Quantum Singularities in Static and Conformally

Static Space-Times

Debbie Konkowski (USNA) & Tom Helliwell (HMC)

Ae100prg – Prague, Czech Republic – June 25-29, 2012

1. Singularities a. Classical b. Quantum2. Static space-times – History and Examples3. Conformally static space-timesa. FRW with Cosmic Stringb. Roberts c. HMN (in progress)d. Fonarev (in progress)4. Discussion5. Acknowledgements

Outline

o smooth, C∞ , paracompact, connected Hausdorff manifold M

o Lorentzian metric g

Space-time (M,g)

Space-time is smooth!

“singular” point must be cut out of space-time

⇒ leaves hole ⇒ incomplete curves ⇒ boundary points

Singularity????

Complete: ST ( M, g) + Boundary (∂M)

Cauchy completeness only works with Riemannian metric, not Lorentzian.

How do we complete ST?

How do we define a boundary ∂Mto space-time????

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))

Singularities as Boundary Points

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

Classical Singularity

Singular point q⇓

Does Rabcd or its kth derivative diverge in some PPON frame?⇙⇘

no yes (Ck quasiregular (Ck curvature singularity) singularity) ⇓ Does a scalar in gab, nabcd and Rabcd or its kth derivative diverge?

⇙⇘ no yes (Ck nonscalar (Ck scalar

curvature sing. ) curvature sing.)

Singularity Types (Ellis and Schmidt (1977))

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

Quantum Wavesvs.

Classical Geodesics

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.

Horowitz and Marolf (1995):

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).

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-adjoint if (i) is met and (ii) can be met by expanding the domain of the operator or its adjoint so that it is true.

Essentially Self- Adjoint

1. 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.

Tests for Essential Self-Adjointness

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 ofWeyl (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.

Technique

Basic References:

1. R.M. Wald, J. Math. Phys. 21, 2802 (1980)

2. G.T. Horowitz and D. Marolf, Phys. Rev. D 52, 5670 (1995)

3. A. Ishibashi and A. Hosoya, Phys. Rev. D 60, 104028 (1999)

Quantum Singularity History

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)

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)

cont.

  Classical and quantum properties of a two-sphere singularit

y. 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]  

 

Previous Work on Quantum Singularities – Static STs

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    

And more…

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-Civita Spacetimes 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)

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-Civita Spacetimes 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-Civita spacetimes Authors: D.A. Konkowski, T.M. Helliwell, C. Wieland Journal-ref: Class.Quant.Grav. 21 (2004) 265-272

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”

Conformally Static Space-Times

is

|g|-1/2 (|g|1/2 gαβ Φ,β),α = ξ R Φ

where

ξ is the coupling constant and

R is the scalar curvature

Klein-Gordon w/ general coupling

“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)/2 φ.Since Tc

uv = C2-d Tcuv , the corresponding inner

product is conformally invariant.” I and H (1999)

Conformally Static Spacetimes

4 Examples:

1. FRW with Cosmic String

2. Roberts Space-time

3. HMN Space-time (in progress)

4. Fonarev Space-time (in progress)

Conformally Static Space-times

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).

FRW with Cosmic String

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.

Classical Singularity Structure

Klein-Gordon Equation with general coupling:

|g|-1/2 (|g|1/2 gαβ Φ,β),α = (M2 + ξ R) Φ

With mode solutions

Φ ≈ T(t) H(r) eimφ eikz

where the T-eqn alone contains M and R,

Quantum Singularity Structure

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.

Schrodinger Form of Radial Equation

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

Quantum Singularity Structure

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.

Roberts Space-time

Klein-Gordon Equation with general coupling:

|g|-1/2 (|g|1/2 gαβ Φ,β),α = (M2 + ξ R) Φ

With mode solutions

Φ ≈ T(t) H(r) Ylm(θ, ϕ)

where the T-eqn alone contains M,

Quantum Singularity Structure

QM singular if ξ < 2

QM non-singular if ξ ≥ 2

Therefore,

1. Minimally coupled ξ =0 is QM singular

2. Conformally coupled ξ =1/6 is also QM singular

Quantum Singularity Structure

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],

where1. 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.

HMN Space-time (in progress)

Klein-Gordon Equation with general coupling:

|g|-1/2 (|g|1/2 gαβ Φ,β),α = (M2 + ξ R) Φ

With mode solutions

Φ ≈ T(t) H(r) Ylm(θ, ϕ).

Quantum Singularity Structure

Massless and Minimally coupled – QM SINGULAR

Massive, generally coupled and, in particular, conformally coupled – IN PROGRESS

Quantum Singularity Structure

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.

Fonarev Space-time (in progress)

Klein-Gordon Equation with general coupling:

|g|-1/2 (|g|1/2 gαβ Φ,β),α = (M2 + ξ R) Φ

With mode solutions

Φ ≈ T(n) F(r) Ylm(θ, ϕ)..

Quantum Singularity Structure

Massless and minimally coupled – QM SINGULAR

Massive, generally coupled and, in particular, conformally coupled – IN PROGRESS

Quantum Singularity Structure

Horowitz and Marolf Technique to study

STATIC spacetimes for

Quantum Singularities is easily extended to

CONFORMALLY STATIC spacetimes.

DISCUSSION

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.

Acknowledgements