Missing step in the derivation of the Robertson-Walker metric

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Discussion Overview

The discussion revolves around the derivation of the Robertson-Walker metric in cosmology, specifically addressing the steps involved in ruling out "dtdr" terms when transitioning from a spatially homogeneous and isotropic metric to the general form of the Robertson-Walker metric. Participants explore the implications of spatial homogeneity and isotropy, the choice of coordinates, and the conditions under which certain terms can be eliminated.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants assert that the metric can be expressed as ##ds^2 = - dt^2 + a^2 (t) d \sigma^2## before investigating forms of ##d\sigma^2##, while others challenge this order of steps.
  • There are discussions about the ability to eliminate certain metric terms through coordinate choices, with references to synchronous coordinates.
  • Some participants express skepticism about the universality of choosing such a gauge, questioning whether it can be applied to all stationary metrics, particularly in relation to the Kerr metric.
  • Concerns are raised regarding the implications of spatial homogeneity and isotropy for the structure of the metric, particularly whether it can be shown that the ##\partial_t## vector is orthogonal to spatial sections.
  • One participant notes that the metric is never stationary in certain coordinate systems, while another counters that this is an overstatement, suggesting exceptions exist in the absence of matter.
  • There are references to Carroll's lecture notes and Weinberg's work, with participants seeking a clearer understanding of the implications of these references on the derivation process.
  • Some participants express a desire for a more straightforward explanation of why the Robertson-Walker metric does not contain a "dtdr" term.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the steps involved in the derivation of the Robertson-Walker metric, with multiple competing views on the implications of homogeneity and isotropy, the choice of coordinates, and the nature of stationary versus static metrics.

Contextual Notes

Discussions include limitations regarding the assumptions made about the choice of coordinates and the implications of symmetry in the context of the Robertson-Walker metric. The complexity of proving certain theorems is acknowledged, with references to specific equations and discussions in the literature.

  • #31
Roy_1981 said:
Thats actually not true, you can put a dtdr term, it does not violate spatial isotropy or homogeneity. dtdφ of course does, which is why I did not put in the metric I mentioned in my original post.

This is false. An irremovable r dependence violates homogeneity, an irremovable angular dependence violates isotropy. Thus they are on equal footing.

Just as an aside, homogeneity (everywhere) does not imply isotropy, but isotropy (everywhere) implies homogeneity.
 
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  • #32
Just to tie up a couple of my earlier posts:

(#21) https://www.physicsforums.com/threads/missing-step-in-the-derivation-of-the-robertson-walker-metric.980564/#post-6262656 links a paper which is based on and subsumes Weinberg's theorem. This paper actually implements and makes rigoruous the approach I described (#16) https://www.physicsforums.com/threads/missing-step-in-the-derivation-of-the-robertson-walker-metric.980564/#post-6262586. It first (beginning of proof of theorem 5.1) establishes that synchronous coordinates are achievable in a neighborhood without any homogeneity or isotropy assumptions. Then, in several steps, reaches the global further simplified metric as the most general form given isotropy and homogeneity.
 

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