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Differentiability and other properties of the cosmological volume function
1Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
Letters in Mathematical Physics
|June 29, 2026
Summary
The regular cosmological volume function (τV) is shown to be a continuously differentiable temporal function in many cases. This discovery enables a canonical splitting of the metric tensor and induces a Riemannian metric.
Area of Science:
- Cosmology
- General Relativity
- Differential Geometry
Background:
- The regular cosmological time function is a standard tool in cosmology.
- Previous work introduced the regular cosmological volume function (τV) as an alternative.
- The properties and applications of τV require further investigation.
Purpose of the Study:
- To demonstrate that the regular cosmological volume function (τV) is continuously differentiable.
- To explore the consequences of this differentiability for metric tensor splitting.
- To investigate the induced Riemannian metric and provide further examples.
Main Methods:
- Building on the work of Chruściel, Grant, and Minguzzi.
- Mathematical analysis of the regular cosmological volume function (τV).
- Investigating metric tensor splitting and induced Riemannian metrics.
Main Results:
- The regular cosmological volume function (τV) is proven to be continuously differentiable in many relevant cases.
- This differentiability allows for a canonical splitting of the metric tensor.
- A canonical "Wick-rotated" Riemannian metric is induced.
Conclusions:
- The continuous differentiability of τV offers a robust framework for cosmological analysis.
- The induced Riemannian metric provides new tools for studying spacetime structure.
- Further results and examples are presented to illustrate the utility of τV.
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