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Numerical simulations of classical and quantum spikes

Numerical simulations of classical and quantum spikes. David Garfinkle Banach Center conference Warsaw, June 28, 2016. Classical: work in progress with Frans Pretorius. Quantum: work in progress with Ewa Czuchry, John Klauder, and Wlodek Piechocki. Singularities and BKL conjecture

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Numerical simulations of classical and quantum spikes

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  1. Numerical simulations of classical and quantum spikes David Garfinkle Banach Center conference Warsaw, June 28, 2016

  2. Classical: work in progress with Frans Pretorius. • Quantum: work in progress with Ewa Czuchry, John Klauder, and Wlodek Piechocki.

  3. Singularities and BKL conjecture • Spikes and their resolution • Quantum destruction of spikes?

  4. The nature of spacetime singularities Singularity theorems tell us that spacetime singularities form in gravitational collapse, but give us very little information about their properties. A blow up of curvature might lead to simple behavior dominated by a few terms in the equations of motion. BKL conjecture that time derivatives are more important than space derivatives.

  5. Scale invariant variables Variables are scale invariant commutators of tetrad (Uggla et al, DG, Gundlach) e0 = N-1 dt e = ei di [e0,e]= u e0 – (H + ) e [e,e]=(2 a[] +  n) e

  6. Scale invariant variables {dt , Ei di} = {e0,e}/H {,A,N}={,a,n}/H

  7. The vacuum Einstein equations become evolution equations and constraint equations for the scale invariant variables. Choose CMC slicing to asymptotically approach the singularity.

  8. In CMC slicing H=(1/3)e-t CabcdCabcd=e-4t F[,N,…]

  9. Results of simulations Spatial derivatives become negligible At each spatial point the dynamics of the scale invariant variables becomes a sequence of “epochs” where the variables are constant, punctuated by short “bounces” where the variables change rapidly

  11. N

  12. Describing and resolving spikes • Spikes occur where  gets stuck in the old phase due to vanishing of N. • BKL dynamics described by ODE, which can be solved parametrically in closed form. • Eigenvectors of  and N are asymptotically constant, so we only need the behavior of their invariants.

  13. Invariant of N

  14. Invariant of 

  15. 2D invariant of 

  16. Close up of spike and computation: 

  17. Close up of spike and computation: N

  18. Quantum destruction of spikes? • Spikes involve special points staying at an unstable equilibrium. • Quantum mechanics tends to forbid unstable equilibrium dynamics (e.g. upside down harmonic oscillator) • Therefore quantum effects may destroy spikes.

  19. AHS system • Quantum treatment requires a Hamiltonian system. • Spacetime singularities like system of scale invariant variables. • AHS give a scale invariant system of loop quantum gravity type variables that form a Hamiltonian system. • BKL ansatz allows treatment of system at a single spatial point.

  20. A wave packet centered around the exceptional behavior of the spike point may spread to be predominantly non-exceptional behavior. • Imposition of the Hamiltonian constraint becomes a wave equation in a potential.

  21. Conclusions • Spikes in singularities are a numerical challenge. • However, through a combination of numerical and analytic methods, we now understand them. • But quantum effects may destroy spikes.

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