Naked singularities: classification, general solution, quantum effects, etc

1. Naked singularities: classification, general solution, quantum effects, etc Prof. Serge Parnovsky Taras Shevchenko National University of Kyiv, Ukraine Workshop “Singularities of general relativity and their quantum fate” Warsaw 27/06/16

2. What is singularity? So, we drop out all pathological and directional singularities. Spacelikesingularities Examples: cosmological singularities, Schwarzschild singularity.

3. Timelikesingularities • We have two possibilities: • There is an even number of horizons • around a timelike singularity. • Example: the Reisner-Nordstroem black hole No mystery. Ordinary black hole with ordinary horizon on the outside. Observer cannot see a singularity from the outside

4. 2) There is no horizon around a timelike singularity. This is a mysterious naked singularity! Distant observer can see it. It could inject radiation, matter and information from... No one knows from what source. Some romantics could say it is a door to our space-time. At least a postern or a small window pane. A possible gate for little green men’s invasion.

5. This leads to a serious problem for science. One cannot formulate a Cauchyproblem for our space-time if there is at least one naked singularity without setting boundary condition on it. The panic solution: The Cosmic Censorship Principle. R.Penrose. Gravitational collapse: the role of general relativity // Riv. Nuovo Cim. -1969 – Vol.1, special number, P.253-276. All singularities produced by a collapse must be inside the horizons! Theoretically, it tolerates naked singularities produced by the Big Bang. By during the inflation they must fly away from the visible part of the Universe. Practical conclusion: There is no naked singularities in our Universe. Nevertheless it is just a hypothesis!

6. The Cosmic Censorship Principle • The only way to prove or disprove it is through studying • a process of the collapse, • A long ambiguous way possibly leading to nothing. • 2) the properties of the naked singularities in the General Relativity. • This way I advise. Nowadays we believe in existence of cosmological constant or dark energy, which acts similar to it. Its influence on a metric vanishes in the vicinity of time-like singularities. So, one can neglect cosmological constant or dark energy when studying naked singularities.

7. Types, examples and hierarchy of naked singularities. The simplest example with the metric depending only on one spatial coordinate x is the spatial Kasner solution with In the paper S.Parnovsky, I.Khalatnikov. On the motion of particles in the field of a naked Kasner-type singularity// Physics Let. v.66A,№6,1978, p. 466-468 it was identified as the metric around the infinitely long straight thread with the constant linear mass density. Let us start from the Weil metric describing static axial-symmetric space-times:

8. is a harmonic axial-symmetric function. In the conditional Here a function flat space with the cylindrical coordinate set it describes the Newtonian potential of some axial-symmetric mass distribution. If this source is the thread r=0 with the constant linear mass density m, then one can get after the transformation the Kasner metric with But if m>1, we get some problems. To solve it let’s consider the case of the source in the form of the finite thread r=0 with the constant linear mass density m and the length L. Using the oblate spheroid coordinate set v, u, φ,one can simply obtain the function ν=2μ ln(tanh(v/2))and, after substitution into the Weil metric, get the Zipoy-Voorhees metric z L Naked singularity corresponds to v=0. Its mass is M=μL/2. But what is the type of this singularity? Is it linear, or point-like or something else?

9. Let us draw the diagrams, describing the simplest properties of the space-time. For the Minkowski space-time let us consider the point source r=0 in the spherical coordinates, the line source r=0 in the cylindrical coordinates and the plane source x=0 in the Cartesian coordinates. One get the corresponding diagrams point line plane Now let’s draw the diagrams for the Zipoy-Voorhees metric for different m Point-like singularity with negative mass The new type of singularity, named paradox-like with positive mass Additional directional singularities on the “ends” v=0, u=±π/2 Linear singularity with positive mass

10. S. L. Parnovskii, “Type and structure of timelike singularities in the general theoryof re-lativity: from the gamma metric to the general solution", Sov. Phys. JETP 61,1139 (1985). At v→0 we have Lφ(v,u) – circumference v,u=const. Lu(v,φ) – length of curve v,φ=const. S(v) - surface area v=const. N - norm of rotational Killing vectorξ=(0,0,1,0).

11. After the transformation r=L cosh2(v/2) one gets the metric Some special cases of the Zipoy-Voorhees metric for special m values If m=1 we get the Schwarzschild metric with the mass M=L/2>0. The fictitious singularity v=0 corresponds to the horizon of the black hole. If μ=-1 we get the Schwarzschild metric with the mass M=-L/2<0 after the transformation r=L sinh2(v/2).The real singularity v=0 corresponds to point-like source with negative mass.

12. Generalization (m ≠ const): m(z) – Weil singularities, m(z,t) – “Simple line sources” (W.Israel. Line sources in general relativity // Phys. Rev.D – 1977 – Vol.15 – P. 935-941), m(z,t,φ) – generalized spatial Kasner metric (Parnovsky S.L Gravitation fields near the naked singularities of the general type // Physica – 1980. – V.104A. – P.1423-1437). All these solutions are approximate at x→0 or r→0. Their properties were analyzed in S. L. Parnovskii, “Type and structure of timelike singularities in the general theory of relativity: from the gamma metric to the general solution", Sov. Phys. JETP 61,1139 (1985).

13. The general solution of Einstein equations near time-like singularity It was found and analyzed in the paper Parnovsky S.L Gravitation fields near the naked singularities of the general type // Physica – 1980. – V.104A. – P.1423-1437. It is an oscillating solution (naturally, approximate one) very similar to the well-known BKL solution near space-like singularities (Belinsky V. A., Khalatnikov I. M., Lifshitz E. M. Oscillatory approach to a singular point in relativistic cosmology // Adv. Phys. – 1970. – Vol. 19, № 80. – P. 525).In order to get a general solution near the arbitrary singularity it had to be matched with BKL solution. This was done in the paper Parnovsky S.L. A general solution of gravitational equations near their singularities // Clas. and Quant. Gravit. – 1990. – V.7. – P.571-575. Nowadays this solution was studied by P. Klinger, “Timelike singularities and Hamiltonian cosmological billiards,“arXiv:1512.03302 [gr-qc] and E. Shaghoulian and H. Wang, “Timelike BKL singularities and chaos in AdS/CFT,“arXiv:1601.02599 [hep-th] using the Iwasawa decomposition. Results differ. This solution for the case of Bianchi IX space-time was studied by S. Parnovsky and W. Piechocki, “Classical dynamics of the Bianchi IX model: spacelike and timelike singularities” arXiv:1605.03696.

14. Typical BKL dynamics

16. A distance between the initial point ξ= 0 and oscillating singularity at ξ=∞is finite and a distance between the initial point and the singularity at ξ=ξ0 isinfinite. This singularity located infinitely far from the oscillating timelike singularity.

19. Rotating linear singularities They were described and analyzed in the paper S. L. Parnovskii Gravitational field of rotating bodies // JETP, Vol. 73, No 5, p. 788-796. It was shown that there must be a small region with closed time-like geodesic lines near it. This is a hint of a causality violation. Nevertheless this solution is not so bad, because the size of this region is less that the Plank length. An influence of non-gravitational fields was analyzed in the papers S. L. Parnovskii, “Effects of electric and scalar fields on timelike singularities", Sov.Phys. JETP 67, 2400 (1988), Parnovsky S.L., Gaydamaka O.Z.A generalization of the Zipoy-Voorhees metric in the presence of a conformally invariant scalar field // UPJ– 2004.– V49,№3.– P. 205-209 and some other ones.

20. Quantum effects near naked singularities If classical collapse leads to the formation of the naked singularity, it could cause a strong radiation due to the quantum pairs production and changing of a vacuum polarization. Its backreaction could slow the collapse in such a way, that it forms a black hole instead of naked singularity. So we need to calculate the mass loss due to quantum radiation during a formation of the naked singularity. A simple model with massive shell, shrinking up to the Plank length was used. Conclusions: mass loss is very small at the formation of the linear singularity (Parnovsky S.L. Quantum particle production in the formation of naked Kasner-type singularities // Physics Let. – 1979. – v.73A, №3. – P.153-156), but very large at the formation of theReisner-Nordstroem singularity with Q>M (Parnovsky S.L. Can Reisner-Nordstrom singularities exist? //Gen. Rel. and Grav. – 1981. – V.13, №9. – P.853-863).

21. Also the already formed naked singularities could be “dressed up” due to quantum radiation and its backreaction. E.g. naked Kerr singularity with a>m acts in the same way (S.L. ParnovskiiQuantum emission of Kerr-type naked singularities //JETP, Vol. 53, No 4, p. 645-649). α=a/m, β2= α2 - 1 During the time of existence of the Universe singularities with solar mass and with parameters β< 10-28 can "dress themselves” through the emission of massless scalar particles; here, the particles can be assumed to be either bosons or fermions. In addition, for bosons, which true scalar particles are, we obtain one further interval of parameters in which a singularity can "dress itself” through exponential increase in the number of particles in the levels. It extends from β ~ 10-20 to α≈ 4.

22. Brief conclusions concerningThe Cosmic Censorship Principle Some types of naked singularities cannot be formed by collapse. Point-like singularities repulse collapsing matter. Paradox-like singularities must have a linear density exceeding the critical value. General oscillation singularities must have a strong influence of the quantum effects (IMHO). But the linear singularities have no such problems. So, in order to prove or disproveThe Cosmic Censorship Principle one has to study a collapse with the formation of the linear naked singularity.

23. Thank you for attention!