FFrom Schwinger Balls to Curved Space
Davood Allahbakhshi ∗ School of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM), P.O.Box 19395-5531, Tehran, Iran
It is shown that the Reissner-Nordstrom black hole is also a gravitational Schwinger ball. It is alsoshown that both massless and massive-particle gravitational Schwinger balls are thermodynamicsystems by deriving the first law of thermodynamics for them. Inconsistency between classicalgeometrical and microscopic definitions of the horizon is discussed. We propose a new metric, moreconsistent with microscopic picture of black hole, as gravitational Schwinger ball, by speculations.It has some interesting features.
I. INTRODUCTION
Hawking radiation [1] showed a conceptual conflict be-tween the classical geometrical definition of black holesand local quantum field theories. Thermal evaporationof the black hole with no memory of the initial state ofthe system raised the problem of information paradox[2]. Bekenstein entropy [3] with no explanation in classi-cal picture is another problem. Many attempts, to solvethese problems with no real success, have raised seriousdoubts about the correctness and physical acceptabilityof the classical definition of black holes itself [4]. It seemsthat this classical picture is not completely physical andthere should be a new definition of black holes with morephysical characteristics.Recent graviton ball picture of black holes [5] has shedlight on problems of these mysterious objects. Par-ticularly the
Schwinger ball picture can produce andexplain many different characteristics of Schwarzschildblack hole as well as problems like information paradox[6]. Bekenstein entropy, black hole thermodynamics, therelation between Schwarzchild radius and the mass of theblack hole can be derived from this picture. Also it isshown that the Hawking radiation is not consistent withSchwarzschild state except for eternal black hole [7].If black holes are realy Schwinger balls, we shouldbe able to explain other types of black holes thanSchwarzschild black hole in this picture. The first onein the list is Reissner-Nordstrom black hole. In presentpaper we show that the Reissner-Nordstrom black holeis also a gravitational Schwinger ball. The only point isincluding the energy of the electromagnetic field appro-priately. At the same time this explanation reveals an in-consistency between the microscopic and classical macro-scopic definitions of the black hole. While in general rela-tivity the black hole is the space-time behind the horizonas a coordinate singularity, in the microscopic picture itis a region of significant Schwinger effect. Favouring themicroscopic Schwinger ball picture leads us to a new defi-nition of the classical geometry which carries facts aboutthe quantum theory of gravity. ∗ [email protected] II. THE FORM OF A BLACK HOLE METRIC
The relation between the size and the mass of theSchwarzschild black hole is R d − = 16 πGMc ( d − d − (1)and in previous work [6], we found that the number ofgravitons inside the Schwinger ball (black hole) is N = d − π AL d − P , (2)where A is the area of the ball, as is also disscusedin [5], which states that the relation between the en-tropy and the number of gravitons of a Graviton ballis S = 2 πN/ ( d − R d − ± = Gµ (cid:32) ± (cid:115) − q µ (cid:33) , (3)where µ = 8 πMc ( d − d − q = 8 πQ c ε ( d − d − d − . (4)How can we derive it from Schwinger effect, like whatwe did for Schwarzschild black hole [6]? Although theanswer to this question is so simple, it leads us to a newworld of physical concepts.The Schwarzschild radius above is the root of the met-ric function of the black hole f ( R ) = 1 − GµR d − + Gq R d − . (5)The question is this: How can the root of the metricfuction be related to Schwinger effect?The answer to the question above is interesting! Let usstart by Schwarzschild black hole. The metric functionis f ( R ) = 1 − GµR d − . (6) a r X i v : . [ phy s i c s . g e n - ph ] A ug We know that the root of this function is where theSchwinger effect becomes significat. In fact we canrewrite the function above in this form f ( R ) = 1 − a ( R ) a g ( R ) , (7)where a ( R ) = Gµc R d − (8)is the Newtonian acceleration (or surface gravity in thiscase) at radius R and a g ( R ) = c R (9)is the acceleration, needed to create a graviton whichcan live in a sphere of radius R , by Schwinger effect .It means that the metric function is a criterion whichdetermines that how much we are away from Schwingerlimit. Also for Reissner-Nordstrom black hole we can writethe metric function in the form of 7, but this time wehave a ( R ) = Gµ ∗ ( R ) c R d − , (10)where µ ∗ ( R ) = 8 πM ∗ ( R ) c ( d − d − M ∗ ( R ) = M − Q c ( d − ε Ω d − R d − . (11) M ∗ ( R ) is the mass/energy inside the sphere of radius R , since from elementary physics we know that just theenergy inside this region contributes to the gravitationalforce but not the mass/energy outside it. So we need tosubtract the electromagnetic energy, outside the sphere,from the total mass/energy. To see this we note that M ∗ can be written in this form M ∗ = M b + E ( R, c = M − E ( ∞ , R ) c , (12)where E ( x, y ) = − ε Ω d − (cid:90) yx Q r d − dr (13)is the energy of the spherically symmetric electromag-netic field between radii x and y . M b is the bare mass ofthe black hole which absorbs the infinite electromagnetic Remember that the acceleration needed to create a quanta ofmass m by gravitational Schwinger effect is mc / (cid:126) [6]. energy in E ( R,
0) to produce the physical mass M . From12 we simply conclude that M = M b + E ( ∞ , c , (14)as expected.Note that the acceleration 10 is the Newtonian accel-eration at radius R , not the surface gravity ! III. THE FIRST LAW OF THERMODYNAMICS
The size of a gravitational Schwinger ball which cre-ates particles of mass m can be derived by equiating the Newtonian acceleration to Schwinger limit a c = mc / (cid:126) mc (cid:126) = Gµ ∗ ( R ) c R d − , (15)or in terms of M ∗ it is mc (cid:126) = 8 πGM ∗ ( R )( d − d − R d − . (16)But A = Ω d − R d − is the area of the Schwinger ball, sowe have ( d − c πG (cid:126) mA = M ∗ ( R ) . (17)Varying this relation gives the first law of black hole ther-modynamics. In next two parts we find this law for agraviton Schwinger ball and a massive-particle gravita-tional Schwinger ball. A. Graviton Schwinger Ball
The variation of the equation 17 for a graviton of mass m = (cid:126) / (2 cR ) is( d − c πG (cid:126) ( dm A + m dA ) = dM ∗ ( R ) . (18)After some simple algebra and using the relations dm = − (cid:126) cR dRdA = ( d − d − R d − dR. (19)It is easy to see that dE = c κ π dA G + V dQ, (20)where E = M c is the energy of the Schwinger ball, and κ = ( d − c (cid:20) R − q R d − (cid:21) (21)is nothing but the surface gravity at the horizon and V ( R ) = Q ( d − ε Ω d − R d − (22)is the electric potential at the surface of the Schwingerball. B. Massive-Particle Schwinger Ball
For massive-particle Schwinger ball, which createsmassive particles instead of gravitons , the calculation isexactly like we did in previous part but this time since themass of the massive particle is fixed, dm = 0 in equation18 and we have dE = c κ π dA G + V dQ, (23)with κ = c λ + ( d − c (cid:20) λ − q R d − (cid:21) , (24)where λ is the Compton wavelength of the particle ofmass m , and R can be derived from equation 17. Somassive-particle Schwinger balls also obey the first lawof thermodynamics but with different temperature.So we see that the Reissner-Nordstrom black hole isalso a gravitational Schwinger ball , but this time weshould include the energy of the electromagnetic field.The very important question here is that why the metricfunction is of the form of 7? And why does it have a rootat the Schwinger limit? In the next section we discussthese questions. IV. ABOUT CURVED SPACE
As mentioned, the metric function of Reissner-Nordstrom and so Schwarzschild metrics can be writtenin the form f ( r ) = 1 − a ( r ) a g ( r ) . (25)It means that the criterion for defining the effective timeor length intervals is the Schwinger limit. Consider proper time-like and space-like intervals in a black holegeometry dT = (cid:113) − a/a g dtdX = dx (cid:112) − a/a g . (26)These expressions are very similar to time-dilation andlength-contraction in special relativity , but this time con-traction and dilation coefficients are made of (cid:112) − a/a g In fact such Schwinger balls also produce gravitons and theSchwinger radius of gravitons (massless particles) is generallymuch larger than the Schwinger radius of massive particles . Sothe massive-particle Schwinger balls are hidden inside the gravi-ton Schwinger balls. To become more familiar with massive-particle Schwinger balls see [6]. instead of (cid:112) − v /c . It means that the accelerationis responsible for contraction and dilation of the spa-tial and temporal intervals of (effective and emergentcurved) space-time and the reference acceleration is the Schwinger limit !On the other hand, the metric function is zero at thehorizon and makes a coordinate singularity, from theviewpoint of the static observer. In microscopic picture,the horizon is where the Schwinger effect becomes sig-nificant but, in geometrical picture it is where the spaceends! On the other hand the Schwinger limit is a smoothlimit at which the probability for a particle to be pro-duced, is larger than e − after that, not a definite pointat which something special happens sharply!From the Schwinger effect viewpoint, the coordinatesingularity at the horizon, that the metric is time-dependent beyond it, can be considered as the tool thatclassical effective theory, general relativity, uses to pro-duce particles. This sigularity is similar to Hagedorntemperature in statistical bootstrap model of hadrons.In Hagedorn’s model, the Hagedorn temperature is thelargest achievable temperature by the hadron gas. Butsimultaneusly it shows signs of a phase transition. Laterwe found that at Hagedorn temperature the hadron gasexperiences a phase transition to deconfined matter. Butthe Hagedorn’s model can not explain beyond this tem-perature because is not capable to do it. Also we foundthat at low baryonic chemical potentials this transitionis in fact a smooth transition, a crossover. In general rel-ativity, as a classical effective theory, the horizon is thedeepest achievable place in the space, by static observeroutside the black hole but, from microscopic point of viewit is where a smooth transition occures from where theSchwinger effect is not significant to where it is. In otherwords the microscopic theory of gravitons does not needa horizon to produce gravitons through Schwinger effect,since in this picture the Schwinger effect is nothing butdecays of gravitons to gravitons (or other particles).The discussion above shows that, from microscopicpoint of view, where we have included the quantum ef-fects of gravitons , the horizon is not a special point but asmooth region and we need a smooth metric there, evenfrom the viewpoint of the static observer. Such smoothnon-singular metric should be derived emergently from aquantum theory of gravity, but here we propose a metricjust by speculations. Since the weight of the Schwingereffect is e − a g /a , one may propose a metric of the form ds = − f ( r ) dt + dr f ( r ) + r d Ω d − , (27)with the metric function f ( r ) = e − a ( r ) /a g ( r ) . (28)When we are far from the Schwinger limit (horizon) andso a (cid:28) a g , the function can be expanded f ( r ) ≈ − a ( r ) a g ( r ) , (29) FIG. 1: The metric function of semi black brane and ourproposed fuction. which is nothing but the function 7. Note that the func-tion 28 can produce the same results of 7, in any practicalcase that we have tested the general relativity. Here wehave considered g rr = 1 /g tt , but more generally g rr canbe e χ /g tt in which χ is another function. This metricobviously is not a solution of the pure Einstein-Hilbertaction but another unknown one which should be found.We do not have any more evidence for correctness ofthe metric above but, there is just an interesting similar-ity between this metric and one we found in [8]. For thisto be more clear we note that for an AdS black brane,similar to the metric above, we can again propose a met-ric of the form ds = 1 z (cid:18) − f ( z ) dt + dr f ( z ) + d(cid:126)x d − (cid:19) , (30)with f ( z ) = e − z d /z dh . (31)In figure 1 we have plotted the function above and whatwe found for semi black brane in [8].For justifying our proposed metric we can claim thatincluding the effect of quantum fluctuations of gravitonsabout a classical geometry in a mean field approximation may lead to an effective action similar to what we foundin [8]. Such effective action generally can include theSchwinger effect and may have a semi black hole solution,like what we proposed above. This claim of course needsinvestigations. V. SUMMARY AND DISCUSSION
We saw that the Reissner-Nordstrom metric can alsobe described as a gravitational Schwinger ball, but weneed to include the energy of the electromagnetic field appropriately. One may ask why does the Reissner-Nordstrom metric just include gravitational Schwingereffect but not electric Schwinger effect ? The answeris obvious. This metric is the solution of the Einstein-Maxwell action which does not include the quantum ef-fects of QED. If we use an electromagnetic effective actionlike Heisenberg-Euler action, in addition to the effectiveinteraction terms between gravity and electromagnetism,then its solution should include both gravitational andelectric Schwinger effects.Inspired by Schwinger weight, we also proposed a met-ric, consistent with microscopic picture of black holes asSchwinger balls. This metric has interesting features.First of all it produces usual gravitational effects. Sec-ondly it is similar to semi black brane that we found inour previous work. It also has another interesting fea-ture; If we calculate the surface gravity of this metric wefind κ = GMr d − e − GM/r d − , (32)which can be written in the form κ = GMr d − e − m ( r ) r , (33)with m ( r ) = 2 GMr d − . (34) m ( r ) can be considered as the dynamical running mass of the graviton, produced by the background mean fieldat radius r . In fact it is the energy of the graviton whichcan be produced through Schwinger effect at radius r by Newtonian acceleration . The surface gravity 32 is aYukawa like force produced by a massive particle. Butthere is a very important difference. The effect of themass in ordinary Yukawa theories is important just atIR but not UV, while the gravitational acceleration 32 isNewtonian at IR limit but, completely different at UV.In fact the proposed dynamical mass of the graviton 34 is infinite at UV, which makes the theory
Renormalizable .On the other hand the acceleration 32 is zero at r → d >
3. It means that the hypothetical quan-tum theory of gravity, which produces such geometry inclassical effective limit, is asymptotically free .If our speculations are nearly correct then we canhope that the simple quantized general relativity is non-renormalizable because its UV behaviour is not correct,since it is probably a truncated effective action .More investigations of course give more interesting re-sults. [1] S. W. Hawking, “Particle Creation by Black Holes,” Com-mun. Math. Phys. , 199 (1975) Erratum: [Commun. Math. Phys. , 206 (1976)]. doi:10.1007/BF02345020[2] S. W. Hawking, “Breakdown of Predictability in Grav- itational Collapse,” Phys. Rev. D , 2460 (1976).doi:10.1103/PhysRevD.14.2460[3] J. D. Bekenstein, “Black holes and entropy,” Phys. Rev.D , 2333 (1973). doi:10.1103/PhysRevD.7.2333[4] S. W. Hawking, “Information Preservation and WeatherForecasting for Black Holes,” arXiv:1401.5761 [hep-th].[5] G. Dvali, “Non-Thermal Corrections to Hawking Radia-tion Versus the Information Paradox,” Fortsch. Phys. ,106 (2016) doi:10.1002/prop.201500096 [arXiv:1509.04645[hep-th]].G. Dvali and C. Gomez, “Black Hole’s Quan-tum N-Portrait”, Fortsch.Phys. 61 (2013) 742-767,arXiv:1112.3359 [hep-th].“Black Hole’s 1/N Hair,” Phys. Lett. B719