Static Black Hole and Vacuum Energy: Thin Shell and Incompressible Fluid

With the back reaction of the vacuum energy-momentum tensor consistently taken into account, we study static spherically symmetric black-hole-like solutions to the semi-classical Einstein equation. The vacuum energy is assumed to be given by that of 2-dimensional massless scalar fields, as a widely used model in the literature for black holes. The solutions have no horizon. Instead, there is a local minimum in the radius. We consider thin shells as well as incompressible fluid as the matter content of the black-hole-like geometry. The geometry has several interesting features due to the back reaction of vacuum energy. In particular, Buchdahl's inequality can be violated without divergence in pressure, even if the surface is below the Schwarzschild radius. At the same time, the surface of the star can not be far below the Schwarzschild radius for a density not much higher than the Planck scale, and the proper distance from its surface to the origin can be very short even for very large Schwarzschild radius. The results also imply that, contrary to the folklore, in principle the Boulware vacuum can be physical for black holes.


Introduction
What is a black hole? The notion of event horizon used to play an important role in our understanding of black holes. Nowadays, the event horizon is considered by many as an inappropriate concept for physical black holes, as its experimental verification takes an infinitely long time. In fact, even the necessity of apparent horizon for black holes has been questioned [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. In this paper, we will refer to back-hole-like objects simply as black holes.
Recently, by taking into account self-consistently the back reaction from Hawking radiation through the semi-classical Einstein equation, it was shown [7][8][9][10][11][12][13] that, if the vacuum energy near the Schwarzschild radius is dominated by Hawking radiation, neither the event nor the apparent horizon forms during a gravitational collapse. Later, it was also shown [17] that even for static black holes, for which there is no Hawking radiation, the vacuum energy-momentum tensor is capable of removing the horizons. 1 Different models of quantum vacuum energy leads to different near-horizon geometries.
For some of the models of vacuum energy, there is no horizon around the Schwarzschild radius and the near-horizon geometry is replaced by a (traversable) wormhole-like structure, that is, a local minimum of the radius r. 2 In Ref. [17], only the vacuum solution is considered. In a more realistic model, the wormhole-like structure does not continue to another open space, and the vacuum solution only applies to the exterior of a star of finite radius. The geometry of the vacuum solution should be put in junction with that for a matter distribution in the star, where the radius continues to zero at the center of the star.
In this paper, we study the static geometry inside the neck (local minimum of the radius) of the wormhole-like structure as well as the internal space of the star, assuming spherical symmetry. Using non-perturbative methods, we uncover novel features of the black-hole geometry that cannot be captured in the perturbative approach.
We first consider a star composed of a spherical thin shell whose energy distribution is proportional to a delta function of the radial coordinate. Next, we consider a star which consists of incompressible fluid, as a simple example of continuous distribution of matter. Since it is difficult to solve the semi-classical Einstein equation exactly, we analyze the solution in different regions with approximate analytic as well as numeric solutions.

The model
In this paper, we focus on the 4-dimensional semi-classical Einstein equation, 3 G µν = κ T µν , (2.1) where G µν is the classical Einstein tensor, while the expectation value of the energymomentum tensor T µν contains the quantum effect. We assume that the energymomentum tensor can be separated as where T m µν is the classical energy-momentum tensor of matter and T Ω µν represents the quantum effect. Typically, the latter is calculated as the vacuum expectation value of the energy-momentum operator of certain quantum fields.
Here, we consider only static spherically symmetric configurations and so a generic metric can be given in the form Defining the tortoise coordinate r * by dr * = dr F (r) , (2.4) we can express the metric as ds 2 = −C(v − u)du dv + r 2 (v − u)dΩ 2 (2.5) in terms of the null coordinates defined by Following Refs. [3,[18][19][20][21][22][23], we consider the model for the vacuum energy-momentum tensor T Ω µν defined by where T (2D) µν is the vacuum expectation value of the energy-momentum tensor operator of N 2-dimensional massless scalar fields [25] obtained through a spherical reduction of the 4-dimensional space-time. T (2D) µν is completely determined by the Weyl anomaly and conservation law up to the initial (or boundary) conditions. In terms of the null coordinates, the energy-momentum tensor of the 2-dimensional scalars is given by (2.10) The single-variable functions T uu (u) and T vv (v) are the integration "constants" arising from solving the conservation law. They should be fixed by the initial (boundary) conditions. Here, we focus on the static configurations without any incoming or outgoing energy flow at the spatial infinity (r → ∞). Choosing the gauge in which C → 1 in the limit r → ∞, we have The quantum state corresponding to this boundary condition is called the Boulware vacuum [26]. It is suitable for describing static configurations.
As the lowest order approximation of a perturbation theory, the energy-momentum tensor T Ω µν of the Boulware vacuum was calculated for the Schwarzschild background [18], and found to diverge at the horizon. Hence, conventionally, the Boulware vacuum is used only for static stars whose radii are larger than the Schwarzschild radius. However, it was recently shown [17] that the perturbation theory breaks down at the horizon, and nonperturbatively T Ω µν is non-singular within a certain range around the Schwarzschild radius.
In the case of a static star of radius R much larger than its Schwarzschild radius, the conventional wisdom is that the Boulware vacuum is the appropriate quantum state for a quantum field in this background. Now imagine that we shrink the star adiabatically, with every particle in the star moving extremely slowly towards the origin. At any given time, the geometry outside the star can be arbitrarily well approximated by a static solution of the semi-classical Einstein equation, assuming that the time scale of the change in radius can be arbitrarily long. The assumption that the time scale of changes can be arbitrarily long would break down if there is a horizon. However, as we have proven analytically in Ref. [17], the geometry has no horizon due to the back reaction of the vacuum energy defined above. It is therefore theoretically possible that the Boulware vacuum can be applied to a star hidden below the Schwarzschild radius. The goal of this paper is to understand the geometry of a static star in the Boulware vacuum in more detail, even when the star is submerged below the surface at the Schwarzschild radius.
3 Geometry of vacuum outside the star In Ref. [17,23], it was shown that there is no horizon for a static star with spherical symmetry in the Boulware vacuum due to the back reaction of the vacuum energymomentum tensor for the model described in the previous section. While the geometry outside the Schwarzschild radius remains almost identical to the Schwarzschild solution, the horizon is deformed to a (traversable) wormhole-like geometry. More specifically, there is a turning point where the radius r is at a local minimum. We will refer to this point (a 2-sphere) of local minimum in r as the "throat" or "neck" of the wormholelike structure, while it is also called a "turning point" or a "bounce" in the literature [23]. The radius r = a of the neck will be called the "quantum Schwarzschild radius". At large distance, the geometry approaches to a Schwarzschild solution with a certain Schwarzschild radius r = a 0 , which will be referred to as the "classical Schwarzschild radius". Sometimes we will just say "Schwarzschild radius" if it does not matter which one we are referring to, as the difference between the values of a and a 0 is extremely small for a large a.
It should be noted that the total mass of the system is not related to the quantum Schwarzschild radius but to the classical Schwarzschild radius. Since the quantum Schwarzschild radius is defined at the neck of the wormhole-like structure, it does not include the effects of the vacuum energy outside the neck, while the classical Schwarzschild radius is related to the total mass of the system since it is defined by the asymptotic structure of the spacetime.
Let us review the exterior geometry of a static star in vacuum. The classical energymomentum tensor for matter is zero outside the star: For static, spherically symmetric configurations, the semi-classical Einstein equation (with the vacuum energy-momentum tensor given in the previous section) gives the following differential equations for C(r) and F (r) in the metric (2.3): where C and F depend only on r and a prime indicates the derivative with respect to r. The parameter α is defined by , (3.4) where N is the number of massless scalar fields.
In the semi-classical Einstein equations, √ α characterizes the length scale of the quantum correction. For N of order 1, √ α is of the order of Planck length p . For a very large N , we can have √ α p , so that the quantum effect of the matter fields becomes important at a sub-Planckian scale when the effect of quantum gravity is still suppressed.
Eqs.(3.2) and (3.3) are equivalent to a single differential equation [17]: with C and F given in terms of ρ as Eq.(3.5) is a second order differential equation. Assuming asymptotic Minkowski space, the solution space has 2 parameters. One of the two parameters is the mass parameter (or the Schwarzschild radius) of the approximate Schwarzschild solution at distance. The other is just a scaling parameter corresponding to a constant scaling of the coordinates (t, r * ). In other words, given the Schwarzschild radius of the asymptotically Schwarzschild solution, there is a unique spherically symmetric solution to the Einstein equation for the vacuum energy (2.8) -(2.10) with zero flux (2.11) at spatial infinity.

The neck
Naively, one expects that the Einstein equation (3.5) can be solved as a perturbative expansion in powers of κ, or equivalently α. However, this perturbative expansion is not valid around the Schwarzschild radius. If we expand ρ in terms of α as the 0-th and 1st order solutions ρ 0 and ρ 1 for eq.(3.5) are where a 0 , c 0 and c 1 are integration constants. The constant a 0 is the classical Schwarzschild radius. The divergence in ρ 0 at r = a 0 implies that C(r) goes to zero at r = a 0 in the classical limit. But the divergence in ρ 1 implies that the quantum correction for C(r) diverges at the horizon, where the perturbative expansion breaks down. One should resort to non-perturbative approaches. It was analytically proven [17] that C cannot go to zero at finite r, due to the nonperturbative nature of eq.(3.5). Let us briefly review the prove here. While the expansion (3.8) is valid outside the Schwarzschild radius, ρ increases indefinitely as r decreases. For sufficiently large ρ , the first and second term in eq.(3.5) can be neglected, and the equation is then approximated by Note that the first term would be absent if α = 0, i.e., if there were no quantum correction to the vacuum energy.
According to this equation, the function ρ continues to increase as r decreases until ρ diverges at some point, say, r = a. We call this radius the quantum Schwarzschild radius. It is the radius of the "neck" of the wormhole-like structure [17]. The solution of ρ can be expanded around r = a as where the constant k is given by and c 0 is an integration constant. We shall always assume that a 2 α.
The expression (3.12) is a good approximation for while the perturbative expansion (3.8)-(3.10) is good outside the Schwarzschild radius As these two approximation schemes are supposed to meet around the points where r − a ∼ O(α/a), the quantum Schwarzschild radius a and the classical Schwarzschild radius a 0 differ by 16) and the order of magnitude of the constant c 0 can be roughly estimated as [17] Around the quantum Schwarzschild radius r = a, the metric is approximately given by [17] The proper length of the throat region approximated by this metric (for 0 ≤ r − a α a ) is of the order of magnitude of In terms of the tortoise coordinate r * , the metric (3.18) becomes where a * is the value of the tortoise coordinate r * when r = a. Examining the coefficient of the term dΩ 2 , we see that r = a is the minimum of the radius r. (The radius r is defined such that the area of a symmetric 2-sphere equals 4πr 2 .) This geometry resembles a traversable static wormhole whose "neck" or "throat" is a local minimum of r. It is certainly not a genuine wormhole, as the vacuum inside the neck terminates on the surface of the star, instead of leading to another open spacetime.
Due to the back reaction of vacuum energy, the horizon at the classical Schwarzschild radius a 0 is replaced by a wormhole-like geometry with the neck at the quantum Schwarzschild radius r = a. Notice that even though ∂ v r = 0 at r = a, suggesting that the outgoing null vectors normal to the neck are non-expanding, it is not an apparent horizon since it is not the boundary of a trapped region. It is simply a local minimum of r.
The energy density around the neck of the wormhole is estimated as This is of the same order as the naive non-relativistic average mass density for a star of radius a; mass volume ∼ 4πa κ 4π but it is much smaller than the mass density of the matter in a solution to the semiclassical Einstein equation, which is of order O(κ −1 α −1 ). (See eq.(6.7) below.) Its contribution to the total mass is negligible because of an additional redshift factor;

Behind the neck
The metric (3.18) (or (3.20)) is valid in a small region around the neck when the radius r satisfies (3.14). In this subsection, we assume that the surface of the star is further deeper down the neck, where (3.14) is no longer satisfied, and we study the geometry with back reaction from the vacuum energy.
As we move down the neck towards the star, the radius r increases, and the magnitudes of ρ and ρ decrease according to eq.(3.12), with ρ decreasing faster than ρ , until we reach the surface of the star. When the condition (3.14) is no longer valid, eq.(3.5) is dominated by the ρ 2 -term and the ρ 3 -term: and so we find the approximate solution for the vacuum space below the neck of the wormhole-like geometry when r − a α/a. 4 The behavior (3.25) can be better understood as follows. First, we note that eq.(3.25) is a small deviation to the exact solution to eq.(3.5) at large r (below the neck). The function F (r) is singular for this solution (3.26), but we will consider small perturbations of this solution so that F is regular.
While eq.(3.5) is a second order differential equation of ρ, it is a first order differential equation of ρ . Hence we expect to find solutions with an integration constant as deviations of the special solution (3.26) in the form: Substituting this into eq.(3.5) and expanding to the linear order in δρ (r), the correction term δρ (r) is solved as ok where f 0 is an integration constant. Note that the factor e −2r 2 /α is extremely small so that the value of f 0 can be very large while keeping the magnitude of the correction δρ (r) sufficiently small for a valid perturbative expansion. The condition for δρ to be a small perturbation is Since the left hand side quickly decreases with increasing r, the condition only needs to be checked near the neck. Let the approximation (3.28) be valid for r ≥ r 1 , where r 1 − a ∼ O(α/a), we assume that the parameter f 0 satisfies the condition The deviation δρ (r) goes to zero as r increases, as a class of solutions approaching to the same attractor solution (3.26). This explains the robust linear bahvior (3.25)) observed in ρ at large r. Eq.(3.28) implies that 31) and the metric in this region is approximately Assuming that this metric can be connected to eq.(3.18) around r − a ∼ O(α/a), we estimatec 0 and f 0 as The scalar curvature R − 8α f 0 r 7 e 2r 2 /α (3.34) diverges in the limit r → ∞ under the neck. This singularity at infinite r is in fact within finite proper distance from the neck. But this divergence is irrelevant in physical situations as it is a divergence of the vacuum solution. In the more realistic case, there (3.35) where we have used the condition (3.30). In other words, the radius r increases from a to ∞ within the proper distance of order √ α. Schematically, the near-neck geometry looks like Fig.1. Since the near-neck region approximated by the metric (3.18) also has a proper length of order O( √ α) (3.19), the maximal length between the neck and the surface of the star is only of the order of √ α.
The energy density − T 0 0 for the vacuum in this region grows exponentially with r from O(a −2 −2 p ) at r = r 1 to infinity as r → ∞. (Recall that r 1 is the radius of the surface below which the approximation in this subsection is valid.) Note that the energy density at r = r 1 is much smaller than the Planck scale due to (3.30). But in order for the density to be sub-Planckian at r s as well, we need the surface radius r s of the star to satisfy For the sake of curiosity, in the case when there is no star under the neck, the contribution of the vacuum energy density (3.36) to the total mass is negative and of the same order as the total mass of the black hole; Therefore, the vacuum contribution to mass is always smaller than that of matter.
Let us summarize the geometrical features of the space outside a static star with spherical symmetry when the back reaction of the vacuum energy is taken into account.
While the space is foliated by 2-spheres, the radius of the 2-sphere decreases as one moves towards the star from distance, until one reaches the neck of the wormhole-like geometry at r = a, which is a local minimum of r. The radius starts to increase behind the point r = a as we move further towards the origin, until we reach the surface of the star. In the hypothetical case when there is no star and the vacuum energy is the only source of gravity, there must be a singularity at the "center" (the limit r → ∞ under the neck -see eq.(3.34)). The more physical situation is that there is a star with positive mass and an outer surface of finite radius r s . The geometry discussed above should be limited to the region outside the outer surface of the star. The proper distance between the neck and the surface of the star is of order √ α or less.
In the following, we will study the geometry inside the star. The radius of the 2-sphere is expected to decrease again after passing through the surface of the star, until the radius goes to zero at the origin. Schematically, the geometry is depicted in Fig. 2, which is reminiscent of Wheeler's "bag of gold" [27]. Notice that this geometry is consistent with the numerical simulation of Ref. [20] for a dynamical black hole for the same model of vacuum energy, including Hawking radiation. We will discuss this interior geometry in the next section.

Thin shell
In this section, we consider the toy model of a star consisting of a static spherical thin shell. The space inside the thin shell is assumed to be Minkowski space, which has zero vacuum energy. The matter distribution of the shell is given by a delta function, and the geometry is obtained by matching across the thin shell the external geometry of vacuum discussed in the previous section with the flat spacetime inside the shell.
Patching two geometries on the two sides of a thin shell imposes junction conditions.
The first junction condition is that the induced metric on the shell must be identical for the bulk metrics on both sides of the shell. The metric is expressed in the general form where C and r outside the shell are given by the solution we have discussed in the previous section. For flat spacetime inside the shell, C and r are given by a constant and linear function of r * , respectively. The continuity condition of the metric for a shell at r * = r sh * and the regularity condition at r 0 then determine C and r in flat spacetime (r * ≤ r sh * ) to be Here, the radius of the shell is denoted as r sh . The function C has discontinuity in its first derivative, and the second derivative of C involves a delta function.
The other junction condition is that the energy-momentum tensor on the shell must match with the discontinuity in the Einstein tensor to satisfy the Einstein equation.
The energy-momentum tensor on the shell contributes to delta-function terms in the full energy-momentum tensor T µν : The first term involves a delta function of , which is a function of r * defined such that the position of the shell is at = 0 and that it gives the normal vector of the shell by with n µ normalized as n 2 = 1. (Basically, is the distance function from the shell.) According to Einstein's equations, the tensor S µν in eq.(4.5) can be expressed in terms of the discontinuity of the extrinsic curvature as where γ µν is the induced metric on the shell, and K µν , K are the extrinsic curvature and its trace, respectively. Not only the thin shell but also the vacuum energy contribute to the tensor S µν . Since there is a discontinuity in the curvature, the vacuum energymomentum tensor (2.8)-(2.10) also contains delta-function terms.
We identify the delta-function terms in the energy-momentum tensor as There is necessarily pressure in the tangential directions to support the thin shell from collapsing.
In order to obtain the energy-momentum tensor of the shell itself, the energymomentum tensor of the vacuum should be subtracted from (4.8)-(4.11). By substituting the discontinuity of the second derivative of C into (2.7)-(2.10), we find that the delta-function term in the vacuum energy-momentum tensor is while T vac rr = T vac θθ = T vac φφ = 0. By subtracting this vacuum energy-momentum tensor from the total energy-momentum tensor, we find the energy-momentum tensor of the matter shell: T shell r * r * = 0 , (4.14) Depending on the location of the shell, the metric functions C and r have different analytic approximations, so we discuss different situations separately in the following.
First, if the shell is located well outside the Schwarzschild radius, the geometry is well approximated by the Schwarzschild solution, and this situation is already well understood. In the following, we consider two cases: (1) the case with the shell very close to the neck so that (3.14) is satisfied and the metric is approximately given by (3.20), and (2) the case when the shell is deeper down the neck so that (3.14) is violated and the metric is approximately described by (3.32).

Shell close to the neck
Consider a shell close to the neck of the wormhole-like geometry, i.e., r sh ∼ a and the condition (3.14) is satisfied. In this case, the metric is approximately given by eq.(3.20), from which one can read off the functions C(r * ) and r(r * ). Substituting this solution into eqs.(4.13)-(4.16), we find the energy-momentum tensor of the shell: Here, we have introduced a constantc 0 by The energy density m 0 and the (angular) pressure P on the shell are found to be The density m 0 is almost the same as the classical case without vacuum energy for a thin shell at the Schwarzschild radius. The quantum Schwarzschild radius a, which is defined as the radius of the neck of the wormhole, is only slightly larger than the classical Schwarzschild radius a 0 .
In terms of the Planck length p , the mass density m 0 and the pressure P behave as The surface energy density is of the order of a −1 −2 p as expected, as the total mass is of order a/κ and the area of order a 2 . On the other hand, the pressure is very large in order to support the shell against gravity from collapsing. In fact, it diverges in the classical case as the shell gets too close to the Schwarzschild radius. By taking into account the quantum effects, the pressure is regularized, but is close to the Planck scale unless N 1.

Shell deep under the neck
Next, we consider the scenario when the shell is deeper inside the wormhole (that is, the condition (3.14)) is violated), so that the metric is approximated by eq.(3.32). Then the energy-momentum tensor of the shell can be calculated as Note that the second term in T shell tt has an exponential factor that is sensitive to r and blows up quickly for a tiny increase in r. Starting with a positive energy density m 0 of the shell at some radius r where (3.20) is a good approximation, we find that the energy density of the shell decreases very quickly if we move the shell further down the throat.
The density m 0 becomes negative beyond a certain point. For even larger radius of the shell (further deep down the throat), the density m 0 is negative with a larger magnitude.
This means that, for a physical shell with positive energy density m 0 > 0, the radius r of the shell (r sh ) is bounded from above, and the location of the shell from the neck is bounded from below, that is, it cannot be located too deep under the neck. Due to the exponential factor e −2r 2 /α , the deviation of the shell from the neck of the wormhole-like structure cannot be much larger than the order of r − a ∼ O((α/a) log(a 2 /α)). That is, a shell under the neck is always very close to the neck.

Classical divergence
In order to appreciate the difference due to the vacuum energy, we first review several facts of the classical case. For the classical case (i.e. when the vacuum energy T Ω µν is absent), the pressure is solved as where r s is the radius at the surface of the matters. In order for this pressure to be positive at r = 0, it must satisfy the condition Otherwise, the denominator of (5.13) becomes zero at some r ≥ 0. That is, the pressure diverges if the density of the fluid is too large for a given radius of the surface, or equivalently, if the radius of the star is too small.
From the junction condition at the surface r s , we see that the radius of the surface r s should be not only larger than the Schwarzschild radius a 0 , but Buchdahl's theorem says that it should satisfy the inequality [28] r s > 9 8 a 0 . (5.15) Therefore, classically, a star must collapse to a singularity under gravitational force when the condition (5.14) or (5.15) is violated.

Regularization by quantum effect
Contrary to the classical case, the pressure no longer diverges if the vacuum energy is taken into account. Eq. (5.11) implies that the pressure diverges only when C goes to zero. But one can prove that C can never go to zero for r α 1/2 , in a way similar to how we have proved the absence of horizon for the vacuum solution [17].
Here we sketch the proof that C can never go to zero. We first assume that C = 0 at some point r = r d . This implies that ρ → −∞ as r → r d , and thus ρ also diverges there. Therefore, in a sufficiently small neighborhood of r = r d , terms proportional to ρ 3 in the 3rd line dominate over the first 2 lines in eq.(5.12). Furthermore, in the coefficients of ρ 3 and ρ in eq.(5.12), the terms proportional to higher powers of e ρ are negligible in comparison with those proportional to lower powers of e ρ . As a result, the differential equation can be approximated by This implies that the divergence of ρ must have the form Up to a constant factor, this is the same solution as the vacuum solution (3.12), for which C does not go to zero.
Hence we conclude that the vacuum energy regularizes the pressure P such that it never diverges at finite r.

Numerical results
In order to see more detailed structure of the geometry for the star of incompressible fluid, we resort to numerical methods. Since C and F are not single-valued functions of r, it is convenient to use the tortoise coordinate r * defined in (2.4) to see the whole structure of the solution. The semi-classical Einstein equation gives the following two differential equations 0 = κP 0 r 2 (r * )e ρ(r * ) + 2r(r * )r (r * ) − 4r(r * )r (r * )ρ (r * ) + 2αρ (r * ) − 2αρ 2 (r * ) , (6.1) for the two functions ρ(r * ) and r(r * ). (A prime refers to the derivative with respect to r * .) Here, we have used m(r * ) = m 0 and (5.11).
Since we are considering the Boulware vacuum, there is no incoming or outgoing energy at the spatial infinity r → ∞. The quantum effects are expected to approach to zero at large distance, and the geometry should be an asymptotically Schwarzschild space. Hence the boundary conditions for ρ(r * ) and r(r * ) is that they are approximated by the Schwarzschild solution with a given classical Schwarzschild radius r = a 0 at large r * .
With the asymptotically Schwarzschild boundary condition, one can solve the Einstein equation for m = P = 0 at large r * , and then turn on the energy density m and the pressure P at the radius r = r s (or in terms of the tortoise coordinate, r * = r * s ) of the surface of the perfect fluid. Since the pressure must vanish at the surface of the star, the parameter P 0 is fixed by the condition P (r * s ) = 0. In principle, the relation between 3 parameters m 0 , r * s and a 0 can be derived by comparing the classical Schwarzschild radius a 0 and the total mass calculated from m 0 and r s . However, in order to calculate the total mass from m 0 and r * s , we need to know details on the geometry. Therefore, we would not know beforehand the suitable value of m 0 for given a 0 and r * s , until we obtain the solution for the geometry. That is, even though conceptually m 0 can be viewed as a function m 0 =m 0 (a 0 , r * s ) of a 0 and r * s , numerically, this function can only be found by trial and error, namely, by solving the differential equation for diverse values of m 0 for fixed a 0 and r * s . Therefore, in our numerical calculation, all 3 physical parameters a 0 , r * s and m 0 are treated as free parameters that we can fix arbitrarily by hand. The value of a 0 has to be fixed first when we impose the asymptotically Schwarzschild boundary conditions on ρ(r * ) and r(r * ) at large r * . Then the value of r * s needs to be fixed by hand to determine where to turn on the energy-momentum tensor of the incompressible fluid.
After that, a range of different values of m 0 are explored. In general, an arbitrary choice of m 0 would produce a singularity at small r * . We know that we have the suitable value m 0 =m 0 (a 0 , r * s ) only when the geometry is found to be regular everywhere. If the density is too small, i.e. m 0 <m 0 (a 0 , r * s ), the singularity should have positive mass.
The vacuum solution with singularity (3.34) in the limit r → ∞ is an example of the special case for m 0 = 0, which is obviously too small for a non-zero a. Similarly, if the density is too large, i.e. m 0 >m 0 (a 0 , r * s ), the singularity should have a negative mass.

Star inside the Schwarzschild radius
We first consider those stars whose surfaces lie inside the neck of the wormhole-like structure. Since r = a is a minimum of the radius, we have the radius of the surface of the star r s > a. We shall label the tortoise coordinate r * for the star's surface at r = r s as r * = r * s , and that for r = a as r * = a * . That is, r(r * s ) = r s and r(a * ) = a. In our numerical simulations, we take α = 0.05 and a 0 = 10. This corresponds to a relatively small star but it is more convenient to emphasize the quantum effects. Although the correction from the quantum effects would be quantitatively much smaller for a blackhole-like object in the real world, the qualitative behavior is expected to be the same. The neck of the wormhole is then found to be located at r * = a  The radius r decreases as r * decreases for r * > a * , but starts to increase for r * a * . It starts to decrease again from r * −60.003, which is slightly inside the surface r * s . On the right: The radius r starts to increase again with decreasing r * for r * < −1097.  Fig.4. The radius r corresponds to that in the horizontal direction while the distance along the 2D surface in the picture is given by the change in the r * coordinate. Both pictures show the radius r on the same scale, but the range of r * shown in the picture on the right is much larger than that on the left. The radius r → ∞ in both limits r * → ±∞. Notice that the distance along the 2D surface is different from the proper distance, which cannot be embedded in 3D flat space.
behavior as the vacuum solution we described in Sec. 3.2 without incompressible fluid (m 0 = 0). Fig. 4 shows that the radius r decreases as r * decreases for r * > a * , but r starts to increase with decreasing r * for r * a * , where a * is the tortoise coordinate at the neck.
The radius r starts to decrease again with decreasing r * from the 2nd turning point at r * −60.003, which is slightly inside the surface r * s . (Fig. 6 shows a magnified view of two small regions in Fig. 4 around r * = a * and r = r * s .) In Fig. 4, we notice that r starts to increase again with decreasing r * from the 3rd turning point at r * −1097,  Figure 6: Magnified view of the plot of r vs. r * around r * = a * −56.685 (left) and that around r * r * s (right) of Fig. 4. Note that r(r * ) is in fact a decreasing function between −56.685 < r * < −60.003 (r increases as r * decreases).  Figure 7: The factor C and pressure P for κm 0 = 50, r * s = −60 < a * , a 0 = 10 and α = 0.05.
On the left: The factor C is always positive and non-zero. On the right: The pressure P is zero at both the outer surface r * = −60, and the inner surface r * −102.086, and it is positive in between.
and never goes to zero but r → ∞ in r * → −∞. This is also compatible with the vacuum solution described in Sec. 3.2.
The resemblance of the limit r * → −∞ of the solution with the vacuum solution in Sec. 3.2 implies that there is a singularity for positive mass in the limit r * → −∞. We deduce that the density κm 0 = 25 is too small to have a regular solution (for a 0 = 10 and r * s = −60). . The factor C becomes very small near and inside the neck, but is still finite as in the previous case. Fig. 8 shows that the radius r increases as r * decreases just inside the neck, but starts to decrease from a point r * −60.003, which is almost on top of but slightly inside the surface of the star (r * s = −60). In contrast to the previous case, there is no third turning point of r, and r continues to decrease. Then the pressure becomes zero at r * = r inner * −102.086. The second turning point r * −60.003 is expected to be larger than the previous case, since the density of the mass is larger. But the difference is too small to be distinguished. The behavior around this point is basically the same as the previous case in Fig. 6, but the behavior for r * < −60.003 is a faster decrease in r.
If one continues the calculation to smaller values of r * , assuming that there is no more matter, there would be a singularity at the origin. We believe that it is a singularity with negative mass. The perfect fluid is a shell with finite thickness in the range r inner * < r * < r * s surrounding the singularity with vacuum in between. We explain the stability of the matter shell against gravitational collapse by the repulsive force from the negative mass at the singularity. We conclude that the density κm 0 = 50 is too large for a 0 = 10 and r * s = −60.  Regarding the geometry under the neck, when the density is too small (m 0 <m 0 ), the radius r does not approach to zero but approaches to infinity; when the density is too large (m 0 >m 0 ), r approaches to zero, but the pressure P goes to zero first when r is still positive.
Due to numerical instability, we could not obtain the numerical results with sufficient accuracy for r smaller than a certain value r e , which is always found to be of the order of √ α. The numerical results are reliable only for r larger than r e . The idealistic regularity condition that P is finite and positive at r = 0 for the case m 0 =m 0 is thus replaced by the practical condition that P is positive and finite at r = r e , the boundary of numerical analysis.
The instability in our numerical simulation occurs at r * −1571, where the radius is r = r e 0.22. (Note that numerically r 2 e α.) As we will check later, the value of r e depends on α and is smaller for smaller α. In fact, r e is always approximately √ α.
Hence, we believe that the numerical instability of the Einstein equation at small r is a reflection of the properties of the vacuum energy. It merely imposes a bound on the resolution of our analysis around the origin that does not affect our interpretation about the numerical results for the regions of r r e where the numerical results are reliable.
We will provide more evidence later to support this conjecture. Fig. 9 shows that C becomes small around the star but is still nonzero as in the previous cases, and the pressure P is nonzero between r e and the outer surface r * s . As depicted in Fig. 10, the behavior of r around the neck resembles Fig. 6, again. The value of r appears to approach to zero up to an uncertainty of r e . This result corresponds to the choice of a approximately appropriate density m 0 m 0 for a regular geometry. It is the transition point between a density too small and  Figure 10: The radius r for κm 0 = 25.2, r * s = −60 < a * , a 0 = 10 and α = 0.05. The behavior near the neck is the same as Fig. 6. The radius r increases as r * decreases from r * = a * −56.685, but starts to decrease from r * −60.003. The instability of the numerical analysis appears around r * −1571 where the radius is r = r e 0.22. Figure 11: Schematic 3D picture for the case of Fig.10. The radius r corresponds to that in the horizontal direction while the vertical axis is chosen such that the distance along the 2D surface in the picture is given by the change in the r * coordinate. a density too large (or equivalently, a positive or negative mass singularity). As a numerical simulation, the determination of the valuem 0 is of course not exact, and there is a small range of m 0 with similar behavior (e.g., at κm 0 = 30).
We understand its numerical uncertainty as follows. When m 0 is too small (m 0 < m 0 ), there is a third turning point of r under the neck. However, if the radius at the third turning point is as small as r e , we will not be able to distinguish the behavior of the solutions outside the third turning point from that for the regular solution (m 0 =m 0 ).
If m 0 is too large (m 0 > m * ), the perfect fluid is a thick shell with an inner boundary.
However, if the radius of the inner boundary is as small as r e , we cannot distinguish it from that for the regular solution (m 0 = m * ).
In this work, we do not plan to pursue the physics in the tiny region around the origin of scale √ α, and we aim to focus on geometry at a larger scale. As we will see later, the instability of the numerical calculation in the region of scale r e around the origin appears even when the surface of the star is well above the neck and satisfies the condition (5.14) for which the geometry is regular and horizonless in the classical limit.
It should be clear that the potential singularity hidden by the numerical instability is of a different nature from the singularity at the origin in the classical black hole geometry. We now comment on the relation between the value ofm 0 (a 0 , r * s ) and r * s , with a 0 fixed. Numerically, we estimate the value ofm 0 by the smallest value of m 0 for which r does not have the third turning point. The resulting relation is shown in Fig. 12. While r increases as r * decreases under the neck, d /dr decreases, where is the proper distance. The volume of the star is smaller for larger r s (with smaller r * s ) by a factor of C 1/2 , and hence larger densitym 0 (a 0 , r * s ) is necessary to keep a 0 unchanged. Furthermore, the classical Schwarzschild radius a 0 is proportional to the mass observed at the infinity, to which the local mass density m 0 contributes through the redshift factor C 1/2 . With both effects combined, the total mass a 0 is related to the density m 0 with a factor of C. This implies that the density m 0 is proportional to C −1 (r s ). The factor C is exponentially suppressed as r becomes larger, and hencem 0 (a 0 , r * s ) increases exponentially as r s increases. Consistent with this qualitative picture, Fig. 12 shows that the densitym 0 appears to increase exponentially as r s increases. Since a much large densitym 0 is required for a slightly larger r s , it is natural to assume that in a generic case the radius of the surface r s is only slightly larger than the radius of the neck r = a unless the density is trans-Planckian.
The Komar mass of the star is well approximated by the density and pressure of the fluid, and contribution from the negative vacuum energy is very small. For a given classical Schwarzschild radius a 0 (a given total mass observed at spatial infinity), Fig. 13 shows the contribution of the perfect fluid to the Komar mass: where the integration is from r e to r s , assuming that the contribution from r < r e is negligible. The fluid's contribution to the Komar mass is almost equal to but slightly larger than the total mass 4πa 0 /κ, due to the negative vacuum energy.
The entropy of the fluid also approximately agrees with the Bekenstein-Hawking entropy. The entropy density s of the fluid is estimated by using the local thermodynamic relation as where T is the local temperature which is related to the Hawking temperature T H by Here, we assume that the Hawking temperature is simply given by that for the classical Schwarzschild black hole, and then, the entropy of the fluid is calculated as where h is the induced metric on the time-slice. The mass M fluid and entropy S fluid of the fluid as functions of r s for a 0 = 10 and α = 0.05 is shown in Fig. 13. We will show their dependence on a 0 later in Fig. 18.

Star above the neck
We have discussed above the situation when the surface of the star hides behind the neck of the wormhole-like geometry. As we have seen in Section 5.1, the geometry has the singularity in the classical case even if the surface of the star is outside the Schwarzschild radius, r s > a 0 . Here, we consider the class of solutions with the surface of the star above the neck, i.e. r * s > a * , to see that the singularity in the classical case is modified by the negative vacuum energy. We shall consider separately two cases: the case when r * s is slightly larger but very close to a * , for which the singularity appears in the classical case (r < 9 8 a), and the case when r * s is much larger than a * , for which the geometry is regular in the classical case (r > 9 8 a). First we consider the case when r * s is slightly larger but close to a * so that r < 9 8 a. As the condition (5.15) is violated, there is no static regular solution for the classical Einstein equations with zero vacuum energy. After taking into account the quantum effects, those stars which break this condition (5.15) can also exist.
There are again three types of solutions. In these semi-classical solutions, there is no turning point of r if the density m 0 is the same as or larger than the densitym 0 for regular geometry (m 0 =m 0 or m 0 >m 0 ). On the other hand, there is one turning point of r, as the case for the vacuum solution if the density m 0 is smaller thanm 0 (m 0 <m 0 ).
The turning point for this case m 0 <m 0 is slightly smaller than the radius r = a of the neck for the case when all the matter resides behind the neck. This is because the size of the turning point is determined by the total mass inside the turning point, and the latter is smaller as some of the matter resides outside the turning point.
For example, for the surface radius at r * s = −50, where r s = 10.0194, the radius for regular geometry is around κm =m 0 10, and κm 0 = 5 is too small while κm 0 = 20 is too large. For κm 0 = 5, the turning point is around r * = −243, where r = 8.36. Fig. 14 shows the behavior of r for κm 0 = 5 and κm 0 = 10. The behavior of r for κm 0 = 20 is almost the same as that for κm 0 = 10, but the pressure P vanishes where r is much larger than r e . Let us now consider the case when the surface of the star is well above the neck, i.e. r * s a * so that the condition r > 9 8 a (5.15) is satisfied, and the geometry is well defined without divergence even in the classical limit.
There are again 3 types of solutions. If the density m 0 is smaller thanm 0 , there is a turning point of r. For example, for the surface at r * s = 50, with r s 23.65, the densitŷ m 0 for regular geometry is approximately κm 0 κm 0 0.00227, and κm 0 = 0.00226 is too small while κm 0 = 0.00228 is too large. Notice that they approximately satisfy the classical relation between the total mass and the density,  for a 0 = 10. The densitym 0 shown in Fig. 17 is around the minimum of this range.
While the minimum of the range may not properly reflect the a 0 -dependence of the exact value ofm 0 , it is clear from Fig. 17 thatm 0 is insensitive to changes in a 0 even though it appears to slightly increase as a 0 increases.
Although the densitym 0 is almost independent of the classical Schwarzschild radius  Fig. 18.
It should be noted that the classical Schwarzschild radius approximately equals to the surface radius since a ∼ a 0 and we took r s = a. Clearly the classical contribution to energy dominates over its quantum counterpart.
The relation betweenm 0 and α for fixed a 0 is depicted in Fig. 19. Here, we take In terms of the Planck length p , the density behaves asm 0 ∼ N −1 −4 p . We assume in this paper that N is very large so that the density is much lower than the Planck scale.
It should be noted that in principle all fields in the theory should contribute to an effective value of N through their contribution to the vacuum energy, including those which are not present in the classical matter of the star. Hence the number of the fields in the theory could be as large as O(10 2 ), in the standard model, for example. In a hypothetical UV-complete theory in which the number of fields is arbitrarily large, √ α defines a characteristic length scale that is arbitrarily larger than the Planck length. This would allow us to avoid Planck-scale physics in our discussions.
Finally, Fig. 20 shows the radius where we face the technical difficulty, r = r e . The radius r e is, in fact, proportional to √ α, and hence the problem is only in a small region around the origin.  Figure 20: The radius r = r e below which the numerical calculation is unreliable due to numerical instability for a 0 = 10 but different values of α. The surface of the star r * s is defined by the relation r(r * s ) = a. The radius r e behaves as r e α 1/2 .

Conclusion and discussions 7.1 Summary
Solving the semi-classical Einstein equation, we studied the geometry of a static, spherically symmetric configuration with the back reaction of vacuum energy taken into consideration. The vacuum energy is given by a toy model that is often used in the study of black holes in the literature. For the Boulware vacuum (no incoming or outgoing energy flux at the spatial infinity), the vacuum has negative energy and the black-hole geometry has no event horizon. Around the classical Schwarzschild radius r = a 0 , the geometry has a local minimum of the radius. The local minimal radius r = a is called the quantum Schwarzschild radius. It differs from the classical Schwarzschild radius a 0 by a − a 0 O(α/a 0 ), corresponding to a proper distance of order O( √ α). This difference is associated with the negative energy outside the quantum Schwarzschild radius, which is only of order O(a −1 0 ). Assuming a large number N of fields contributing to the vacuum energy, the quantity α = κN/24π defines a characteristic length scale √ α of vacuum geometry that is much longer than the Planck length p = √ κ. This is hence a theoretical model in which the quantum gravitational effect is suppressed while the quantum effect in low energy effective theories plays an important role.
A star with its surface below the neck at r = a would appear very much like a black hole to a distant observer due to the huge blue-shift factor of order O(a/ √ α) at the neck. The space behind the neck has an even larger blue-shift factor, but it is still causally connected with the space outside, in the absence of horizon. Furthermore, if the surface of the star is below the neck, its proper distance from the neck is at most of order O( √ α), This is a result of the peculiar property of the geometry in vacuum described in Sec.3.2, and is thus independent of the matter content of the star.
The mass density of the vacuum around the neck is only of order O(1/a 2 2 p ). It is much smaller than the mass density of the incompressible fluid, which is of order O(1/κα) or larger, if the surface of the star is around the neck or lower. While the total energy is dominated by the matter, the vacuum energy plays an important role in modifying the geometry so that the surface of the star is always close to the Schwarzschild radius.
We studied thin shells and incompressible fluid as simple models of black-hole-like objects. Due to the effect of the vacuum energy, classically forbidden configurations are regularized. There are static configurations with smooth distribution of energy and pressure which are regular everywhere (up to an uncertainty within a region of scale √ α around the origin).
In the classical case, it is well known that the pressure diverges for incompressible fluid if the surface radius is smaller than 9 8 of the classical Schwarzschild radius, or equivalently, if the density κm 0 is larger than 8 3r 2 s . Therefore, for a given incompressible fluid (with given mass density m 0 ), the classical theory predicts its collapse into a black hole for sufficiently large radius r s , even if r s is initially larger than a 0 .
By turning on the vacuum energy of a quantum field, the pressure no longer diverges.
There is no bound on the radius or the density. This may be surprising to the reader because the vacuum energy can be extremely small if the radius r s is not very close to a 0 . Yet a small correction to the energy-momentum tensor is in fact capable of significant modifications. We have given an analytic proof of the regularity of the pressure in a similar fashion as our proof for the absence of horizon in the vacuum solution.

Comments
One may wonder how a tiny vacuum energy can have the significant effect on the geometry as described in this paper. Note that the Hawking radiation is also extremely weak as part of the vacuum energy-momentum tensor, but it can lead to the complete evaporation of an arbitrarily large black hole, making crucial difference to the global causal structure of space-time. Note also that any radiation, however weak it is, can appear arbitrarily strong for an observer moving towards the source close to the speed of light. Whether an energy-momentum tensor is strong or weak depends on the specific problem we focus on.
While the spacetime outside the Schwarzschild radius is very well approximated by the Schwarzschild solution, the modification of the geometry around and below the neck seems dramatic at first sight. For instance, the horizon disappears completely due to the vacuum energy. However, the event horizon is defined by the condition that anything inside the horizon takes an infinite amount of time to get out. If the geometry is deformed such that anything inside the horizon takes an extremely long time (say, 10 100 times the age of the universe) to come out, even though the event horizon is completely removed, it can still be viewed a small deformation for physicists.
For the model of vacuum energy we have considered, the geometry is modified such that the surface of the star can never be over a few Planck lengths behind the Schwarzschild radius, where the horizon is replaced by a turning point in the radius.
This feature is reminiscent of the fuzzball scenario [2] and other proposals [29] in which quantum effects modify the near-horizon geometry. This model serves as a new class of order-1 corrections to the horizon geometry, which is needed In the dynamical process of gravitational collapse for unitarity [30].

Outlook
It is not very clear whether this model defined by 2D massless scalars properly represents the qualitative features of the 4D vacuum energy of the real world, despite its frequent appearance in the literature. It was shown in Ref. [17] that different models of vacuum energy lead to quite different models of static black holes. We study this model as a case study of the various possibilities. To say the least, it provides a concrete toy model of black holes for the sake of discussions, among others that were proposed in the literature [31]. It points out an interesting new possibility about how vacuum energy modifies the black-hole geometry in a significant way.
In fact, it was shown in [17] that a vacuum state with negative energy would in general lead to the wormhole-like structure (a local minimum of r) instead of a horizon. If the vacuum energy is negative, the geometry is expected to share some of the qualitative features as those described in this paper. Black holes with back reaction from other models of vacuum energy will be studies in more detail in the near future.
Based on the static solutions, we speculate on what to expect in a dynamical process of gravitational collapse for the same model of vacuum energy. Typically one assumes that the initial state is the Minkowski vacuum with zero vacuum energy in the infinite past. The vacuum energy becomes negative and increases in magnitude during the collapse because of the increment of curvature. If there is no energy exchange with matters, the conservation law for the vacuum energy implies an outgoing (positive) energy flow for energy conservation. This is a generalization of the notion of Hawking radiation. Even an arbitrarily slow collapse without the formation of horizon involves the generalized Hawking radiation.
For simplicity, we made the assumption that the energy-momentum tensors are conserved separately for the fluid and the vacuum. As a result, the Hawking radiation cannot take away the energy of the fluid. The fluid remains there even when the total energy becomes zero (the magnitude of the negative vacuum energy equals the fluid energy). At the end of such a "complete evaporation", the Schwarzschild radius (both a and a 0 ) goes to zero, or to the Planck scale, with the liquid persisting in a large space behind the vanishing neck, resembling Wheeler's bag of gold [27]. We have shown in this paper that this cannot happen for static configurations with realistic density of the fluid. However, this can be realized for dynamical configurations. This type of solution has already been observed in numerical analysis [20].
In a fundamental theory including quantum gravity such as the string theory, there are always (direct or indirect) interactions among different fields. The quantum fields in vacuum and the collapsing matter are allowed to exchange energy, and the mass of the fluid decreases during evaporation, leading to a reduction in size of the interior space under the neck. The question now is whether the neck shrinks slower than the interior size of the bag, so that there is no remnant in the end of a complete evaporation. For example, if the evaporation of the black hole is sufficiently slow and the solution can be approximated by the static solution in this paper at each moment, the size of the neck and that of the interior are almost the same and the neck cannot shrink to zero with a large bag behind. On the other hand, if the evaporation is very fast and the fluid inside cannot settle down to almost static configurations, the "bag of gold" with a large interior and a Planckian-sized neck would appear as in some numerical analyses [20]. In order to see more details on this process, time-dependent solutions of the semi-classical Einstein equation should be considered. This is left for future studies.