[Paper Review] Energy and Angular Momentum of Dilaton Black Holes

Marcelo Samuel Berman|arXiv (Cornell University)|Apr 6, 2008

Relativity and Gravitational Theory13 references3 citations

TL;DR

This paper derives the energy and angular momentum of rotating dilaton black holes by generalizing the Kerr-Newman solution to include a scalar (dilaton) field, showing that the scalar field reduces gravitational and electromagnetic contributions by a factor of (1−β²), indicating a non-trivial interaction that obscures these fields when the scalar field is strong.

ABSTRACT

Following a prior paper, we review the results for the energy and angular momentum of a Kerr-Newman black hole, and then calculate the same properties for the case of a generalised rotating dilaton of the type derived, without rotation, by Garfinkle, Horowitz, and Strominger (1991; 1992). We show that there is, as far as it refers only to the energy and angular momentum, an interaction among the fields, so that, the gravitational and electromagnetic fields may be obscured by the strength of the scalar field.

Motivation & Objective

• To extend the energy and angular momentum calculations for Kerr-Newman black holes to include a dilaton (scalar) field.
• To investigate how the presence of a scalar field modifies the gravitational and electromagnetic contributions to energy and angular momentum.
• To determine whether the scalar field induces an effective neutralization of gravity and electromagnetism in rotating black hole solutions.
• To derive a new metric for the rotating dilaton black hole that reduces to known solutions (Kerr-Newman, Reissner-Nordström, Garfinkle-Horowitz-Strominger) in limiting cases.
• To examine the role of the parameter β in modulating the strength of the scalar field's influence on energy and angular momentum.

Proposed method

• Adapts the action principle from Garfinkle, Horowitz, and Strominger (1991, 1992) to include a dilaton field Φ and electromagnetic tensor Fμν, with coupling parameter β.
• Derives field equations for the scalar field and electromagnetic field, and solves them in the slow-rotation approximation (a ≪ r).
• Applies the Abbott-Deser formalism to compute energy and angular momentum in the weak-field, slow-rotation limit.
• Constructs a new metric ansatz (Eq. 33) that satisfies boundary conditions: reduces to Kerr-Newman when β=0, to Garfinkle-Horowitz-Strominger when a=0, and to Reissner-Nordström when both β=0 and a=0.
• Uses perturbative expansion in powers of (a/ϱ) up to third order to derive expressions for energy and angular momentum.
• Identifies gravitomagnetic contributions ΔE and ΔJ as corrections proportional to (1−β²), showing suppression of self-energy by the scalar field.

Experimental results

Research questions

• RQ1 How does the inclusion of a dilaton field alter the energy and angular momentum of a rotating black hole compared to the Kerr-Newman case?
• RQ2 What is the role of the coupling parameter β in modifying the contributions of gravitational and electromagnetic fields to the total energy and angular momentum?
• RQ3 Can the scalar field effectively shield or reduce the influence of gravity and electromagnetism in the black hole's energy-momentum content?
• RQ4 What metric structure is required to consistently describe a rotating dilaton black hole that reduces to known solutions in limiting cases?
• RQ5 How do the gravitomagnetic contributions ΔE and ΔJ emerge in the slow-rotation limit, and what is their dependence on β?

Key findings

• The energy and angular momentum of the dilaton black hole are reduced by a factor of (1−β²) compared to the Kerr-Newman case, indicating suppression of gravitational and electromagnetic self-energies.
• The gravitomagnetic contributions to energy and angular momentum are given by ΔE ≈ −(M²a²)/(3R³)(1−β²) and ΔJ ≈ −(2M²a)/(3R)(1−β²), showing a negative correction due to the scalar field.
• The scalar field contributes a positive energy term β²(M²+Q²+P²)/(2r), but its energy density is negative: −[β²(M²+Q²+P²)]/(8πR⁴), a hallmark of scalar field behavior.
• The derived metric (Eq. 33) satisfies all required limiting cases: it reduces to Kerr-Newman when β=0, to Garfinkle-Horowitz-Strominger when a=0, and to Reissner-Nordström when both β=0 and a=0.
• The results do not match those of Chamorro and Virbhadra (1996) except in the special case M=P=0, due to the inclusion of rotation and the scalar field's dynamic role.
• The scalar field induces a relative interaction between matter, charge, and the scalar field, suggesting a possible mechanism for effective gravitational screening at small scales via (1−β²) suppression.