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Quantum Singularities in Static and Conformally Static Space-Times

Quantum Singularities in Static and Conformally Static Space-Times . Debbie Konkowski (USNA) & Tom Helliwell (HMC) Ae100prg – Prague, Czech Republic – June 25-29, 2012. Outline. 1. Singularities a. Classical b. Quantum 2. Static s pace-times – History and Examples

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Quantum Singularities in Static and Conformally Static Space-Times

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  1. Quantum Singularities in Static and Conformally Static Space-Times Debbie Konkowski (USNA) & Tom Helliwell(HMC) Ae100prg – Prague, Czech Republic – June 25-29, 2012

  2. Outline 1. Singularities a. Classical b. Quantum 2. Static space-times – History and Examples 3. Conformally static space-times a. FRW with Cosmic String b. Roberts c. HMN (in progress) d.Fonarev (in progress) 4. Discussion 5. Acknowledgements

  3. Space-time (M,g) • smooth, C∞, paracompact, connected Hausdorff manifold M • Lorentzian metric g

  4. Singularity???? • Space-time is smooth! • “singular” point must be cut out of space-time ⇒ leaves hole ⇒ incomplete curves ⇒ boundary points

  5. How do we define a boundary ∂Mto space-time???? Complete: ST ( M, g) + Boundary (∂M) Cauchy completeness only works with Riemannian metric, not Lorentzian. How do we complete ST?

  6. Singularities as Boundary Points a (abstract) - boundary (Scott and Szekeres (1994)) b (bundle) - boundary (Schmidt (1971)) c (causal) - boundary (Geroch, Kronheimer and Penrose (1972)) g (geodesic) – boundary (Geroch (1968))

  7. Classical Singularity “ a singularity is indicated by incomplete geodesics or incomplete curves of bounded acceleration in a maximal space-time” (Geroch (1968))

  8. Singularity Types(Ellis and Schmidt (1977)) Singular point q ⇓ Does Rabcdor its kthderivative diverge in some PPON frame? ⇙⇘ no yes (Ckquasiregular (Ck curvature singularity) singularity) ⇓ Does a scalar in gab, nabcd and Rabcd or its kth derivative diverge? ⇙⇘ no yes (Cknonscalar(Ck scalar curvature sing. ) curvature sing.)

  9. Quantum Wavesvs.Classical Geodesics What happens if instead of classical particle paths (timelike and null geodesics) one used quantum mechanical particles (QM waves) to identify singularities????

  10. Horowitz and Marolf (1995): A space-time is QM nonsingular if the evolution of a test scalar wave packet, representing a quantum particle, is uniquely determined by the initial wave packet, manifold and metric, without having to put boundary conditions at a classical singularity.

  11. TECHNICALLY: A static ST is QM-singular if the spatial portion of the Klein-Gordon operator is not essentially self-adjoint on C0∞(Σ) in L2 (Σ).

  12. Essentially Self- Adjoint An operator, A, is called self-adjoint if (i) A = A* (ii) Dom(A) = Dom(A*) where A* is the adjoint of A. An operator is essentially self-adjointif (i) is met and (ii) can be met by expanding the domain of the operator or its adjoint so that it is true.

  13. Tests for Essential Self-Adjointness • von Neumann criterion of deficiency indices: study solutions to A*Ψ = ± iΨ, where A is the spatial K-G operator and find the number that are self-adjoint for each i. 2. Weyl limit point-limit circle criterion: relate essential self-adjointness of Hamiltonian operator to behavior of the “potential” in an effective 1D Schrodinger Eq., which in turn determines the behavior of the scalar wave packet.

  14. Technique To study the quantum particle propagation in spacetimes, we use massive scalar particles described by the Klein–Gordon equation and the limit circle–limit point criterion of Weyl (1910). In particular, we study the radial equation in a one-dimensional Schrodinger form with a ‘potential’ and determine the number of solutions that are square integrable. If we obtain a unique solution, without placing boundary conditions at the location of the classical singularity, we can say that the solution to the full Klein–Gordon equation is quantum-mechanically nonsingular. The results depend on spacetime metric parameters and wave equation modes.

  15. Quantum Singularity History Basic References: • R.M. Wald, J. Math. Phys. 21, 2802 (1980) • G.T. Horowitz and D. Marolf, Phys. Rev. D 52, 5670 (1995) • A. Ishibashi and A. Hosoya, Phys. Rev. D 60, 104028 (1999)

  16. Review Articles on Quantum Singularities 1. • Quantum Singularities in Static Spacetimes. • Joao Paulo M. Pitelli, Patricio S. Letelier (Campinas State U., IMECC). Oct 2010. 14 pp. • Published in Int.J.Mod.Phys. D20 (2011) 729-743 • e-Print: arXiv:1010.3052 [gr-qc] Includes: • Math Review • Cosmic String • Global Monopole • BTZ Space-time (massive field – QM singular; massless field – QM non-singular)

  17. cont. • 2. • Quantum singularities in static and conformally static space-times • D.A. Konkowski, T.M. Helliwell • Comments: 16 pages, 8th Friedmann Seminar, Rio de Janeiro, Brazil, 30 May - 3 June, 2011 • Journal-ref: International Journal of Modern Physics A, Vol.26, No.22 (2011) 3878-3888 arXiv:1112.5488 (gr-qc)

  18. Previous Work on Quantum Singularities – Static STs • Classical and quantum properties of a two-sphere singularity. • T.M. Helliwell (Harvey Mudd Coll.), D.A. Konkowski (Naval Academy, Annapolis). 2010. 10 pp. • Published in Gen.Rel.Grav. 43 (2011) 695-701 • e-Print: arXiv:1006.5231 [gr-qc] • Quantum singularities in (2+1) dimensional matter coupled black hole spacetimes. • O. Unver, O. Gurtug (Eastern Mediterranean U.). Apr 2010. 21 pp. • Published in Phys.Rev. D82 (2010) 084016 • e-Print: arXiv:1004.2572 [gr-qc]

  19. And more… • arXiv:hep-th/0602207 • Title: Scalar Field Probes of Power-Law Space-Time Singularities • Authors: Matthias Blau, Denis Frank, Sebastian Weiss • Comments: v2: 21 pages, JHEP3.cls, one reference added • Journal-ref: JHEP0608:011,2006 • arXiv:0911.2626 • Title: Quantum Singularities Around a Global Monopole • Authors: João Paulo M. Pitelli, Patricio S. Letelier • Comments: 5 pages, revtex • Journal-ref: Phys.Rev.D80:104035,2009 • arXiv:0805.3926 • Title: Quantum singularities in the BTZ spacetime • Authors: João Paulo M. Pitelli, Patricio S. Letelier • Comments: 6 pages, rvtex, accepted for publication in PRD • Journal-ref: Phys.Rev.D77:124030,2008 • arXiv:0708.2052 • Title: Quantum Singularities in Spacetimes with Spherical and Cylindrical Topological Defects • Authors: Paulo M. Pitelli, Patricio S. Letelier • Comments: 7 page,1 fig., Revtex, J. Math. Phys, in press • Journal-ref: J.Math.Phys.48:092501,2007

  20. arXiv:1006.3771 Title: "Singularities" in spacetimes with diverging higher-order curvature invariants Authors: D.A. Konkowski, T.M. Helliwell Comments: 3 pages, no figures, submitted to the Proceedings of the 12th Marcel Grossmann Meeting on General Relativity and Gravitation, Paris, July 13-18, 2009 arXiv:1006.3743 Title: Quantum particle behavior in classically singular spacetimes Authors: D.A. Konkowski, T.M. Helliwell Comments: 3 pages, no figures, submitted to Proceedings of the 12th Marcel Grossmann Meeting on General Relativity and Gravitation, Paris, July 13-18, 2009 arXiv:gr-qc/0701149 Title: Quantum healing of classical singularities in power-law spacetimes Authors: T. M. Helliwell, D. A. Konkowski Comments: 14 pages, 1 figure; extensive revisions Journal-ref: Class.Quant.Grav.24:3377-3390,2007 arXiv:gr-qc/0412137 Title: Mining metrics for buried treasure Authors: D.A. Konkowski, T.M. Helliwell Comments: 16 pages, no figures, minor grammatical changes, submitted to Proceedings of the Malcolm@60 Conference (London, July 2004) Journal-ref: Gen.Rel.Grav. 38 (2006) 1069-1082 arXiv:gr-qc/0410114 Title: Classical and Quantum Singularities of Levi-CivitaSpacetimes with and without a Positive Cosmological Constant Authors: D.A. Konkowski, C. Reese, T.M. Helliwell, C. Wieland Comments: 14 pages, no figures, submitted to Proceedings of the Workshop on Dynamics and Thermodynamics of Blackholes and Naked Singularities (Milan, May 2004)

  21. arXiv:gr-qc/0408036 Title: Quantum Singularities Authors: D.A. Konkowski, T.M. Helliwell, C. Wieland Comments: 5 pages, no figures, references current Journal-ref: Gravitation and Cosmology: Proceedings of the Spanish Relativity Meeting 2002, ed. A. Lobo (Barcelona, Spain: University of Barcelona Press, 2003) 193 arXiv:gr-qc/0402002 Title: Are classically singular spacetimes quantum mechanically singular as well? Authors: D.A. Konkowski, T.M. Helliwell, V. Arndt Comments: 3 pages, no figures, submitted to the Proceedings of the Tenth Marcel Grossmann Meeting on General Relativity, Rio de Janeiro, July 20-26, 2003 arXiv:gr-qc/0401040 Title: Definition and classification of singularities in GR: classical and quantum Authors: D.A. Konkowski, T.M. Helliwell Comments: 3 pages, no figures, submitted to Proceedings of the Tenth Marcel Grossmann Meeting on General Relativity, Rio de Janeiro, July 20-26, 2003 arXiv:gr-qc/0401038 Title: "Singularity" of Levi-CivitaSpacetimes Authors: D.A. Konkowski, T.M. Helliwell, C. Wieland Comments: 3 pages, no figures, submitted to Proceedings of the Tenth Marcel Grossmann Meeting on General Relativity, Rio de Janeiro, July 20-26, 2003 arXiv:gr-qc/0401009 Title: Quantum singularity of Levi-Civitaspacetimes Authors: D.A. Konkowski, T.M. Helliwell, C. Wieland Journal-ref: Class.Quant.Grav. 21 (2004) 265-272

  22. Conformally Static Space-Times Test for Quantum Singularity using Conformally Coupled Scalar Field and Associated Inner Product Idea comes from Ishibashi and Hosoya (1999) who used it for “wave regularity”

  23. Klein-Gordon w/ general coupling is |g|-1/2 (|g|1/2 gαβΦ,β),α = ξ RΦ where ξ is the coupling constant and R is the scalar curvature

  24. Conformally Static Spacetimes “As is well-known, in the conformally coupled scalar field case, that is ξ = (d – 2)/4(d -1) for any spacetime dimension d, the field equation is invariant under the conformal transformations of the metric and the field, guv(x) guv(x) = C2(x) guv(x), φ φ= C(2-d)/2φ. Since Tcuv= C2-dTcuv, the corresponding inner product is conformally invariant.” I and H (1999)

  25. Conformally Static Space-times 4 Examples: • FRW with Cosmic String • Roberts Space-time • HMN Space-time (in progress) • Fonarev Space-time (in progress)

  26. FRW with Cosmic String Model by Davies and Sahni (1988) ds2 = a2(t) (-dt2 + dr2 + β2 r2 dφ2 + dz2) where β = 1 – 4μ and μ is the mass per unit length of the cosmic string. This metric is conformally static (actually conformally flat).

  27. Classical Singularity Structure a(t) = 0 : Scalar curvature singularity β2 ≠ 1 : Quasiregular singularity The latter is in the related static metric, it is timelike, and we will investigate its quantum singularity structure.

  28. Quantum Singularity Structure Klein-Gordon Equation with general coupling: |g|-1/2 (|g|1/2 gαβΦ,β),α= (M2 + ξ R) Φ With mode solutions Φ ≈ T(t) H(r) eimφeikz where the T-eqn alone contains M and R,

  29. Schrodinger Form of Radial Equation The radial equation is H’’ + (1/r) H’ + (-k2 – q – (m2/β2r2)) H = 0. Let r = x and H = x u(x) to get correct inner product and 1D Schrodinger Equation, u’’ + (E – V(x))u = 0 Where E = -k2 – q and V(x) = (m2– β2/4)/β2x.

  30. Quantum Singularity Structure Near zero, V(x) is limit point if m2/β2 ≥ 1. The m = 0 mode is clearly limit circle so the singularity is Quantum Mechanically Singular

  31. Roberts Space-time The Roberts (1989 ) metric is ds2 = e2t (-dt2 + dr2 + G2(r) dΩ2) where G2(r) = ¼[ 1 + p – (1 – p) e-2r](e2r -1). It is conformally static, spherically symmetric and self-similar. Classical scalar curvature singularity at r = 0 for 0 < p < 1 that is timelike.

  32. Quantum Singularity Structure Klein-Gordon Equation with general coupling: |g|-1/2 (|g|1/2 gαβΦ,β),α= (M2 + ξ R) Φ With mode solutions Φ ≈ T(t) H(r) Ylm(θ, ϕ) where the T-eqn alone contains M,

  33. Quantum Singularity Structure QM singular if ξ < 2 QM non-singular if ξ≥ 2 Therefore, • Minimally coupled ξ=0 is QM singular • Conformally coupled ξ=1/6 is also QM singular

  34. HMN Space-time (in progress) The HMN (Husain-Martinez-Nunez, 1994 ) metric is ds2= (at+b) [-(1-2c/r)α dt2 + (1-2c/r)-α dr2 + r2(1-2c/r)1-α dΩ2], where 1. c=0: non-static, conformally flat, no time-like singularity 2. c≠0 and a=0: static, time-like singularity 3. c≠0 and a=-1: non-static, BH, time-like singularity at r=2.

  35. Quantum Singularity Structure Klein-Gordon Equation with general coupling: |g|-1/2 (|g|1/2 gαβΦ,β),α= (M2 + ξ R) Φ With mode solutions Φ ≈ T(t) H(r) Ylm(θ, ϕ).

  36. Quantum Singularity Structure Massless and Minimally coupled – QM SINGULAR Massive, generally coupled and, in particular, conformally coupled – IN PROGRESS

  37. Fonarev Space-time (in progress) The Fonarev (1994) metric is ds2 = a2(n) ( -f2(r) dn2 + f-2(r) dr2 + S2(r) dΩ2), where f(r) = (1-2w/r)α/2, S(r) = r2(1-2w/r)1-α, α = λ/(λ2+2)1/2, a(n) = |n/n0|2/(c-2), c=λ2, with 0< λ2≤ 6 and λ2≠2. There is a time-like singularities at r=0 and r=2w in this conformally static space-time. If α2=3/4, w=1, and n0=1, then this space-time overlaps with HMN.

  38. Quantum Singularity Structure Klein-Gordon Equation with general coupling: |g|-1/2 (|g|1/2 gαβΦ,β),α= (M2 + ξ R) Φ With mode solutions Φ ≈ T(n) F(r) Ylm(θ, ϕ). .

  39. Quantum Singularity Structure Massless and minimally coupled – QM SINGULAR Massive, generally coupled and, in particular, conformally coupled – IN PROGRESS

  40. DISCUSSION Horowitz and Marolf Technique to study STATIC spacetimes for Quantum Singularities is easily extended to CONFORMALLY STATIC spacetimes.

  41. Acknowledgements Thanks to Queen Mary, University of London where much of this work was carried out, Professor Malcolm MacCallum for useful discussions and suggestions, and two USNA students B.T. Yaptinchay and S. Hall for discussions.

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