Numerical simulations of gravitational singularities

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Numerical simulations of gravitational
singularities
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• Gravitational collapse
• Singularity theorems
• BKL conjecture
• Gowdy spacetimes
• General spacetimes

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Star in equilibrium between gravity and pressure.
Gravity can win and the star can collapse
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Singularity theorems

• Once a trapped surface forms
• (given energy and causality conditions)
• Some observer or light ray ends in a
• Finite amount of time
• Very general circumstances for
• Singularity formation
• Very little information about
• The nature of singularities

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Approach to the singularity

• As the singularity is approached, some terms in
  the equations are blowing up
• Other terms might be negligible in comparison
• Therefore the approach to the singularity might
  be simple

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BKL Conjecture

• As the singularity is approached time derivatives
  become more important than spatial derivatives.
  At each spatial point the dynamics approaches
  that of a homogeneous solution.
• Is the BKL conjecture correct? Perform numerical
  simulations and see

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Gowdy spacetimes

• ds2 e(lt)/2 (- e-2t dt2 dx2)
• e-t eP (dyQdz)2 e-P dz2
• P, Q and l depend only on t and x
• The singularity is approached as t goes to
  infinity

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Einstein field equations

• Ptt e2P Qt2 e-2t Pxx e2(P-t)Qx20
• Qtt 2 Pt Qt e-2t (Qxx 2 Px Qx) 0
• (subscript means coordinate derivative)
• Note that spatial derivatives are multiplied by
  decaying exponentials

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• As t goes to infinity
• P P0(x)tv0(x)
• Q Q0(x)
• But there are spikes

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P
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Q
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General case

• Variables are scale invariant commutators of
  tetrad
• e0 N-1 dt ea eai di
• e0,ea ua e0 (Hdab sab) eb
• ea,eb(2 aadbg eabd ndg) eg

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Scale invariant variables

• dt , Eai di e0,ea/H
• Sab,Aa,Nabsab,aa,nab/H

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• The vacuum Einstein equations become evolution
  equations and constraint equations for the scale
  invariant variables.

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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

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S
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N
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u
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Conclusion

• BKL conjecture appears to be correct
• What remains to be done
• Matter
• (2) Small scale structure