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\begin{document}
\preprint{ }
\title[BH Thermodynamics and GW150914]{Test of the Second Law of Black Hole Thermodynamics with the LIGO event GW150914}
\author{Unnikrishnan.~C.~S.}
\affiliation{Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India}
\keywords{Gravitational Waves, LIGO, Binary Black Holes, Area theorem, Black hole entropy}
\pacs{PACS number}
\begin{abstract}
The recent LIGO discovery of the binary black holes merging and forming a
single Kerr black hole provides the first and unique opportunity to test the
black hole area-entropy theorem or the second law of black hole
thermodynamics. We discuss the test of the entropy law \ using the mass and
spin estimates from the LIGO event GW150914. Because both the initial and
final states consist only of black holes with high entropy and coherent
gravitational waves with very low entropy, the test is essentially geometrical
and ideal. However, the precision and the test itself are limited by
interdependencies and errors in parameter estimation. Future studies on
similar BBH events are critical precision tests of black hole area-entropy theorem.
\end{abstract}
\date{11 February 2016: Tribute to J. D. Bekenstein (1947-2015)}
\maketitle
\section{Black hole area-entropy theorem}
The relation between the classical area of the black hole horizon and to the
entropy in black hole thermodynamics is well known \cite{Bekenstein}. The
second law of black hole thermodynamics states that the area-entropy
$S_{bh}=A/4l_{p}^{2}$, where $A$ is the area of the black hole horizon and
$l_{p}$ the Planck length , cannot decrease in any classical physical process.
The generalized second law then is that the sum of the entropies of the
thermodynamical system outside the black hole and that of the black hole
cannot decrease in any physical process that includes interactions of the two.
For a real astrophysical Kerr black hole with spin (and relatively negligible
charge as we expect), the entropy law includes the angular momentum, with the
area given by
\begin{equation}
A=4\pi(r_{h}^{2}+r_{j}^{2})
\end{equation}
where $r_{h}=r_{M}+\left( r_{M}^{2}-r_{j}^{2}\right) ^{1/2}$ with
$r_{M}=GM/c^{2}$ and $r_{j}=J/Mc$ is the specific intrinsic angular momentum.
The dimensionless spin parameter $a=r_{j}/r_{M}.$
For the special case of two black holes in a binary system spiralling in
emitting gravitational waves, the initial entropy is purely geometrical, that
of the two black holes, with no other form of entropy present. Since the final
state also is a black hole, the final entropy is also purely geometrical, but
for the entropy in the emitted gravitational waves. However, in this specific
case of binary black holes or neutron stars, the gravitational waves are
coherent and quasi monochromatic, with a chirp waveform that increases in
frequency and amplitude all the way to merger. Since the gravitational wave
coherent state is quantum mechanically a pure state \cite{Unni-GTG-GWoptics},
its entropy is zero. Even if the actual wave is mildly multi-modal, the
entropy is negligible, at most the logarithm of number of modes. This then is
the most ideal situation to test the area-entropy law for black holes and we
expect that the sum of the geometrical entropies of the binary black holes is
nearly equal to that of the final black hole. This puts fairly severe
constraints on the spin of the final black hole, especially if the initial
spins are insignificant compared to the total orbital angular momentum.
Remarkably, the recent LIGO gravitational wave event GW150914
\cite{LIGO-detection} is exactly such a geometrically pure astrophysical event
in which to stellar mass black holes with comparable masses coalesced into a
single Kerr black hole emitting about 5\% of the total gravitational energy in
gravitational waves. We discuss how well this event tests the second law of
black hole thermodynamics and show that such events will provide precision
tests of the area-entropy law as the sensitivity of multiple detectors improves.
\section{LIGO event GW150914}
The LIGO event GW150914 is the centre of attention at present and has been
discussed extensively. For our analysis we only need the set of parameters
estimated from the analysis of the chirp waveform reconstructed from the
signals detected in the aLIGO detectors at Hanford and Livingston
\cite{LIGO-detection, LIGO-PE}. The masses are determined to about 10\%
accuracy. These are, for the BH in the binary, $M_{1}/M_{\odot}=36.3_{-4.5}%
^{+5.3}$, $M_{2}/M_{\odot}=28.3_{-4.2}^{+4.4}$ and $M/M_{\odot}=62.0_{-4.0}%
^{+4.4}.$ The initial spins are not well-determined, except as an upper limit
$a_{1}<0.65$ for the heavier BH in the binary. The actual parameter estimates
are consistent with nearly zero or spin smaller than $a\simeq0.1.$ The spin of
the final BH is much better determined because it is dominated by the orbital
angular momentum of binary, which is well determined as $a=0.67_{-0.08}%
^{+0.06}$ from the chirp waveform and mass parameters.
\section{Test of BH area-entropy theorem}
The validity of the black hole area-entropy theorem and the second law of BHT
demands that $S_{bh}+S_{GW}\geq S_{1}+S_{2}.$ We have already reasoned that
$S_{GW}$ is negligible for the coherent quasi-monochromatic waves. Therefore
we test for $S_{bh}\geq S_{1}+S_{2}$ noting that $S_{GW}$, if any, makes the
inequality more valid. This is equivalent to testing $A_{f}\geq A_{1}+A_{2}.$
The area of the horizon of a Kerr black hole is given by%
\begin{equation}
A=4\pi(r_{h}^{2}+r_{j}^{2})
\end{equation}
with
\begin{equation}
r_{h}=\frac{GM}{c^{2}}+\left[ \left( \frac{GM}{c^{2}}\right) ^{2}-\left(
\frac{J}{Mc}\right) ^{2}\right] ^{1/2}=r_{M}+\left( r_{M}^{2}-r_{j}%
^{2}\right) ^{1/2}%
\end{equation}
We get
\begin{equation}
A=4\pi(r_{h}^{2}+r_{j}^{2})=8\pi r_{M}^{2}\left( 1+\frac{\left( r_{M}%
^{2}-r_{j}^{2}\right) ^{1/2}}{r_{M}}\right) =8\pi r_{M}^{2}\left(
1+(1-a^{2})^{1/2}\right)
\end{equation}
For the Schwarzschild BH without spin, $a=0,$ and horizon radius
$r_{S}=2GM/c^{2},$ \ the area $A=16\pi r_{M}^{2}=4\pi r_{S}^{2}.$
For GW150914, we have $A_{1}\leq16\pi\left( GM_{1}/c^{2}\right) ^{2}$ and
$A_{2}\leq16\pi\left( GM_{2}/c^{2}\right) ^{2},$ where $M_{1}=\left(
29\pm4\right) M_{\odot}$ and $M_{1}=\left( 36\pm5\right) M_{\odot}.$ For
the final black hole, the estimated parameters for mass $M_{1}=\left(
62\pm4\right) M_{\odot}$ and spin $a\leq0.7$. In principle, and ideally, the
mass of the final black could be estimated from the ring-down part of the
detected waveform, which makes it independent of the chirp part of the
waveform. The frequency of the lowest mode (n=0,l=2), is inversely
proportional to the mass, and one expects about 300-350 Hz for the
gravitational waves for the spinning 60 solar mass black hole \cite{QNM-llr} .
However, for this particular first detection, detailed numerically relativity
fits with general relativistic continuity of the whole waveform, with
parameters of the binary, had to be used to obtain accurate estimate of the
parameters of the final black hole \cite{LIGO-PE}. \ This already implies an
agreement with the area-entropy law \cite{Badri}. However, the stand taken in
this paper is that the area-entropy law as a law of black hole thermodynamics
is a general deep feature of gravity, independent of general relativity and
that it could be as fundamental as the conservation laws. In this sense, the
law is a fundamental constraint on all theories of gravity (there could be
small quantum corrections, irrelevant for astrophysical situations). For
example, one can think of estimating the\ allowed mass-spin parameter space of
the final black hole using only the thermodynamical relation and not the
explicit general relativistic calculations and fits, when the final part of
the waveform is not available. We assume that future improvements in
sensitivity will allow fairly good estimates of the mass of the final black
hole from ring down and does not use the masses of the initial black holes in
an essential way, allowing genuine precision tests. To indicate how the test
goes, we nevertheless use the mass and spin estimates for the final black hole
as estimated using numerical relativity fits \cite{LIGO-PE} and get
\begin{equation}
A_{f}=8\pi r_{0}^{2}\left[ 1+(1-a^{2})^{1/2}\right] \geq\left(
3.61\pm0.47\right) \times10^{11}m^{2}%
\end{equation}
Similarly we can estimate the sum of the areas of the (non-spinning) black
holes in the binary using the their mass estimates and the knowledge that spin
values are consistent with $a<0.1$ as,
\begin{equation}
A_{1}+A_{2}<\left( 2.36\pm0.48\right) \times10^{11}m^{2}%
\end{equation}
This is very interesting because the fairly close (blurred by the statistical
uncertainties) boundaries show that the the final black hole could not be
spinning much faster if the smaller two had no significant spin to start with.
For example, with both initial spins near zero, the final black hole spinning
at $a>0.95$ may indicate a violation of the second law of black hole
thermodynamics. In other words, the second law of black hole thermodynamics is
useful to constrain spin value of the final black hole, with only partial
knowledge of its other relevant parameters. We can already see that if the
spin parameter is known, the mass parameter indeed can be estimated using the
second law alone because with only black holes and waves in the initial and
final states, we expect the area-entropy theorem satisfied as a near equality.
There is another interesting issue that could be raised. In such a clean
system with only geometric entropies, one does not expect the final black hole
to have significantly larger entropy than the sum of the two initial
entropies. This suggests that the central value of the final black hole could
be somewhat smaller or that the spin could be slightly larger than the
estimated value. A proper treatment of this should estimate the relveant
probabilities for the agreement with the area law and this has been addressed
in the context of the test of the area theorm for GW150914 by W. del Pozzo
and S. Vitale \cite{del Pozzo}. However, a good test requires higher precision
in parameter estimates and this is expected in some future such events, when
the detector is operating at its full sensitivity. Yet, it is clear that the
first detection by LIGO has shown the opportunity to test the second law of
black hole thermodynamics for the first time in a real astrophysical event.
\section{Concluding comments}
The LIGO event GW150914 is remarkable in that apart from the double discovery
of action of gravitational waves on a terrestrial detector, it also discovered
the first stellar mass binary black hole system that spiralled in and merged
to form a single Kerr black hole, promptly ushering in a new wave in
astronomy. While straightforward, it is truly interesting that the first
observation already allows physical tests only imagined so far in highly
theoretical and hypothetical scenarios. This by itself does not prove that
ideas of black hole thermodynamics and its hypothetical, yet essential,
quantum basis are correct. However, this seems to be a first step with its own
deep significance. No doubt, future BBH events in aLIGO and other future
detectors in the network will allow precision tests of the second law of black
hole thermodynamics, and perhaps more, and a violation would of course be
highly significant and paradigm changing.
\begin{thebibliography}{9} %
\bibitem {Bekenstein}J. D. Bekenstein, Phys.\ Rev. D \textbf{7}, 2333 (1973).
\bibitem {Hawking}S. W. Hawking, Phys.\ Rev. Lett. 26, 1344 (1971).
\bibitem {Unni-GTG-GWoptics}C. S. Unnikrishnan and G. T. Gillies, Class.
Quantum Grav. \textbf{32,} 145012 (2015).
\bibitem {LIGO-detection}B. P. Abbott, \textit{et al., }LIGO Scientific
Collaboration and Virgo Collaboration, Phys. Rev. Lett. \textbf{116}, 061102 (2016).
\bibitem {LIGO-PE}B. P. Abbott, \textit{et al}., LIGO Scientific Collaboration
and Virgo Collaboration, \textit{Properties of the binary black hole merger
GW150914}, LIGO document P1500218 (2016).
\bibitem {QNM-llr}K. D. Kokkotas and B. G. Schmidt, \textit{Quasi Normal Modes
of Stars and Black Holes}, Living Reviews in\ Relativity, lrr-1999-2.
\bibitem {Badri}I thank Badri Krishnan for emphasizing and pointing this out
to me.
\bibitem {del Pozzo}W. del Pozzo \textit{et al}., GW150914: experimental
verification of Hawking's area theorem, LIGO Document P1600008-v2 (22/01/2016).
\end{thebibliography}
\end{document}