This document discusses whether causal dynamical triangulation (CDT) has a Newtonian limit similar to general relativity. It presents a method of approximating static, axisymmetric solutions using a Weyl metric that leads to a "strut" and stress tensor that can reproduce the Newtonian force law when the masses are identified with the quasilocal energy. Computational methods for measuring distance and stress in CDT are also outlined. The work is based on an existing CDT implementation and is in early stages.

1. The Newtonian Limit in CDT
Adam Getchell (UC Davis) acgetchell@ucdavis.edu
Sunday, April 14, 13

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 fall
Sunday, April 14, 13

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. 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. 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. 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. 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 discussions
Sunday, April 14, 13