[Paper Review] Towards the rotating scalar-vacuum black holes

Yu. F. Pirogov|arXiv (Cornell University)|Jun 20, 2013

Pulsars and Gravitational Waves Research1 references3 citations

TL;DR

This paper critically examines the validity of proposed rotating scalar-vacuum black hole solutions in General Relativity and Brans-Dicke theory, arguing that metrics claimed via the Newman-Janis algorithm—such as Kerr-like and Demianski-like solutions with a massless scalar field—are mathematically inconsistent. The key result is that these metrics fail to satisfy the vacuum field equations except in special cases (e.g., α=1 or a=0), leaving the search for a correct stationary, axisymmetric scalar-vacuum black hole solution unresolved.

ABSTRACT

The problem of merging in black holes the rotation and a massless scalar field is treated. It is argued in the comment that the Kerr-like and Demianski-like metrics in General Relativity with a free massless scalar field or in the free Brans-Dicke theory, claimed so far in the literature as an application of the Newman-Janis algorithm, prove to be invalid. Finding the missing stationary axisymmetric scalar-vacuum solution(s), containing the Kerr one, is still urgent.

Motivation & Objective

• To assess the validity of rotating black hole solutions with a massless scalar field in General Relativity and Brans-Dicke theory.
• To investigate whether the Newman-Janis algorithm correctly generates stationary, axisymmetric scalar-vacuum solutions.
• To identify the conditions under which proposed Kerr-like and Demianski-like metrics satisfy the vacuum field equations.
• To establish that the claimed metrics are invalid for generic values of rotation (a) and scalar coupling (α), leaving the problem of finding a correct solution open.

Proposed method

• Application of the Newman-Janis algorithm to generate a stationary, axisymmetric metric from a static, spherically symmetric scalar-vacuum solution.
• Explicit construction of the proposed Kerr-like metric in null coordinates using parameters r₀ (mass), a (angular momentum), and α (scalar coupling index).
• Computation of the Ricci tensor component R_ur to test consistency with vacuum field equations R_μν = 0, except where influenced by the scalar field.
• Analysis of the condition R_ur = 0 to derive constraints on the parameters, particularly identifying when the metric satisfies the field equations.
• Use of symbolic computation (assisted by computer algebra) to verify the Ricci tensor components and the derived consistency conditions.
• Comparison of the results across different frames (Einstein and Jordan) in Brans-Dicke theory, confirming the invalidity is frame-independent.

Experimental results

Research questions

• RQ1 Does the Newman-Janis algorithm produce a valid stationary, axisymmetric solution for a rotating black hole coupled to a massless scalar field in General Relativity?
• RQ2 Under what conditions does the proposed Kerr-like metric satisfy the vacuum Einstein field equations?
• RQ3 Why do the claimed scalar-vacuum solutions fail when α ≠ 1 and a ≠ 0, despite their formal derivation?
• RQ4 Is the invalidity of the proposed metrics specific to a particular frame (e.g., Einstein or Jordan) in Brans-Dicke theory?
• RQ5 What are the necessary and sufficient conditions for a stationary, axisymmetric scalar-vacuum solution to exist that reduces to the Kerr metric in the absence of the scalar field?

Key findings

• The proposed Kerr-like metric fails to satisfy the vacuum field equations R_μν = 0 for arbitrary values of rotation parameter a and scalar coupling α, except in special cases.
• The Ricci tensor component R_ur vanishes only if F = 0 (implying a = 0) or α = 1, which corresponds to the standard Kerr metric without a scalar field.
• The condition R_ur = 0 leads to a nontrivial algebraic equation that is only satisfied when α = 1, confirming the metric's inconsistency for general α.
• Marginal cases such as r₀ = 0 or α = 0 reduce to flat or spherically symmetric solutions, not valid rotating scalar-vacuum black holes.
• The invalidity of the Kerr-like metric in the Einstein frame of Brans-Dicke theory implies the same failure in the Jordan frame, confirming frame independence of the result.
• The Demianski-like metric, which includes the Kerr-like solution as a special case, is also invalid for the same reasons, leaving the problem of finding a correct solution open.