Newtonian limit in cdt

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Newtonian limit in cdt

  1. 1. The Newtonian Limit in CDT Adam Getchell (UC Davis) acgetchell@ucdavis.eduSunday, April 14, 13
  2. 2. Does CDT have a Newtonian Limit? • CDT looks like GR at cosmological scales, does it have a Newtonian limit? Gm1 m2 F = r2 • At first glance, this is hard: • CDT is not well suited for approximating smooth classical space-times • We don’t have the time or resolution to watch objects fallSunday, April 14, 13
  3. 3. A trick from GR • The static axisymmetric Weyl metric ds2 = e2 dt2 e2(⌫ ) dr2 + dz 2 r2 e 2 d 2 • With two-body solutions q µ1 µ2 2 (r, z) = , Ri = r2 + (z zi ) R1 R2  2 µ2 r 2 1 µ2 r 2 2 4µ1 µ2 r + (z z1 ) (z z2 ) ⌫(r, z) = 4 4 + 2 1 R1 R2 (z1 z2 ) R1 R2 • Leads to a “strut” 4µ1 µ2 ⌫(0, z) = 2 (z1 z2 ) • With a stress 1 ⇣ ⌫(r,z) ⌘ Tzz = 1 e 2⇡ (r) 8⇡G • That can be integrated to get the Newtonian force Z 1 ⇣ ⌫(r,z) ⌘ Gm1 m2 F = Tzz dA = 1 e = 2 for µi = Gmi 4G (z1 z2 )Sunday, April 14, 13
  4. 4. Measuring Mass in CDT Mass ➔ Epp quasilocal energy Z p ⇣p p ⌘ 1 ¯ ¯2 EE ⌘ d2 x | | k2 l2 k2 l 8⇡G ⌦ µ⌫ µ⌫ l⌘ lµ⌫ k⌘ kµ⌫ • In 2+1 CDT, extrinsic curvature at an edge is proportional to the number of connected tetrahedra • In 3+1 CDT, extrinsic curvature at a face is proportional to the number of connected pentachorons (4-simplices)Sunday, April 14, 13
  5. 5. Measuring Mass in CDT Mass ➔ Epp quasilocal energy Z p ⇣p p ⌘ 1 ¯ ¯2 EE ⌘ d2 x | | k2 l2 k2 l 8⇡G ⌦ µ⌫ µ⌫ l⌘ lµ⌫ k⌘ kµ⌫ • In 2+1 CDT, extrinsic curvature at an edge is proportional to the number of connected tetrahedra • In 3+1 CDT, extrinsic curvature at a face is proportional to the number of connected pentachorons (4-simplices)Sunday, April 14, 13
  6. 6. Some Computational Methods • Distance • Calculate single-source shortest path between the two masses using Bellman- Ford algorithm • Modify allowed moves in a sweep to not permit successive moves which increase or decrease distance • Stress • Hmmm ...Sunday, April 14, 13
  7. 7. On-Going work • Based on Rajesh Kommu’s CDT implementation • arxiv.org/abs/1110.6875 • Code is on GitHub (warning: messy!) • https://github.com/ucdavis/CDT • Work in early stages, results “soon” • Thanks to Markus Afshar, Professor Steve Carlip, Joshua Cooperman, Colin Cunliff, Henrique Gomez, Rajesh Kommu, Jonah Miller, and Charles Pierce for many helpful discussionsSunday, April 14, 13

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