In the 3+1 formulation of GR we have the following basic variables:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]g_{ij} = \textrm{metric on a spatial surface}[/tex]

[tex]\pi^{ij} = \textrm{momentum conjugate to }g_{ij}[/tex]

[tex]N^i = \textrm{shift vector}[/tex]

[tex]N = \textrm{lapse function}[/tex]

Both [itex]N[/itex] and [tex]N^i[/itex] are purely gauge variables, so are essentially unimportant. This means that the phase space for GR is essentially the space of pairs [itex](g_{ij},\pi^{ij})[/itex] over a given spatial manifold [itex]\Sigma[/itex]. (I know this isn't really the phase space since GR has constraints, but that's irrelevant for my current question.)

The point is that, in most treatments of which I am aware, the spatial manifold [itex]\Sigma[/itex] is taken to be closed, i.e., compact and without boundary. However, if we extend these ideas by setting [itex]\partial\Sigma\ne0[/itex] then we know that the action for GR also requires a boundary term. Does this then imply that the phase space for GR should be extended to the set

[tex]\{g_{ij},\pi^{ij},\gamma_{AB},p^{AB}\}[/tex]

where [itex]\gamma^{AB}[/itex] is a two-dimensional metric on [itex]\partial\Sigma[/itex] and [itex]p^{AB}[/itex] is its conjugate momentum?

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Phase space for GR

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads for Phase space | Date |
---|---|

B Does the phase of light change as it travels? | Feb 17, 2018 |

B Could a black hole involve a phase transition? | Jun 13, 2016 |

Phase space in special relativity | May 4, 2014 |

Is there a phase space in GR? | Aug 2, 2011 |

Phase, Geodesics, and Space-Time Curvature | Jul 3, 2010 |

**Physics Forums - The Fusion of Science and Community**