Foliating Spacetime: Understanding the Curves and Metrics

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SUMMARY

The discussion centers on the concept of foliating spacetime into spatial hypersurfaces, denoted as \(\Sigma_t\), using a global time function \(t\). It highlights the existence of orthogonal vectors \(n_\mu\) that generate a flow on spacetime, with a specific focus on whether these curves are geodesics. The conclusion drawn is that for the choice \(N=1\), these curves indeed represent geodesics, as referenced in Gourgoulhon's work, although it is noted that this choice may not be universally applicable across all spacetime configurations.

PREREQUISITES
  • Understanding of general relativity concepts, particularly spacetime and geodesics.
  • Familiarity with the mathematical notation used in differential geometry.
  • Knowledge of the intrinsic metric and its implications in the context of spacetime.
  • Basic comprehension of global time functions and their role in foliating spacetime.
NEXT STEPS
  • Study Gourgoulhon's paper on spacetime foliations, particularly the section before Eq 4.77.
  • Explore the implications of different choices for \(N\) in the context of spacetime geometry.
  • Research the concept of intrinsic metrics and their applications in general relativity.
  • Investigate the mathematical foundations of geodesics in curved spacetime.
USEFUL FOR

This discussion is beneficial for physicists, mathematicians, and students of general relativity who are interested in the geometric structure of spacetime and the mathematical intricacies of foliating spacetime into hypersurfaces.

naima
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Hi all,

I began to read this paper :http://www.physics.adelaide.edu.au/mathphysics/abstracts/ADP-95-11-M28.html"
On page 9 the author foliates spacetime into spatial hypersurfaces, [tex]\Sigma_t[/tex], labeled by a global time function, t.
for each point of [tex]\Sigma^t[/tex] there is a vector [tex]n_\mu[/tex] orthogonal to it
These vectors generate a flow on spacetime.
Physically what are these curves? are they geodesics?
Have you links using his notations for the intrinsic metric?

thanks.
 
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For the choice N=1, they are geodesics. See the discussion just before Eq 4.77 of Gourgoulhon's http://arxiv.org/abs/gr-qc/0703035. His discussion also says it's not always possible to make this choice over the entire spacetime.
 
Great

This is exactly what I was looking for.
 

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