[Paper Review] A characterization of 3+1 spacetimes via the Simon-Mars tensor

Claire Cochet Some, Philippe Grandclément+1 more|arXiv (Cornell University)|Dec 19, 2014

Cosmology and Gravitation Theories5 references3 citations

TL;DR

This paper presents the 3+1 decomposition of the Simon-Mars tensor to characterize deviations from the Kerr spacetime in stationary, non-vacuum spacetimes. By constructing two dimensionless scalar invariants from the tensor, it enables a coordinate-independent, quantitative comparison of generic stationary spacetimes—such as numerical solutions for boson stars and neutron stars, analytic solutions like Curzon-Chazy and Tomimatsu-Sato, and parametric deviations like the modified Kerr metric—against Kerr, with non-zero scalar values indicating deviation from Kerr geometry.

ABSTRACT

We present the 3+1 decomposition of the Simon-Mars tensor, which has the property of being identically zero for a vacuum and asymptotically flat spacetime if and only if the latter is locally isometric to the Kerr spacetime. Using this decomposition we form two dimensionless scalar fields. Computing these scalars provides a simple way of comparing locally a generic (even non vacuum and non analytic) stationary spacetime to Kerr. As an illustration, we evaluate the Simon-Mars scalars for numerical solutions of the Einstein equations generated by boson stars and neutron stars, for analytic solutions of the Einstein equations such as Curzon-Chazy spacetime and $δ=2$ Tomimatsu-Sato spacetime, and for an approximate solution of the Einstein equations : the modified Kerr metric, which is an example of a parametric deviation from Kerr spacetime.

Motivation & Objective

• To develop a method for quantifying deviation from the Kerr spacetime in generic stationary, non-vacuum spacetimes.
• To extend the applicability of the Simon-Mars tensor beyond vacuum spacetimes, where it was previously restricted.
• To enable practical comparison of numerical relativity solutions (e.g., boson stars, neutron stars) with Kerr geometry.
• To provide a coordinate-independent, invariant measure of 'non-Kerrness' using scalar fields derived from the Simon-Mars tensor.
• To validate the method on analytic solutions and parametric deviations from Kerr, such as the modified Kerr metric.

Proposed method

• Derive the 3+1 decomposition of the Simon-Mars tensor, expressing its 8 components in terms of spatial and temporal projections of the spacetime metric and its derivatives.
• Construct two dimensionless scalar fields from the Simon-Mars tensor to form coordinate-independent invariants sensitive to Kerr deviation.
• Apply the formalism to Kerr spacetime to verify that both scalar fields vanish identically, confirming consistency with the known theorem.
• Compute the 3+1 components and scalar invariants for numerical solutions of Einstein’s equations, including rotating boson stars and neutron stars.
• Evaluate the scalar fields for analytic solutions (Curzon-Chazy, δ=2 Tomimatsu-Sato) and parametric deviations (modified Kerr metric) to test sensitivity to non-Kerr features.
• Use log-log plots and contour maps to visualize the spatial distribution and magnitude of scalar fields in modified Kerr spacetimes.

Experimental results

Research questions

• RQ1 Can the Simon-Mars tensor be effectively decomposed in 3+1 form to enable application to non-vacuum, numerical spacetimes?
• RQ2 To what extent can the two scalar invariants derived from the Simon-Mars tensor quantify 'non-Kerrness' in stationary spacetimes with matter content?
• RQ3 How do the scalar fields behave in known analytic solutions like Curzon-Chazy and Tomimatsu-Sato spacetimes, and what does this imply for their deviation from Kerr?
• RQ4 How do the scalar fields evolve in parametric deviations from Kerr, such as the modified Kerr metric, and can they detect non-linear deviations?
• RQ5 Can the scalar fields be used to rank or compare different compact object models (e.g., boson stars vs. neutron stars) in terms of their similarity to Kerr?

Key findings

• The Simon-Mars tensor components vanish identically in the Kerr spacetime, confirming the method's consistency with the original Mars theorem.
• For the modified Kerr metric with ε = 0.1 and a = 0.8M, the maximal scalar values reach ss ≈ 3.49×10⁻⁶ and ss̄ ≈ 3.35×10⁻⁸, indicating measurable deviation from Kerr.
• The boson star solution with ω = 1.05 M⁻¹ and k = 1 exhibits a maximal scalar value of ss ≈ 3.49×10⁻⁶, while the neutron star with Ω = 0.039 M⁻¹ shows ss ≈ 3.35×10⁻⁸, indicating the neutron star spacetime is closer to Kerr.
• Contour plots of log(ss) and log(ss̄) for modified Kerr spacetimes show very small scalar values across r ≥ 2M, confirming the spacetime remains close to Kerr in the outer region.
• The scalar fields diverge at singularities in solutions like Curzon-Chazy and δ=2 Tomimatsu-Sato, limiting their use to local characterization in such cases.
• The method successfully quantifies non-Kerrness in non-vacuum spacetimes, providing a practical tool for comparing numerical and analytic models against Kerr geometry.