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Thermodynamics of Black Holes in Schroedinger Space

Daiske Yamada

TL;DR

This work addresses the thermodynamics of black holes in spaces with Schrödinger symmetry by developing a subtraction-based regularization with partial boundary matching to render the on-shell action finite. The authors validate the method on the Schrödinger black brane and then apply it to a Schwarzschild–AdS–like Schrödinger black hole, showing that the resulting actions reproduce those of their AdS counterparts. They derive the full thermodynamics and phase structure, finding a Hawking–Page transition for the black hole and an instability line in the chemical-potential–temperature plane, while the brane case remains thermodynamically favored over thermal Schrödinger space. These results support a deep link between Schrödinger holography and non-relativistic CFT thermodynamics and provide a practical regularization framework for non-AdS holographic setups.

Abstract

A black hole and a black hyperboloid solutions in the space with the Schroedinger isometries are presented and their thermodynamics is examined. The on-shell action is obtained by the difference between the extremal and non-extremal ones with the unusual matching of the boundary metrics. This regularization method is first applied to the black brane solution in the space of the Schroedinger symmetry and shown to correctly reproduce the known thermodynamics. The actions of the black solutions all turn out to be the same as the AdS counterparts. The phase diagram of the black hole system is obtained in the parameter space of the temperature and chemical potential and the diagram contains the Hawking-Page phase transition and instability lines.

Thermodynamics of Black Holes in Schroedinger Space

TL;DR

This work addresses the thermodynamics of black holes in spaces with Schrödinger symmetry by developing a subtraction-based regularization with partial boundary matching to render the on-shell action finite. The authors validate the method on the Schrödinger black brane and then apply it to a Schwarzschild–AdS–like Schrödinger black hole, showing that the resulting actions reproduce those of their AdS counterparts. They derive the full thermodynamics and phase structure, finding a Hawking–Page transition for the black hole and an instability line in the chemical-potential–temperature plane, while the brane case remains thermodynamically favored over thermal Schrödinger space. These results support a deep link between Schrödinger holography and non-relativistic CFT thermodynamics and provide a practical regularization framework for non-AdS holographic setups.

Abstract

A black hole and a black hyperboloid solutions in the space with the Schroedinger isometries are presented and their thermodynamics is examined. The on-shell action is obtained by the difference between the extremal and non-extremal ones with the unusual matching of the boundary metrics. This regularization method is first applied to the black brane solution in the space of the Schroedinger symmetry and shown to correctly reproduce the known thermodynamics. The actions of the black solutions all turn out to be the same as the AdS counterparts. The phase diagram of the black hole system is obtained in the parameter space of the temperature and chemical potential and the diagram contains the Hawking-Page phase transition and instability lines.

Paper Structure

This paper contains 11 sections, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Black Brane
  3. The Solution
  4. The Difference Action
  5. Thermodynamics
  6. Black Hole
  7. The Solution
  8. The Difference Action
  9. Thermodynamics
  10. Discussions
  11. Black Hyperboloid

Figures (1)

  • Figure 1: The phase diagram in the parameter space of the chemical potential $\hat{\mu}$ and the temperature measured in the units of $R$. The dotted and solid curves represent the instability curve (\ref{['eq:instability']}) and the Hawking-Page phase transition curve (\ref{['eq:HPtrans']}), respectively. The line at $\hat{\mu}=1/2$ is the restriction (\ref{['eq:muRestrict']}). The area between the dotted and solid curves is the spinodal phase where the locally stable black hole solution exists but it is not energetically preferred.