Critical phenomena in gravitational collapse of Husain-Martinez-Nunez scalar field

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 some 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 $c\neq 0$, 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 constructed from a different scalar field with continuous self-similarity, we obtain the same value of $\gamma$. However, we show that the solution with $c\neq 0$ is not self-similar. Therefore, we provide a rare example that a scalar field without self-similarity also possesses the features of critical collapse.


Introduction
Gravitational collapse is the main reason of various galactic structures and it remains one of the most interesting and fundamental problems in general relativity. The end state of gravitational collapse could be a black hole, naked singularity or flat spacetime. In a seminal work by Choptuik [1], some intriguing and universal properties concerning the formation of black holes from massless scalar fields were found. This is called the critical phenomenon. Particularly, near the threshold, the black hole mass can always be expressed in the form of power-law: for p > p * , where p is a parameter of the initial data to the threshold of black hole formation. Numerical simulations have shown that that the critical exponent γ is equal to 0.5 for solutions with continuous self-similarity (CSS) and γ ≈ 0.37 for solutions with discrete self-similarity (DSS). Details about the critical phenomenon can be learned in [2,3]. In addition to numerical calculation, analytical models were also built to explore the critical phenomena. Patrick R. Brady [4] studied an exact one parameter family of scalar field solutions which exhibit critical behaviours when black hole forms. J. Soda and K. Hirata [5] 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 γ = 0.5 for n = 4. A. Wang et al [6] 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 γ = 0.5. A. Wang et al [7] also analytically studied the gravitational collapse of a massless scalar field with conformal flatness. They showed that the mass of the 2 Husain-Martinez-Nunez (HMN) spacetime The Husain-Martinez-Nunez spacetime [15] satisfies the Einstein-scalar field equations The spherically symmetric solution is given by 1 where α = ± √ 3 2 . From the Ricci scalar we see that the curvature singularities are located at r = 2c (timelike singularity) and t = −1/a(spacelike singularity). Using Θ± to label the expansion of null geodesics, we have where √ h = R 2 sin θ = (at + 1)r 2 1 − 2c r 1−α sin θ, and λ± is the affine parameter of the null geodesics. The tangent to the null geodesic with affine parameter is We can get The apparent horizon satisfies Θ+ = 0, Θ− < 0, which is located at The detailed analyses about the HMN sapcetime can be found in [18]. We shall focus on the case −∞ ≤ t ≤ − 1 a and a < 0 because it corresponds to a black hole solution. When a = 0 and α = √ 3 2 , the apparent horizon in the HMN spacetime has "S-curve" shape. However, this does not make differences in our results. Therefore, in this paper, we just study the case α = − 2 . The MisnerSharp mass [34] is defined by where R denotes the areal radius. From [19], we know that in spherically symmetric spacetimes, the apparent horizon satisfies g ab ∇aR∇ b R = 0 .
Therefore, on the apparent horizon, the Misner-sharp mass becomes 3 Critical behaviour of HMN scalar field with conformal flatness (c = 0)

Matching at a timelike boundary
To study the critical phenomenon of HMN massless scalar field, we start with the simple case c = 0, where the spacetime is conformally flat [15]. First, we need to join the HMN solution with an outgoing Vaidya solution such that the resulting spacetime is asymptotically flat. In this section, we assume that the boundary connecting the two solutions is a timelike hypersurface. We shall use "−" to label the inner HMN spacetime and "+" to label the exterior Vaidya spacetime (see Fig. 1). Figure 1: a = 0, c = 0 timelike hypersurface The interior spacetime is described by Eq. (4) for c = 0 The exterior spacetime is described by the outgoing Vaidya metric where f+ = 1 − 2m(U )

R
. We choose ξ i = {λ, θ, φ} as the intrinsic coordinates on the hypersurface. Σ is determined by functions t(λ), r(λ) from the interior and T (λ), R(λ) from the exterior. The induced metric on Σ is given by Here the do"." means the derivative with respect to λ. We use the Darmois junction conditions to match the solutions across Σ where k ab is the extrinsic curvature of Σ.
Denote the coordinates of the four-dimensional spacetime by {x µ }. Σ is determined by the functions {x µ (ξ i )}. Then the components of k ab can be calculated from where na is the spacelike normal to Σ. Computing kij from the interior and exterior, respectively, we obtain the nonvanishing components ar 2ṙ + 2(at + 1)ṫr Substituting Eqs. (19) and (20) into Eq. (21), we have Substituting Eqs. (26) and (27) into Eq. (22), with the help of Eq. (29), we obtain Eq. (28) yieldsṘ Therefore, one can solve Eqs. (29)-(31) and obtainṡ U = 2(at + 1) Thus, To proceed, we calculate the following derivatives: wheret = t + a −1 . Substitute the above results into Eq. (25) and according to Eq. (22),let the right-hand side of Eq. (24) be equal to the right-hand side of Eq. (25). After a lengthy calculation, we obtain the following result, which is surprisingly simple Obviously, the solution isṙ =ṫ orṙ = −ṫ. But this means that the hypersurface is null, inconsistent with our assumption. Therefore, we conclude that the two spacetimes cannot be matched through a timelike hypersurface if the continuity of the extrinsic curvature is required.

Matching at a null hypersurface
Matching the two solutions at a null hypersurface is more complicated than at a timelike hypersurface. We shall follow the method in [6] and [26]. First we use the coordinate transformation v = t + r to replace the coordinate r in Eq. (4) and obtain the metric in the interior where Let Σ be the null hypersurface v = v0. The normal to Σ is where s is a negative arbitrary function such that n a − is a future directed vector. We can introduce a transverse null vector Na by requiring Without loss of generality, we assume that N − a = Nvdva + Nrdra. Then it is easy to show that ∂ξ a as given in the [6]. Thus Similarly to Eq. (23), the transverse extrinsic curvature for the null surface is given by [26].
By straightforward calculation, we find (48) On the other hand, we need to join the HMN solution to the exterior Vaidya spacetime (18) at Σ, as shown in Fig. 2. Assume that the null surface Σ can be described by U = U0(R) from the Vaidya solution. It follows from (18) .
The spacetime coordinates {x µ + } can be expressed as functions of ξ i : Define e +µ (i) ≡ The normal to Σ is given by where β is a negative function which will be determined later. Then the transverse null vector N a in Eq. (42) is The continuity condition on Σ requires [26] N + µ e +µ This also guarantees that the normal vectors n a defined on both sides are the the same.

Mass of the black hole
From Eq. (14), one can calculate the Misner-Sharp mass for the HMN spacetime described by metric (17) Note that the null surface is determined by Eqs. (64) and (65) immediately gives the coordinates at the intersection of Σ and the apparent horizon: Since r > 0 and a < 0, from Eq. (66), we see that the existence of the intersection requires v0 > |a| −1 Therefore, the Misner-Sharp mass at the intersection is As is known, the event horizon coincides with the apparent horizon in the outgoing Vaidya spacetime as shown in Fig. 3. It is also known that the mass function m in the Vaidya metric is constant alone the event horizon [35]. Thus, it is natural to take the mass in Eq. (69) to be the mass of the black hole.
To investigate the critical behavior as a → 0, we impose the condition that ri in Eq. (67) does not change with a. This means that v0 must take the form where V is a positive constant independent of a. Thus, Eq. (69) gives the mass of black hole: Eq. (71) shows that the mass of black hole can be put in the form of Eq. (1) and the scaling exponent is γ = 0.5. Obviously, as a approaches zero, the mass of the black hole vanishes and the spacetime becomes Minkowski.

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

Matching to an outgoing Vaidya solution at a null hypersurface
Under the coordinate transformation Eq. (4) can be rewritten in the form (73) Here, the function h(r) satisfies The areal radius R takes the form Similarly to section 3, we match the solution with an outgoing Vaidya solution at the null hypersurface v = v0 (see Fig. 4). Substitution of v = v0 into Eq. (75) yields the function r = r(R). By the method in section 3.2, the extrinsic curvature can be calculated as where It is easy to see that k + ab takes the same form as Eq. (57). Then k − ab = k + ab yields Thus, by our construction, the metric of the resulting spacetime is continuous and the extrinsic curvature of the null hypersurface is also continuous. Therefore, we have shown that the general HMN spacetime ( a = 0, c = 0) and Vaidya spacetime can be matched at a null hypersurface as showed in Fig. 4. 2 ) matching with an outging Vaidya solution. There are two singularities at t = | 1 a | and r = 2c, where r = 2c is a naked singularity.

Mass of the black hole
By the argument in section 3.3, one can show that m(r) in Eq. (79) is exactly the Misnersharp mass for the metric in Eq. (73). Therefore, from Eqs. (13), (16) and (75), we can obtain the mass on the apparent horizon mAH (r): The ingoing null hypersurface boundary is defined by From the Eq. (13) and the Eq. (80), we can get the coordinates (ri, ti) at the intersection of the apparent horizon and the null hypersurface v = v0, which satisfies v0 + Similarly, we choose the null hypersurface which intersects with the apparent horizon at a fixed radius ri = r0, i.e., independent of a. Again, we take the Misner-Sharp mass at the intersection as the black hole mass. Then, Eq. (79) gives the mass of the black hole where Eq. (83) shows clearly that the black hole mass satisfies the power law with γ = 0.5. However, the spacetime for a < 0 possesses a naked singularity r = 2c (see Fig. 4), in violation of the cosmic censorship conjecture. To remove the naked singularity, we join another outgoing Vaidya spacetime at v = v1(v1 < v0), as shown in Fig. 5. No naked singularity exists in this new spacetime. When we study the relation between the mass and the parameter a, we treat c as a constant. We see that M bh → 0 as a → 0. When a = 0, there is no black hole but a naked singularity as shown in section 4.3.
To calculate the apparent horizon of the spacetime, we first choose two families of radial null vector fields where λ ± is the affine parameter of the null geodesic. According to Eq. (11) Therefore, the spacetime has no apparent horizon when r > 2c. According to Eq. (6), the spacetime possesses a naked singularity at r = 2c (see Fig. 6). Figure 6: Penrose diagram for the HMN spacetime(a = 0, c = 0). There is a naked singularity at r = 2c.
Since the spacetime is asymptotically flat, we calculate the ADM mass and find When a = 0, the mass of the black hole is given by Eq. (83), which shows clearly that M → 0 as a → 0. However, when a = 0, as we just discussed, the spacetime is not Minkowski and its ADM mass is nonzero. Therefore, there exists a mass gap between the black hole solution (a = 0) and its limiting spacetime (a = 0).

Conclusion
In this paper, we have used the "cut and paste" method to construct analytical models and study 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 = 0, we have joined the scalar field onto two pieces 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 γ = 0.5. This is consistent with previous results in the literature. When c = 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 [7], 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 = 0 as discussed in section 4. 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 the universal features in gravitational collapse.

A Self-similarity of HMN spacetime
In this appendix, we will prove that the HMN spacetime is CSS (continuous self similar) only when c = 0 and a = 0. A spacetime is continuous self-similar if there exists a conformal Killing vector field ξ a satisfying ∇ (a ξ b) = g ab . (91) As a result of the spherical symmetry, we can write Here, x and y are functions of r and t. Substituting this expression into Eq. (91), we find that yR,r + xR,t = R , yν,r + xν,t + y,r = 1 , yλ,r + xλ,t + x,t = 1 , Here, Obviously, C0 must be a constant independent of r and t. So the only solution is and consequently D(t) = D0 .
Now Eqs. (98) and (99) become Thus, we have proven that the HMN spacetime is continuous self-similar only for c = 0 and a = 0.