# Critical phenomena in gravitational collapse of Husain–Martinez–Nunez scalar field

## Abstract

We construct analytical models to study the critical phenomena in gravitational collapse of the Husain-Martinez-Nunez massless scalar field. We first use the cut-and-paste technique to match the conformally flat solution (\(c=0\) ) onto an outgoing Vaidya solution. To guarantee the continuity of the metric and the extrinsic curvature, we prove that the two solutions must be joined at a null hypersurface and the metric function in Vaidya spacetime must satisfy certain constraints. We find that the mass of the black hole in the resulting spacetime takes the form \(M\propto (p-p^*)^\gamma \), where the critical exponent \(\gamma \) is equal to 0.5. For the case \(c\ne 0\), we show that the scalar field must be joined onto two pieces of Vaidya spacetimes to avoid a naked singularity. We also derive the power-law mass formula with \(\gamma =0.5\). Compared with previous analytical models which were constructed from a different scalar field with continuous self-similarity, we obtain the same value of \(\gamma \). However, we show that the solution with \(c\ne 0\) is not self-similar. Therefore, we provide a rare example that a scalar field without self-similarity also possesses the features of critical collapse.

## 1 Introduction

*p*is a parameter of the initial data to the threshold of black hole formation. Numerical simulations have shown that the critical exponent \(\gamma \) is equal to 0.5 for solutions with continuous self-similarity (CSS) and \(\gamma \approx 0.37\) for solutions with discrete self-similarity (DSS). Details about the critical phenomenon can be learned in [9, 10].

In addition to numerical calculation, analytical models were also built to explore the critical phenomena. Patrick R. Brady [11] studied an exact one parameter family of scalar field solutions which exhibits critical behaviours when black hole forms. Soda and Hirata [12] analytically studied the collapse of continuous self-similar scalar field in higher dimensional spacetimes and found a general formula for the critical exponents which agrees with the exponent \(\gamma =0.5\) for \(n=4\). Wang et al. [13] constructed an analytical model by pasting the BONT model (a massless scalar field) with the Vaidya model. They demonstrated that the black hole mass obeys the power law with \(\gamma =0.5\). Wang et al. [14] also analytically studied the gravitational collapse of a massless scalar field with conformal flatness. They showed that the mass of the black hole without self-similarity turns on with finite nonzero values. Recent developments regarding the critical phenomenon can be found in Refs. [15, 16, 17, 18, 19, 20, 21, 22].

Previously, the studies of critical phenomena concerns black holes in non-dynamical backgrounds. It would be interesting to investigate solutions of Einstein’s equation in a cosmological background. The McVittie spacetime [23] discovered in 1933 is regarded as the first analytical solution interpreted as describing a central object embedded in an FLRW universe. In this paper, we investigate the critical phenomena associated with an exact scalar field solution discovered by Husain-Martinez-Nunez (HMN) [24], which represents a black hole in an FLRW universe [25, 26, 27]. It was pointed out that the HMN solution brings in new phenomenology(S-curve) of apparent horizon [28]. Moreover, the conformally transformed HMN spacetime can be an inhomogeneous vacuum solution in Brans-Dick theory [29, 30, 31].

Since the HMN solution is not asymptotically flat, we need join it onto an exterior solution with asymptotic flatness. The Vaidya spacetime is a possible candidate because it describes a dynamic black hole and the free function *m*(*U*) in the metric can be used to guarantee the continuity of the resulting spacetime. Following the treatment in [13], we match the HMN solution onto an outgoing Vaidya solution along a null hypersurface. Usually, the hypersurface connecting the two parts of the spacetime is a thin shell, i.e., the extrinsic curvature across the hypersurface is discontinuous. By applying the Darmois-Israel formula [33], one can find the relationship between the jump of the extrinsic curvature and the surface stress-energy tensor of the thin shell. However, by properly choosing the function *m*(*U*) in the Vaidya metric, we find that the extrinsic metric can be continuous across the null surface. Therefore, no thin shell forms and the resulting spacetime can be at least \(C^1\).

After the matching, we calculate the apparent horizon and define the black hole mass as the Komar mass at the intersection of the apparent horizon and the null hypersurface. There are two parameters, *a* and *c* in the HMN metric, whose values determine the staticity and homogeneity features of the solution. We first study the case \(c=0\) where the HMN solution is conformally flat and has CSS. The mass of the black hole is found in the power-law form \(M\propto \sqrt{-a}\) for \(a<0\), which means \(\gamma =0.5\). When \(a=0\), the black hole disappears and the spacetime becomes Minkowski.

The case of \(c\ne 0\) is more complicated. This solution has no self-similarity. We still find \(M\propto \sqrt{-a}\) for \(a<0\). Differing from the case of \(c=0\), we show that the limiting spacetime (\(a=0\)) possesses a naked singularity with non-zero ADM mass. Therefore, there is a mass gap between the black hole spacetime (\(a<0\)) and the spacetime with naked singularity \(a=0\).

The paper is organized as follows. In Sect. 2, we briefly introduce the Husain-Martinez-Nunez (HMN) scalar field solution. In Sect. 3, by using the cut-and-paste method, we match the conformally flat HMN solution (\(c=0\) with CSS) onto an outgoing Vaidya spacetime at a null hypersurface. This matching guarantees the continuity of the metric and extrinsic curvature across the surface. Then, we use this analytical model to study the critical phenomenon and derive the mass formula. In Sect. 4, we join a general HMN (\(a\ne 0,c\ne 0\)) with two outgoing Vaidya spacetimes. We show that the mass of the black hole approaches zero for \(a<0\). We also find that the critical spacetime (\(a=0\)) possesses a naked singularity with nonzero ADM mass. Concluding remarks are given in Sect. 5. In Appendix A, we show that the HMN solution and the Vaidya solution cannot be matched through a timelike hypersurface. In Appendix B, we prove that the HMN solution has CSS only when \(a\ne 0\) and \(c=0\).

## 2 Husain–Martinez–Nunez (HMN) spacetime

^{1}

*R*denotes the areal radius. From [28], we know that in spherically symmetric spacetimes, the apparent horizon satisfies

## 3 Critical behaviour of HMN scalar field with conformal flatness (\(c\ne 0\))

To study the critical phenomena of HMN massless scalar field, we start with the simple case \(c=0\), where the spacetime is conformally flat [24]. First, we need to join the HMN solution onto an outgoing Vaidya solution such that the resulting spacetime is asymptotically flat. In Appendix A, we have shown that the two spacetimes cannot be matched at a timelike boundary. In the following subsection, we shall replace the timelike hypersurface with a null hypersurface.

### 3.1 Matching at a null hypersurface

*r*in Eq. (4) and obtain the metric in the interior

*s*is an arbitrary negative function such that \(n^a_-\) is a future directed vector. We can introduce a transverse null vector \(N_a\) by requiring

*r*(

*R*) is given by Eq. (18) with \(v=v_0\). We define \(e_{(a)}^{-\mu }\equiv \dfrac{\partial x^{\mu }_{-}}{\partial \xi ^a}\) as given in the [13]. Thus

*m*(

*r*) as

*m*(

*U*) in the Vaidya solution. Therefore, we have matched the conformally flat spacetime with Vaidya spacetime at the null hypersurface.

### 3.2 Mass of the black hole

*m*(

*r*) in Eq. (39) is just the Misner-Sharp mass at \(v=v_0\). The apparent horizon determined by Eq. (13) takes the simple form for \(c=0\):

*m*in the Vaidya metric is constant alone the event horizon [35]. Thus, it is natural to take the mass in Eq. (47) to be the mass of the black hole.

*a*. This means that \(v_0\) must take the form

*V*is a positive constant independent of

*a*. Thus, Eq. (47) gives the mass of black hole:

*a*approaches zero, the mass of the black hole vanishes and the spacetime becomes Minkowski.

## 4 Collapse of the general HMN scalar field

In this section, we shall investigate the gravitational collapse associated with a general HMN scalar field (\( a \ne 0, c \ne 0\)).

### 4.1 Matching to an outgoing Vaidya solution at a null hypersurface

*h*(

*r*) satisfies

*R*takes the form

### 4.2 Mass of the black hole

*m*(

*r*) in Eq. (57) is exactly the Misner-sharp mass for the metric in Eq. (51). Therefore, from Eqs. (13), (16) and (53), we obtain the mass on the apparent horizon:

*a*. Again, we take the Misner-Sharp mass at the intersection as the black hole mass. Then, Eq. (57) gives the mass of the black hole

However, the spacetime possesses a naked singularity \(r=2c\) (see Fig. 3), in violation of the cosmic censorship conjecture. To remove the naked singularity, we join another outgoing Vaidya spacetime at \(v=v_1(v_1<v_0)\), as shown in Fig. 4. In the new spacetime, the HMN solution is configured in between two Vaidya solutions and no naked singularity exists.

*a*, we treat

*c*as a constant. We see that \(M_{bh}\rightarrow 0\) as \(a\rightarrow 0\). When \(a=0\), there is no black hole but a naked singularity as discussed in Sect. 4.3.

### 4.3 “Critical spacetime”: \(a=0\) and \(c\ne 0\)

## 5 Conclusion

In this paper, we have used the “cut and paste” method to construct analytical models and studied the critical phenomena of the HMN scalar filed. We have shown that the HMN solution with conformal flatness (\(c= 0\)) can be matched with the Vaidya solution along a null hypersurface, but not a timelike hypersurface. We have derived the differential equation which specifies the metric function in the Vaidya solution. For \(c\ne 0\), we have joined the scalar field onto two patches of Vaidya spacetimes to avoid the naked singularity.

We have studied the gravitational collapse for the HMN scalar field and shown that black hole mass satisfies the power law with \(\gamma =0.5\). This is consistent with previous results in the literature. When \(c\ne 0\), the HMN spacetime has no CSS and the black hole also turns on at infinitely small mass. The result is different from the model in [14], which shows that the formation of black holes may turn on at finite mass when the gravitational collapse has no self-similarity. On the other hand, the mass gap exists between the black hole and the naked singularity during the gravitational collapse of HMN scalar field when \(c\ne 0\) as discussed in Sect. 4.

Previous studies on the critical phenomena are associated black holes in static backgrounds. The HMN black hole is embedded in a cosmological background, which is dynamical. Many properties of black holes could change due to the dynamical background. This is why the model we constructed is special and gives some different results. Our work suggests that critical collapse can be studied from analytical models which are constructed by known solutions. More models should be investigated in the future in order to test universal features in gravitational collapse.

## Footnotes

## Notes

### Acknowledgements

This research was supported by NSFC Grants No. 11775022, 11575286 and 11731001.

## Supplementary material

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