[Paper Review] Spherical Domain Wall formed by Field Dynamics of Hawking Radiation and Structure Near Horizon

Yukinori Nagatani|ArXiv.org|Jul 30, 2003

Black Holes and Theoretical Physics3 citations

TL;DR

This paper proposes that a spherical domain wall forms around a black hole due to field dynamics of Hawking radiation in the gauge-Higgs-Yukawa theory, even when the Hawking temperature is below the critical temperature for a thermal phase transition. Using a general relativistic ballistic model, it shows that radiation pressure from particles tunneling from the horizon stabilizes a domain wall near the horizon, restoring symmetry on the horizon and separating a symmetric phase from a broken phase outside, with the wall's structure arising from field dynamics rather than thermal equilibrium.

ABSTRACT

The Hawking radiation in the vacuum of the spontaneous symmetry breaking in the gauge-Higgs-Yukawa theory is investigated by a general relativistic formulation of the the ballistic model. The restoration of the symmetry on the horizon and the formation of the spherical domain wall around the black hole are shown even if the Hawking temperature is lower than the critical temperature of the phase transition in the gauge-Higgs-Yukawa theory. When the Hawking temperature is much lower than the critical temperature, the domain wall closely near the horizon is formed. The wall is formed by the field dynamics rather than the thermal phase transition.

Motivation & Objective

• To investigate whether Hawking radiation can induce a phase transition and domain wall formation in the gauge-Higgs-Yukawa theory near a black hole.
• To determine if domain walls can form when the Hawking temperature is below the critical temperature for a thermal phase transition.
• To develop a general relativistic formulation of the ballistic model to describe particle trajectories and field dynamics near the horizon.
• To analyze the effective Higgs potential and vacuum expectation value (vev) structure under full general relativistic effects.
• To clarify whether the domain wall arises from field dynamics or requires thermal equilibrium.

Proposed method

• Formulate a general relativistic ballistic model where Hawking-radiated particles follow geodesics in Schwarzschild spacetime.
• Derive the effective Higgs potential including full general relativistic effects, depending on the differential particle-density distribution $ dE \times \mathcal{N}(E,r) $.
• Model the Higgs vev profile $ |\langle\phi(r)\rangle| $ as a function of radial distance $ r $, treating it as a domain wall structure.
• Solve the field equation for the Higgs profile near the horizon using a linearized differential equation in terms of $ s = r_{\text{BH}} + d_{\text{DW}} s $, with boundary conditions ensuring finiteness at the horizon and asymptotic approach to 1 at infinity.
• Use hypergeometric functions to solve the radial profile equation and match solutions numerically to determine coefficients like $ C_+ $.
• Analyze the balance of radiation pressure, wall tension, and potential force to stabilize the domain wall structure.

Experimental results

Research questions

• RQ1 Can a domain wall form around a black hole even when the Hawking temperature is below the critical temperature for a phase transition in the gauge-Higgs-Yukawa theory?
• RQ2 What is the role of general relativistic effects in the formation and stability of the Higgs vev wall near the horizon?
• RQ3 Does the domain wall arise from field dynamics or require local thermal equilibrium and thermal phase transition?
• RQ4 How does the Higgs vev profile behave in the near-horizon limit, and does symmetry restoration occur on the horizon?
• RQ5 What determines the thickness and position of the domain wall, and how do the forces (radiation pressure, tension, potential) balance to stabilize it?

Key findings

• A spherical domain wall forms around the black hole due to field dynamics of Hawking radiation, even when the Hawking temperature is below the critical temperature for a phase transition.
• Symmetry is restored on the horizon, as the Higgs vev vanishes ($ f(0) = 0 $), while the Higgs vev remains finite and non-zero at large distances.
• The domain wall is stabilized by a balance between radiation pressure (from Hawking particles), wall tension, and Higgs potential force.
• Near the horizon, the Higgs profile behaves as $ f(s) \simeq C_+ s^{\sqrt{A^2/2}} $ for small $ A^2 $, with $ C_+ \approx 0.90 $ for $ A^2 = 0.001 $ and a wall extremely close to the horizon.
• The wall thickness $ d_{\text{DW}} $ is determined by the field dynamics and the particle density, with the wall forming even at low Hawking temperatures.
• The solution matches numerical results, confirming that $ C_+ \approx 1.00 $ for $ d_{\text{DW}} = r_{\text{BH}} $ and $ C_+ \approx 1.14 $ for larger walls.