Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Thermodynamic Volume Product in Spherically Symmetric and Axisymmetric
  Spacetime   In this Letter, we have examined the thermodynamic volume products for
spherically symmetric and axisymmetric spacetimes in the framework of
\emph{extended phase space}. Such volume products usually formulated in terms
of the outer horizon~(${\cal H}^{+}$) and the inner horizon~(${\cal H}^{-}$) of
black hole ~ (BH) spacetime. Besides volume product, the other thermodynamic
formulations like \emph{volume sum, volume minus and volume division} are
considered for a wide variety of spherically symmetric spacetime and
axisymmetric spacetimes. Like area~(or entropy) product of multihorizons, the
mass-independent~(universal) feature of volume products are sometimes also
\emph{fail}. In particular for a spherically symmetric AdS spacetimes the
simple thermodynamic volume product of ${\cal H}^{\pm}$ is not
mass-independent. In this case, more complicated combinations of outer and
inner horizon volume products are indeed mass-independent. For a particular
class of spherically symmetric cases i.e. Reissner Nordstr\"om BH of Einstein
gravity and Kehagias-Sfetsos BH of Ho\v{r}ava Lifshitz gravity, the
thermodynamic volume products of ${\cal H}^{\pm}$ is indeed \emph{universal}.
For axisymmetric class of BH spacetime in Einstein gravity all the combinations
are \emph{mass-dependent}. There has been no chance to formulate any
combinations of volume product relation is to be mass-independent.
Interestingly, \emph{only the rotating BTZ black hole} in 3D provides the
volume product formula is mass-independent i.e. \emph{universal} and hence it
is quantized.

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The categories of this paper are: comp-gas
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The categories of this paper are: econ
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The categories of this paper are: math-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: alg-geom
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The categories of this paper are: cs
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The categories of this paper are: math
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The categories of this paper are: eess
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The categories of this paper are: astro-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-bio
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ex