# Attempt to explain black hole spin in X-ray binaries by new physics

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## Abstract

It is widely believed that the spin of black holes in X-ray binaries is mainly natal. A significant spin-up from accretion is not possible. If the secondary has a low mass, the black hole spin cannot change too much even if the black hole swallows the whole stellar companion. If the secondary has a high mass, its lifetime is too short to transfer the necessary amount of matter and spin the black hole up. However, while black holes formed from the collapse of a massive star with solar metallicity are expected to have low birth spin, current spin measurements show that some black holes in X-ray binaries are rotating very rapidly. Here we show that, if these objects are not the Kerr black holes of general relativity, the accretion of a small amount of matter (\(\sim \)2 \(M_\odot \)) can make them look like very fast-rotating Kerr black holes. Such a possibility is not in contradiction with any observation and it can explain current spin measurements in a very simple way.

## Keywords

Black Hole Kerr Black Hole Radiative Efficiency Spin Parameter Kerr Solution## 1 Introduction

When a star exhausts all its nuclear fuel, it shrinks to find a new equilibrium configuration. For very massive stars, there is no known mechanism capable of balancing their own weight: these objects undergo a complete gravitational collapse and the final product is a black hole (BH). It is thought that in our Galaxy there are about \(10^7\) BHs formed from the gravitational collapse of massive stars. Despite this huge number, we only know about 20 stellar-mass BH candidates [1]. They live in X-ray binaries and from the study of the orbital motion of the stellar companion it is possible to infer that the mass of the compact object exceeds 3 \(M_\odot \). This is the maximum mass for a neutron or a quark star [2], and therefore a compact object exceeding this limit is classified as a BH candidate.

In four-dimensional general relativity, an uncharged BH is described by the Kerr solution and it is completely specified by only two parameters, corresponding to the mass \(M\) and the spin angular momentum \(J\) of the object. A fundamental limit for a Kerr BH is the bound \(|a_*| \le 1\), where \(a_* = J/M^2\) is the dimensionless spin parameter.^{1} For \(|a_*| > 1\) there is no event horizon in the Kerr metric and the spacetime has a naked singularity [3]. If we can measure both \(M\) and \(a_*\) of a Kerr BH, we know all the properties of the spacetime geometry. The effect of the accretion disk on the background metric is indeed negligible [4]. However, it is not easy to estimate the BH spin: the spin has no effects in Newtonian gravity and therefore it is necessary to probe the spacetime close to the object. At present, the spin parameter has been measured only for about 10 stellar-mass BH candidates [5, 6].

It is commonly thought that the spin of stellar-mass BHs in X-ray binaries is mainly natal and that the effect of the accretion process is negligible [7] (but see Ref. [8]). The argument can be summarized as follows. Stellar-mass BH candidates have a mass around 10 \(M_\odot \). If the stellar companion is a few solar masses, the BH cannot significantly change its mass and spin angular momentum even swallowing the whole star. If the stellar companion is heavy, its lifetime is too short: even if the BH accretes at the Eddington rate, there is not the time to transfer the necessary amount of matter to significantly spin the BH up. In the end, a BH cannot swallow more than a few \(M_\odot \) from the companion star, and for a \(10\) \(M_\odot \) object this is not enough to significantly change its spin parameter \(a_*\) [7].

BH binaries can be grouped into two classes. Low-mass X-ray binaries are systems in which the stellar companion is not more than a few solar masses (\(\lesssim \)3 \(M_\odot \)) and the mass transfer occurs for Roche lobe overflow. These systems are transient X-ray sources because the mass transfer is not continuous. High-mass X-ray binaries are systems in which the stellar companion is massive (\(\gtrsim \)10 \(M_\odot \)) and the mass transfer from the companion star to the BH is due to the wind of the former. These systems are persistent X-ray sources. If the BH spin is mainly natal, its value should be explained by studying the gravitational collapse of massive stars. While there are still uncertainties in the angular momentum transport mechanisms of the progenitors of stellar-mass BHs, it is widely accepted that the gravitational collapse of a massive star with solar metallicity cannot create fast-rotating remnants [9, 10]. The birth spin of these BHs is expected to be very low (see e.g. [8] and references therein). However, this is not what we observe. Assuming the Kerr metric, BH spin measurements show that some of these objects have a spin parameter close to 1. In the case of low-mass X-ray binaries, the BH candidate in GRS 1915+105 has \(a_* > 0.98\) [11] and \(M = 12.4 \pm 2.0\) \(M_\odot \) [12], while the stellar-companion’s mass is \(M = 0.52 \pm 0.41\) \(M_\odot \). In the case of high-mass X-ray binaries, the BH candidate in Cygnus X-1 has \(a_* > 0.98\) [13, 14] and \(M = 14.8 \pm 1.0\) \(M_\odot \), while the stellar wind from the companion is not an efficient mechanism to transfer mass. Both spin constraints are at 3 \(\sigma \). While BHs in low- and high-mass X-ray binaries form in different environments, in both cases the origin of spin values so high is puzzling: the birth spin is expected to be low and accretion can spin a BH up only by transferring a significant amount of matter.

In this paper, we show that current spin measurements can easily be explained if BH candidates in X-ray binaries are not the Kerr BHs of general relativity. In particular, an initially non-rotating BH can look like a fast-rotating Kerr BH after accreting a small amount of matter (\(\sim \)2 \(M_\odot \)) if it is more prolate than a Kerr BH. Strictly speaking, this does not necessary mean that the BH must be prolate, but simply that it must be less oblate than the Kerr one. Here the key point is the innermost stable circular orbit (ISCO), which depends on the background metric. A BH more prolate than a Kerr one can look like a very fast-rotating Kerr BH when its spin parameter is much lower, which can be acquired after accreting a modest amount of mass. While the scenario is speculative and requires new physics, it is not in contradiction with any observation or theoretical argument [15, 16], and it provides a simple explanation to current spin measurements.

## 2 Kerr black holes

## 3 Non-Kerr black holes

If BH candidates in X-ray binaries were not the Kerr BHs of general relativity, current spin measurements would be wrong (because obtained assuming the Kerr metric) and the evolution of the spin parameter as a result of mass transfer from the stellar companion would be different. A number of studies have clearly shown that there is a fundamental degeneracy between the spin and possible deviations from the Kerr solution, with the result that a non-Kerr BH may be interpreted as a Kerr BH with a different value of \(a_*\) [20, 21, 22, 23]. The main technique to estimate the spin parameter of stellar-mass BH candidates is the so-called continuum-fitting method, namely the study of the thermal spectrum of thin disks [5, 6]. As a first crude approximation, the approach measures the radiative efficiency \(\eta \) [24], which is then translated into a spin measurement under the assumption of the Kerr background, exploiting the fact there is a one-to-one correspondence between \(\eta \) and \(a_*\). It turns out that BHs more prolate than the Kerr ones have a higher radiative efficiency for a lower value of \(a_*\) and they may thus look like very fast-rotating Kerr BHs after acquiring a relatively small amount of mass from the stellar companion. The result is very general, but it is useful to see this with some specific example.

The Johannsen–Psaltis metric is a phenomenological metric and does not describe any known solution of spinning BHs in modified gravity. It can be used as a toy model to described non-Kerr BHs assuming that particles follow the geodesics of its spacetime, namely that it can be obtained as a solution (or approximated solution) of some metric theory of gravity. In this case, it is possible to study the evolution of the spin parameter of these non-Kerr objects [27, 28, 29]. The master equation is still (3), but \(E_\mathrm{ISCO}\) and \(L_\mathrm{ISCO}\) are different because they are determined by the background metric.

(i) the equilibrium spin parameter is much lower than 1, and

(ii) an initially non-rotating BH reaches a high radiative efficiency very quickly, after a modest amount of mass transfer. This is a preliminary indication that mass accretion onto a non-Kerr BH may explain the observation of fast-rotating Kerr BHs in X-ray binaries.

It should not be difficult to find more efficient non-Kerr models, namely BHs that can look like very-fast rotating Kerr BHs after accreting a smaller amount of matter. However, it has to be noted that the Johannsen–Psaltis and Cardoso–Pani–Rico spacetimes are phenomenological metrics to parameterize generic deviations from the Kerr solution. Some spinning BH solutions in well-motivated alternative theories of gravity are known [33, 34, 35, 36]. Generally speaking, if we consider a specific alternative theory of gravity, it is possible that its BH solutions are not sufficiently more prolate than Kerr for any choice of the parameters of the theory. Moreover, the deformation parameters in the Johannsen–Psaltis and Cardoso–Pani–Rico spacetimes are only constrained by observations sensitive to the metric (assuming geodesic motion), not by the field equations of the theory (which are not given). If we have a theory, there may be independent bounds coming from the field equations (e.g. emission of gravitational waves from a binary pulsar). After satisfying these constraints, it is not obvious that their BHs can do the job proposed in the present paper. Every theory is different and it should be analyzed by itself. There are also scenarios like the one suggested in [37, 38], in which general relativity holds up to quite strong gravitational fields, but BHs do not have the usual properties (even the concept of metric breaks down on the surface of these objects) and, in many aspects, they behave like compact stars made of exotic matter. In this case, constraints can only be obtained by BH observational data and the current bounds on possible deviations from the Kerr solution are weak [24].

## 4 Concluding remarks

In this paper, we showed that current spin measurements of BHs in X-ray binaries may easily be explained if these objects are more prolate than the predictions of general relativity. The point is that similar objects can look like very fast-rotating Kerr BHs with a lower value of the spin parameter, which can be acquired after a modest mass transfer from the stellar companion. In the case of Kerr BHs, the required amount of mass stripped from the stellar companion is too high and therefore it is not understood the origin of so high spins for some BHs in X-ray binaries. It is at least intriguing that even other observations seem to require that stellar-mass BH candidates are more prolate than the Kerr ones, namely the power of steady jets [39] and quasi-periodic oscillations [40, 41]. At this stage, the proposal that BH candidates are not the Kerr BHs of general relativity is a very speculative possibility, but it is not in contradiction with any observation. It is probably difficult to test this scenario with a more detailed study of the origin of the BH spin, but future observational facilities will be hopefully able to test the Kerr nature of astrophysical BH candidates [42, 43, 44].

## Footnotes

- 1.
Throughout the paper, we use units in which \(G_\mathrm{N} = c = 1\).

## Notes

### Acknowledgments

This work was supported by the NSFC grant No. 11305038, the Shanghai Municipal Education Commission grant for Innovative Programs No. 14ZZ001, the Thousand Young Talents Program, and Fudan University.

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