1 Black holes do not exist Jean-Pierre Petit1
Key words : black hole, space bridge, giant black hole, neutron star, Kerr metric ____________________________________________________________________________________________________ Abstract We reconsider classical features of Schwarzschild and Kerr metrics, which are the fundamental basis of the black hole model, through new space and time coordinates which transform the object into a space bridge linking two folds of the universe, PT symmetrical. In addition, such change gives a fast mass transfer through such spheroidal throat, one way membrane, which makes the black hole model questionable. Back to TOV model we assume that the physical criticity at r = 0,942 Rs, occuring before geometrical criticity, corresponds to real physical phenomenon. We associate this to the fast establishment of a small space bridge between the two folds, through which matter in excess would be immediatly transfered, transformed into negative mass, and consquently spread away. So that such space bridge would correspond to an instant picture of a fast transient phenomenon, that would keep neutron stars’mass above 3 solar masses. The discussion is extended to so-‐called « giant black holes ». ____________________________________________________________________________________________________ Introduction Consider the metric : (1) dr 2 d Σ2 = + r 2 dϕ 2 Rs 1− r Its signature is ( + , + ). It tends to euclidian metric when r tends to infinite. This signature is modified if r < Rs and becomes ( -‐ , + ). Consider lenght measured along a radial path ( ϕ = 0 ). If r > Rs this last is real. If r < Rs it becomes imaginary. Let us show that we are out of the surface. Introduce the change of coordinates : (2) r = Rs ( 1+ Log ch ρ ) The metric becomes : (3) ⎡ ( 1 + Log ch ρ ) 2 ⎤ d Σ 2 = Rs2 ⎢ th ρ d ρ 2 + ( 1 + Log ch ρ ) 2 dϕ 2 ⎥ Log ch ρ ⎣ ⎦
1 Former Research Director at the french CNRS. Private mail :
[email protected]
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r = Rs corresponds to ρ = 0 . At the point the determinant of the metric is no longer zero : (4) 2 4 ( 1 + Log ch ρ ) det g = Rs th 2 ρ Log ch ρ The metric is well defined for all values of ρ . The determinant does not tend to zero when ρ → ± 0 . If we imbed the surface in a 3d eucliean space we can define the meridian, corresponding to : (5) dr 2 2 dΣ = + d z 2 Rs 1− r we immediatly get the meridian : (4) r z2 2 z = ± 2 Rs −1 r = Rs + Rs 4 Rs The surface is a space bridge, linking two 2D euclidean surfaces.
Fig.1 : The surface, imbedded in 3
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Everything becomes real. From Lagrange equations we may compute the geodesics in [ ρ , ϕ ] coordinate system. If imbedded in 3 , the surface holds a throat circle whose perimeter is 2 π Rs . We can extend this in 3D, introducing the metric : (5) dr 2 d Σ2 = + r 2 ( d θ 2 + sin 2 θ dϕ 2 ) Rs 1− r which is enclidean at infinite. Introducing the same coordinate change (2) we can imbed this 3D hypersurface into a 4D euclidean space. Then it becomes a space bridge linking two 3D eucliean hypersurfaces through a throat sphere whose area is 4 π Rs2 . If we think about this surface throught [ r , θ , ϕ ] coordinates , when r > Rs the length is real, but when r < Rs this length becomes imaginary. It just means that we are out of the hypersurface. The question « what is inside a sphere whose radius is Rs » has no physical meaning. The radial coordinate r just corresponds to a wrong choice, which does not fit the real nature of the hypersurface. The topology associated to the Schwarschild solution Consider now the Schwarzschild metric [1] : (6) R dr 2 d s 2 = ( 1 − s )(d x°)2 − − r 2 ( d θ 2 + sin 2 θ dϕ 2 ) Rs r 1− r It is similar to the metric (5). Introduce the change of coordinate (2), we get : (7) Log ch ρ d s2 = (d x°)2 1 + Log ch ρ
⎡ ( 1 + Log ch ρ ) 2 − Rs2 ⎢ th ρ d ρ 2 + ( 1 + Log ch ρ ⎣
⎤ Log ch ρ ) 2 ( dθ 2 + sin 2 dϕ 2 ) ⎥ ⎦
The metric is still Lorentian at infinite. It is well defined for all values of the variable ρ . The structure is similar and evokes some sort of space bridge linking two 4D Lorentian spaces. Classically, the concept of black holes appears when we write : (8) x° = c t c being a constant. Then if we compute the free fall times we find distinct values if we express it in term of proper time s of in term of time t , associated to « an external observer ». We refind all these features with the Kerr metric [2].
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(9) 2mρ ρ 2 + a 2 cos 2 θ 2 ds = (1 − 2 )(dx°) − 2 d ρ 2 − ( ρ 2 + a 2 cos 2 θ ) dθ 2 2 2 2 ρ + a cos θ ρ + a − 2mρ 2
⎡ 2 m a 2 sin 4 θ ⎤ 2 4mρa − ⎢ ( ρ 2 + a 2 )sin 2 θ + 2 dϕ − 2 d x°dϕ ⎥ 2 2 ρ + a cos θ ⎦ ρ + a 2 cos 2 θ ⎣
The presence of a crossed term d x°dϕ introduces an azimutal frame dragging. As a consequence, when computing the speed of light along a circle whose axis is the one of the system one finds two different values if the test photon circles with this rotating object or against this movement. Back to the Schwarzschild metric, introduce the Eddington time-‐marker change [4] : (10) r x° = x − Rs Log − 1 Rs and write : (11) x = c t with c constant. In [ x , r , θ , ϕ ] the line element becomes : (12) R R R d s 2 = ( 1 − s ) c 2 dt 2 − ( 1 + s ) dt 2 − 2 s c d t d r − r 2 ( d θ 2 + sin 2 θ dϕ 2 ) r r r When r = Rs the determinant of the metric no longer vanishes. When r tends to infinite this metrics tends to Lorentzian. If we compute the geodesics, due to the crossed term dx dt we find a radial frame-dragging effect. The speed of light along radial paths has distinct values, depending if the movement is directed inwards or outwards. For an extermal observer the free fall time becomes finite, while it takes an infinite time to escape from what still behaves like a one-‐way membrane. Each metric solution goes with an isometry group. Classically SO3 x R is the isometry group associated to Schwarzschild metric. It corresponds to isotropy and time-‐ independance conditions. Similar conditions for axisymmetric Kerr solution. Now we can extend this solution to the two folds cover of a manifold M4, with two (+ ) (− ) metrics that we call gµν and gµν , corresponding, after A.Sakharov [3], to two « twin universes », enantiomorphic, with opposite arrows of time ( PT symmetric ). Then we can link the two folds through a throat sphere, using the coordinate changes : (13) ⎡ ch ρ ⎤ r = Rs ( 1+ Log ch ρ ) x° = ct + δ Log ⎢ Log with δ = ± 1 Rs ⎥⎦ ⎣ where δ = -‐1 corresponds to the fold ( or « sector » ) F (+ ) (ours) and δ = +1 to the « twin fold » (or « sector ») F (− ) . Then the metrics become :
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(14) ds 2 =
Log ch ρ 2 2 2 + Log ch ρ 2 c dt − th ρ d ρ 2 1 + Log ch ρ 1 + Log ch ρ
+
2 δ c dt d ρ − Rs2 ( 1 + Log ch ρ )2 ( dθ 2 + sin 2 θ dϕ 2 ) with δ = ± 1 1 + Log ch ρ
We can express the geodesics in different coordinate sets. It is easier to keep the variable r, and write : (15) r x° = c t + δ Rs Log ( −1) with δ = ± 1 and r > Rs Rs Then the coupled Schwarzschild metrics are : (16) R R R d s 2 = (1 − s ) c 2 dt 2 − (1 + s ) r 2 + 2 δ s c dt dr − r 2 ( d θ 2 + sin 2 d ϕ 2 ) r r r For r > Rs we refind the classical pattern of quasi-‐Keplerian paths with associated parameters : (17) dr dϕ = ± 2 λ − 1 Rs 1 Rs 2 r + 2 − 2+ 3 h2 hr r r Questionable black hole Let’s focus on radial paths. The value ν = + 1 corresponds to centrifugal paths, while ν = -‐ 1 corresponds to centripetal paths. δ = + 1 corresponds to « our fold of our universe » F (+ ) , and δ = -‐ 1 to the « twin fold » F (− ) : (18) R λ r − δ ν Rs λ 2 − 1 + s r dr dt = R c ν ( r − Rs ) λ 2 − 1 + s r The Schwarzschild sphere behaves like a one-‐way membrane. Il our fold a test particule following a centrifugal path paths, needs a infinite time to escape. Conversely, the centripetal free fall time is finite (and short). The situation is reversed in the twin fold. So that such hypertoric space bridge may swallow matter from our fold, and send it to the twin fold (or « sector ») in a finite and short time. There is no more time freezing for an external observer, so that the classical interpretation of the Schwarzschild metric as a « black hole » becomes questionable. By the way, while a little bit more complicated,
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similar results can be obtained from Kerr metric. This asymmetry is easy to see, considering the situation close to the one-‐way membrane ( r Rs ) : (19) λ ν r − δ ν Rs dt dr c ( r − Rs ) This justifies the forms of the metrics (14), to ensure the continuity of movements. We have considered a space bridge linking two PT-‐symmetrical hypersurfaces. This corresponds to joint bimetric geometries, derived from the action : (20) J = ∫ ( R(+ ) + R(− ) )d 4 x δ J = 0 D4
satisfied for : (21) R(+ ) = − R(− )
(+ ) (− ) Rµν = Rµν = 0
A positive mass test-‐particule, if located in , F (+ ) is attracted by the space bridge, and repelled if located in F (− ) . Following Jean-‐Marie Souriau [5], time-‐inversion goes with energy and mass inversion. The fate of a neutron star that overcomes its limit of stability. If we couple the external Schwarzschild metric solution (6) to the following internal Scharzschild metric solution : (22)
we get a geometrical representation of a sphere, whose radius is rn , filled by constant density ρ material , and surrounded by void. The stability condition is : (23) 8π G ρ 3 3 Rs = rn < rn < = Rˆ 2 3c 8π G ρ corresponding to figure 2.
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Fig.2 : Subcritical neutron star, schematical In constant density growth the two radius becomes equal to the radius of the star, so that we have a double geometrical criticity. See figure 3.
Fig.3 : Double geometrical criticity. The famous TOV equation [6] gives the pressure versus radius in such object. (24) ⎛ ⎛ p⎞ ⎛ 4π G p r3⎞ p ⎞ ⎛ Rs 4 π G p r 3 ⎞ ρ + m + ρ + + ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ c 2 ⎟⎠ ⎜⎝ c4 c 2 ⎟⎠ ⎜⎝ 2 c4 dp =− =− dr r ( r − 2m ) r ( r − Rs ) When integrating this differential equation, we get figure 4.
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Fig.4 : Physical criticity in a neutron star For rn = 0.9428 Rs a physical criticity appears, before geometrical criticity. Classically one considers the two values as just quite close. If we assume this corresponds to a physical phenomenon, it means that something happens in the center of the star when this physical critical condition is reached. In former papers we developped the concept of joints variations for the so-called constants of physics ( [7], [8], [9] ). Then such very important increase of pressure would go with the increase of c to infinite and create a bridge between the two sectors, the one for positive energy and mass particles, the other for negative energy and mass particles. The challenging decoding of Schwarzschild metric would then correspond to some instant picture of a fast process, to be built from the set of field equations introduced in [10]. A control mechanism for neutron stars’masses. The universe contains a very large number of objects, at distance, of even in our galaxy. When some theoretician predicts the existence of some object, if he is right, this last will be discovered late or soon. As an example Fritz Zwicky predicted the phenomenon of supernova in 1931. He observed the first in 1937, then he found one more four months later. Three years after he had evidenced 12. Now we have thousands. Same thing for neutron stars, exo-planets. The existence of black hole was conjectured in 1963, when Kerr built his axisymmetrical metric solution. Half century have passed, and we have very few questionable stellar black holes candidates. In fact, they are more black hols in poplar journals that observed in the sky, in spite of remarkable progress of observational techniques. A control mechanism for neutron
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stars could justify such absence. According to this model, when a neutron star receives matter from a companion star, and get a pressure jump at its center, a small space bridge opens there. Matter flows in at large density and relativistic density. The sign of such mass is inverted, so that it escapes the star. It does not any longer interfer with this dense material for matters with opposite signs only interact through gravitational force. Subsidiarily it becomes invisible to our eyes and telescopes. Then the mass of neutron stars should be limited to 3 solar masses, instead 3.3 . The proper time of transfered matter is not distorted and the particle, although its « apparent mass » is inverted, still cruises from pas to future for it does not turn around on its geodesic path. What about « giant black holes » ? Measurements of velocities of fast stars, orbiting close to the center of galaxies indicate that objects whose masses would range between 107 and 109 solar masses should be located there [13]. The scientific community decided to call it « giant black holes ». But are they black holes ? How do they form ? I have suggested a scenario in [11]. From my point of view, the universe is filled by positive and negative masses. In 1957 H.Bondi [12] showed that this cohabitation was impossible, due to the preposterous runaway phenomenon. If the universe is considered as a M4 manifold with a single metric, solution of Einstein’s equation, and if two masses with opposite signs encounter the positive one escapes while the negative one runs after it. This corresponds to the alleged « runaway phenomenon ». Shifting to a bimetric representation ([8], [9], [10] ) we find different the gravitational interaction laws: masses with same signs mutually attract, through Newton’s law. Masses with opposite signs mutually repel through anti-Newton’s law. This model fully challenges the dark matter model. Joint gravitational instability explains VLS, galaxies’ confinement, flatness of their rotation curves and large gravitationel lens effects, mainly imputed to negative lensing [8]. On cosmological scales it explains the observed acceleration of the universe, without need to mysterious « dark energy ». Coupled field equations system may produce joint metric oscillations during the expansion process. It would modify the strenght of the confinement of galaxies, due to their environment of invisible negative mass. If much weakened, the confinement becomes no longer ensured and we get irregular galaxies. If strongly reinforced it creates an annular density wave which focus towards the center of the galaxy. By the way, the density wave triggers young stars’ birth. They ionize the gas and create high magnetic Reynolds number conditions, so that the ring collects the magnetic lignes as a peasant gathers corn ears. This reinforces the magnetic field. When the ring reaches the center, it forms a ball of hot gas in which Lawson conditions occurs in the bulk. Fusion debris are ejected along two diametrally opposite lobes. We get a QSO. The magnetic field gradient acts as a natural particle accelerator and the object is the source of high energy particles, cosmic rays. Soft process would cause gas loss, if the confinement is weakly reduced. Soft reinforcement would send gas to the center of the galaxy, at moderate velocity. This gas would. The mass control mechanism evoked above would limit its growth and would place it in sub-critical configuration R ≤ 0.942 Rs . If some matter would pass by, when captured, its electrons, accelerated, would emit radiation, X-rays . But the wavelength of the radiation emitted by the object would be greatly enlaged by gravitational redshift effect. On another hand, such supermassive objects may have been formed very early by gravitational instability, at the very begining of the life of galaxies. According to the classical theory that would give giant black holes. Considering this new point of view it would give sub-critical objects, due to the evoked mass control process.
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Conclusion Starting from the model of black hole, based on Schwarzschild and Kerr metrics we introduce two successive changes of coordinates. The first reveals the true underlying topology of this four dimensional hypersurface, eliminating singularity, showing that this last comes from a wrong choice of space coordinate and is not an intrinsic attribute. The horizon, is still a one-‐way membrane which becomes the mid part of a space bridge linking the two CPT symmetrical Minkowskian sectors of a bimetric structure. When trespassing, matter gets negative mass. A second change of time-‐marker, inspired by the one introduced by Eddington keeps the metric Lorentzian at infinite. To the azimutal Kerr’s frame-‐dragging effect it adds radial frame-‐dragging. The free fall time of a test particle towards the Schwarzschild throat sphere become finite, which makes the black hole model questionable. . The TOV model predicts that when the radius of a neutron star reaches 0.942 Rs the pressure at the centre becomes infinite. We assume it corresponds to a physical criticity, which would appears before geometrical criticity and prevent the birth of a singularity. We conjecture that this pressure rise goes with a rise of the local speed of light to infinity and cause a geometrical surgery which modifies, during a short time the geodesic design with local mass inversion process. This mechanism would limit the neutron star’s mass to 3 solar masses. We suggest that similar mechanism could take place at the center of galaxies and draining off all matter in excess, transforming it into negative mass and preventing the birth of a singularity. . References [1] Schwarzschild K. : Über das Gravitational eines Massenpunktes nach der Einsteineschen Theory, Sitzber. Preuss. Akad. Wiss. Berlin, 1916, p.189-196 [2] Kerr, R. P. Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics. Phys. Rev. Let. 11, 237-‐238, 1963. [3] A.Sakharov : "CP violation and baryonic asymmetry of the Universe". ZhETF Pis'ma 5 : 32-35 ( 1967) ; Traduction JETP Lett. 5 : 24-27 (1967) [4] Eddington A.S. : A comparizon of Withehead's and Esinstein's formulæ Nature 113 : 192 (1924) [5] J.M.Souriau : Structure of dynamical systems, Birkhauser Ed, 1998 and Editions Dunod (french) 1974. [6] Oppenheimer J.R. and H.Snyder (1939) : On continued Gravitational Contraction, Phys. Rev. 56 : 455 [7] J.P.Petit : An interpretation of cosmological model with variable light velocity. Modern Physics Letters A, Vol. 3, n°16, nov 1988, p.1527 [8] J.P.Petit : Twin Universe cosmology. Astronomy and Space Science 1995, 226 pp. 273-‐307 [9] J.P. Petit, P.Midy & F.Landsheat : Twin matter against dark matter. Intern. Meet. on Atrophys. and Cosm. "Where is the matter ? ", Marseille 2001 june 25-‐29
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[10] J.P.Petit : The missing mas problem. Il Nuevo Cimento B, Vol. 109 july 1994 pp 697-‐ 710 [11] J.P.Petit : On a perdu la moitié de l’Univers. Ed. Albin Michel, 1997 [12] H.Bondi : Negative mass in General Relativity, Rev. of Mod. Phys. 29 n°3, july 1957, pp. 423-428. [13] Schödel R et al. And : A star in a 15.2 orbit around a supermassive black hole at the centre of the Milky Way, Nature, 419, 17 oct. 694-696 (2002)
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Annex : Didactic model It can be useful to give some images of the process.
Fig.5 : The two surfaces figures the gravitational potential, « in mirror ».
Fig. 6 : Stellar wind brings matter
Fig.7 : Matter in excess is evacuated. Its mass is inverted and it spreads away.
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Fig.8 : Stand by in sub-critical conditions
