1. by Debbie Konkowski (USNA) and Tom Helliwell (HMC) MG12 July 12 -18, 2009

2. T.M. Helliwell and D.A. Konkowski, “Quantum healing of classical singularities in power-law spacetimes,” Class. Quantum Grav. 24 (2007) 3377–3390 K. Lake, “Scalar polynomial singularities in power-law spacetimes,” Gen. Rel. Grav. 40 (2008) 1609-1617

3.  Space-time is smooth!  “singular” point must be cut out of space-time ⇒ leaves hole ⇒ incomplete curves

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

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

6. 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.

8. 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. 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.

9. We consider the class of space-times which can be written with the power-law metric form ds 2 = −r α dt 2 + r β dr 2 + C -2 r γ dθ 2 + r δ dz 2 in the limit of small r, where α, β, γ, δ and C are constant parameters and the variables have the usual ranges. We are particularly interested in the metrics at small r, because we suppose that if the space-time has a classical singularity (and nearly all of these do), it occurs at r = 0.

10. Eliminating α by scaling r results in two metric types: Type I: ds 2 = r β (−dt 2 + dr 2 ) + C -2 r γ dθ 2 + r δ dz 2, if α ≠ β + 2. Type II: ds2 = −r β+2 dt 2 + r β dr2 +C -2 r γ dθ 2 + r δ dz 2, (7) if α = β + 2.

11.  Except for isolated values of β, γ, δ, C all of these power-law space-times have diverging scalar polynomial invariants if and only if β > −2.  Lake (2008) has shown that: (1) Type II space-time singularities at r=0 are null, naked and at finite affine distance ⇒ CLASSICALLY SINGULAR, while those in (2) Type I space-times ‘singularities’ at r=0 are timelike, naked and at finite affine distance if and only if β > −1 ⇒ CLASSICALLY SINGULAR at r=0 only if β > −1..

12. To study the quantum particle propagation in these 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. To study the quantum particle propagation in these 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.

13. Type II space-times are globally hyperbolic; the wave operator in that case must be essentially self-adjoint, so these space-times contain no quantum singularities. It is easy to verify this conclusion directly by checking essential self-adjointness using the wave operator.

14. Bowl Dimensions: Bottom: a β = -2 base plane Sides: (1) two vertical planes: γ+δ = 6 and γ+δ = -2 and (2) two tilted planes: δ = β+2 and γ = β+2. Points within the bowl are QM singular; points outside are QM non-singular.

15.  Type I space-times with −2 < β ≤ −1 are entirely eliminated if either the WEC or the DEC is invoked. (These space-times are non-singular anyway, since they are geodesically complete.)  For any other finite value of β there is a range of parameters γ, δ which satisfies the WEC, and a (nearly always smaller but non-zero) range of parameters which satisfies the DEC. Although either energy condition is helpful in eliminating quantum singularities from power-law space-times, neither is entirely successful.  Of course, neither condition is necessarily valid anyway, since the Casimir effect illustrates that local negative energy densities are possible.

16. A large class of the classically-singular power-law space-times we have examined is quantum-mechanically non-singular. A large class of the classically-singular power-law space-times we have examined is quantum-mechanically non-singular.

17. References: T.M. Helliwell and D.A. Konkowski, “Quantum healing of classical singularities in power-law spacetimes,” Class. Quantum Grav. 24 (2007) 3377–3390 K. Lake, “Scalar polynomial singularities in power-law spacetimes,” Gen.Rel.Grav. 40 (2008) 1609-1617