[Paper Review] Ultra-spinning Chow's black holes in six-dimensional gauged supergravity and their properties

Di Wu, Shuang‐Qing Wu|arXiv (Cornell University)|Jun 27, 2021

Black Holes and Theoretical Physics57 references17 citations

TL;DR

This paper constructs ultra-spinning charged black hole solutions in six-dimensional N = 4 gauged supergravity by applying an ultra-spinning limit to Chow’s rotating charged black holes with equal charge parameters. It demonstrates that thermodynamical quantities satisfy the first law and Bekenstein-Smarr formula, and shows that the reverse isoperimetric inequality (RII) can be violated or obeyed depending on mass and charge parameters, leading to either super-entropic or sub-entropic behavior.

ABSTRACT

By taking the ultra-spinning limit as a simple solution-generating trick, a novel class of ultra-spinning charged black hole solutions has been constructed from Chow's rotating charged black hole with two equal-charge parameters in six-dimensional $\mathcal{N} = 4$ gauged supergravity theory. We investigate their thermodynamical properties and then demonstrate that all thermodynamical quantities completely obey both the differential first law and the Bekenstein-Smarr mass formula. For the six-dimensional ultra-spinning Chow's black hole with only one rotation parameter, we show that it does not always obey the reverse isoperimetric inequality, thus it can be either sub-entropic or super-entropic, depending upon the ranges of the mass parameter and especially the charge parameter. This property is obviously different from that of the six-dimensional singly-rotating Kerr-AdS super-entropic black hole, which always strictly violates the RII. For the six-dimensional doubly-rotating Chow's black hole but ultra-spinning only along one spatial axis, we point out that it may also obey or violate the RII, and can be either super-entropic or sub-entropic in general.

Motivation & Objective

• To construct ultra-spinning charged black hole solutions in six-dimensional N = 4 gauged supergravity via the ultra-spinning limit of Chow’s rotating charged black holes.
• To investigate the thermodynamical properties of these ultra-spinning solutions using conformal Ashtekar-Magnon-Das (AMD) and Abbott-Deser (AD) methods.
• To examine the validity of the reverse isoperimetric inequality (RII) in both singly- and doubly-rotating ultra-spinning cases.
• To determine whether the black holes are super-entropic (RII violated) or sub-entropic (RII obeyed), depending on solution parameters.
• To compare the behavior of ultra-spinning charged black holes with known uncharged super-entropic black holes, particularly in terms of RII violation.

Proposed method

• Apply the ultra-spinning limit to Chow’s six-dimensional rotating charged black hole solution with two equal electric charges in N = 4, SU(2) gauged supergravity.
• Use coordinate transformations to reduce the doubly-rotating solution to a singly-rotating case for initial analysis.
• Employ the conformal Ashtekar-Magnon-Das (AMD) method to compute the thermodynamical mass and angular momentum via boundary stress-energy tensor.
• Use the Abbott-Deser (AD) method to independently calculate the mass and angular momenta, ensuring consistency with AMD results.
• Compute the horizon area and entropy using the Bekenstein-Hawking formula, and verify the differential first law of thermodynamics.
• Evaluate the isoperimetric ratio R = V^{1/3}/A^{1/2} to test the reverse isoperimetric inequality (RII), comparing R to 1 to classify the black hole as super- or sub-entropic.

Experimental results

Research questions

• RQ1 Do the thermodynamical quantities of the ultra-spinning charged black hole in six-dimensional gauged supergravity satisfy the differential first law of thermodynamics?
• RQ2 Is the Bekenstein-Smarr mass formula satisfied for the ultra-spinning charged black hole solutions derived from Chow’s solution?
• RQ3 Does the reverse isoperimetric inequality (RII) always hold for ultra-spinning charged black holes, or can it be violated or obeyed depending on parameters?
• RQ4 How does the electric charge parameter influence the entropic nature (super- or sub-entropic) of the ultra-spinning black hole?
• RQ5 How does the behavior of the RII differ between the singly-rotating and doubly-rotating ultra-spinning cases?

Key findings

• All thermodynamical quantities—mass, angular momentum, entropy, and temperature—computed via both the AMD and AD methods are consistent and satisfy the differential first law of black hole thermodynamics.
• The Bekenstein-Smarr mass formula is fully satisfied for both the singly- and doubly-rotating ultra-spinning charged black hole solutions.
• For the singly-rotating ultra-spinning charged black hole, the reverse isoperimetric inequality (RII) is not universally violated; the black hole can be either super-entropic (R > 1) or sub-entropic (R < 1), depending on the values of the mass and charge parameters.
• In the doubly-rotating case, the RII can be either obeyed or violated, and the black hole can be either super-entropic or sub-entropic depending on the ranges of the rotation and charge parameters.
• The behavior differs significantly from the uncharged singly-rotating Kerr-AdS6 black hole, which always strictly violates the RII, whereas the charged ultra-spinning solution shows parameter-dependent entropic behavior.
• Numerical analysis confirms that for certain charge and rotation values (e.g., q = 0.75, a = 0.9), the isoperimetric ratio R exceeds 1.23, indicating strong super-entropic behavior, while for smaller charges and rotations, R can drop below 1, indicating sub-entropic nature.