Download Konkowski_Deborah - Relativity and Gravitation – 100 years

Debbie Konkowski (USNA)
& Tom Helliwell (HMC)
Ae100prg – Prague, Czech Republic – June 25-29, 2012
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
o
o
smooth, C∞ , paracompact,
connected Hausdorff manifold M
Lorentzian metric g
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Space-time is smooth!
“singular” point must be cut out of
space-time
⇒ leaves hole
⇒ incomplete curves
⇒ boundary points
Complete: ST ( M, g) + Boundary (∂M)
Cauchy completeness only works with
Riemannian metric, not Lorentzian.
How do we complete ST?
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))
“ a singularity is indicated by
incomplete geodesics or
incomplete curves of bounded
acceleration in a maximal
space-time” (Geroch (1968))
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.)
What happens if instead of classical
particle paths (timelike and null
geodesics) one used quantum
mechanical particles (QM waves)
to identify singularities????
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.
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.
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.
1.
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.
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 quantummechanically nonsingular.
The results depend on spacetime metric parameters and wave equation
modes.
Basic References:
1.
1.
1.
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)
1.
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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)
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2.
Quantum singularities in static and conformally static spacetimes
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)
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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]
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arXiv:hep-th/0602207
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Title: Scalar Field Probes of Power-Law Space-Time Singularities
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Authors: Matthias Blau, Denis Frank, Sebastian Weiss
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Comments: v2: 21 pages, JHEP3.cls, one reference added
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Journal-ref: JHEP0608:011,2006
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arXiv:0911.2626
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Title: Quantum Singularities Around a Global Monopole
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Authors: João Paulo M. Pitelli, Patricio S. Letelier
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Comments: 5 pages, revtex
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Journal-ref: Phys.Rev.D80:104035,2009
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arXiv:0805.3926
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Title: Quantum singularities in the BTZ spacetime
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Authors: João Paulo M. Pitelli, Patricio S. Letelier
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Comments: 6 pages, rvtex, accepted for publication in PRD
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Journal-ref: Phys.Rev.D77:124030,2008
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arXiv:0708.2052
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Title: Quantum Singularities in Spacetimes with Spherical and Cylindrical Topological Defects
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Authors: Paulo M. Pitelli, Patricio S. Letelier
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Comments: 7 page,1 fig., Revtex, J. Math. Phys, in press
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Journal-ref: J.Math.Phys.48:092501,2007
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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”
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
“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 Tcuv = C2-d Tcuv , the corresponding inner
product is conformally invariant.”
I and H (1999)
4 Examples:
1.
FRW with Cosmic String
2.
Roberts Space-time
1.
HMN Space-time (in progress)
1.
Fonarev Space-time (in progress)
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).
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.
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,
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.
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
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.
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,
QM singular if ξ < 2
QM non-singular if ξ ≥ 2
Therefore,
1.
Minimally coupled ξ =0 is QM singular
1.
Conformally coupled ξ =1/6 is also
QM singular
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.
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).
Massless and Minimally coupled –
QM SINGULAR
Massive, generally coupled and, in
particular, conformally coupled – 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.
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).
.
Massless and minimally coupled –
QM SINGULAR
Massive, generally coupled and, in
particular, conformally coupled – IN
PROGRESS
Horowitz and Marolf Technique
to study
STATIC spacetimes
for
Quantum Singularities
is easily extended to
CONFORMALLY STATIC spacetimes.
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.