[Paper Review] Galactic Halos and Black Holes in Non-Canonical Scalar Field Theories

Ratindranath Akhoury, C. Gauthier|ArXiv.org|Apr 22, 2008

Cosmology and Gravitation Theories27 references7 citations

TL;DR

This paper investigates static, spherically symmetric solutions in non-canonical scalar field theories coupled to gravity to model galactic dark matter halos and black holes. It finds that scalar field configurations with negative energy density near the center can reproduce observed rotation curves with linear rise at small radii, while maintaining positive total mass; it also re-evaluates black hole no-hair theorems, showing loopholes when energy density is negative or boundary conditions are not asymptotically flat.

ABSTRACT

We consider static spherically symmetric solutions of a general scalar field theory with non-standard kinetic energy coupled to gravity with a view to explain dark matter halos as a coherent state of the scalar field. Consistent solutions are found with a smooth scalar profile which can describe observed rotation curves. Some of the solutions have negative scalar energy density near the origin though the total energy is positive definite. The solutions with positive energy density everywhere have a steeper rotation curve near the origin than those that don't. We also reconsider the no scalar hair theorems for black holes with emphasis on asymptotic boundary conditions and super-luminal propagation. Some modifications and extensions of previous analyses are discussed.

Motivation & Objective

• To model galactic dark matter halos as coherent scalar field configurations in non-canonical scalar field theories coupled to gravity.
• To determine whether such scalar field solutions can reproduce observed rotation curves, particularly the linear rise at small radii.
• To re-express and extend the scalar no-hair theorems for black holes in the context of non-standard kinetic energy terms.
• To assess the physical viability of solutions with negative energy density, especially regarding superluminal propagation and causality.
• To examine the coexistence of black holes and scalar halos in a unified framework, particularly under non-asymptotically flat boundary conditions.

Proposed method

• Formulates a general scalar field Lagrangian depending on the scalar field $\phi$ and the kinetic invariant $X = g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi$, without assuming quadratic or separable kinetic terms.
• Derives the energy-momentum tensor and scalar field equations of motion from the action $S = \int d^4x \sqrt{-g} L(X, \phi)$, using variational principles.
• Imposes spherical symmetry and staticity to reduce the Einstein-scalar field equations to a system of ordinary differential equations.
• Analyzes boundary conditions, including asymptotic flatness and behavior at the origin, to classify solutions with positive or negative energy density.
• Applies constraints from energy conditions (e.g., weak energy condition) and examines implications for superluminal signal propagation.
• Revisits the no-hair theorems by analyzing the conditions under which scalar hair can be non-trivial, particularly when energy density is negative or boundary conditions are relaxed.

Experimental results

Research questions

• RQ1 Can non-canonical scalar field theories produce static, spherically symmetric solutions that reproduce observed galactic rotation curves with a linear rise at small radii?
• RQ2 What are the physical and consistency conditions for scalar field configurations with negative energy density near the origin, and can such configurations still have positive total mass?
• RQ3 Under what conditions do scalar no-hair theorems for black holes break down in non-canonical scalar field theories?
• RQ4 Is superluminal signal propagation in these models necessarily inconsistent with causality, especially when the total energy is positive?
• RQ5 Can black holes and scalar halos coexist dynamically in a single solution, particularly when the scalar field has negative energy density in some regions?

Key findings

• Solutions exist with a smooth scalar profile that reproduce observed rotation curves, showing $v_c \sim r$ at small radii, characteristic of a linear rise in velocity.
• Solutions with negative energy density near the origin are classically allowed and can have positive total mass, even though the local energy density is negative.
• The total energy of such configurations remains positive definite, supporting their physical viability despite negative local energy density.
• Superluminal propagation does not necessarily imply causality violation, and such solutions are not ruled out a priori by causality constraints.
• Loopholes in the scalar no-hair theorems are identified when the weak energy condition is violated or boundary conditions are not asymptotically flat.
• Explicit solutions with negative energy density and positive total mass are consistent with known results from other physical systems, such as the Scharnhorst effect and string theory models.