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A black universe without phantoms

A black universe without phantoms. K.A. Bronnikov (VNIIMS, Moscow; PFUR, Moscow) E.V. Donskoy (PFUR, Moscow, Russia) A.V. Michtchenko ( ESIME-SEPI-IPN, Mexico ). Regular black holes and the notion of a black universe Black universes as solutions of GR with a phantom scalar field

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A black universe without phantoms

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  1. A black universe without phantoms K.A. Bronnikov (VNIIMS, Moscow; PFUR, Moscow) E.V. Donskoy (PFUR, Moscow, Russia) A.V. Michtchenko (ESIME-SEPI-IPN, Mexico) Regular black holes and the notion of a black universe Black universes as solutions of GR with a phantom scalar field Alternative to a phantom: gravity on the brane A black universe on the brane: examples

  2. Regular BH geometries • BHs with a regular center: • ρ→0,r≈ const·ρ , A > 0, A(dr/dρ)2 ≈ 1+O(r2). RN-like global structures Bardeen, 1968 Dymnikova, 1992 and later works (vacuum de Sitter core, variable Λ) K.B., 2000(magnetic BHs and monopoles in nonlinear ED)

  3. 2. BHs withouta center, having second-order horizons of infinite area (so-called cold BHs due to TH=0) Such configurations were obtained in STT with phantom scalar fields (K.B., G. Clement, C. Constantinidis, J.C. Fabris, 1998) They are very special and do not seem realistic

  4. 3. BHs withtwo spatial infinities (causal structure of a non-extreme Kerr BH but without asingular ring) Such solutions were obtained for effective 4D gravity equations in RS2 type brane worlds (R. Casadio, A. Fabbri and L. Mazzacurati, 2002; K.B, H. Dehnen, V.N. Melnikov, 2003). They are generic parts of families of solutions, also including symmetric WHs. Types 2 and 3 of regular BHs were recently obtained by Balakin, Sushkov and Zayats (2007) with nonminimally coupled YM firelds

  5. 4. Regular BHs with a Schwarzschild-like causal structure but expanding cosmology instead of a singular centre(“black universes”).K.B., J.C. Fabris, 2005 Obtained as generic solutions to the Einstein-scalar equations for minimally coupled phantomscalar fields with certain potentials, STT with phantom fields and many kinds of k-essence,

  6. The first paper on black universes

  7. Scalar fields in GR, spherical symmetry: where Ris the scalar curvature, ε = +1corresponds to ausual scalar field with positive kinetic energy and ε = -1to a phantom field. The interest in phantom fields is inspired by cosmology - acceleration, eq. of state with w < -1. If such matter exists, it should also show itself in local phenomena, e.g., in BH physics. We use the general static, sphericallysymmetric metric and ф= ф(ρ). Killing horizons, if any, correspond to regular zeros of thefunction A(ρ): The metric:whereA(ρ)> 0 (in R regions) –static, sph. symmetric; whereA(ρ)< 0(in T regions)–homogeneous, anisotropic Kantowski-Sachs (KS) cosmology; ρis a time coordinate,

  8. Field equations: (scalar) (00) (00) – (11) (00) – (22) The latter may be integrated: Our aim: determine possible regular configurations. Phantom fields violate the known energy conditions => there can be solutions of interest for BH physics and cosmology, impossible with usual matter.

  9. A search for globally regular configurations If no pathology at intermediate values of ρ, regularity is provided by the asymptotics. 4 kinds of regular asymptotics: flat, dS, AdS, r→ const asρ→±∞ Assume that, as ρ→∞, there isMink., dS or AdS; r ~ ρ at large ρ Normal fields, ε = +1: r’’ ≤ 0 => inevitably r-> 0 at some ρ This is a center, which may be regular in some solutions. It is theonly kind of regular configurations with ε = +1. Phantom fields, ε = -1: r’’ ≥ 0 => possible to haveρ→ -∞, another asymptotic at which all kinds of regular behavior can occur. Solutions with a regular center are also not excluded. => a variety of regular models

  10. Regular phantom configurations Table:rows- asymptotic behavior as ρ ->+∞, columns– the same for ρ -> -∞. “sym”:combinations obtained from others by symmetryρ<->- ρ. Cases labelled KS* comprisethree types of solutions each: there can be two simple horizons, one double horizon or no horizons between two dS asymptotics. 13 variants in the table + 3 kinds of PLS with a regular center => 16 classes of regular solutions

  11. Examples of each behavior may befound in an algorithmic manner by properly choosing the function r(ρ)and invoking the inverse problem method. A, ф and V are then found from the field equations: (1) (2) 2) (3) => (4) Specify r(ρ); (4) =>B(ρ), A(ρ); (3) =>ф(ρ); (2) =>V(ρ). If r’’ ≠ 0, then ф(ρ) is monotonic => V(ф) is well defined. One should choose: r’’ < 0 for normal fields, r’’ > 0 for phantom fields. In regular solutions, V(ф) tends to extrema at the asymptotics.

  12. Example We choose: We then obtain: The solution behavior is controlled by two integration constants: c moves B(ρ) up and down, and ρ0shows the maximum of B(ρ). Both r(ρ) and B(ρ)are even functions if ρ0= 0, otherwise the solution loses thissymmetry.

  13. 1. Simplest case:ρ0= c = 0:Ellis wormhole,V =0 andA= 1. • 2.Solutions with ρ0= 0 but c ≠ 0:symmetric structures: • c > 0:wormholes with twoAdS asymptotics Two dS asymptotics if c < 0: 0 > c > -1: R region bounded by two simple horizons (SdS structure), at c = -1they merge. c < -1: pure KS cosmology

  14. 3.ρ0≠ 0: asymptotics ρ= ∞ and ρ= - ∞are different. In solutions asympt.flat atρ= ∞it holds2bc = -πρ0. Schwarzschild mass:m =ρ0/3. For ρ0> 0, when V(- ∞) > 0, there is a regular BHwith m > 0 and a dS asymptotic far beyond the horizon (a black universe). A Schwarzschild-like causal structure, but thesingularityr = 0inthe Carter-Penrose diagram is replacedby a dS asymptotic r = ∞. r = ∞ (dS) r = ∞ (flat) r = ∞ (flat) r =∞ (dS)

  15. Regular phantom BHs, or “black universes” They combine the properties of BHs, whose main feature is a horizon, andwormholes, whose main feature is a throat (minimum of r, here r = b). If the BH mass (m)dominates over thescalar charge(b), the throat is invisible to a distant observer, and the BH looks almost as usualin GR. However, a possible BH explorer now gets a chance to survive for anew life in an expanding KS universe. Opportunity: our Universe could appear from collapse to a phantom BH in another, mother universe and undergo isotropization (e.g., due to particle creation) soon after crossing the horizon. The KS nature of our Universe is not excluded observationally if its isotropization happened early enough, before the last scattering epoch (at redshifts z about 1000).

  16. What’s wrong? The phantom nature of the scalar field. Negative kinetic energy: never observed, possible quantum instabilities etc. (NB: phantom behavior in cosmology may be obtained without a phantom field, e.g., due to nonminimal coupling in scalar-tensor gravity – Starobinsky et al., 2006) Another possible source of NEC violation, necessary for the existence of wormholes and black universes: brane worlds, where the “tidal EMT” leads to both wormholes and regular BHs

  17. Brane world models Matter is confined or trapped on the brane while gravity propagates in the bulk brane Suggested in the 80s: K. Akama (1982); V.A. Rubakov, M.E. Shaposhnikov (1983); M. Visser (1985); I. Antoniadis (1985) M. Pavsic (1986) [gr-qc/0101075]; E.J. Squires, Phys. (1986); G.W. Gibbons, D.L. Wiltshire, (1987). Related to M-theory [Horava, Witten (1996)] bulk Recent reviews (among others) V.A. Rubakov. Phys.-Usp, 171, 9, 913 (2001); P. Brax and C. van de Bruck, hep-th/0303095; R. Maartens, Living Rev. Rel.(2004); gr-qc/0312059; P. Kanti, hep-ph/0402168.

  18. 5D, thin branes RS1: two flat branes in compact AdS bulk RS2: a single flat brane in non-compact AdS Simplest models: This work: curved RS2 type branes in 5 dimensions Shiromizu et al. ‘99: equations of 4D gravity on the brane in 5D bulk (1) -”electric” part of 5D Weyl tensor projected on the brane By construction, the E-tensoris traceless. • - not a closed set of eqs: Eμνconnects 5D geometry with brane gravity. • One should either solve the full 5D problem • ortry to extract as much info as possible from the 4D eqs. • Thecorresponding 5D solutions always exist(the Campbell-Magaard theorem),

  19. A search for a black universe on the brane • Desired: 1) flat infinity as ρ →∞ • 2) de Sitter infinity as ρ → -∞ • With R = 0 we cannot have item 2); • with R = 4Λ we cannot have item 1). • Moreover, with R=0, KS cosmologies cannot expand forever. • Thus to have a BU solution, some matter source is necessary. • Approach: • We choose, as a matter source, a scalar field with a potential V(ф) and normalkinetic energy. • We neglect the quadratic part Π of the total EMT. • (Π becomes significant only at densities ≥ 1029 g/cm3)

  20. The `tidal’ tensor is traceless and conservative. We take it in the form Conservation: Other eqs: We use the inverse problem method, specifying r(ρ) and f(ρ). An analytic example of BU solution is obtained under the assumptions: r = (|ρ+c|2 – b2)1/2; c >1 (so that r’’<0 at all ρ≠0), f(ρ) ~ δ(ρ).

  21. Solution with f(ρ) ~ δ(ρ) r = [(|x|+c)2 – 1]1/2; c >1 r’’ has a δ-like singularity, compensated by f(x) Equation for B = A/ρ2: r (x), x = ρ/b Initial cond.: Schwarzschild asymptotic as x ->-∞. Solution for x<0: Here: 2h(x):= ln |(x+1)/(x-1)|. Solution for x>0 is more cumbersome. Result: plot ofB(x) B(x) has a fracture due to δ-like f(x). B -> const <0 as x -> ∞ => dS asymptotic

  22. δ(ρ) replaced with a smooth function |x| > d: r (x) = b[(|x|+1)2-1]1/2; |x|< d: r (x) = ax2 + h, a, h chosen to make r and r’ continuous. f(x) chosen to make ф continuous Equation for B(x) = A/r 2: (r 4B’)’ + 2 + r 2(4P + 3Br 2f ) = 0, f =0 for |x| > d; f = 2C/r 6 for |x|< d; P = CB + C1/r 4 ; C, C1= const. B(x) for x < -d and x>d is found analytically, for -d < x < d- numerically. B -> const <0 as x -> ∞ => dS asymptotic

  23. Conclusion • A self-interacting phantom scalar field can form • many different regular spherical configurations • The most interesting ones: BHs with de Sitter • expansion far beyond the horizon • (black universes) • Possible origin of our Universe from such a BH • Generalizations: STT, k-essence type fields etc. • Possible existence of BU solutions • without phantoms in a brane world

  24. THANK YOU!

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