity, to change the gauge without an alteration of the phys- although the first solution of this type was discovered ical problem in the real domain, i.e. without change of first by Schwarzschild nearly 90 years ago [5]. Moreover, the boundary conditions, as well as without introduc- in GR at present there is no clear understanding both of tion of new singular points, or change of the character the above gauge problem and of its physical significance. of the existing ones. Such special type of regular gauge Here we present some initial steps toward the clarifica- transformations in GR describe the diffeomorphisms of tion of the role of different GR gauges in broad sense for (1,3) the real manifold gµν(x) . This manifold is al- spherically symmetric static space-times with point sin- ready fixed by theM initial{ choice of} the gauge. Hence, the gularity at the center of symmetry and vacuum outside (1,3) real manifold gµν (x) is actually described by a this singularity. class of equivalentM gauges,{ which} correspond to all diffeo- morphisms of this manifold and are related with regular gauge transformations. II. STATIC SPHERICALLY SYMMETRIC The transitions between some specific real manifolds SPACE-TIMES WITH POINT SOURCE OF (3) gmn(r) , which are not diffeomorphic, can be GRAVITY producedM {− by the} use of proper singular gauge transforma- tions. If we start from an everywhere smooth manifold, The static point particle with bare mechanical rest after a singular transformation we will have a new man- mass M0 can be treated as a 3D entity. Its proper frame ifold with some singularities, which are to describe real of reference is most suitable for description of the static physical phenomena in the problem at hand. The inclu- space-time with this single particle in it. We prefer to sion of such singular transformations in our consideration present the problem of a point source of gravity in GR yields the necessity to talk about gauge transformations as an 1D mathematical problem, considering the depen- in a broad sense. They are excluded from present-day dence of the corresponding functions on the only essential standard considerations by the commonly used assump- variable – the radial variable r. This can be achieved in tion, that in GR one has to allow only diffeomorphic map- the following way. pings. The spherical symmetry of the 3D space reflects ad- Similar singular transformations are well known in equately the point character of the source of grav- gauge theories of other fundamental physical interac- ity. A real spherically symmetric 3D Riemannian space (3) (1,3) tions: electromagnetic interactions, electroweak interac- gmn(r) gµν (x) can be described tions, chromodynamics. For example, in the gauge theo- usingM {− standard} spherical ⊂ M coordinates{ } r [0, ), θ ries singular gauge transformations describe transitions [0, π), φ [0, 2π). Then ∈ ∞ ∈ between solutions in topologically different gauge sec- ∈ tors. Singular gauge transformations are used in the the- r = x1, x2, x3 = r sin θ cos φ, r sin θ sin φ, r cos θ . ory of Dirac monopole, vortex solutions, t’Hooft-Poliakov { } { } monopoles, Yang-Mills instantons, etc. See, for example, The physical and geometrical meaning of the radial [3, 4] and the references therein. coordinate r is not defined by the spherical symmetry of The simplest example is the singular gauge transfor- the problem and is unknown a priori [6]. The only clear mation of the 3D vector potential: A(r) A(r) + thing is that its value r = 0 corresponds to the center of ϕ(r) in electrodynamics, defined in Cartesian→ coor- the symmetry, where one must place the physical source dinates∇ x,y,z by the singular gauge function ϕ = of the gravitational field. In the present article we assume α arctan({y/x) (}α = const). Suppose that before the that there do not exist other sources of gravitational field (3) transformation we have had a 3D space gmn(r)= outside the center of symmetry. (3) M {− δmn = δmn . Then this singular gauge transfor- There exists unambiguous choice of a global time t due mation} removesR { the} whole axes OZ out of the Euclidean to the requirement to use a static metric. In proper units 3 it yields the familiar form of the space-time interval [9]: plays the role of a Lagrange multiplier, which is needed in a description of a constrained dynamics. This auxil- 2 2 2 2 2 2 2 ds =gtt(r) dt +grr(r) dr ρ(r) (dθ +sin θ dφ ) (3) iary variable enters the Lagrangian in a nonlinear − GR manner, in a contrast to the case ofL the standard La- with unknown functions gtt(r) > 0, grr(r) < 0, ρ(r). grange multipliers. The corresponding Euler-Lagrange In contrast to the variable r, the quantity ρ has a clear equations read: geometrical and physical meaning: It is well known that 2 2 ρ defines the area Aρ = 4πρ of a centered at r = 0 2ρρ′ ′ ρ′ √ grr +2GN M0δ(r)=0, sphere with ”a radius” ρ and the length of a big circle on √ grr − √ grr − − it l = 2πρ. One can refer to this quantity as an “area − − ρ ′ ρ√gtt ′ ρ′ √gtt ′ radius”, or as an optical ”luminosity distance“, because =0, the luminosity of distant physical objects is reciprocal to √ grr − √ grr − ! − Aρ. 2 2ρρ′ √gtt ′ +(ρ′) √gtt w One has to stress that in the proper frame of refer- √g √ g = 0 (5) √ g − tt − rr ence of the point particle this quantity does not mea- − rr sure the real geometrical distances in the corresponding where the symbol ” =w ” denotes a weak equality in the curved space-time. In contrast, if ρfixed is some arbi- trary fixed value of the luminosity distance, the expres- sense of the theory of constrained dynamical systems. If one ignores the point source of the gravitational field, sion (ρ ρ ) / (1 2M/ρ )1/2 measures the 3D geo- 2 − 1 − fixed thus considering only the domain r > 0 where δ(r) 0, metrical distance between the geometrical points 2 and 1 one obtains the standard solution of this system [9]:≡ on a radial geodesic line in the frame of free falling clocks 2 [7]. Nevertheless, even in this frame the absolute value gtt(r)=1 ρG/ρ(r), gtt(r) grr(r)= (ρ(r)′) , (6) of the variable ρ remains not fixed by the 3D distance − − measurements. where ρG =2GN M is the Schwarzschild radius, M is the In the static spherically symmetric case the choice gravitational (Keplerian) mass of the source, and ρ(r) is of spherical coordinates and static metric dictates the an arbitrary 1 function. C form of three of the gauge fixing coefficients (2): Γ¯t = 0, Γ¯θ = cot θ, Γ¯φ = 0, but the form of the quantity − III. SOME EXAMPLES OF DIFFERENT ¯ √ grr ′ Γr = ln − 2 , and, equivalently, the function ρ(r) √gtt ρ RADIAL GAUGES are still not fixed. Here and further on, the prime de- notes differentiation with respect to the variable r. We In the literature one can find different choices of the refer to the freedom of choice of the function ρ(r) as ”a function ρ(r) for the problem at hand: rho-gauge freedom 1/3 ”, and to the choice of the ρ(r) function 1. Schwarzschild gauge [5]: ρ(r)= r3 +ρ3 . It pro- as ”a rho-gauge fixing” in a broad sense. G duces Γ¯ = 2 ρ(r3 ρ3 ) ρ (r3/2 ρ3 ) / ρ3(ρ ρ ) . In the present article we will not use more general r − r − G − G − G − G 2. Hilbert gauge [10]: ρ(r) = r. It gives Γ¯r = gauge transformations in a broad sense, than the rho- ρ ρ 2 1 G / 1 G . This simple choice of the func- gauge ones. In our 1D approach to the problem at hand r 2r r −tion ρ(−r) is often− related incorrectly with the original all possible mathematical complications, due to the use of such wide class of transformations, can be easily con- Schwarzschild article [5] (See [11]). In this case the co- trolled. ordinate r coincides with the luminosity distance ρ and the physical domain r [0, ) contains an event horizon The overall action for the aggregate of a point particle ∈ ∞ at ρH = ρG. This unusual circumstance forces one to de- and its gravitational field is tot = GR + M0 . Ne- glecting the surface terms oneA can representA theA Hilbert- velop a nontrivial theory of black holes for Hilbert gauge 1 4 – see for example [1, 12] and the references therein. Einstein action GR = d x g R and the me- A − 16πGn | | 3. Droste gauge [13]: The function ρ(r) is given chanical action M0 = M0 ds of the point source with 2 A − R p implicitly by the relations ρ/ρG = cosh ψ 1 and a bare mass M0 as integrals with respect to the time r/ρ = ψ + sinh ψ cosh ψ. The coefficient≥ Γ¯ = t and the radial variable r Rof the following Lagrangian G r 2 ρ 3 ρG / 1 ρG . For this solution the vari- densities: − ρ ρG − 4 ρ − ρ able r has a clear geometrical meaning: it measures the 2 q 1 2ρρ′ √gtt ′ +(ρ′) √gtt 3D-radial distance to the center of the spherical symme- GR = +√gtt√ grr , L 2GN √ grr − try. − ! 2 4. Weyl gauge [14]: ρ(r) = 1 r/ρ + ρ /r M0 = M0√gttδ(r). (4) 4 G G L − 2 ≥ ¯ 2 ρG 2 r ρG ρG and Γr = / 1 2 = p/ p . In this Here GN is the Newton gravitational constant, δ(r) is − r − r − ρG ρG − r (3) the 1D Dirac function [8]. (We are using units c=1.) gauge the 3D-space gmn(r) becomes obviously As a result of the rho-gauge freedom the field vari- conformally flat. TheM coordinate{− r is} the radial variable able √ g is not a true dynamical variable but rather in the corresponding Euclidean 3D space. − rr 4
5. Einstein-Rosen gauge [15]. In the original article ii) The choice of the function ρ(r) can change drasti- 2 [15] the variable u = ρ ρG has been used. To have a cally the character of the singularity at the place of the proper dimension we replace− it by r = u2 0. Hence, point source of the metric field in GR. 2 r+ρG/2 ≥ ρ(r)= r + ρG ρG and Γ¯r = . New interesting gauge conditions for the radial vari- ≥ − r r+ρG 6. Isotropic t-r gauge, defined according to the for- able r and corresponding gauge transformations were in- ρ troduced and investigated in [17]. In the recent articles mula r = ρ+ρG ln 1 , ρ ρG. Then only the com- ρG − ≥ [18] it was shown that using a nonstandard ρ-gauge for bination (dt2 dr2) appears in the 4D-interval ds2 and the applications of GR to the stelar physics one can de- − ¯ 2 ρG scribe extreme objects with arbitrary large mass, density Γr = ρ 1 ρ . − − and size. In particular, one is able to shift the value of 7. One more rho-gauge in GR was introduced by Pu- the Openheimer-Volkoff limiting mass 0.7M of neutron gachev and Gun’ko [9] and, independently, by Menzel ⊙ stars to a new one: 3.8M . This may shed a new light on [16]. In comparison with the previous ones it is simple ⊙ these astrophysical problems and once more shows that and more natural with respect to the quantity Γ¯ . We fix r the choice of rho-gauge (i.e. the choice of the coordinate it using the condition Γ¯ = 2 which is identical with the r r r) can have real physical consequences. rho-gauge fixing for spherical− coordinates in a flat space- Thus we see that by analogy with the classical elec- time. Thus the curved space-time coordinates t,r,θ,φ trodynamics and non-abelian gauge theories of general are fixed in a complete coherent way with the flat space- type, in GR we must use indeed a more refined terminol- time spherical ones. ogy and corresponding mathematical constructions. We Then, absorbing in the coordinates’ units two inessen- already call the choice of the function ρ(r) ”a rho-gauge tial constants, we obtain a novel form of the 4D interval fixing in a broad sense”. Now we see that the different for a point source: functions ρ(r) may describe different physical solutions 2 2 of Einstein equations (1) with the same spherical sym- 2 2ϕ (r) 2 dr r 2 2 2 ds =e N dt dθ +sin θdφ , metry in presence of only one singular point at the center − N(r)4 − N(r)2 of symmetry. As we have seen, the mathematical prop- erties of the singular point may depend on the choice of 1 2ϕ ρG − N where N(r)=(2ϕN ) e 1 ; N(r) 1+ ( r ) – for the rho-gauge in a broad sense. Now the problem is to r − ∼ O r and N(r) – for r +0. In this specific gauge clarify the physical meaning of the singular points with →∞ ∼ ρG → ρ(r)= r/N(r) and ρ(r) r – for r , ρ(r) ρ – for different mathematical characteristics. ∼ → ∞ → G r +0. Some of the static spherically symmetric solutions → The most remarkable property of this flat space-time- with a singularity at the center can be related by reg- coherent gauge is the role of the exact classical New- ular gauge transformations. For example, choosing the ton gravitational potential ϕ (r)= GN M in the above Hilbert gauge we obtain a black hole solution of the vac- N − r exact GR solution. As usual, the component gtt = uum Einstein equations (1). Then we can perform a reg- r 2ϕN ( ) r r e 1+2ϕN ( ) + (ϕN ( )) – for r . But ular gauge transformation to harmonic coordinates, de- ∼ O → ∞ ¯ ¯ ¯ ¯ now gtt has an essentially singular point at r = 0. fined by the relations Γt = 0, Γx1 = 0, Γx2 = 0, Γx3 = 0, The new form of the metric is asymptotically flat preserving the existence of the event horizon, i.e. staying for r . For r +0 it has a limit dθ2 + in the same gauge sector. In the next sections we will sin2θdφ→2. The∞ geometry→ is regular in the whole space- find a geometrical criteria for answering the question: (1,3) time (gµν (x)) because the two nonzero scalars ”When the different representations of static spherically M 4 r 4 1 µνλκ 1 1 2ϕN ( ) symmetric solutions with point singularity describe dif- 48 RµνλκR = 4 (1 g00) = 4 1 e and − 6 − r 6 feomorphic space-times ?” 1 λκ στµν 1 1 2ϕN ( ) 96 RµνλκR στ R = 8 (1 g00) = 8 1 e The definition of the regular gauge transformations al- are finite everywhere, including− − the center− r−= +0. At lows transformations that do not change the number and this center we have a zero 4D volume, because g = the character of singular points of the solution in the real 4ϕ (r) 8 2 | | e N N(r)− r sin θ. At the same time the coordinates domain and the behavior of the solution at the bound- m p x ( , ), m = 1, 2, 3; are globally defined in ary, but can place these points and the boundary at new (1,∈3) −∞ ∞ m N 3 m (gµν (x)). One can check that ∆gx =2 r x . positions. M m 1 m m Hence, ∆ x 2 – for r and ∆ x 2x g r g (r) – for r ∼+0, O i.e., the coherent→ ∞ coordinates∼ xm are∼ Oasymptotically→ harmonic in both limits, but not for finite IV. GENERAL FORM OF THE FIELD values of r ρ . EQUATIONS IN HILBERT GAUGE AND SOME ≈ G At this point one has to analyze two apparent facts: BASIC PROPERTIES OF THEIR SOLUTIONS
i) An event horizon ρH exists in the physical domain only under Hilbert choice of the function ρ(r) r, but According to Lichnerowicz [2, 20, 21], the physical ≡ (1,3) not in the other gauges, discussed above. This demon- space-times gµν (x) of general type must be strates that the existence of black holes strongly depends smooth manifoldsM at{ least} of class C2 and the metric 3 on this choice of the rho-gauge in a broad sense. coefficients gµν (x) have to be at least of class C , i.e. at 5 least three times continuously differentiable functions of the distance to the source goes to zero. This very in- the coordinates x. When one considers gtt(r), grr(r) and teresting new possibility appears in curved space-times ρ(r) not as distributions, but as usual functions of this due to their unusual geometrical properties. It may have class, one is allowed to use the rules of the analysis of a great impact for physics and deserves further careful functions of one variable r. Especially, one can multiply investigation. these functions and their derivatives of the correspond- In contrast, if one accepts the value ρ0 = 0, one has to ing order, one can raise them at various powers, define note that the Hilbert-gauge singularity at ρ = 0 will be functions like log, exp and other mathematical functions space-like, not time-like. This is a quite unusual nonphys- of gtt(r), grr(r), ρ(r) and coresponding derivatives. In ical property for a physical source of any static physical general, these operations are forbidden for distributions field. [8]. The choice of the values of the parameters ρ0 and σ0 In addition, making use of the standard rules for op- was discussed in [22]. There, a new argument in favor erating with Dirac δ-function [8] and accepting (for sim- of the choice ρ0 = ρG and σ0 = 1 was given for differ- plicity) the following assumptions: 1) the function ρ(r) is ent physical problems with spherical symmetry using an a monotonic function of the real variable r; 2) ρ(0) = ρ0 analogy with the Newton theory of gravity. In contrast, and the equation ρ(r) = ρ0 has only one real solution: in the present article we shall analyze the physical mean- r = 0; one obtains the following alternative form of the ing of the different values of the parameter ρ0 0. They Eq.(5): turn out to be related with the gravitational mass≥ defect. The solution of the subsystem formed by the last two 1 d 1 equations of the system (7) is given by the well known gρρ ln ρ 1 = − − √ g d ln ρ g − functions − ρρ − ρρ p 2σ0GN M0 δ(ρ ρ0), − gtt(ρ)=1 ρG/ρ, gρρ(ρ)= 1/gtt(ρ). (8) 2 2 − − d ln gtt 1 d ln gtt 1 d ln gtt d ln gρρ + + 1+ =0, Note that in this subsystem one of the equations is a (d ln ρ)2 2 d ln ρ 2 d ln ρ d ln ρ field equation, but the other one is a constraint. However, d ln gtt w + gρρ +1 =0. (7) these functions do not solve the first of the Eq.(7) for any d ln ρ value of ρ0, if M0 = 0. Indeed, for these functions the left hand side of the6 first field equation equals identically Note that the only remnant of the function ρ(r) in the zero and does not have a δ-function-type of singularity, system Eq.(7) are the numbers ρ0 and σ0 = sign(ρ′(0)), in contrast to the right hand side. Hence, the first field which enter only the first equation, related to the source equation remains unsolved. Thus we see that: 2 of gravity. Here gtt and gρρ = grr/(ρ′) are considered as 1) Outside the singular point, i.e. for ρ>ρ0, the functions of the independent variable ρ [ρ0, ). Birkhoff theorem is strictly valid and we have a stan- ∈ ∞ The form of the Eq. (7) shows why one is tempted to dard form of the solution, when expressed through the consider the Hilbert gauge as a preferable one: In this luminosity distance ρ. form the arbitrary function ρ(r) ”disappears”. 2) The assumption that gtt(r), grr(r) and ρ(r) are One usually ignores the general case of an arbitrary usual C3 smooth functions, instead of distributions, value ρ0 = 0 accepting, without any justification, the yields a contradiction, if M = 0. This way one is not 6 0 6 value ρ0 = 0, which seems to be natural in Hilbert gauge. able to describe correctly the gravitational field of a mas- Indeed, if we consider the luminosity distance as a mea- sive point source of gravity in GR. For this purpose the sure of the real geometrical distance to the point source first derivative with respect to the variable r at least of of gravity in the 3D space, we have to accept the value one of the metric coefficients gµν must have a finite jump, ρ0 = 0 for the position of the point source. Otherwise needed to reproduce the Dirac δ-function in the energy- the δ-function term in the Eq. (7) will describe a shell momentum tensor of the massive point particle. with radius ρ0 = 0, instead of a point source. 3) The widespread form of the Schwarzschild’s solution Actually, according6 to our previous consideration the in Hilbert gauge (8) does not describe a gravitational point source has to be described using the function δ(r). field of a massive point source and corresponds to the The physical source of gravity is placed at the point r =0 case M0 = 0. It does not belong to any physical gauge by definition. There is no reason to change this initial po- sector of gravitational field, created by a massive point sition of the source, or the interpretation of the variables source in GR. Obviously, this solution has a geometrical- in the problem at hand. To what value of the luminosity topological nature and may be used in the attempts to distance ρ0 = ρ(0) corresponds the real position of the reach pure geometrical description of ”matter without point source is not known a priori. This depends strongly matter”. In the standard approach to this solution no on the choice of the rho-gauge. One can not exclude such bare mass distribution was ever introduced. a nonstandard behavior of the physically reasonable rho- The above consideration confirms the conclusion, gauge function ρ(r) which leads to some value ρ0 = 0. which was reached in [19] using isotropic gauge for Physically this means that instead to infinity, the lumi-6 Hilbert solution. In contrast to [19], in the next Sections nosity of the point source will go to a finite value, when we will show how one can include in GR neutral particles 6
with nonzero bare mass M0. The corresponding new so- The variation of the total action with respect to the lutions are related to the Hilbert one via singular gauge auxiliary variableϕ ¯ gives the constraint: transformations. ϕ¯ 2 2 ϕ¯ 2ϕ2 w e (¯ρϕ′ ) +(¯ρϕ′ ) e− e =0. (12) − 1 2 − V. NORMAL COORDINATES FOR One can have some doubts about the correctness of the GRAVITATIONAL FIELD OF A MASSIVE above derivation of the field equations (11), because here POINT PARTICLE IN GENERAL RELATIVITY we use the Weyl’s trick, applying the spherical symme- try directly to the action functional, not to the Einstein equations (1). The correctness of the result of this pro- An obstacle for the description of the gravitational field cedure is proved in the Appendix A. Therefore, if one of a point source at the initial stage of development of prefers, one can consider the Lagrangian densities (4), GR was the absence of an adequate mathematical formal- or the actions (10) as an auxiliary tools for formulating ism. Even after the development of the correct theory of of our one-dimensional problem, defined by the reduced mathematical distributions [8] there still exist an opin- spherically symmetric Einstein equations (1), as a vari- ion that this theory is inapplicable to GR because of the ational problem. The variational approach makes trans- nonlinear character of Einstein equations (1). For exam- parent the role, in the sense of the theory of constrained ple, the author of the article [21] emphasizes that ”the dynamical systems, of the various differential equations, Einstein equations, being non-linear, are defined essen- which govern the problem. tially, only within framework of functions. The function- als, introduced in ... physics and mathematics (Dirac’s δ-function, ”weak” solutions of partial differential equa- VI. REGULAR GAUGES AND GENERAL tions, distributions of Scwartz) are suitable only for linear REGULAR SOLUTIONS OF THE PROBLEM problems, since their product is not, in general, defined.” In the more recent article [23] the authors have consid- The advantage of the above normal fields’ coordinates ered singular lines and surfaces, using mathematical dis- is that when expressed through them the field equations tributions. They have stressed, that ”there is apparently (11) are linear with respect to the derivatives of the un- no viable treatment of point particles as concentrated known functions ϕ (r). This circumstance legitimates sources in GR”. See also [2] and the references therein. 1,2 the correct application of the mathematical theory of dis- Here we propose a novel approach to this problem, based tributions and makes our normal coordinates privileged on a specific choice of the field variables. field variables. Let us represent the metric ds2 (3) of the problem at The choice of the functionϕ ¯(r) fixes the rho-gauge in hand in a specific form: the normal coordinates. We have to choose this func-
2ϕ1 2 2ϕ1+4ϕ2 2ϕ ¯ 2 2 2ϕ1+2ϕ2 2 2 2 tion in a way that makes the first of the equations (11) e dt e− − dr ρ¯ e− (dθ +sin θdφ )(9) − − meaningful. Note that this non-homogeneous equation is quasi-linear and has a correct mathematical meaning if, where ϕ1(r), ϕ2(r)andϕ ¯(r) are unknown functions of the and only if, the condition ϕ (0) ϕ¯(0) < is satisfied. variable r andρ ¯ is a constant – the unit for luminosity | 1 − | ∞ ϕ1+ϕ2 Let’s consider once more the domain r > 0. In distance ρ =ρe ¯ − . By ignoring the surface terms in the corresponding integrals, one obtains the gravitational this domain the first of the equations (11) gives ϕ¯(r) and the mechanical actions in the form: ϕ1(r) = C1 e− dr + C2 with arbitrary constants C1,2. Suppose that the functionϕ ¯(r) has an asymp- R n 1 ϕ¯ 2 2 ϕ¯ 2ϕ2 totics exp( ϕ¯(r)) kr in the limit r +0 (with some GR = dt dr e (¯ρϕ1′ ) +(¯ρϕ2′ ) +e− e , − ∼ → A 2GN − arbitrary constants k and n). Then one easily obtains Z Z n+1 ϕ1(r) ϕ¯(r) C1kr /(n +1)+ n ln r + ln k + C2 – if = dt dr M eϕ1 δ(r). (10) − ∼ AM0 − 0 n = 1, and ϕ1(r) ϕ¯(r) (C1k 1) ln r + ln k + C2 – Z Z for6 n−= 1. Now we− see that∼ one− can satisfy the condi- − Thus we see that the field variables ϕ1(r), ϕ2(r) and tion limr 0 ϕ1(r) ϕ¯(r) = constant < for arbitrary → | − | ∞ ϕ¯(r) play the role of a normal fields’ coordinates in our values of the constants C1,2 if, and only if n = 0. This problem. The field equations read: means that we must haveϕ ¯(r) k = const = for r 0. We call such gauges regular∼ gauges for6 the±∞ prob- → GN M0 ϕ1(r) ϕ¯(r) lem at hand. Then ϕ1(0) = const = . Obviously, the ∆¯ ϕ (r)= e − δ(r), r 1 ρ¯2 simplest choice of a regular gauge6 is±∞ϕ ¯(r) 0. Further on we shall use this basic regular gauge.≡ Other regu- 1 2(ϕ2(r) ϕ¯(r)) ∆¯ ϕ (r)= e − (11) r 2 ρ¯2 lar gauges for the same gauge sector defer from it by a regular rho-gauge transformation which describes a dif- ϕ¯ d ϕ¯ d where ∆¯ r = e− e is related to the radial part feomorphism of the fixed by the basic regular gauge man- dr dr (3) r (3) (3) ij ifold gmn( ) . In terms of the metric components of the 3D-Laplacean, ∆= 1/ g ∂i g g ∂j = M { } 4 4 − | | | | the basic regular gauge condition reads ρ gtt +¯ρ grr =0 gij ∂2 Γ¯ ∂ =∆ +1/ρ2∆ : ∆¯ = g ∆ . and gives Γ¯ = 0. − ij − i j r pθφ r p− rr r r 7
Under this gauge the field equations (11) acquire the gives a unique possibility to derive the admissible real simplest quasi-linear form: domain of the variable r from the dynamical constraint (14), i.e., this dynamical constraint yields a geometrical G M 1 N 0 ϕ1(0) 2ϕ2(r) constraint on the values of the radial variable. ϕ1′′(r)= e δ(r), ϕ2′′(r)= e . (13) ρ¯2 ρ¯2 As a result, in the physical domain the values of the first integrals (16) are related by the standard equation The constraint (12) acquires the simple form: w εtot = ε1 + ε2 =0, (18) 2 2 2ϕ2 w (¯ρϕ1′ ) +(¯ρϕ2′ ) e =0. (14) − − which reflects the fact that our variation problem is in- As can be easily seen, the basic regular gaugeϕ ¯(r) 0 variant under re-parametrization of the independent vari- has the unique property to split the system of field equa-≡ able r. At the end, as a direct consequence of the rela- tions (11) and the constraint (12) into three independent tion (18) one obtains the inequality ε2 = ε1 > 0, be- − relations. cause in the real physical domain r [0, ) we have 1 2 2 ∈ ∞ The new field equations (13) have a general solution ε = ρ¯ ϕ′ (r) = const < 0. 1 − 2 1 For the function ρreg,0(r) 0, which corresponds to GN M0 ϕ1(0) ≥ ϕ (r)= e Θ(r) Θ(0) r+ϕ′ (0) r+ϕ (0), the basic regular gauge, one easily obtains 1 ρ¯2 − 1 1 1 1 r r r r − ρ (r)= ρG 1 exp 4 − ∞ . (19) ϕ2(r)= ln sinh √2ε2 ∞ − . (15) reg ϕ¯=0 − √2ε ρ¯ | − ρG 2 Now one has to impose several additional conditions The first expression in Eq.(15) represents a distribution on the solutions (15): ϕ1(r). In it Θ(r) is the Heaviside step function. Here we i) The requirement to have an asymptotically flat use the additional assumption Θ(0) := 1. It gives a spe- space-time. The limit r r corresponds to the cific regularization of the products, degrees and functions limit ρ . For solutions→ (15)∞ we have the property of the distribution Θ(r) and makes them definite. For ex- → ∞2 2 3 grr(r )/ρ′ (r ) = 1. The only nontrivial asymptotic ample: Θ(r) = Θ(r), Θ(r) = Θ(r),...,Θ(r)δ(r)= ∞ ∞ condition is gtt(r )= 1. It gives ϕ1′ (0) r +ϕ1(0)=0. δ(r), f(rΘ(r)) = f(r)Θ(r) f(0) Θ(r) Θ(0) – for any ii) The requirement∞ to have the correct∞ Keplerian mass − − function f(r) with a convergent Taylor series expansion M, as seen by a distant observer. Excluding the variable 2ϕ1(r) ϕ1(r)+ϕ2(r) around the point r = 0, and so on. This is the only sim- r> 0 from gtt(r)= e and ρ(r)=¯ρe− for ple regularization of distributions we need in the present const solutions (15) one obtains gtt =1+ ρ , where const = article. 2 2 sign(ρ) sign(ϕ2′ (0))ρ ¯ ϕ1′ (0) = 2GN M. The second expression ϕ2(r) in Eq.(15) is a usual iii) The consistency of the previous− conditions with function of the variable r. The symbol r is used 2 ∞ the relation gtt grr + ρ′ = 0 gives sign(r )= sign(ρ)= as an abbreviation for the constant expression r = ∞ ∞ sign(ϕ (0)) = sign(ϕ (0)) = 1. sign (ϕ (0))ρ ¯ sinh √2ε e ϕ2(0) /√2ε . 1′ 2′ 2′ 2 − 2 iv) The most suitable choice of the unitρ ¯ isρ ¯ = The constants GN M = ρG/2. As a result all initial constants become
1 2 2 GN M0 ϕ1(0) functions of the two parameters in the problem – r ε = ρ¯ ϕ′ (r) + ϕ′ (0) e Θ(r) Θ(0) , ∞ 1 1 2 1 r∞ r∞ −2 ρ¯ − and M: ϕ1(0) = , ϕ2(0) = ln sinh , GN M GN M 1 − − 2 2 2ϕ2(r) 1 1 r∞ ε2 = ρ¯ ϕ2′ (r) e (16) ϕ′ (0) = , ϕ′ (0) = coth . 2 − 1 − GN M 2 GN M GN M v) The gravitational defect of the mass of a point par- are the values of the corresponding first integrals (16) of ticle. the differential equations (13) for a given solution (15). Representing the bare mechanical mass M0 of the Then for the regular solutions (15) the condition(14) r∞ point source in the form M0 = 0 M0 δ(r)dr = reads: r∞ 4π g (r) ρ2(r) µ(r)dr, one obtains for the 0 − rr R G M w mass distribution of the point particle the expression ε + ε + N 0 eϕ1(0) Θ(r) Θ(0) =0. (17) R p 1 2 ρ¯2 − µ(r)= M δ(r)/ 4π g (r) ρ2(r) = M δ (r), where 0 − rr 0 g 2 An unexpected property of this relation is that it can- δg(r) := δ(r)/ 4π p grr(r)ρ (r) is the 1D invariant not be satisfied for any value of the variable r ( , ), − ∈ −∞ ∞ Dirac delta function. p The Keplerian gravitational mass because ε1,2 are constants. The constraint (17) can be M can be calculated using the Tolman formula [1]: satisfied either on the interval r [0, ), or on the in- ∈ ∞ r∞ terval r ( , 0). If, from physical reasons we chose it 2 M =4π ρ′(r)ρ (r)µ(r)dr = M0 gtt(0). (20) to be valid∈ −∞ at only one point r [0, ), this relation ∗ Z0 will be satisfied on the whole interval∈ r ∞[0, ) and this p ∈ ∞ Here we use the relation ρ′ = √ gtt grr. As a result we interval will be the physically admissible real domain of − 2ϕ1(0) r∞ the radial variable. Thus one can see that our approach reach the relations: gtt(0) = e = exp 2 = − GN M 8
2 M M0 r [0, r ]. It does not seem to be very convenient to 1 and r = GN M ln 0. (Note that M0 ≤ ∞ M ≥ work∈ with∞ such unusual radial variable r. One can eas- due to our convention Θ(0) := 1 the component gtt(r) ily overcome this problem using the regular rho-gauge is a continuous function in the interval r [0, ) and ∈ ∞ transformation gtt(0) = gtt(+0) is a well defined quantity.) M r/r˜ The ratio ̺ = = gtt(0) [0, 1] describes the M0 r r (25) gravitational mass defect of the point∈ particle as a second → ∞ r/r˜ +1 physical parameter in thep problem. The Keplerian mass M and the ratio ̺ define completely the solutions (15). with an arbitrary scaler ˜ of the new radial variable r For the initial constants of the problem one obtains: (Note that in the present article we are using the same notation r for different radial variables.) This linear frac- 1 ̺2 tional diffeomorfism does not change the number and the ϕ (0) = ln ̺, ϕ (0) = ln − , 1 2 2̺ character of the singular points of the solutions in the − (1) whole compactified complex plane ˜r of the variable r. 1 1 1+ ̺2 The transformation (25) simply placesC the point r = r ϕ1′ (0) = , ϕ2′ (0) = 2 . (21) ∞ GN M GN M 1 ̺ at the infinity r = , at the same time preserving the − initial place of the origin∞ r = 0. Now the new variable Thus we arrive at the following form of the solutions r varies in the standard interval r [0, ), the regular (15): solutions (22) acquire the final form∈ ∞ r/r˜ r Θ(r) ϕ1(r)= ln(1/̺) 1 Θ(r/r˜) , ϕ1(r)= ln(1/̺), − − r/r˜ +1 GN M − 1 1 1 1 r/r˜+1 r/r˜+1 r/GNM r/GNM ϕ2(r)= ln (1/̺) ̺ , ϕ2(r)= ln 1/̺e− ̺e (22) − 2 − − 2 − ϕ¯(r) = 2 ln(r/r˜ + 1) + ln(˜r/r ). (26) and the rho-gauge fixing function ∞ The final form of the rho-gauge fixing function reads: 1 2 − ρreg ϕ¯=0 (r)= ρG 1 ̺ exp 4r/ρG . (23) 2 1 | − ρ (r)= ρ 1 ̺ r/r˜+1 − . (27) reg G − An unexpected feature of this two parametric variety of solutions for the gravitational field of a point particle is The last expression shows that the mathematically ad- that each solution must be considered only in the domain missible interval of the values of the ratio ̺ is the open r [0, GN M ln (1/̺)], if we wish to have a monotonic interval (0, 1). This is so, because for ̺ = 0 and for increase∈ of the luminosity distance in the interval [ρ , ). ̺ = 1 we would have impermissible trivial gauge-fixing 0 ∞ It is easy to check that away from the source, i.e., for functions ρreg(r) 1 and ρreg(r) 0, respectively. ≡ ≡ r > 0, these solutions coincide with the solution (8) in Now we are ready to describe the singular character the Hilbert gauge. Hence, outside the source the solu- of the coordinate transition from the Hilbert form of tions (22) acquire the well known standard form, when Schwarzschild solution (8) to the regular one (26). To represented using the variable ρ. This means that the simplify notations, let us introduce dimensionless vari- solutions (22) strictly respect a generalized Birkhoff the- ables z = ρ/ρg and ζ = r/r˜. Then the Eq. (27) shows orem. Its proper generalization requires only a justifi- that the essential part of the change of the coordinates cation of the physical domain of the variable ρ. It is is described, in both directions, by the functions: remarkable that for the solutions (22) the minimal value 1 2 − of the luminosity distance is z(ζ)= 1 ̺ ζ+1 , and − ρ =2G M/(1 ̺2) ρ . (24) ln(1/̺2) 0 N − ≥ G ζ(z)= 1, ̺ (0, 1). (28) ln z ln(z 1) − ∈ This changes the Gauss theorem and leads to important − − physical consequences. One of them is that one must ap- Obviously, the function z(ζ) is regular at the place of ply the Birkhoff theorem only in the interval ρ [ρ0, ). the point source ζ = 0; it has a simple pole at ζ = and ∈ ∞ an essential singular point at ζ = 1. At the same∞ time the inverse function ζ(z) has a logarithmic− branch points VII. REGULAR GAUGE MAPPING OF THE at the Hilbert-gauge center of symmetry z = 0 andat the INTERVAL r ∈ [0,r∞] ONTO THE WHOLE “event horizon” z = 1. Thus we see how one produces INTERVAL r ∈ [0, ∞] the Hilbert-gauge singularities at ρ = 0 and at ρ = ρH , starting from a regular solution. The derivative As we have stressed in the previous section, the so- ln(1/̺2) lutions (22) for the gravitational field of a point par- dζ/dz = ticle must be considered only in the physical domain z(z 1) (ln z ln(z 1))2 − − − 9 aproaches infinity at these two points, hence the singular constraint (18) and the first of the relations (21), one im- character of the change in the whole complex domain mediately obtains for the total energy of the GR universe of the variables. Thus we reach a complete description with one point particle in it: of the change of coordinates and its singularities in the complex domain of the radial variable. ∞ Etot = totdr = M = ̺M0 M0 . (32) The restriction of the change of the radial variables on H ≤ Z0 the corresponding physical interval outside the source: ζ (0, ) z (1/(1 ̺2), ), is a regular one. This result completely agrees with the strong equiv- ∈The∞ expressions↔ ∈ (26)− and∞ (27) still depend on the alence principle of GR. The energy of the gravitational choice of the units for the new variable r. We have field, created by a point particle is the negative quantity: E = E E = M M = M (1 ̺) < 0. to fix the arbitrary scale of this variable in the form GR tot − 0 − 0 − 0 − 2 M0 The above consideration gives a clear physical explana- r˜ = ρG/ ln(1/̺ )= GN M/ ln to ensure the validity M tion of the gravitational mass defect for a point particle. of the standard asymptotic expansion: gtt 1 ρG/r + 2 ∼ − (ρG/r) when r . O Then the final form→ of ∞ the 4D interval, defined by the IX. INVARIANT CHARACTERISTICS OF THE new regular solutions outside the source is: SOLUTIONS WITH A POINT SOURCE 2 2 2ϕ 2 dr 2 2 2 2 ds =e G dt ρ(r) dθ +sin θdφ . (29) − N (r)4 − A. Local Singularities of Point Sources G Here we are using a modified (Newton-like) gravitational Using the invariant 1D Dirac function one can write potential: down the first of the Eq. (11) in a form of an exact relativistic Poisson equation: GN M ϕ (r; M,M0) := (30) G M0 ∆ ϕ (r)=4πG M δ (r). (33) −r + GN M/ ln( M ) r 1 N 0 g
1 2ϕG This equation is a specific realization of Fock’s idea (see a coefficient NG (r)=(2ϕG )− e 1 , and an optical luminosity distance − Fock in [1]) using our normal fields coordinates (Sec. V). It can be re-written, too, in a transparent 3D form: r+G M/ ln( M0 ) 2ϕ N M (3) r ρ(r)=2GN M/ 1 e G = . (31) ∆ϕ1(r)=4πGN M0 δg ( ). (34) − NG (r) The use of the invariant Dirac δ -function has the These basic formulas describe in a more usual way our g advantage that under diffeomorphisms of the space regular solutions of Einstein equations for r (0, ). (3) g (r) the singularities of the right hand side of They show immediately that in the limit ̺ ∈0 our∞ so- mn theM relativistic{− } Poisson equation (33) remain unchanged. lutions tend to the Pugachev-Gun’ko-Menzel→ one, and Hence, we have the same singularity at the place of the ϕ (0; M,M ) = ln ̺ . In the limit: ̺ 1 we G 0 source for the whole class of physically equivalent rho- obtain for any value of→ the −∞ ratio r/ρ : g (r/ρ ,̺→) 1, g tt G gauges. Then one can distinguish the physically different g (r/ρ ,̺) 1, and ρ(r/ρ ,̺) . Because of→ the rr G G solutions of Einstein equations (1) with spherical symme- last result the→− 4D geometry does not→ ∞have a meaningful try by investigating the asymptotics in the limit r +0 limit when ̺ 1. In this case ϕ (r; M,M0) 0 at → → G → of the coefficient γ(r)=1/ 4πρ(r)2 g (r) in front all 3D space points. Physically this means that solutions − rr without mass defect are not admissible in GR. of the usual 1D Dirac δ(r)-function inp the representation δg(r)= γ(r)δ(r) of the invariant one. For regular solutions (22), (26) the limit r +0 of VIII. TOTAL ENERGY OF A POINT SOURCE this coefficient is a constant → AND ITS GRAVITATIONAL FIELD 1 1 γ(0) = ̺ , (35) 4πρ2 ̺ − In the problem at hand we have an extreme example of G an ”island universe“. In it a privileged reference system which describes the intensity of the invariant δ-function and a well defined global time exist. It is well known and leads to the formula δ (r)= γ(0)δ(r). We see that: that under these conditions the energy of the gravita- g 1) The condition ̺ (0, 1) ensures the correct sign of tional field can be defined unambiguously [1]. Moreover, the intensity γ(0), i.e.,∈ the property γ(0) (0, ), and we can calculate the total energy of the aggregate of a thus a negativity of the total energy E ∈of the∞ gravi- mechanical particle and its gravitational field. GR tational field of a point particle, according the previous Indeed, the canonical procedure produces a total Section III. Hamilton density = Σ πµ ϕ = Htot a=1,2;µ=t,r a a,µ − Ltot 2) For ̺=0 we have γ(0)= , and for ̺=1: γ(0)=0. 1 2 2 2 2 2ϕ2 ϕ1 ∞ ρ¯ ϕ1′ +ρ ¯ ϕ2′ e + M0e δ(r). Using the Hence, we have non-physical values of the intensity γ(0) 2GN − − 10
the physical in these two cases. The conclusion is that ̺W∞ = 1 at the point r = . Hence, it turns out that interval of values of the ratio ̺ is the open interval (0, 1). this solution is the first exact∞ analytical two-particle-like This is consistent with the mathematical analysis of this solution of Einstein equations (1). Unfortunately, as a problem, given in the previous Section. two-point solution, the Weyl solution is physically trivial: It is easy to obtain the asymptotics of the coefficient it does not describe the dynamics of two point particles γ(r) in the limit r 0 for other solutions, considered in at a finite distance, but rather gives only a static state of the Introduction: → two particles at an infinite distance. b) To make these statements correct, one has to com- 1/2 1 ρG (3) r Schwarzschild solution: γS (r) 4πρ2 r ; pactify the comformally flat 3D space gmn( ) ∼ G M {− } 5/2 joining it to the infinite point r = . Then both the i ρG ∞ Hilbert solution: γH (r) 4πρ2 r ; and ∼ G Weyl solution its source will be invariant with re- 1 spect to the inversion r ρ2 /r. Droste solution: γD (r) 4πρ2 ; G ∼ G → 4 5) The isotropic (t-r) solution resembles a regular one, 16 r Weyl solution: γW (r) πρ2 ρ ; but in it x 1.27846 is the only real root of the equation ∼ G G ≈ 1/2 1 ρG x + ln(x 1) = 0 and gives a non-physical value of the Einstein-Rosen solution: γER (r) 4πρ2 r ; − ∼ G parameter ̺I 2.14269, which does not belong to the 1/2 ≈ 1 x physical interval (0, 1). Isotropic (t-r) solution: γER (r) 4πρ2 1 x ; ∼ G − 6) The singularity of the coefficient γ(r) in front of Pugachev-Gun’ko-Menzel 2 the usual 1D Dirac δ-function in the Pugachev-Gun’ko- 1 r ρG/2r Menzel solution is stronger than any polynomial ones in solution: γP GM (r) 4πρ2 ρ e . ∼ G G the other listed solutions. As we can see: 1) Most of the listed solutions are physically differ- ent. Only two of them: Schwarzschild and Einstein- B. Local Geometrical Singularities of the Regular Rosen ones, have the same singularity at the place of the Static Spherically Symmetric Space-Times point source and, as a result, have diffeomorphic spaces (3) gmn(r) , which can be related by a regular rho- It is well known that the scalar invariants of the Rie- Mgauge{ transformation.} mann tensor allow a manifestly coordinate independent 2) As a result of the alteration of the physical meaning description of the geometry of space-time manifold. The problem of a single point particle can be considered as of the variable ρ = r inside the sphere of radius ρH , in Hilbert gauge the coefficient γ(r) tends to imaginary an extreme case of a perfect fluid, which consists of only infinity for r 0. This is in a sharp contrast to the real one particle. According to the article [24], the maximal asymptotic of→ all other solutions in the limit r 0. number of independent real invariants for such a fluid is 3) The Droste solution is a regular one→ and cor- 9. These are the scalar curvature R and the standard invariants r1, w1, w2,m3,m5. (See [24] for their defini- responds to a gravitational mass defect with ̺D = √5 1 /2 0.61803. The golden-ratio-conjugate tions.) The invariants R, r1 and m3 are real numbers − ≈ and the invariants w ,w and m are, in general, com- number √5 1 /2, called sometimes also ”a silver ra- 1 2 5 plex numbers. The form of these invariants for spheri- tio”, appears− in our problem as a root of the equation cally symmetric space-times in normal field variables is 1/̺ ̺ = 1, which belongs to the interval ̺ (0, 1). (The − ∈ presented in Appendix C. Using these formulas we ob- other root of the same equation is √5+1 /2 < 0 and tain for the regular solutions (22) in the basic gauge the does not correspond to a physical− solution.) Hence, the following simple invariants: Droste solution can be transformed to the form (22), or (26) by a proper regular rho-gauge transformation. 4) The Weyl solution resembles a regular solution with 1 (1 ̺2)4 r π(1 ̺2)3 r 1 I1 = 2 −2 δ = − δg = R(r), ̺W = 1 and γW (0) = 0. Actually the exact invariant 8ρ ̺ ρ 2̺ ρ −2 G G G δ-function for this solution may be written in the form I =0, (see the Appendix B): 2 4 2 Θ(r/ρG) 1 2 4r/ρG ρG Θ(r/ρG) 1 I3 = − 1 ̺ e = − , 4 4 2 2 2 2 4 16 r ρG ρG 8ρG̺ − 8̺ ρ(r) δg,W (r)= 2 δ(r)+ δ . (36) 3 πρG ρG r r Θ(r/ρG) 2 4r/ρG ρG Θ(r/ρG) I4 = 2 1 ̺ e = 3 . (37) 4ρG − 4 ρ(r) This distribution acts as a zero-functional on the stan- dard class of smooth test functions with compact sup- As can be seen easily: port, which are finite at the points r =0, . 1) The invariants I1,...,4 of the Riemann tensor are well The formula (36) shows that: ∞ defined distributions. This is in accordance with the gen- a) The Weyl solution describes a problem with a spher- eral expectations, described in the articles [25], where ical symmetry in the presence of two point sources: one one can find a correct mathematical treatment of distri- 0 with ̺W = 1 at the point r = 0 and another one with butional valued curvature tensors in GR. 11
(1,3) As we saw, the manifold gµν (x) for our regu- we have considered together with the Hilbert one, do not lar solutions has a geometricalM singularity{ } at the point, have an event horizon at all. where the physical massive point particle is placed. Indeed, for the point ρH at which gtt(ρH ) = 0 one
Note that the metric for the regular solutions is glob- obtains ρ(ρH ) = ρG. The last equations do not have (1) ally continuous, but its first derivative with respect to any solution ρH for the regular solutions (22), the radial variable r has a finite jump at the point source. (26). The absence∈ of C a horizon in the physical domain This jump is needed for a correct mathematical descrip- r [0, ) is obvious in the representation (29) of the tion of the delta-function distribution of matter of the regular∈ ∞ solutions. massive point source in the right hand side of Einstein For the other classical solutions one obtains in the equations (1). limit r ρ as follows: → H During the calculation of the expressions I3,4 we have ρG used once more our assumption Θ(0) = 1. Schwarzschild solution: grr(ρH ) r , ρH = 0; ∼−ρG Three of the invariants (37) are independent on the Hilbert solution: grr(ρH ) , ρH =ρG; ∼− r ρG real axes r ( , ) and this is the true number of the Droste solution: g (ρ ) −1, ρ = 0; ∈ −∞ ∞ rr H ∼− H independent invariants in the problem of a single point Weyl solution: grr(ρH ) 1, ρH = ρG; ∼− ρG source of gravity. On the real physical interval r [0, ) Einstein-Rosen solution: grr(ρH ) r , ρH = 0 ; ∈ ∞ ∼− r one has I3 = 0 and we remain with only two independent ρG Isotropic (t-r) solution: grr(ρH ) e , ρH = ; invariants. For r (0, ) the only independent invariant Pugachev-Gun’ko-Menzel ∼ −∞ ∈ ∞ r is I4, as is well known from the case of Hilbert solution. ρG 4 − ρG solution: grr(r) r e , ρH = 0. 2) The scalar invariants I1,...,4 have the same form as ∼− in Eq. (37) for the regular solutions in the representation As one can see, only for Hilbert, Schwarzschild and (26), obtained via the diffeomorphic mapping (25). Einstein-Rosen solutions the metric component grr(r) 3) All other geometrical invariants r1(r), w1,2(r), has a simple pole at the point ρH . In the last two m3,5(r) include degrees of Dirac δ-function and are not cases this is a point in the corresponding manifold well defined distributions. Therefore the choice of the (3) gmn(r) , where the center of symmetry is placed, simple invariants I1,...,4 is essential and allows us to use Mnot a{ horizon.} solutions of Einstein equations, which are distributions. To the best of authors knowledge, the requirement to make possible the correct use of the theory of distribu- X. ON THE GEOMETRY OF MASSIVE POINT tions is a novel criteria for the choice of invariants of the SOURCE IN GR curvature tensor and seems to not have been used until now. It is curious to know if it is possible to apply this It is curious that for a metric, given by a regular solu- new criteria in more general cases as opposed to the case tion, the 3D-volume of a ball with a small radius rb <ρG, of static spherically symmetric space-times. centered at the source, is 4) The geometry of the space-time depends essentially on both parameters M and ̺, which define the regular 4 3 12̺ rb rb V ol(rb)= πρG 2 4 + 2( ). solutions (26) of Einstein equations (1). Hence, for differ- 3 (1 ̺ ) ρG O ρG ent values of the two parameters these solutions describe − It goes to zero linearly with respect to rb 0, in con- non-diffeomorphic space-times with different geometry. → 3 trast to the Euclidean case, where V olE(rb) rb . This happens, because for the regular solutions (22),∼ (26) we obtain 3g = ρ(r)2 sin θ ρ sin θ = 0 in the limit C. Event Horizon for Static Spherically Symmetric | | → 0 6 Space-Times with Point Singularity r 0. Nevertheless, limrb 0 V ol(rb) = 0 and this legit- imates→ thep use of the term→ ”a point source of gravity” in the problem at hand: the source can be surrounded by a According to the well known theorems by Hawking, sphere with an arbitrary small volume V ol(r ) in it and Penrose, Israel and many other investigators, the only b with an arbitrary small radius r . solution with a regular event horizon not only in GR, b In contrast, when rb 0 the area of the ball’s surface but in the theories with scalar field(s) and in more gen- →2 4πρG 2 eral theories of gravity, is the Hilbert one [12]. As we has a finite limit: (1 ̺2)2 > 4πρG, and the radii of the − 2πρG have seen, this solution has a pure geometrical nature big circles on this surface tend to a finite number 1 ̺2 > − and does not describe a gravitational field of a point par- 2πρG. ticle with bare mechanical mass M0 = 0. The strong Such unusual geometry, created by the massive point mathematical results like the well known6 no hear theo- sources in GR, may have an interesting physical con- rems, etc, for the case of metrics with an event horizon sequences. For example, space-times, defined by the can be shown to be based on to the assumption that such regular solutions (22), (26), have an unique property: horizon is indeed present in the solution. These mathe- When one approaches the point source, its luminosity matical results do not contradict to the ones, obtained remains finite. This leads to a very important modifica- in the present article, because the other solutions, which tion of Gauss theorem in the corresponding 3D spaces. 12
After all, this modification may solve the well known which depends on only one parameter – the Keplerian problem of the classical divergences in field theory, be- mass M. As shown in Section IX A, this is not the case. cause, as we see, GR offers a natural cut-off parameter As a consequence of Birkhoff theorem, when expressed 2 r˜ = ρG/ ln(1/̺ ) for fields, created by massive point par- in the same variables, the solutions with the same mass ticles. If this will be confirmed by more detailed cal- M indeed coincide in their common regular domain of culations, the price, one must pay for the possibility to coexistence – outside the singularities. In contrast, they overcome the old classical divergences problem will be to may have a different behavior at the corresponding sin- accept the idea, that the point objects can have a nonzero gular points. surface, having at the same time a zero volume and a zero A correct description of a massive point source of grav- size r. ity is impossible, making use of most of these classical On the other hand, as we have seen, the inclusion of solutions. point particles in GR is impossible without such unusual Using novel normal coordinates for the gravitational geometry. The Hilbert solution does not offer any more field of a single point particle with bare mechanical mass attractive alternative for description of a source of grav- M0 we are able to describe correctly the massive point ity as a physical massive point, placed at the coordinate source of gravity in GR. The singular gauge transfor- point ρ = 0, and with usual properties of a geometrical mations yield the possibility to overcome the restriction point. Hence, in GR we have no possibility to introduce to have a zero bare mass M0 for neutral point particles a notion of a physical point particle with nonzero bare in GR [19]. It turns out that this problem has a two- mass without a proper change of the standard definition parametric family of regular solutions. for point particle as a mathematical object, which has One of the parameters – the bare mass M0 of the point simultaneously zero size, zero surface and zero volume. source, can be obtained in a form of surface integral, We are forced to remove at least one of these three usual integrating both sides of Eq. (34) on the whole 3D space requirements as a consequence of the concentration of the (3) g (r) : M {− mn } bare mass M0 at only one 3D space point. From physical point of view the new definition for 1 3 3 M0 = d r g ∆ (ln √gtt)= 3 physical point particle, at which we arrived in this ar- 4πGN | | ticle, is obviously preferable. ZM p 1 2 ij tt Moreover, it seems natural to have an essential differ- d σi g g ∂j g . 3 4πGN ∂ | | ence between the geometry of the “empty” space-time I M p p points and that of the matter points in GR. The same The second parameter – the Keplerian mass M, is de- geometry for such essentially different objects is possi- scribed, as shown in Section VIII, by the total energy: ble only in the classical physics, where the space-time geometry does not depend on the matter. Only under 3r 3 3r (3) r M = d g tot = d g M0 δg ( ). the last assumption there is no geometrical difference be- 3 | | H 3 | | tween pure mathematical space points and matter points. ZM p ZM p The intriguing new situation, described in this article, These two parameters define the gauge class of the deserves further careful analysis. given regular solution. Obviously the parameters M The appearance of the above nonstandard geometry in and M0 are invariant under regular static gauge trans- the point mass problem in GR was pointed at first by formations, i.e., under diffeomorphisms of the 3D space (3) Marcel Brillouin [27]. Our consideration of the regular gmn(r) . It is an analytical manifold with a solutions (22), (26) confirms his point of view on the strongM {− singularity} at the place of the massive point parti- character of the singularity of the gravitational field of cle. For every regular solution both parameters are finite massive point particles in GR. and positive and satisfy the additional physical require- ment 0
2(ϕ1 2ϕ2) APPENDIX C: INDEPENDENT NONZERO e − ϕ1′′ + E3 = 4 (m3/w1 r1) /3, INVARIANTS OF RIEMANN CURVATURE IN − 2(ϕ1 2ϕ2) p NORMAL FIELD COORDINATES e − 2ϕ′′ E E = 2 (4r m /w ) /3 (C4) 1 − 2 − 3 1 − 3 1 p Using the the representation (9) of the metric in nor- are linear with respect to the derivative ϕ1′′. Using proper mal field variables after some algebraic manipulations one linear combinations of these invariants and the scalar 16
2(ϕ1 2ϕ2) 2 2 2ϕ2 2 curvature R one can easily check that the quantities I3 := e − ϕ1′ + ϕ2′ e /ρ¯ /2, 2(ϕ1 2ϕ2) 2(ϕ1 2ϕ2) − − E2 e − , and E3 e − are independent invari- 2(ϕ1 2ϕ2) I := e − ϕ′ ϕ′ ϕ′ /2. ants too. 4 1 1 − 2 Thus, our final result is that, in general, for the metric (9) in the basic regular gaugeϕ ¯ 0 the Riemann curva- ≡ ture tensor has the following four independent invariants: These invariants are linear with respect to the second
2(ϕ1 2ϕ2) derivatives of the functions ϕ1,2 – a property, which is of I1 := e − ϕ1′′, critical importance when we have to work with distribu- 2(ϕ1 2ϕ2) 2ϕ2 2 I := e − ϕ′′ e /ρ¯ , (C5) tions ϕ′′ . 2 2 − 1,2
[1] L. D. Landau, E. M. Lifshitz, The Classical Theory of Grav. 1, 337 (1971); Ll. Bel., J. Llosa, Gen. Rel. and Fields, 2d ed.; Reading, Mass: Addison-Wesley, 1962; Grav. 27, 1089 (1995); Ll. Bel, Gen. Rel. and Grav. 28, V. A. Fock, The Theory of Space, Time and Gravita- 1139 (1996); tion, Pergamon, Oxford, 1964. R. C. Tolman, Reativity, [18] J. M. Aguirregabiria, Ll. Bel, J. Martin, A. Molina, Thermodynamics and Cosmology, Claderon Press, Ox- E. Ruiz gr-qc/0104019, Gen. Rel. and Grav. 33 1809 ford, 1969. S. Weinberg,Gravitation and Cosmology, Wi- (2001); J. M. Aguirregabiria, Ll. Bel, gr-qc/0105043, ley, N.Y., 1972; C. Misner, K. S. Thorne, J. A. Wheeler, Gen. Rel. and Grav. 33, 2049 (2001) ; Ll. Bel, Gravity, W. H. Freemand&Co.,1973. gr-qc/0210057. [2] D. Kramer, H. Stephani, M. Maccallum, E. Herlt, [19] E. Arnowitt, S. Deser, C. M. Misner, Phys. Rev. Lett. Ed. E. Schmutzer, Exact Solutions of the Einstein Equa- D4, 375 (1960); Phys. Rev. 120 321 (1960); in Gravita- tions, Deutscher Verlag der Wissenschaften, Berlin, 1980. tion: an introduction to current research, Ed. L. Witten, H. Stephani, D. Kramer, M. Maccallum, E. Herlt, Exact John Wiley, N.Y., 1962. Solutions of Einstein’s Field Equations, Sec. Ed., Cam- [20] A. Lichnerowicz, Th´eories Relativistes de la Gravitation bridge University Press, 2003. et de l’Electromagn´etisme, Paris, Masson, 1955. [3] M. G¨ockeler, T. Sch¨ucker, Differential Geometry, Gauge [21] Y. Bruhat, Cauchy Problem, in Gravitation: an intro- Theories, and Gravity, Cambridge University Press, duction to current research, ed. L. Witten, John Wiley, Cambridge, 1987. N.Y. , 1962. [4] V. Rubakov, Classical Theory of Gauge Fields, Princeton [22] L. S. Abrams, Phys. Rev. D20, 2474 (1979); University Press, Princeton and Oxford, 1999. Can. J. Phys., 67, 919 (1989); Physica A, 131 (1996); [5] K. Schwarzschild, Sitzungsber. Preus. Acad. Wiss. Phys. Int. J. Theor. Phys. 35, 2661 (1996). Math. Kl., 189 (1916). [23] R. Geroch, J. Traschen, Phys. Rev. D36 1017 (1987). [6] A. S. Eddington, The mathematical theory of relativity, [24] J. Carminati, R. G. McLenaghan, J. Math. Phys. 32, 2nd ed. Cambridge, University Press, 1930 (repr.1963). 3135 (1991). [7] R. Gautreau, B. Hoffmann, Phys. Rev. D17, 2552 (1978). [25] A. H. Taub, J. Math. Phys. 21 1423 (1979). C. K. Raju, [8] L. Schwartz, Th´eorie des distributions I, II, Paris, 1950- J. Phys. A: Math. Gen. 15 1785 (1982). 51; I. M. Gel’fand, G. E. Shilov, Generalized Finc- [26] P. A. M.Dirac, Proc. Roy Soc. (London), A270, 354 tions, N.Y., Academic Press, 1964; H. Bremermann, Dis- (1962); Conference in Warszawa and Jablonna, L. Infeld trinutions, Complex Variables and Fourier Transform, ed., Gauthier-Villars, Paris (1964), pp. 163-175. Addison-Wesley Publ. Co. Reading, Massachusetts, 1965. [27] M. Brillouin, Le Journal de physique et Le Radium 23, [9] J. T. Combridge, Phil. Mag. 45, 726 (1923); H, Janne, 43 (1923). Bull. Acad. R. Belg. 9, 484 (1923). F.J.Belinfante, [28] P. O. Mazur, E. Motolla, Gravitational Condensate Stars: Phys. Rev. 98, 793 (1955). Ya. I. Pugachev and V. D. An Alternative to Black Holes, gr-qc/0109035; M. Visser, Gun’ko, Soviet Phys. Journ., Izvestiya Vysshikh Ucheb- D. L. Wiltshire, Stable Gravistars - an Alternative to nykh Zavednii, Fizika, 10, 46, (1974). Black Holes, gr-qc/0310107. [10] D. Hilbert, Nachr. Ges. Wiss. G¨otingen, Math. Phys. Kl., [29] H. Poincare, Science et M´etode, Book 1, Flammarion, 53 (1917). Paris, 1934. [11] S. Antoci, D.-E. Liebscher, Astron. Nachr. 322, 137 [30] M. A. Abramowicz, W. Klu´zniak, J-P. Lasota, Astron. (2001); S. Antoci, D.-E. Liebscher, L. Mihich, Class. Astrophys., 396, L31 (2002); astro-ph/0207270. Quant. Grav. 18, 3466 (2001). [31] E. W. Mielke, F. E. Schunck, Nucl. Phys. B564, p.185 [12] V. P. Frolov, I. D. Novikov, Black Hole Physics, Kluwer (2000); gr-qc/0001061. R. Narayan, Evedences for the Acad. Publ., 1998. Black Hole Event Horison, astro-ph/0310692. [13] J.Droste, Proc.K.Ned.Acad.Wet., Ser., A19, 197 (1917). [32] S. G. Turyshev, J. G. Williams, K. Nordtvedt, Jr., [14] H. Weyl, Ann. Phys., Leipzig, 54, 117 (1917). M. Shao, T. W. Murphy, Jr., 35 Years of Test- [15] A. Einstein, N. Rosen, Phys. Rev., 48, 73 (1935). ing Relativistic Gravity: Where do we go from here?, [16] D. H. Menzel, M´em. Soc. Roy. Sci. Li´ege, 9, 343 (1976). gr-qc/0311039. [17] Ll. Bel, J. Math. Phys. 10, 1501 (1969); Gen. Rel. and