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Numerical Solution of Einstein Field Equations with Planck Core Boundary Condition

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Numerical Solution of Einstein Field Equations with Planck Core Boundary Condition XXIII Workshop “What Comes Beyond the Standard Models?" Bled, Slovenia, 6-10 July 2020 Numerical solution of Einstein field equations with Planck core boundary condition Igor Nikitin Fraunhofer Institute for Algorithms and Scientific Computing Sankt Augustin, Germany Thanks for hosting: Virtual Institute of Astroparticle physics Short bio Igor Nikitin: ● graduated from Moscow Institute for Physics and Technology 1991 ● worked in Institute for High Energy Physics, Protvino, Russia 1993-1997 ● PhD topic: Anomaly free quantization of string theory in 4 dimensions (special types of motion) ● Doc.Sci topic: Anomaly free quantization of string theory in 4 dimensions (general case) ● Currently working in Fraunhofer Institute for Algorithms and Scientific Computing, Sankt Augustin, Germany Field of work: energetics 2 Content Planck stars Spherically symmetric models with Planck core in the center ● white holes: questions of stability ● RDM-stars: galactic rotation curves and fast radio bursts ● QG-bubbles: one more candidate to the fast radio bursts Hypotheses: What is astrophysical dark matter made of? arXiv: 1701.01569 1811.03368 1812.11801 1903.09972 1906.09074 3 Planck stars ● calculations in quantum gravity (QG) for a cosmological model with a scalar field (Ashtekar et al. 2006) give a correction to the mass density: ρ = ρ (1-ρ/ρ ), ρ ~ ρ X c c P ● ρ = ρ => ρ = 0 at critical density the gravity is switched off c X ● ρ > ρ => ρ < 0 in excess of critical density the effective negative c X mass appears (exotic matter), with gravitational repulsion (a quantum bounce phenomenon) ● Planck star model: Rovelli, Vidotto (2014), Barceló et al. (2015) ● collapse of a star replaced by extension, black hole turns white Planck star ● in this talk we consider a stationary version of Planck star, stabilized under the pressure of the external matter (a Planck core) ● 3 spherically symmetric models with Planck core in the center will be considered ● the question is related with the stability of white holes, the shape of galactic rotation curves and the origin of fast radio bursts 4 General remark on negative masses Evolution of viewpoint: ● Energy conditions (Einstein, Hawking): there are NO negative masses ● 't Hooft (1985): “... negative mass solutions unattractive to work with but perhaps they cannot be completely excluded.“ ● negative masses are needed to create the wormholes (Morris-Thorne 1988), warp drives (Alcubierre 1994), time machines (Visser 1996) ● Barceló, Visser (2002), Twilight for the energy conditions? ● Rovelli, Vidotto (2014), Barceló et al. (2015): negative masses can be obtained effectively by an excess of Planck density ● alternative way: Tippett, Tsang (2017), many interesting solutions can be obtained with non-exotic matter in f(R)-gravity (examples are wormholes and accelerating cosmology) ● … 5 White hole on Penrose diagram collapse of a star into a black hole the eruption of a white hole (Oppenheimer-Snyder model) (Lemaître-Tolman model) r = 0, the singularity t =+∞ t =+∞ Cauchy horizon r = 0 r =+∞ T-reflection event horizon r = 0 star surface loc. time coord radial light rays ±45 ° t =-∞ r = 0, the singularity t =-∞ loc. space coord 6 White hole (it is not as simple) ● Eardley instability (1974) – one should take into account the interaction with external matter 2. initially internal matter can come out of the white hole (in a finite time according to its own clock) 1. Cauchy horizon of the white hole 7 singularity White hole (Eardley instability) 2. initially internal matter can come out of the white hole (in a finite time according to its own clock) 3.external matter (e.g. relict radiation) 1. Cauchy horizon of cannot enter the white hole, in principle the white hole 8 singularity White hole (Eardley instability) the exponential process: (value doubled every minute) r =1.2*1010m, t=13.8 Gyears, s A-1/2~exp(tc/(2r ))~exp(1016)!!! s 4. after a long (13.8 billion years) wait at the Cauchy horizon and superstrong blue shifting the external matter forms a thin super-energetic blue sheet 2. initially internal matter can come out of the white hole (in a finite time according to its own clock) 3.external matter (e.g. relict radiation) 1. Cauchy horizon of cannot enter the white hole, in principle the white hole 9 singularity White hole (Eardley instability) the exponential process: (value doubled every minute) r =1.2*1010m, t=13.8 Gyears, s A-1/2~exp(tc/(2r ))~exp(1016)!!! 5. as a result, a black s hole is formed, under the event horizon 4. after a long (13.8 billion years) wait at the of which the blue sheet Cauchy horizon and superstrong blue shifting falls, together with the the external matter forms a thin super-energetic internal matter that blue sheet didn't had time to escape the white hole => the external matter 2. initially internal matter can come out prevents the eruption of of the white hole (in a finite time the white hole, only a according to its own clock) small part of the internal matter can go outside 3.external matter (e.g. relict radiation) 1. Cauchy horizon of cannot enter the white hole, in principle the white hole 10 singularity White hole (null shell model) ● Ori-Poisson model (1994) ● gravitational interaction of null shells ● M is the mass of the white hole, dm is the mass of the input shell, E is the mass / energy of output shell (G = c = 1), m – remainder 3. after a collision 0 with the blue sheet, ● solution consists of 4 Schwarzschild's a negligible part of patches ABCD the initial energy comes out 0 A = 0 C D m B 2. explodes, completely releasing its mass into a diverging null 1. original white hole shell 11 White hole (null shell model) The proof uses Dray-’t Hooft-Redmount (DTR) relation: efficiency of white hole eruption ~ exp(-1016) ~ exp(-1016) exponential process of approaching of the ingoing null shell to the Cauchy horizon 12 White hole (null shell model) ● so, the black holes are stable, but the white holes aren’t ... what is about T-symmetry? ● the paradox resolution: there is a 5. the negative mass of the T-symmetric scenario according to the core is compensated by the principle “for every incoming blue sheet incoming blue sheet, there is the same outgoing ...” a black hole of the same mass as the original white ● negative mass remainder required hole formed 4.after the collision in solution (exotic matter) of two blue sheets, a shell of the same energy as ingoing one comes out 0 A < 0 C D m 3. leaves B behind the core 2. explodes, emitting of exotic matter more energy than its own mass in the form of a diverging blue sheet 1. original white hole 13 White hole (null shell model) ● calculation with β<0 shows self-consistency of the solution: (100% efficiency) Т-symmetric case: Conclusion1: in Ori-Poisson model, Eardley instability can be eliminated if the system has a core of negative mass 14 Dark Stars ● also known as quasi black holes, boson stars, gravastars, fuzzballs ... ● solutions of general theory of relativity, which first follow Schwarzschild profile and then are modified ● outside are similar to black holes, inside are constructed differently (depending on the model of matter used) ● review of the models: Visser et al., Small, dark, and heavy: But is it a black hole?, arXiv: 0902.0346 ● our contribution to this family: RDM stars (quasi black holes coupled to Radial Dark Matter) t Stationary solution, including T-symmetric supersposition of ingoing and outgoing radially directed flows of dark matter r xyz 15 Galactic model with radial dark matter ● The simplest model of a spiral galaxy ● 30kpc level (for MW) ● Dark matter flows radially converge towards the center of the galaxy ● The limit of weak gravitational fields, one-line calculation: ρ ~ r -2, M ~ r, v 2 = GM / r = Const ● Qualitatively correct behavior of galactic rotation curves (asymptotically flat shape at large distances) ● The orbital velocity of the stars and interstellar gas consisting of Kepler and constant terms ● Q1: How to describe the deviation of rotation curves from the flat shape? (the model of distributed RDM-stars will be considered) ● Q2: What is happening in the center of the galaxy? (the calculation in the limit of strong 16 fields will be done) Typical behaviour of galactic rotation curves (Kepler + constant) (v/c)2 = r /(2r) + ε S-stars s (6e-4...0.01pc, Keplerian) c e s / m SUN position k (DM dominated) , y t i switch from DM-dominated c ε o to Keplerian regime at r~r /(2 )~0.5pc l s e (fine structure considered below ...) v l a t i b r Data: Sofue, Rubin 2001 o 17 distance to center, kpc Rotation curves of galaxies, in details ● Universal Rotation Curve (URC, Salucci et al. 1995-2017) ● represents averaged exp. rotation curves of >1000 galaxies ● before averaging: galaxies are subdivided to bins over magnitude mag ● curves V(R,mag) are normalized to the values at optical radius: V / Vopt, R / Ropt ● the averaging smoothes the individual characteristics of the curves (loc. minima / maxima) ● detailed modeling of rotation curves in RDM model ● based on the assumptions: (1) all black holes are RDM-stars; (2) their density is proportional to the concentration of the luminous matter in the galaxy ● in this case, the dark matter density is given by the integral (Kirillov, Turaev 2006) constant Freeman 1970 model the contribution of one RDM-star optical radius of the galaxy encompassing 83% of the light 18 Rotation curves of galaxies, in details The physical meaning of KT-integral: every element of luminous matter (i.e., RDM-stars contained in it) gives additive contributions to dark matter density, mass, M gravitational field, orbital velocity, gravitational potential..
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