““INFORMATION PARADOX”AND SCHWARZSCHILDIAN GEODESICS
ANGELO LOINGER AND TIZIANA MARSICO
Abstract.
We show that the “Information Paradox” follows from inap-propriate considerations on the geodesics of a Schwarzschildian manifoldcreated by a gravitating point-mass. In particular, we demonstrate thatthe geometric differential equation which gives the radial coordinate asa function of the angular coordinate of the geodesics does not representfully all the consequences following from the metric tensor. We remarkthat: i ) it does not yield the conditions characterizing the circular orbits;(this fact has been ignored in the previous literature); ii ) it “neglects”the space region in which the radial coordinate is minor or equal to twicethe mass of the gravitating point (in suitable units of measure). Summary – . Introduction and r´esum´e of the main theses. – . On Kundt’sphysical explanations of the observational data about the believed BHs. – , , . A precise treatment `a la Hilbert of the geodesics of a Schwarzaschild’s mani-fold. – . Physical meaning of the t -parametrization of the mentioned geodesics. – . Independently of any specific instance, the formal structure of GR excludes theexistence of any “Information Paradox”. – . Legenda regarding De Jans’ diagramsof geodesics at the end of the present paper. –
Appendix A : Computative verifica-tions of the inadequacy of geometric eq. (7) in the treatment of the circular orbits.–
Appendix B : On binaries composed of two mass-points according approximatecalculations of Numerical Relativity founded on a (3 + 1)-decomposition of Einsteinfield equations. –Diagrams of Schwarzschildian geodesics. –PACS 04.20 – General relativity. – A glance over the recent literature about the Einsteinian gravitatingpoint-mass shows that the subject is still interesting [1]. The diffuse inter-pretation of Schwarzschild’s solution which has given origin to the notionof black hole (BH) is unfounded, see [2]. A unpleasant consequence of thisnotion is the belief in the so-called “Information Paradox”, according towhich GR would be in contradiction with the time reversibility. Indeed, ithas been affirmed that in the instance of Schwarzschild’s manifold createdby a gravitating mass-point the test-particles and the light-rays go beyondthe space surface R ( r ) = 2 m – where R ( r ) is the radial coordinate [3] and m is the mass of the gravitating point ( c = G = 1) – and disappear from the“external” world for ever, with an irreversible process. We shall show in this a r X i v : . [ phy s i c s . g e n - ph ] A p r ANGELO LOINGER AND TIZIANA MARSICO
Note that if one takes into account all the assumptions which characterizethe deduction of the geometric differential equation of the geodesic trajec-tories [ R ( r ) = a function of ϕ, (0 ≤ ϕ ≤ π )] of test-particles and light-rays,one obtains a confirmation of the dynamical results by Droste [4] and DeJans [5]: the geodesics that arrive on the surface R ( r ) = 2 m find here theirend: the “Information Paradox” does not exist .Of course, this conclusion is implicitly contained in [2], but we think usefulto give an explicit and detailed proof of the erroneousness of a widespreadbelief. – Astrophysics is an observational and experimental science. All phe-nomena that the current “ δ ´ oξα ” ascribes to BHs can be actually explainedin quite physical ways – see, e.g. , Kundt [6]. According to this Author: i )the believed stellar-mass BHs are neutron stars inside accretion disks; ii )the central engine of an Active Galactic Nucleus (AGN) is a nuclear-burningdisk. – Schwarzschild’s manifold of a gravitating point-mass m is characterizedby the following d s :(1)d s = R ( r ) R ( r ) − α [d R ( r )] +[ R ( r )] (d ϑ +sin ϑ d ϕ ) − R ( r ) − αR ( r ) d t ; ( c = G = 1) , where α ≡ m , and R ( r ) is a regular function of r such that the d s becomes Minkowskian if r → ∞ . (In the standard solution R ( r ) ≡ r , in theoriginal Schwarzschild’s solution R ( r ) = ( r + α ) / , in Brillouin’s solution R ( r ) ≡ r + α , etc. ). The geodesics of test-particles and light-rays are planetrajectories and obey the following equations – see [7], eqs. (41) ÷ (44):(2) RR − α (cid:18) d R d p (cid:19) + R (cid:18) d ϕ d p (cid:19) − R − αR (cid:18) d t d p (cid:19) = A ( ≤
0) ;(3) R (cid:18) d ϕ d p (cid:19) = B ;(4) R − αR d t d p = C ;(5)dd p (cid:18) RR − α d R d p (cid:19) + α ( R − α ) (cid:18) d R d p (cid:19) − R (cid:18) d ϕ d p (cid:19) + αR (cid:18) d t d p (cid:19) = 0 .A, B, C are integration constants (which respect to the affine parameter p ); A is zero for the light-rays, negative for the test particles; we can put INFORMATION PARADOX” AND SCHWARZSCHILDIAN GEODESICS 3 C = 1, by virtue of the arbitrariness of p . The Lagrangean eq. (5) for R isconnected with eqs. (2), (3), (4); indeed, we have the identity:(6) d[2]d p − ϕ d p d[3]d p + 2 d t d p d[4]d p = d[ R ]d p [5] , where the brackets denote the left sides of eqs. (2), (3), (4), (5). Theelimination of d p and d t from (2), (3), (4) gives the geometric differentialequation of the geodesics:(7) (cid:18) d (cid:37) d ϕ (cid:19) = 1 + AB − A αB (cid:37) − (cid:37) + α(cid:37) , where (cid:37) := 1 /R . Since for the circular orbits d R/ d p = 0, in this caseidentity (6) is not a consequence of (2), (3), (4). Consequently, as it is easyto verify, eq. (7) does not give the correct restrictions on the above orbits (see Appendix A). – Of course, eq. (2) implies R > α , but eq. (7) “neglects” this condition .The substitutions ( R − α ) (cid:29) − t give for R ≤ α a non -static metric forwhich the temporal and the radial coordinates interchange their roles; inparticular, eq. (7) becomes:(7 (cid:48) ) (cid:18) d[1 / ( α − t )]d ϕ (cid:19) = 1 + AB − A αB α − t ) − α − t ) + α ( α − t ) . This is not , however, a significant result, because the geodesic parametriza-tion with p – or with the proper time s – and the parametrization with ϕ of eq. (7) – would give a geodesic surpassing of R = α with the originalcoordinates R and t .For the circular orbits eq. (5) gives:(5 (cid:48) ) − R (cid:18) d ϕ d p (cid:19) + αR (cid:18) d t d p (cid:19) = 0 ;from which the circular velocity v :(5 (cid:48)(cid:48) ) v = (cid:18) R d ϕ d p (cid:19) = α R .
For the test-particle geodesics, we have from eq. (2) – with
A < (cid:48) ) that(8)
R > α , (8 (cid:48) ) v < √ . ANGELO LOINGER AND TIZIANA MARSICO
And for the light-rays ( A = 0):(9) R = 32 α ;(9 (cid:48) ) v = 1 √ , The restrictions (8), (8 (cid:48) ) and (9), (9 (cid:48) ) are not deducible from the geometricequation (7). – The metric generated in the space domain R ( r ) ≤ α by the substi-tutions R ( r ) − α (cid:29) − t is a non-static metric for a static problem. From thestandpoint of the differential geometry, this fact does not represent a diffi-culty. But from the physical point of view, things stand otherwise, becausethe non-static character implies clearly the existence of transport forces, thatare extraneous to our problem. This means that the above metric is onlya formal trick, which cannot give a physical significance to space domain R ( r ) ≤ α , that in reality does not belong to Schwarzschild’s manifold. Re-mark that for the radial coordinates by Schwarzschild and by Brillouin (cf.sect. ) this space domain is reduced to a singular point.We emphasize finally that also the well-known metric of Kruskal andSzekeres is non-static, and therefore introduces transport forces in a staticproblem. – As it was emphasized by von Laue [8], in Schwarzschild’s manifold ofa gravitating material point the Systemzeit t has a clear physical meaning,as it is specially attested by the red-shift of the spectral lines.Now, with the t -parametrization of the dynamical evolution – which isprivileged by Droste [4] –, we have that the velocities and the accelerationsof the geodesics at R = α are equal to zero : Hilbertian repulsion by theevent horizon R ( r ) = α . [7]. – Back to the “Information Paradox”. We could affirm a priori , i.e. without the detailed examination of the geodesics in a Schwarzschildianmanifold of a gravitating point-mass, that it cannot have a real existence ina theory as the GR, that has been devised in a manner which is independentof the directions of the spacetime coordinates, and in particular independentof the direction of the temporal coordinate.Any contradiction to this fact must be ascribed to an erroneous interpre-tation of a given aspect of the formalism. INFORMATION PARADOX” AND SCHWARZSCHILDIAN GEODESICS 5 – At the end of paper [5 b )], De Jans emphasizes that the solutions ofeq. (7) can be divided into four categories, that he illustrates with somediagrams. For each figure he gives the values of σ ≡ − α A/B and τ ≡ α /β , where A, B are the constants of our eqs. (2) and (3). For the radialand circular geodesics we have no figure. It is remarkable that for no geodesicthere is the surpassing of the space surface R = α . Categorie A . – Orbits with pericentre: periodic orbits (Figs. 1a and 1b);limiting orbits; open orbits (Figs. 2a and 2b).
Categorie B . – Finite orbits, with apocentre and without pericentre (Figs.3a and 3b).
Categorie C . – Finite orbits, with apocentre and without pericentre (Fig.4); infinite orbits without apsides, without asymptote; infinite orbits withoutapsides, with an asymptote (Figs. 5a and 5b); radial orbits.
Categorie D . – Transition orbits between categories B and C (cf. Figs. 3);orbits with apocentre and internal asymptotic circle (Fig. 6); orbits withoutapocentre, with internal asymptotic circle, without asymptote; orbits with-out apocentre, with internal asymptotic circle, with an asymptote (Fig. 7);orbits without pericentre and with internal asymptotic circle (Fig. 8).The subdivision into the above categories depends on the discriminant ∆of Weierstraß’ elliptic function P which gives the general solution of eq. (7):(10) α R = P ( ϕ + K ) + 112 , where K is a constant of integration; ∆ is given by the following equality:(10 (cid:48) ) ∆ := g − g , where:(11) g := 112 + α AB . (11 (cid:48) ) g := 1216 (cid:18) − α AB − α B (cid:19) . We have: ∆ > Categorie A , Categorie B ; ∆ < Categorie C ;∆ = 0 for
Categorie D and for the circular orbits. –See the diagrams of [5 b )] at the end of the present paper; they are referredto the standard radial coordinate R ( r ) ≡ r . APPENDIX A
To show the inadequacy of eq. (7) in the treatment of the circular geodesics,it is sufficient to consider the instance of the light-rays, for which A = 0.Eq. (7) becomes: ANGELO LOINGER AND TIZIANA MARSICO (A1) (cid:18) d(1 /R )d ϕ (cid:19) = 1 B − R + αR . For a circular orbit we must have:(A2) 1 B = 1 R − αR . Let us put:(A3) R = k α , with k ≥ . then:(A4) 1 B = 1 k α (4 k − > k = 3:(A5) 1 B = 127 α , which is the unique value prescribed by the dynamical solution. We see,however, that the geometric eq. (7) allows all the trajectories for which k > k = (3 − η ), with η > ≤
3, the right-hand side of (A4) becomes − η ) α (1 − η ). We see that for η ≥ APPENDIX B
The Numerical Relativity investigates, in particular, existence and behaviourof binaries composed of two mass-points with approximate calculations thathardly can have an exact counterpart [9].
A fortiori , this considerationholds for the three believed supermassive BHs residing in a quasar triplet[10]. Clearly, also for these instances no “Information Paradox” exists.The numerical computations make use of (3 + 1)decompositions of Ein-stein field equations. Now, a (3 + 1)-decompositions is not fully equivalentto Einstein gravitational theory, which considers also reference frames cor-responding to metrics not belonging to the class of metrics characterizedby any (3 + 1)-decomposition. The (3 + 1)-decompositions are an obviousgeneralization of the Gaussian frames, which were devised by Hilbert in1916 [7]. In general, the metrics of GR must only satisfy the well-knownconditions that g be negative and the quadratic form with the coefficients g αβ , ( α, β = 1 , , INFORMATION PARADOX” AND SCHWARZSCHILDIAN GEODESICS 7 ANGELO LOINGER AND TIZIANA MARSICOINFORMATION PARADOX” AND SCHWARZSCHILDIAN GEODESICS 9
Par suite d’une erreur, le nombre 6 a ´et´e omis dans la num´erotation des figures[5 b )], p.91.– References [1] See, e.g. : A.E. Broderick et alii , Ap. J. , (2011) 110; A.E. Broderick et alii , Ap.J. , (2011) 57; A.E. Broderick et alii , Ap. J. , (2009) 1357; Etc. .[2] See, e.g. , sects. and in A. Loinger and T. Marsico, arXiv:1205.3158 [physics.gen-ph] 13 May 2012.[3] T. Levi-Civita, The Absolute Differential Calculus (Calculus of Tensors) – (DoverPubls., Inc., Mineola, N.Y.) 1977, p. 408 and seqq . (Originally published in 1926 byBlackie and Son Limited, London and Glasgow).[4] J. Droste,
Proc. Roy. Acad. Amsterdam , (1917) 197.[5] C. De Jans: a ) M´em. Acad. Roy. Belg. (Sciences) , (1922) 1-41; b ) Ibidem , (1923)1-98; c ) Ibid. , (1924) 96-117.[6] W. Kundt, Astrophysics (Springer-Verlag, Berlin, etc. ) 2007, pp. 190 and 187; and passim .[7] D. Hilbert,
G¨ott Nachr. , Erste Mitteilung (20. Nov. 1915); zweite Mitteilung (23. Dez.1916) –
Mathem. Annalen , (1924) 1; also in Gesammelte Abhandlungen , DritterBand (J. Springer, Berlin) 1935, p.258.[8] M. v. Laue,
Die allgemeine Relativit¨atstheorie , 4. neubearbeitete Auflage (Friedr.Vieweg und Sohn, Braunschweig) 1956, p.112.[9] A. Loinger and T. Marsico, arXiv:1211.6152 [physics.gen-ph] 18 Nov 2012.[10] E.P. Farina et alii , , arXiv:1302.0849 [astro-ph.CO] 4 Feb 2013.
A.L. – Dipartimento di Fisica, Universit`a di Milano, Via Celoria, 16 - 20133Milano (Italy)T.M. – Liceo Classico “G. Berchet”, Via della Commenda, 26 - 20122 Milano(Italy)
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