How the Tiger got its Stripes

By Nicolas Yunes, Stephon Alexander, and Kent Yagi

Nature is sometimes lazy and messy, like a 4 year-old that likes to play with all of the toys and put none of them away, always increasing the degree of disorder. In physics, we quantify disorder through the concept of entropy. The tendency of systems to always increase their entropy is encoded in the second law of thermodynamics. Aside from a kid’s disorganised bedroom, another place in the universe with a tremendous amount of entropy is a black hole. Bekenstein and Hawking [1-4] proved that the entropy of a black hole in General Relativity is proportional to its area. For astrophysical black holes of stellar-mass, this yields a value of entropy of about 1056 Joules per Kelvin. The reason for this large value is their incredibly small temperature, about 10-9 Kelvin for a stellar-mass black hole, which in fact cannot be any smaller for an object of its size due to Heisenberg’s uncertainty principle.


Yunes is an Associate Professor of Physics at Montana State University and co-founder of the eXtreme Gravity Institute.


Alexander is a Professor of Physics at Brown University.


Yagi is an Assistant Professor of Physics at the University of Virginia.

The entropy area scaling is surprising because the entropy of extensive systems in statistical mechanics is proportional to their volume, not their area. In fact, Bekenstein’s and Hawking’s observation was a strong motivator for the concept of holography. The typical rationalisation of the area result is that black holes are special objects because, by definition, they possess a horizon. The argument is then that the entropy is proportional to the area because somehow the internal degrees of freedom of a black hole are imprinted on the surface area associated with its horizon. The consequence of this argument is that the only objects in nature whose entropy scales with their surface area are those with event horizons. In particular, the entropy of stars, be them compact or not, should scale with their volume and not their area.

But certain relativistic stars, neutron stars to be exact, have recently been shown to possess approximately universal relations that are similar to the black hole no hair theorems [5,6]. More precisely, the quadrupole moment, the moment of inertia and the tidal Love number (which characterises the tidal deformability of an object due to some external tidal field) of neutron stars are related to each other in a way that is very insensitive to their internal composition or equation of state (hence the name “universal I-Love-Q” relations), which is largely unknown [7], even if the latter includes phase transitions and quark-gluon plasmas in the inner core. Moreover, Yagi and Yunes recently showed that the more compact the neutron star, the more insensitive these relations are, and the closer the quadrupole moment, the moment of inertia and the tidal Love number become to those of a black hole [8]. Of course, this statement cannot be made arbitrarily precise, because a sequence of neutron stars parameterised by their central density can only go up to the Buchdahl limit (the stellar compactness being 4/9=0.444…) at most, leaving a gap before reaching compactnesses appropriate to a black hole. Nonetheless, exploring this gap through a sequence of neutron stars with anisotropic pressure, one finds that indeed the moment of inertia, the quadrupole moment and the tidal Love number flow to their black hole limits smoothly.

These intriguing results have direct applications to gravitational wave astrophysics, nuclear physics and experimental relativity. For example, the LIGO collaboration recently used the I-Love-Q relations, together with another version of these universal relations between tidal parameters of neutron stars that Yagi and Yunes discovered [9], to constrain the equation of state of neutron stars in a model-independent way with gravitational waves emitted by a pair of merging neutron stars [10]. Similarly, the relations also allow for model-independent tests of General Relativity that are insensitive to the equation of state, and thus, do not suffer from degeneracies with the star’s unknown internal structure. Recently, however, we began to ponder whether these relations also had implications on fundamental theoretical physics that perhaps could allow us to better understand the transition from a neutron star to a black hole.

This brings us back to entropy. The entropy of a typical neutron star is about 1040 Joules per Kelvin, many orders of magnitude lower than that of a black hole, because neutron stars are much hotter at temperatures of about 106 Kelvin. Moreover, neutron stars have a surface instead of a horizon, so one expects this entropy to scale with their volume. Contrary to all of this, we have found that a certain dimensionless combination of the entropy, the temperature and the mass of a neutron star scales with the area and approaches the corresponding value for a black hole, in a manner that is insensitive to the star’s equation of state [11]. Said a different way, the entropy, the mass and the temperature scale together in such a way that the result depends only on the compactness of the object, with all other dependencies essentially effacing away. And this effacement holds over a tremendously large number of scales, from QCD scales for a neutron star to the Planck scale for a black hole.

Consider the difference in temperatures, for example. As Roberto Emparan pointed out when we discussed our work with him [12], the Hawking temperature of a black hole scales inversely with its mass, which corresponds to its size, and thus, variations in its thermal radiation are controlled by its thermal wavelength, which depends on the entire black hole all at once. On the other hand, the thermal wavelength of a neutron star is much, much smaller, depending only on local physics, on scales much smaller than the size of the neutron star. Despite this, our results indicate that for sufficiently compact stars, their temperature does scale inversely with their mass near the threshold of gravitational collapse, but only after integrating over the entire star and combining effects from the micro-physics and from relativistic gravity. Such a relation, natural perhaps for black holes, is highly non-trivial and seemingly non-local for neutron stars.

And if it walks like a duck and quacks like a duck, it’s probably not a chicken. Time and again, universality of this type has been found to be associated with the emergence of hidden symmetries. In the I-Love-Q case, Yagi and Yunes (plus collaborators) proposed that the approximate universality may be generated by the emergence of self-similarity in the elliptical isodensity contours of neutron stars [13]. Much like the layers of an onion, such isodensity contours have indeed been shown to be the lowest energy configuration for stable and cold neutron stars [14]. Perhaps then, the cause of the universality in the dimensionless entropy is also related to the emergence of self-similarity near the threshold of gravitational collapse. The physics that governs neutron stars, however, is still very different from that which governs black holes. Therefore, the reason for why the scaling exponent of the entropy, temperature and mass is smooth in the neutron star to black hole flow, instead of becoming massively discontinuous, remains a mystery.


[1] J. D. Bekenstein, Phys. Rev. D7, 2333 (1973).

[2] J. D. Bekenstein, Phys. Rev. D9, 3292 (1974).

[3] S. W. Hawking, Euclidean quantum gravity, Commun. Math. Phys. 43, 199 (1975).

[4] S. W. Hawking, Nature 248, 30 (1974).

[5] W. Israel, Nuovo Cimento B44, 1-14 (1966).

[6] S. W. Hawking, Commun. Math. Phys. 25, 152-166 (1972).

[7] K. Yagi and N. Yunes, Science 341, 365 (2013), arXiv:1302.4499 [gr-qc].

[8] K. Yagi and N. Yunes, Phys. Rev. D91, 103003 (2015), arXiv:1502.04131 [gr-qc].

[9] K. Yagi and N. Yunes, Class. Quant. Grav. 33, 9, 095005 (2016), arXiv: 1601.02171.

[10] The LIGO Collaboration, “GW170817: Measurements of neutron star radii and equation of state” arXiv: 1805.11581 [gr-qc].

[11] “An Entropy-Area Law for Neutron Stars Near the Black Hole Threshold” by Alexander, Stephon; Yagi, Kent; Yunes, Nicolas [i.e. this paper]. Article reference: CQG-105329.R1

[12] Private communication

[13]  K. Yagi, L. C. Stein, G. Pappas, N. Yunes, and 
T. A. Apostolatos, Phys. Rev. D90, 063010 (2014), arXiv:1406.7587 [gr-qc].

[14] D. Lai, F. A. Rasio and S. L. Shapiro, Astrophys.J. 420, 811-829 (1994), astro-ph/9304027.

Read the full article in Classical and Quantum Gravity:
An entropy-area law for neutron stars near the black hole threshold
Stephon H Alexander et al 2019 Class. Quantum Grav. 36 015010

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