Newtonian limit in cdt

The Newtonian Limit in CDTAdam Getchell (UC Davis) [email protected]

Sunday, April 14, 13

mailto:[email protected]

mailto:[email protected]

Does CDT have a Newtonian Limit?

• CDT looks like GR at cosmological scales, does it have a Newtonian limit?

• 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

F =Gm1m2

r2


A trick from GR• The static axisymmetric Weyl metric

• With two-body solutions

• Leads to a “strut”

• With a stress

• That can be integrated to get the Newtonian force

ds2 = e2�dt2 � e2(⌫��)�dr2 + dz2

�� r2e�2�d�2

�(r, z) = � µ1

R1� µ2

R2, Ri =

qr2 + (z � zi)

2

Tzz =1

8⇡G

⇣1� e�⌫(r,z)

⌘2⇡�(r)

F =

ZTzzdA =

1

4G

⇣1� e�⌫(r,z)

⌘=

Gm1m2

(z1 � z2)2 for µi = Gmi

⌫(0, z) =4µ1µ2

(z1 � z2)2

⌫(r, z) = �µ21r

2

R41

� µ22r

2

R42

+4µ1µ2

(z1 � z2)2

r2 + (z � z1) (z � z2)

R1R2� 1

�


Measuring Mass in CDT

Mass ➔ Epp quasilocal energy

• 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)

EE ⌘ � 1

8⇡G

Z

⌦d

2x

p|�|

⇣pk

2 � l

2 �p

k̄

2 � l̄

2⌘

l ⌘ �µ⌫ lµ⌫ k ⌘ �µ⌫kµ⌫


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 ...


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


Newtonian limit in cdt

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