[Paper Review] Static forms of the Robertson-Walker spacetimes

M. Ibison|arXiv (Cornell University)|Apr 24, 2007

Geophysics and Sensor Technology5 references3 citations

TL;DR

This paper proves that only maximally-symmetric spacetimes—such as Minkowski, de Sitter, and anti-de Sitter—can be expressed in both Robertson-Walker (expanding) form and static form. It exhaustively lists all possible static forms of these spacetimes and generalizes the result to show that only maximally symmetric spacetimes can simultaneously be written in orthogonal-time isotropic form and static form.

ABSTRACT

It is shown that only the maximally-symmetric spacetimes can be expressed in both the Robertson-Walker form and in static form - there are no other static forms of the Robertson-Walker spacetimes. All possible static forms of the metric of the maximally-symmetric spacetimes are presented as a table. The findings are generalized to apply to functionally more general spacetimes: it is shown that the maximally symmetric spacetimes are also the only spacetimes that can be written in both orthogonal-time isotropic form and in static form.

Motivation & Objective

• To determine which Robertson-Walker spacetimes can also be expressed in static form.
• To identify whether any non-maximally-symmetric spacetimes admit both forms.
• To systematically enumerate all possible static forms of maximally-symmetric spacetimes.
• To generalize the duality between static and orthogonal-time isotropic forms to broader classes of spacetimes.
• To clarify the geometric constraints that allow a spacetime to be both static and isotropic in time-orthogonal coordinates.

Proposed method

• Analyzes the metric structure of Robertson-Walker spacetimes under coordinate transformations to static form.
• Applies the condition that a spacetime is static if it admits a timelike Killing vector field with vanishing twist.
• Uses the requirement of spatial isotropy in orthogonal-time coordinates to constrain the metric components.
• Derives the full set of solutions to the Einstein field equations under the combined assumptions of staticity and orthogonality to spatial isotropy.
• Compares the resulting metric forms with the standard Robertson-Walker form to identify overlaps.
• Classifies all solutions as either Minkowski, de Sitter, or anti-de Sitter spacetimes based on curvature and signature.

Experimental results

Research questions

• RQ1 Which Robertson-Walker spacetimes can be written in a static form?
• RQ2 Are there any non-maximally-symmetric spacetimes that admit both static and Robertson-Walker forms?
• RQ3 What are all possible static metric representations of maximally-symmetric spacetimes?
• RQ4 Under what geometric conditions can a spacetime be both static and isotropic in orthogonal-time coordinates?
• RQ5 Is the duality between static and orthogonal-time isotropic forms exclusive to maximally-symmetric spacetimes?

Key findings

• Only maximally-symmetric spacetimes—Minkowski, de Sitter, and anti-de Sitter—can be expressed in both Robertson-Walker and static forms.
• All possible static forms of the maximally-symmetric spacetimes are explicitly listed in a comprehensive table within the paper.
• The paper establishes that the only spacetimes admitting both static and orthogonal-time isotropic forms are the maximally-symmetric ones.
• The result implies a unique geometric rigidity: the combination of staticity and spatial isotropy in orthogonal time coordinates forces maximal symmetry.
• No non-maximally-symmetric spacetime can satisfy both the Robertson-Walker and static metric forms simultaneously.
• The classification confirms that the static forms of these spacetimes are uniquely determined by their curvature and signature.