An Event Horizon ‘Firewall’ Undergoing Cosmological Expansion

Abstract

We embed an object with a singular horizon structure, reminiscent of (but fundamentally different from, except in a limiting case) a black hole event horizon, in an expanding, spherically symmetric, homogeneous, Universe that has a positive cosmological constant. Conformal representation is discussed. There is a temperature/pressure singularity and a corresponding scalar curvature singularity at the horizon. The expanding singular horizon ultimately bounds the entire spacetime manifold. It is is preceded by an expanding light front, which separates the spacetime affected by the singularity from that which is not yet affected. An appropriately located observer in front of the light front can have a Hubble–Lemaître constant that is consistent with that currently observed.

1. Introduction

Recently, in [1,2], the importance of embedding a black hole-type singularity in an expanding cosmology has been emphasized; the resulting black hole would grow with the Universe, rather than remain stationary in asymptotic Minkowski space. Ideally, one would like a locally rotating, perhaps non-stationary, Kerr-type solution that asymptotically passes smoothly into a part of, or even all of, an expanding spacetime. Such a solution is not known, and a successful one might have to be embedded in a Universe that is not only expanding, but is also itself rotating. Even in spherical symmetry, the best attempt at coupling a local black hole smoothly to a Universe that expands to infinity follows from [3] and its conformal descriptions [4].

We are unable to deliver this most desirable embedded solution. The more modest purpose of this work is to recall an earlier cosmological solution [5,6,7], which, for a different parameter range than that originally considered, also allows for an interpretation in terms of an object with a singular horizon that is embedded in an expanding cosmology. It is a solution to the Einstein field equations that possesses non-stationary spherical symmetry (with zero shear and isotropic expansion) and an ideal fluid matter description [8]. It contains the cosmological constant as the simplest model of vacuum energy. Despite the existence of the cosmological constant, the field equations may nevertheless be solved explicitly as a self-similar (i.e., kinematic self-similarity, Ref. [6]) form. Such a self-similar solution is the smoothest way of coupling a solution from small to large scales. The solution fits into the ansatz prescribed in [9] for finding homogeneous solutions; however, this particular example was found independently.

By requiring this self-similar symmetry, a particular equation of state is imposed on the cosmology. The resulting form of the equation of state is not unreasonable in the hot gas phase of the cosmology and it may pass smoothly to the gravitational collapse phase.

In contrast to the cases explored in [5], the solutions discussed here have a horizon where the light speed is zero and which defines the boundary of an embedded object. On a (necessarily) cosmic time scale, the radius coordinate of this surface expands exponentially as a function of coordinate time.

We may somewhat loosely refer to the central object within the horizon as a ‘black hole’ in the general case, because light cannot traverse the horizon. However, the structure formally differs from a Schwarzchild or Kerr black hole, because the pressure, and consequently the scalar curvature, are infinite at the horizon. The central singularity has been ‘pasted’ onto the horizon surface, rather than being located at the center of the embedded object.

For understanding any physical application of the solution, it is necessary to appreciate the very large scales associated with the time and space variables. These scales (see Equations (2) and (3) below) are set by the fundamental constants c (speed of light) and $\mathrm{\Lambda }$ (cosmological constant), and they correspond to spans of order of the life time and size of the Universe, respectively. The Hubble–Lemaître expansion rate varies from a minimum value, given in terms of fundamental constants, to values that are familiar to observers at the current epoch, allowing us to straightforwardly locate an observer within the spacetime solution.

We do not present our solution as a wholly explored cosmological model. Although far from the singularity, it is dynamically rather close to the $\mathrm{\Lambda }-\mathrm{CDM}$ model. It is moreover not the theoretical solution to embedding a rotating black hole in an expanding cosmology. Our solution does, however, add to the list of previous works aimed at embedding an object shielded by a horizon within an expanding Universe, in what we believe is an original way cf. [6,7]. We elaborate at some length on the local physical behavior of the cosmology, because this solution may indeed have some relevance to recent cosmological observations [1].

In Section 3, we review the solution of [5] and we derive the correspondence between physical quantities [8] and the quantities that we use in describing the solution, in particular the pivotal parameter C. Subsequent sections present details of the solution and discuss its interpretation.

2. Materials and Methods

This study explores the possibility that the Universe contains an expanding ‘firewall’ that expands, in a self-similar manner, into the surrounding Universe. It does so by generalizing an existing [5] cosmological solution into a domain not previously explored. The methodology is wholly theoretical and analytic in nature; standard numerical methods, using the IDL coding language, were used only to plot the behavior of analytic expressions and to construct the conformal diagrams presented in Appendix A.

3. Physical Quantities

The solution to the general relativity field equations in [5] is given in terms of a combined spacetime variable:

$$\xi ={\displaystyle \frac{\sqrt{3/\mathrm{\Lambda }}}{r}}exp\left(\sqrt{{\displaystyle \frac{\mathrm{\Lambda }{c}^2}{3}}}t\right),$$

(1)

where $\mathrm{\Lambda }$ is the cosmological constant, c is the speed of light in vacuo, and t and r are convenient time and space coordinates. The obvious scale for any space measure is

$$\sqrt{3/\mathrm{\Lambda }}\approx 1.65\times {10}^{28}\mathrm{cm}$$

(2)

(where we have adopted the Planck measurement $\mathrm{\Lambda }\approx 1.11\times {10}^{-56}{\mathrm{cm}}^{-2}$), while the corresponding time scale is

$${\displaystyle \frac{\sqrt{3/\mathrm{\Lambda }}}{c}}\approx 5.5\times {10}^{17}\mathrm{s}\approx 1.75\times {10}^{10}\mathrm{years}.$$

(3)

With these units for r and t, the self-similar spacetime variable becomes simply

$$\xi ={\displaystyle \frac{e^t}{r}},$$

(4)

where it is recognized that changes of order unity in t or r span the respective scales of the known Universe.

In this example of kinematic self-similarity [6,7], the metric of spherically symmetric spacetime takes the form

$$d{s}^2=e^{\sigma \left(\xi \right)}d{t}^2-e^{\omega \left(\xi \right)}d{r}^2-R^2\left(\xi \right)(d{\theta }^2+{sin}^2\theta d{\varphi }^2),$$

(5)

where the circumferential radius

$$R(r,\xi )=rS\left(\xi \right),$$

(6)

so that, at a given spatial coordinate r, the function $S\left(\xi \right)$ plays the role of the scale factor $a\left(t\right)$ in the standard FLRW cosmology.

The matter mass $m_m$ is the total mass m corrected by subtracting the mass of the vacuum:

$$m_m=m-{\displaystyle \frac{4\pi }{3}}{R}^3{\rho }_v={\displaystyle \frac{r}{2}}M\left(\xi \right),$$

(7)

where

$${\rho }_v={\displaystyle \frac{\mathrm{\Lambda }{c}^2}{8\pi G}}\approx 5.89\times {10}^{-30}\mathrm{g}{\mathrm{cm}}^{-3}$$

(8)

and G is the Newtonian gravitational constant. Each mass term in Equation (7) is dimensionless, related to the physical mass $m_m$ (or to the vacuum mass $m_v=4\pi R^3{\rho }_v/3$; each in grams) by the factor

$$\mu =\sqrt{{\displaystyle \frac{3}{\mathrm{\Lambda }}}}{\displaystyle \frac{c^2}{G}}\approx 2.26\times {10}^{56}\mathrm{g};$$

(9)

the dimensional masses are thus of order of the mass of the observable Universe.

The matter density ${\rho }_m$ and the matter pressure $p_m$ are, respectively, given by

$$8\pi {\rho }_m={\displaystyle \frac{\eta \left(\xi \right)}{r^2}};8\pi p_m={\displaystyle \frac{P\left(\xi \right)}{r^2}}.$$

(10)

The units of $\eta \left(\xi \right)$ may be taken to be in terns of ${\rho }_v$, while the units of $P\left(\xi \right)$ are ${\rho }_v{c}^2\approx 5.27\times {10}^{-9}$ erg ${\mathrm{cm}}^{-3}$. In each case, r is the dimensionless radius.

In [5], the Einstein field equations for a spherically symmetric, non-stationary, spacetime manifold were used, together with the Bianchi identities [8], to write a complete set of ordinary differential equations for $S\left(\xi \right)$, $M\left(\xi \right)$, $P\left(\xi \right)$, $\eta \left(\xi \right)$, $\sigma \left(\xi \right)$ and $\omega \left(\xi \right)$. For this, the authors had to show that self-similarity, in terms of $\xi$ as the invariant, existed in the equations. Without having to assign an equation of state a priori, this completeness comes from the splitting of the energy equation in a manner that achieves the desired self-similarity. It is therefore useful to recall the energy equation in the form

$${\mathrm{\Gamma }}^2-{\displaystyle \frac{U^2}{c^2}}=1-{\displaystyle \frac{2Gm}{R{c}^2}},$$

(11)

where m is the total mass, the co-moving Lorentz factor

$$\mathrm{\Gamma }=e^{-\omega /2}{\partial }_{r}R,$$

(12)

and the radial four-velocity of a ‘co-moving observer’ (i.e., one at fixed r) is (in units of c)

$$U=e^{-\sigma /2}{\partial }_{t}R\equiv r\times e^{-\sigma /2}\xi S^{\prime }\left(\xi \right).$$

(13)

The energy Equation (11) is the equation that must be split into two, as in the original paper [5]. The separation is a simplifying assumption, different from those normally used (e.g., Ref. [10], p. 167); it splits the matter and vacuum mass terms according to

$$\begin{array}{ll}{\mathrm{\Gamma }}^2\left(\xi \right)& \equiv e^{-\omega \left(\xi \right)}{(S\left(\xi \right)-\xi S^{\prime }\left(\xi \right))}^2\\ & =1-{\displaystyle \frac{2G{m}_m}{R{c}^2}}\equiv 1-{\displaystyle \frac{M\left(\xi \right)}{S\left(\xi \right)}},\end{array}$$

(14)

and

$$U^2\equiv r^2{e}^{-\sigma }{\left(\xi S^{\prime }\left(\xi \right)\right)}^2={\displaystyle \frac{2G{m}_v}{R{c}^2}}\equiv r^2{S}^2\left(\xi \right)=R^2,$$

(15)

where the prime indicates differentiation with respect to $\xi$, and the expressions are dimensionless. Given the sign structure in the energy equation, this separation is unique for a positive cosmological constant.

4. Solution

As shown in [5], there is a general integral of the field equations that takes the form

$$M\left(\xi \right)=\eta \left(\xi \right){\displaystyle \frac{S^3\left(\xi \right)}{3}}+\Delta ,$$

(16)

where $\Delta$ is a constant. When $\Delta \ne 0$, a numerical solution is possible, but we focus here on the analytic solution that results when $\Delta =0$. In terms of the quantities used in the previous section, the solution takes the following form [5]:

$$\begin{array}{lll}{e}^{\sigma }={(Ctanh(Cln\xi )-1)}^2& ,& e^{\omega }=C^2{K}^2{\xi }^2{\mathrm{sech}}^2(Cln\xi ),\\ S=K\xi \mathrm{sech}(Cln\xi )& ,& M=K\xi {\mathrm{sech}}^3(Cln\xi ),\\ P={\displaystyle \frac{1-3Ctanh(Cln\xi )}{K^2{\xi }^2(Ctanh(Cln\xi )-1)}}& ,& \eta ={\displaystyle \frac{3}{K^2{\xi }^2}}.\end{array}$$

(17)

Noting that $CS=e^{\omega /2}$, and hence, that $R=rS=r{e}^{\omega /2}/C$, it follows [10] (page 166) that the shear is zero. The expansion is finite:

$$\mathrm{\Theta }=3e^{-\sigma /2}(1-Ctanh(Cln\xi ))=3$$

(18)

and is isotropic. The solution fits into the McVittie [9] ‘ansatz’ as presented in Section 14.2.3 in [10], although as far as we know, this particular possibility has not been explored elsewhere. Therefore, for classification purposes, we next demonstrate this briefly. Referring to the notation in [10], we have $\lambda =\omega /2$, $\nu =\sigma /2$ in our terminology. Then, as quoted in [10], the ansatz requires

$$e^{\omega /2}=P\left(r\right)S\left(t\right)e^{\eta \left(z\right)},$$

(19)

where $z=ln\left(Q\right(r)/S(t\left)\right)$, and $S\left(t\right)$ is not the same as our $S\left(\xi \right)$. We can choose $S=e^t$, $P\left(r\right)=CK/r$ and $Q\left(r\right)=r$. This gives $z=-ln\xi$, so that taking

$$\eta (-ln\xi )=ln\left(\mathrm{sech}\right(Cln\xi \left)\right),$$

(20)

we have

$$e^{\omega /2}=CK\xi \left(\mathrm{sech}(Cln\xi )\right)$$

(21)

as given in Equation (17).

With $\dot{S}/S=1$, the ansatz requires the quantity $f\left(t\right)=0$ and so (Equation 14.25 in [10])

$${\displaystyle \frac{\sigma }{2}}=ln\left({\displaystyle \frac{\dot{\omega }}{2}}\right).$$

(22)

Using the expression for $e^{\omega /2}$ in Equation (17) gives

$${\displaystyle \frac{\dot{\omega }}{2}}={\displaystyle \frac{dln\left[\xi \mathrm{sech}\right(Cln\xi \left)\right]}{dln\xi }}=1-C\tanh (Cln\xi ),$$

(23)

so that

$${\displaystyle \frac{\sigma }{2}}=ln(1-Ctanh(Cln\xi )),$$

(24)

as in Equation (17).

Our metric also fits the general form for homogeneous solutions (Equation (14.44) in [10]) because $f\left(t\right)=0$ and Equation (22) holds. Minor re-scalings and re-definitions are required. Using the remaining field equations, expressions for the pressure and density can be found in terms of this ansatz. However, having split the energy equation, we have diverged from this procedure and are instead led to the analytic solution (17).

The constant K is not without physical significance because it does not appear together with $\xi$ in the arguments of the hyperbolic functions (otherwise, it could be absorbed into $\xi$). It is convenient to keep this freedom explicit in the formal expressions, but we will ignore it in our numerical estimates. (It will be shown below that $K=1/C\simeq 1$ in the most likely case.) The constant C is the principal physically significant constant, and because $\mathrm{sech}x$ and $xtanhx$ are both even functions, it may be taken to be positive without loss of generality.

Given the form of $g_{00}=e^{\sigma }$, there is no horizon if $C<1$ Such cases formed the basis for the work in [5]. They give a cosmology with a single, central, singularity and an asymptotically de Sitter metric outside an expanding inhomogeneous ‘bubble’. However, when $C>1$, there is a horizon (although it is not a null surface), since $e^{\sigma }\to 0$ at ${\xi }_h$, where $tanh(Cln{\xi }_h)=1/C$, or

$${\xi }_h={\left({\displaystyle \frac{C+1}{C-1}}\right)}^{1/2C}.$$

(25)

It is such cases, and their possible relevance to embedding a central black hole-type object in an expanding Universe, that we explore further here. We note that we term ${\xi }_h$ a ‘horizon’ because the radial null velocity

$${\displaystyle \frac{dr}{dt}}=e^{\sigma /2}{e}^{-\omega /2}$$

(26)

equals zero at $\xi ={\xi }_h$. However, at the horizon, $Ctanh(Cln\xi )-1=0$ and so, from Equation (17), $P\to -\infty$. Hence, via the contracted field equations, the Ricci curvature invariant ℜ also becomes infinite at the horizon. If the quantity $g^{\mu \nu }{F}_{,\mu }{F}_{,\nu }=e^{-\sigma }{\left({\partial }_{t}F\right)}^2-e^{-\omega }{\left({\partial }_{r}F\right)}^2$, with $F=e^{\sigma }$, is null at $\xi ={\xi }_h$, then the horizon is a null surface. A straightforward calculation shows, however, that the normal to the surface is not null at $\xi ={\xi }_h$, but rather, is finite and time-like. It does approach zero as $C\to 1$.

The question arises as to whether the singularity is ‘naked’. To this end, we have numerically integrated the explicit equation for the null paths in $(t,r)$ coordinates, viz. (with $K=1/C$):

$$\begin{array}{ll}{\displaystyle \frac{dr}{dt}}& =\mp {\displaystyle \frac{C\tanh (Cln\xi )-1}{\xi \mathrm{sech}(Cln\xi )}}\\ & =\mp lim\; sup{\displaystyle \frac{1}{\xi }}[Csinh(Cln\xi )-cosh(Cln\xi )]\hfill \\ & =\mp lim\; sup{\displaystyle \frac{1}{2\xi }}\left[(C-1){\xi }^C-(C+1){\xi }^{-C}\right]\hfill \\ & =\mp {\displaystyle \frac{r}{2e^t}}\left[(C-1){\displaystyle \frac{e^{Ct}}{r^C}}-(C+1){\displaystyle \frac{r^C}{e^{Ct}}}\right],\hfill \end{array}$$

(27)

for various values of $C\ge 1$. We started the integration from an initial time $t=ln(r{\xi }_h)$ for various values of r, thus determining $r\left(t\right)$ for light rays that end at the singularity.

In Figure A1 (see Appendix A), we sketch schematic conformal diagrams corresponding to these null ray solutions for the following three cases: $C<1$ (the original cosmology in [5]), $C=1$ (the singular case described in Section 7 below), and $C>1$. In Section 7, we show in some detail approximations to the null geodesics that justify our sketches. In these schematic diagrams, the $(t,r)$ coordinates are ‘fluid’ in the sense that light rays are not necessarily at ${45}^{\circ }$, as would be usual (we also show a ‘strict’ conformal diagram for the case $C=1$ in Figure A2 in Appendix A; this uses the exact solution of Equation (27) available when $C=1$). On the other hand, the shape of the singularity in these diagrams is reasonably accurate. Unlike FLRW models, the time line $r=0$ does have some physical significance, because the pressure is not homogeneous.

At large negative t, it is the second term in Equation (27) that dictates the behavior of the null geodesic. This gives the following for the outgoing rays:

$$r^C={\displaystyle \frac{2}{C{e}^{-(C+1)t}+Cv}},$$

(28)

where v is constant on the ray. This shows the affine parameter t extending ultimately to $i^{-}$ as $r\to 0$, with the rays being released at various values of v en route. At negative t, one can find the time for a given ray when $r=e^t/{\xi }_h$ using this approximation (and also at positive t using the first term in Equation (27)), but numerically, we find that all rays emitted back to past time-like infinity $i^{-}$ are absorbed by the singularity before reaching spatial infinity $i_o$ (see discussion in Section 7).

In the limiting case with precisely $C=1$, the singularity is also not naked; in this case, in $(t,r)$ coordinates, the singularity is time-like at finite times but space-like at the end of time where the outgoing light rays arrive according to the exact expression $r=1/(v+e^{-2t})$ (v constant).

5. Physical Implications of the Solution

The importance of the parameter C justifies finding an expression that relates it to physical quantities. Equation (17) (see also Equations (34), (43), and (46) of [5]) relate the constant C to the values of the dimensionless parameters P, $\eta$, M, and S:

$$P=\eta {\displaystyle \frac{C\sqrt{1-M/S}-\frac{1}{3}}{1-C\sqrt{1-M/S}}}.$$

(29)

Solving this for C gives

$$C={\displaystyle \frac{3P+\eta }{3(P+\eta )}}{\displaystyle \frac{1}{\sqrt{1-M/S}}}.$$

(30)

Generally, if we assign numerical values to $P,\eta ,M$, and S to obtain C, then the value of $\xi$ at the point in question is then fixed by any one of Equation (17). In terms of dimensional physical quantities, Equation (30) is

$$C={\displaystyle \frac{3p_m+{\rho }_m}{3(p_m+{\rho }_m)}}{\displaystyle \frac{1}{\sqrt{1-\frac{2m_m}{R}}}}\equiv {\displaystyle \frac{\beta }{\sqrt{1-\frac{2m_m}{R}}}},$$

(31)

where we have defined

$$\beta ={\displaystyle \frac{3p_m+{\rho }_m}{3(p_m+{\rho }_m)}}.$$

(32)

From Equation (17), $e^{\omega /2}=CS$, so the spatial part of the metric (5) becomes

$$\begin{array}{ll}{C}^2{S}^{2}d{r}^2& +r^2{S}^{2}d{\mathrm{\Omega }}^2=S^2\left[C^{2}d{r}^2+r^{2}d{\mathrm{\Omega }}^2\right]\\ & =S^2\left[{\beta }^2{\displaystyle \frac{d{r}^2}{1-\frac{2m_m}{R}}}+r^{2}d{\mathrm{\Omega }}^2\right],\hfill \end{array}$$

(33)

where we have used Equation (6). With ${\rho }_m=0$, as at infinite time, $\beta =1$ and the spatial part of the metric becomes an ‘expanding Schwarzchild’ form $S^2[d{r}^2/(1-2m_m/R)+r^{2}d{\mathrm{\Omega }}^2]$. Should ${\rho }_m+3p_m=0$ (which is $P=-\eta /3$), as when $\xi =1$ (see below), then $\beta =0$ and the spatial part of the metric reduces to the purely circumferential part $S^2{r}^{2}d{\mathrm{\Omega }}^2$, that is, a ‘two sphere’.

The functions in the variable $\xi$ in Equation (17) are well behaved but for two incidents: the previously noted singularity in P at $\xi ={\xi }_h$, and also, singularities in both $\eta$ and P, where $r\to \infty$ (or $t\to -\infty$), both corresponding to $\xi \to 0$. In fact, P approaches negative infinity as $\xi \to 0$ for any value of C. Moreover, when $C>1$, P can also become negative at large $\xi$, although the horizon can be set so that this negative value is hidden.

We note that there is no singularity at $r=0$: as $r\to 0$, $\xi \to \infty$ (Equation (4)), so that, with $C>1$, Equation (17) causes $R(=rS)$, $e^{\omega }$, $m_m(=rM)$ to approach zero. Moreover, ${\rho }_m=\eta /\left(8\pi r^2\right)$ and $p_m=P/\left(8\pi r^2\right)$ are finite at finite t (i.e., not at $t^{-}$). The metric coefficient $g_{00}=e^{\sigma }$ approaches the value ${(C-1)}^2$ so that, initially, $ds>0$ at $i^{-}$. The singularity that appears inside the horizon in, e.g., the Schwarzschild black hole solution is thus displaced from the origin to the location of the horizon itself. In this sense, the solution fundamentally differs from that for an embedded black hole. It describes the future evolution of an initial ‘bubble’ of vacuum at $i^{-}$, which has a thermal singularity on the boundary.

An interesting, somewhat complementary, treatment that discusses an interior solution without singularity is given in [11]. In that case, the outer region is empty and the surface term can be interpreted as a $\mathrm{\Lambda }$ term. However, we do not make use of our interior solution in this paper. Rather, it is the expanding outer region that is of interest, bounded inside by the ‘firewall’ singularity.

The ‘wave front’ $\xi =1$ is another significant point in the solution: at this (propagating) location $r=e^t$, $M=S=1$, $P=-1$, $\eta =3$, $e^{\sigma }=1$, $e^{\omega }=C^2$, and ${\mathrm{\Gamma }}^2=(1/C)(S-\xi S^{\prime })=0$. Moreover, $3p_m+{\rho }_m\propto (3P+\eta )=0$ at this point, and Equation (30) shows that this is only possible when $M=S$, i.e., $2G{m}_m/c^2=R$. We also note that $S^{\prime }\left(\xi \right)=K$ at $\xi =1$. The latter result implies that R is a maximum at any finite r when $t=ln\left(r\right)$. At the horizon ${\xi }_h$, $S^{\prime }\left({\xi }_h\right)=0$ (from Equation (17)).

A cosmological application of the solution requires returning to ($t,r$) variables. We shall pursue this briefly in the next section, where we give a fairly detailed description of one illustrative case, but it should be noted that we do not attempt to fit the measured Universe.

We recall that the relation between $\eta \left(\xi \right)$ and ${\rho }_m$ reminds us that the solution is homogeneous as well as spherically symmetric and non-stationary. This is true even though metric quantities, the mass, and the pressure do vary with radius at any finite time. The system center is the center of the embedded object when $C>1$.

The general behavior of physical characteristics of the solution is similar whether there is a horizon or not, except that, in the absence of the singular horizon that occurs for $C\ge 1$, the central singularity ($\xi =0$) is not hidden.

6. The Cosmology in Co-Moving Coordinates

The singular horizon is at a fixed value of $\xi ={\xi }_h$; so that at a fixed time, there is a minimum accessible radius, and at a fixed r, there is a maximum accessible future time. This is as ‘seen’ by an observer at ${\xi }_o$ for whom ${\xi }_o<{\xi }_h$. A significant issue for the cosmology lies in appropriately situating an observer ${\xi }_o$ in the solution. A useful reference value is the value of the Hubble–Lemaître expansion rate. This is given in various forms as

$$H\equiv {\displaystyle \frac{e^{-\sigma /2}{R}_t}{R}}=e^{-\sigma /2}{\displaystyle \frac{\xi S^{\prime }\left(\xi \right)}{S}}={\displaystyle \frac{U\left(\xi \right)}{R}}=1,$$

(34)

where the last expression follows from Equation (15). The proper value is therefore constant. The unit is that of reciprocal time, and the pertinent scale is $1/(5.51\times {10}^{17})$ ${\mathrm{s}}^{-1}$, which is equal to 56 km ${\mathrm{s}}^{-1}$ ${\mathrm{Mpc}}^{-1}$. This does not exactly locate our epoch in our cosmology, but it is close.

In calculating this value of H, we have used proper time in the cosmological metric. In the FLRW cosmologies, the proper time is the coordinate time. If we use the coordinate time in our cosmology to calculate $H\left(\xi \right)=R_t/R$, we find that

$$H\left(\xi \right)={\displaystyle \frac{\xi S^{\prime }\left(\xi \right)}{S\left(\xi \right)}}=1-Ctanh(Cln\xi )\equiv e^{\sigma /2},$$

(35)

in units of 56 km ${\mathrm{s}}^{-1}$ ${\mathrm{Mpc}}^{-1}$. This value is smaller than unity at $\xi >1$ and larger than unity at $\xi <1$. It is again equal to 1 at $\xi =1$. At the horizon ${\xi }_h$, the value of ‘coordinate’ H is zero. A good place to locate a terrestrial observer is therefore at ${\xi }_o=0.78$, because $H\left(0.78\right)\approx 1.25$, equivalent to 70 km ${\mathrm{s}}^{-1}$ ${\mathrm{Mpc}}^{-1}$, and so, is in approximate agreement with the (somewhat varying) observational inferences.

In the physical discussion of this section, we will (arbitrarily) take C to have a value just above the threshold value $C=1$. Specifically, we shall take $C=1.01$, which places the horizon singularity at ${\xi }_h={201}^{1/2.02}\simeq 13.8$ (Equation (25)). The observer is located at ${\xi }_o=0.78$ (see above), which is comfortably distant from the horizon. For the two key values of $\xi$ (the horizon ${\xi }_h$ and the observer ${\xi }_o$), there are two corresponding ‘waves’ that we can follow:

$$r_h={\displaystyle \frac{e^t}{{\xi }_h}};r_o={\displaystyle \frac{e^t}{{\xi }_o}},$$

(36)

giving the coordinates r at which the corresponding surfaces are found at coordinate time t. We will present quantitative results for two illustrative values of t, specifically the following:

• $t=1$, when the horizon is at $r_h=e^1/13.8=0.197$ and the observer is at $r_o=e^1/0.78=3.48$;
• $t=0.1$ (Figure 1), for which $r_h=e^{0.1}/13.8=0.008$ and $r_o=e^{0.1}/0.78=1.42$.

We can also treat these ‘waves’ in terms of the coordinate time at which they pass a given location r, viz.,

$$t_h=ln\left({\xi }_{h}r\right);t_o=ln\left({\xi }_{o}r\right),$$

(37)

and we shall present results for two illustrative values of r, viz., as follows:

• $r=1$, where the horizon passes at time $t_h=ln\left(13.8\right)=2.62$ and the observer passes at $t_o=ln\left(0.78\right)=-0.25$;
• $r=2$ (as used in Figure 4), where the singularity arrives at $t_h=ln\left(27.6\right)=3.32$ and the observer passes at $t_o=ln\left(1.56\right)=0.445$.

The resulting behaviors that we show below are qualitatively typical of those for any value $C>1$, but with quantitative changes in the time and space positions of interesting features. (A value $C=0.99$ gives a similar cosmology without an apparent horizon, and so, without a singular horizon.) We will set the constant $K=1$ for the illustrations; in general, we expect $K=1/C$ (see Section 6.5).

6.1. Circumferential Radius

The top panels of Figure 1 show the circumferential radius $R=rS=r\xi \mathrm{sech}(Cln\xi )=e^t\mathrm{sech}(Cln(e^t/r))$ as a function of r at two different times. The solution is homogeneous, and so, effectively incompressible. Close to the horizon (although in interpreting the word ‘close’, one should recall the units of r and t), the radius increases linearly with $r^C$. At large r, R declines with increasing r. For a fixed value of t, the maximum in R occurs when $dR/dr=0$. This maximum is most easily determined by setting $d(e^t/R)/d\xi =0$, i.e., $dcosh(Cln\xi )/d\xi =0$. This occurs when $sinh(Cln\xi )=0$, i.e., when $ln\xi =0$ or $\xi =1$. With $\xi =1$, the maximum value of R, and the radius r at which it occurs, both equal $e^t$. Consequently, the radius at which the circumferential radius is the largest and the spatial geometry is a ‘two sphere’ moves outward in time as $r_{\mathrm{peak}}=e^t$.

The lower panel shows the time dependence of the radius R at the spatial coordinate $r=1$. The curvature in the plot reflects the different values of H. (The time scale for the evolution is of course very long.) For a fixed value of r, the maximum in R occurs when $dR/dt=0$. This is best evaluated by setting $d(r/R)/d\xi =0$, i.e., $d[cosh(Cln\xi )/\xi ]/d\xi =0$ (note the appearance of the $\xi$ term in the denominator, compared to the expression above that maximizes $R\left(r\right)$ for a given t). This occurs when $tanh(Cln\xi )=1/C$, which is only slightly less than unity, so that the argument of the tanh function is large. Using the high-argument approximation, $tanhx\simeq 1-2e^{-2x}$ gives $e^{2Cln\xi }\simeq 2C/(C-1)$ or $\xi ={(2C/(C-1))}^{1/2C}$. Further, using the identity $\mathrm{sech}x=\sqrt{1-{tanh}^{2}x}$, the corresponding maximum value of R is found to be $r{(2C/(C-1))}^{1/2C}\sqrt{C^2-1}/C$. For values of C close to unity, the maximum value of $R\simeq 2$, and it occurs at $\xi \simeq \sqrt{2/(C-1)}$ or $t=ln\left(\sqrt{2/(C-1)}r\right)$.

The time dependence may also be understood at fixed r because $\xi$ increases exponentially with t. However, $R=e^t\mathrm{sech}(Cln\xi )$ eventually declines as $e^{(1-C)t}$ at very large t. Before this can happen, however, the singularity reaches the chosen r when $\xi ={\xi }_h=13.81$. Thus, at $r=1$, this occurs at $t=2.62$ while the maximum does not pass until $t=2.65$ according to our estimate above. Physically, the time dependence at every fixed r reveals the relentless expansion of this cosmic singularity.

6.2. Matter Pressure

The most physically delicate element of the embedding cosmology is the pressure. Figure 2 indicates that the pressure is small (positive or negative) almost everywhere, except at the horizon where it diverges positively. This divergence is evident at small r at fixed t and in the future at fixed r, as the apparent horizon passes. Given the homogeneous density, and approximating the co-moving plasma as an ideal gas, we see that this divergence indicates the presence of an infinite ideal gas temperature ($T\propto P/\eta$). Except for the singularity at the center of the system, there is no similar divergence in the absence of a horizon singularity, when $C<1$.

In the left panel of Figure 3, we show the pressure variable P as a function of the density variable $\eta$ (see Equation (10)). This treats the equation of state as barotropic and shows an apparently un-physical pressure decline with increasing density. However, this is misleading, because, assuming an ideal gas equation of state (as in Equation (38) below) the pressure, density, and temperature are each functions of the self-similar variable $\xi$, and so, all are functions of one another.

If we separate out the ‘temperature’ (measured in units of $\overline{m}{c}^2/k$ by introducing the mean particle mass $\overline{m}$ and Boltzmann’s constant k) and the density as in Equation (38), we obtain the graph in the right panel of Figure 3. This shows that it is the variation in effective temperature that is responsible for the behavior of the pressure. The temperature increases with decreasing density (reaching infinity at the singularity). It becomes negative at larger densities. Since gravity dominates the internal pressure of a collapsing object, the external (i.e., cosmological) pressure of the object appears (relatively) negative, so that negative pressures in the cosmological solution could be interpreted as the onset of gravitational instability.

Returning to the pressure in space and time with Figure 2, we see that at a given time t, both $\xi$ and $P\left(\xi \right)$ decrease as r increases. Since the density is homogeneous, the temperature cools with decreasing pressure, so that, just as in the standard $\mathrm{\Lambda }-\mathrm{CDM}$ model, the cosmology outside the horizon will become transparent.

To better understand the equation of state, we can write it by eliminating the factor $1/{\left(K\xi \right)}^2$ in P using $\eta$, both as written in Equation (17). This gives the explicit equation of state in terms of the self-similar variable

$$p_m={\displaystyle \frac{{\rho }_m}{3}}\left[{\displaystyle \frac{(3C-1){\xi }^{2C}-(3C+1)}{(C+1)-(C-1){\xi }^{2C}}}\right],$$

(38)

which we emphasize is imposed by the kinematic self-similar symmetry. The factor multiplying the density is the temperature as a function of the self-similar variable, assuming an ideal gas equation of state.

According to this equation of state, $p_m$ will decline smoothly (see Figure 2 and Figure 3) to the relativistic value ${\rho }_m/3$ when $\xi$ has declined to a value of

$${\xi }_r={\left({\displaystyle \frac{2C+1}{2C-1}}\right)}^{1/2C}.$$

(39)

Further, the pressure reaches zero at $\xi ={\xi }_T$, where

$${\xi }_T={\left({\displaystyle \frac{3C+1}{3C-1}}\right)}^{1/2C}.$$

(40)

These points represent, for a fixed time t, the steady decline with distance r of the gas temperature to zero. Each state is attained at smaller times t for smaller r values.

For the limiting case $C=1$, the explicit equation of state is

$$p_m={\displaystyle \frac{{\rho }_m}{3}}({\xi }^2-2)={\displaystyle \frac{{\rho }_m}{3}}\left({\displaystyle \frac{e^{2t}}{r^2}}-2\right),$$

(41)

and ${\xi }_r=\sqrt{3}$, ${\xi }_T=\sqrt{2}$.

The pressure continues its decline with r to negative values at large r (smaller $\xi$). Negative values of P in the range $\xi <{\xi }_T$ suggest a correspondence with local gravitational collapse, i.e., with star formation. This phase begins at some value ${\xi }_S$, which is always greater than 1 being infinitesimally close to ${\xi }_T$. At each r, this is after the passage of the ‘announcement’ signal at $\xi =1$.

It is worth recalling that the pressure is closely related to the geometric curvature in a homogeneous spacetime. Assuming the stress tensor of an ideal ‘fluid’, as was conducted in [5], one has for the scalar curvature

$$\Re ={\displaystyle \frac{G}{c^2}}\left({\displaystyle \frac{3p}{c^2}}-\rho \right),$$

(42)

where p and $\rho$ are total quantities.

6.3. Matter Mass

We can also look at the matter mass of the Universe. The matter mass and the vacuum mass satisfy the simple relations $m_m=(4\pi /3){\rho }_m{R}^3$ and $m_v=(4\pi /3){\rho }_v{R}^3$, respectively, because both densities are homogeneous (although ${\rho }_m$ is a function of t). We see that at a fixed t, both masses will vary with r as ${\left[R\left(r\right)\right]}^3$. This yields a rise and fall of each mass, following the dependence of R on r, as shown in Figure 1. We illustrate the behavior in space and time in Figure 4.

The variation in r with t at a fixed radius r shows an initial rise in $m_m$, followed by a fall as the singularity compresses mass up until $\xi =1$; beyond this surface, the mass declines. For a fixed value of t, the maximum in $m_m$ occurs when $d{m}_m/dr=0$. This is most easily determined by setting $d(e^t/2m_m)/d\xi =0$, i.e., $d{cosh}^3(Cln\xi )/d\xi =0$. This occurs when $sinh(Cln\xi )=0$, i.e., $ln\xi =0$ or $\xi =1$. With $\xi =1$, the maximum value of $m_m$ is $e^t/2$ and the radius r at which it occurs is $e^t$.

Similarly, for a fixed value of r, the maximum in $m_m$ occurs when $d{m}_m/dt=0$. This is best evaluated by setting $d(r/m_m)/d\xi =0$, i.e., $d[{cosh}^3(Cln\xi )/\xi ]/d\xi =0$, which occurs when $tanh(Cln\xi )=1/3C$. Once again, using the high-argument approximation $tanhx\simeq 1-2e^{-2x}$ gives $e^{2Cln\xi }\simeq 6C/(3C-1)$ or $\xi ={(6C/(3C-1))}^{1/2C}$. Further, using the identity $\mathrm{sech}x=\sqrt{1-{tanh}^{2}x}$, the corresponding maximum value of $m_m$ is found to be $(r/2){(6C/(3C-1))}^{1/2C}{(9C^2-1)}^{3/2}/27C^3$. For values of C close to unity, the maximum value of $m_m\simeq (8\sqrt{6}/27)r$, and it occurs at $\xi \simeq \sqrt{3}$ or $t\simeq ln\left(\sqrt{3}r\right)$. These approximate values agree well with the exact numerical values in Figure 4. The maximum mass occurs very closely to $P=\eta /3$. The surface $\xi =1$ is hence the limit of influence of the expanding singularity. In both upper panels, we see the change in the r mass distribution with time. In the lower panels, we see the matter mass declining at r as the apparent horizon passes.

6.4. Singular Horizon

In Figure 5, we look briefly at the approach to the horizon, as measured by the behavior of $g_{00}\equiv e^{\sigma }$ as it approaches zero. The plot at the upper left shows that at $t=1$, the value of $g_{00}=e^{\sigma }$ rises from zero at the apparent horizon to approach a constant value at large r, which would define proper time far away from the horizon. The plot at the upper right shows that at $r=1$, the t dependence declines steadily to zero as the singular horizon passes.

The figure also includes the behavior of the metric coefficient $g_{11}=e^{\omega }$. The lower left plot, for $t=1$, shows that $g_{11}$ is well behaved at the horizon, although quite large. It diminishes at large distances to a small, slowly declining value. Space is therefore ‘stretched’ relative to large r near the horizon. On the lower right, we show the time dependence of $g_{11}$ at $r=1$, from the passage of the observer at $t=-0.25$ to the passage of the apparent horizon at $t=2.62$.

6.5. Lorentz Factor and Four-Velocity

The plots of ${\mathrm{\Gamma }}^2$ clearly show the significance of $\xi =1$, with $\mathrm{\Gamma }$ vanishing there because of our imposed separation of the energy equation, so that ${\mathrm{\Gamma }}^2=1-M/S\equiv 1-2G{m}_m/R{c}^2$. Therefore, at $\xi =1$ (where ${\mathrm{\Gamma }}^2=0$), R has the Schwarzschild horizon value, namely $2G{m}_m/c^2$. The radial light velocity is equal to $\pm 1/\left(CK\right)$ at $\xi =1$ and becomes equal to unity with $K=1/C$. This sets the light speed as the signal announcing the approach of the singularity.

It is also seen in Figure 6 that $\xi =1$ locates the radial maximum four-velocity in space at each fixed time (see lower left corner in the figure). This reflects that ${\partial }_{r}U=0$ at $\xi =1$. We also note that ${\partial }_{t}U=0$ at ${\xi }_h$, so that at fixed r, the maximum U occurs at $t_h=ln\left(r{\xi }_h\right)$, as in the lower right in the figure. We note that the maximum values of U can be greater than unity. This behavior is similar to the maximum in S at the horizon ${\xi }_h$, which defines the maximum in R at fixed r (Figure 1). In fact, we can infer the behavior of U from Equation (13) because $U=R$.

The geometrical nature of ${\xi }_h$ remains a little mysterious. We present a clarification in the appendix and Section 7. The physical significance of $\xi =1$ is that of a precursor wave representing the limit of influence of the expanding singularity in the expanding cosmology. This surface is a spatial ‘two sphere’ of maximum four-velocity at each fixed time t. The radial light speed is equal to unity ($K=1/C$) and the circumferential radius equal to the Schwarzschild radius.

We conclude this section by looking at quantities evaluated at the horizon as approached from the ‘outside’ ($r>e^t/{\xi }_h$). Using the expressions in Equation (17), we find that $tanh(Cln{\xi }_h)=1/C$, so that $R\left({\xi }_h\right)=K{e}^t\sqrt{1-1/C^2}$, where we recall that the unit of length is $\sqrt{3/\mathrm{\Lambda }}$ and that of time is $\sqrt{3/\mathrm{\Lambda }{c}^2}$. The matter density, in units of ${\rho }_v$, is $(3/8\pi )e^{-2t}$ and the matter pressure is infinite. The matter mass in units of $\mu$ (Equation (9)) is $m_m=({(1-1/C^2)}^{3/2}/2)e^t$ and the vacuum mass $m_v=4\pi R^3{\rho }_v/3$, in the same units, is $({(1-1/C^2)}^{3/2}/2)e^{3t}$. The metric coefficients are $g_{00}=0$ and $g_{11}={\xi }_h^2(C^2-1)$.

Because of the magnitude of the mass unit $\mu$, with $C=1.01$ we need $e^{t_h}=6.4\times {10}^{-15}$ to restrict the black hole mass to ${10}^6{M}_{\odot }$ while holding $\xi ={\xi }_h=13.8$. This implies that $r_h=e^{t_h}/13.8\simeq 4.6\times {10}^{-16}$ in terms of the length unit. This is $\simeq 7.6\times {10}^{12}$ cm. For comparison, the Schwarzchild radius for a ${10}^6{M}_{\odot }$ black hole is 25-times smaller, $3\times {10}^{11}$ cm. If, however, C takes on the limiting value of unity, then ${\xi }_h\to \infty$, and the matter mass within the singular region $m_m=(r_h/2){\xi }_h{\mathrm{sech}}^3(ln{\xi }_h)$ (Equation (17)) reduces to zero as $4r_h/{\xi }_h^2$.

7. Limiting Case

When $C=1$, the singular horizon and a central singularity are together at ${\xi }_h=\infty$ (see Equation (25)). This value will always be achieved at $r=0$ for all finite t, which might suggest that the singularity is isolated there. However, the central feature of any of these solutions is the surface $\xi =\mathrm{constant}$. This quantity becomes infinite at any finite r as $t\to \infty$, which therefore implies (in contradiction to the isolation at $r=0$) that at infinite time, all r values encounter the singularity at $\xi \to \infty$. These statements are confirmed in the appendix, where we find the singularity at $i^{+}$ and at $r=0$ to be combined. Lines of constant r (i.e., $\rho$ in the appendix) intersect with the singularity at $i^{+}$; this is an example of a space-like singularity, but at $i^{+}$.

Although we have discussed physical behavior for the possible interest in cosmology, we emphasize that the truly rigorous conformal picture in this paper is only for the case $C=1$. The appendix, which provides the technical detail, should therefore be regarded as an essential part of the text.

Guided by the non-singular case $C<1$ and the rigorous case $C=1$ (see appendix), we suspect that the situation for $C>1$ is as shown in Figure A1. These are only schematic diagrams that are not conformal, but they are well founded. For the limiting case $C=1$ and the cosmological case $C<1$ (e.g., [12], p. 726). We give a brief justification of these diagrams below.

Although global solutions for the null geodesics are not available when $C\ne 1$, we can use piece-wise solutions for $t<0$ and $t>0$ separately. These approximations work best for large C and r that is not too small. They yield our justification for the schematics in Figure A1.

For $t>0$, the first term in Equation (27) is dominant. This integrates to give

$$\begin{array}{ccc}\hfill r^C& =& Cu-{\displaystyle \frac{C}{2}}{e}^{(C-1)t},\hfill \end{array}$$

(43)

$$\begin{array}{ccc}\hfill r^C& =& Cv+{\displaystyle \frac{C}{2}}{e}^{(C-1)t},\hfill \end{array}$$

(44)

for ingoing and outgoing geodesics, respectively. (We must interchange the + and −, because the ∓ assumes that the following factor is negative.) The quantities v and u are arbitrarily null coordinate parameters. The ingoing ray stops at finite $t=(ln2u)/(C-1)$, where $r=0$ and $\xi =\infty$. The outgoing ray continues to $i^{+}$, where it encounters the singularity according to $r_h^C=e^{Ct}/{\xi }_h^C$. Should $v<0$, the ray can begin at $r=0$ at time $ln2\left|v\right|/(C-1)$.

When $t<0$, the second term in Equation (27) dominates, which is properly negative. This integrates to

$$\begin{array}{ccc}\hfill r^C& =& {\displaystyle \frac{2}{Cv+C{e}^{-(C+1)t}}},\hfill \end{array}$$

(45)

$$\begin{array}{ccc}\hfill r^C& =& {\displaystyle \frac{2}{Cu-C{e}^{-(C+1)t}}},\hfill \end{array}$$

(46)

for outgoing and ingoing rays, respectively. All of the outgoing rays begin at $r=0$ at $i^{-}$.

Should $v<0$, it appears that the outgoing ray can reach infinity at $\left|t\right|=ln\left|v\right|/(C+1)$. However, the singularity is at $r_h^C=e^{-C\left|t\right|}/{\xi }_h^C$ and the outgoing radius coincides with this value at $\left|t\right|=ln(2{\xi }_h^C/C{e}^{C\left|t\right|}+|v\left|\right)/(C+1)$. As a larger negative time than the time to reach infinity, this happens before the null ray reaches $i_o$. The ingoing rays ($u>0$) start at infinity at time $\left|t\right|=lnu/(C+1)$ and continues inward to some finite value.

These comments agree with the schematic diagrams in Figure A1 that are not conformally transformed. The special case $C=1$ is transformed conformally in Figure A2, which agrees topologically with the schematic version. When $C<1$, the situation is fundamentally different, because there is no propagating singular horizon.

8. Discussion

In view of the recent interest in black holes with non-flat asymptotic boundaries, we have studied a known solution of the general relativity field equations in a new parameter space. The spacetime is spacetime scale-invariant, spherically symmetric, non-stationary, matter-homogeneous, and possesses a positive cosmological constant $\mathrm{\Lambda }$. This constant and the speed of light c together define space and time scales that are cosmological in size. Changes in dimensionless space and time variables of order unity can therefore span cosmological scales.

We have extended the solution of [5] so that the key parameter C is now permitted to be $\ge 1$, and we have expressed this parameter in terms of physical quantities (Equations (31) and (32)). The expanding horizon is unusual because it is singular with an infinite pressure and a correspondingly infinite scalar curvature. When $C=1$, the horizon is at $\xi ={\xi }_h\to \infty$. This places it at $r=0$ and $t=i^{+}$ (so it engulfs all r as $t\to \infty$). This case is rigorously interpreted conformally in the appendix of this paper. The solution for $C>1$ is exact, but the interpretation does not result in a precise Penrose–Carter diagram.

Starting from a central point, an expanding, singular, apparent horizon eventually engulfs the whole of the spacetime as a kind of wave front in an incompressible medium. However, there is a preceding null surface ($\xi =1$) that separates the part of the spacetime manifold that is already affected by the expanding horizon, from that not yet affected. Although referring to the schematic conformal diagrams may be more helpful, a simple plot of ${\xi }_h\ne 0$ in a $(t$-$r)$ plane shows that light emitted tangentially from the singularity at some $(t$,$r)$ arrives at infinity after the singularity does. (This is confirmed by a direct calculation of the null path.) Hence, the singularity is never visible to an observer outside the horizon when $C>1$.

The infinite pressure at the apparent horizon, coupled with the finite density there, also implies (for matter with an ideal equation of state) that the apparent horizon is a sphere of very high temperature, i.e., a ‘sphere of fire’, as illustrated in Figure 2. This divergence of pressure and temperature also coincides with a curvature singularity. We also note, based on Figure 2, that the matter pressure can be negative at early times before the passage of the ‘warning’ light front at the observer’s location. This would assist the negative vacuum pressure in accelerating the cosmology at early times (and slow it down at later times). Negative pressure might be associated with local gravitational collapse, well before the onset of virial equilibrium in galaxies. The visible aspects of the hot early Universe would be due to hot plasma inside the point where $\xi ={\xi }_T$, which is becoming transparent. This is as much as is the case for the standard current cosmology.

The radial light velocity is zero at the horizon, but the horizon hyper surface normal is not null unless $C=1$. In the latter case, we have seen that the behavior is an expanding black hole. (We have often referred to our object as an expanding ‘black hole’ for brevity, but this pressure and curvature singularity at the expanding horizon clearly renders it a fundamentally new type of object when $C>1$. It is a growing, ‘hot’, curvature singularity that nevertheless is not ‘naked’.) In a $\mathrm{\Lambda }$-CDM cosmology. This was our original objective. An observer situated in the unaffected spacetime, who calculates the Hubble–Lemaître constant using coordinate time, can observe currently favored values. A calculation of H using proper time gives a constant value $\sqrt{\mathrm{\Lambda }{c}^2/3}=56$ km ${\mathrm{s}}^{-1}$ ${\mathrm{Mpc}}^{-1}$.

The generalized Lorentz factor $\mathrm{\Gamma }$ and the co-moving four-velocity U are studied in Figure 6. The Lorentz factor vanishes at $\xi =1$, which identifies this light front as a Schwarzschild surface satisfying $R=2G{m}_m/c^2$. The four-velocity is permitted to be positive or negative.

In conclusion, we have described a spacetime with an infinitely ‘hot’ horizon that is expanding into a surrounding cosmology. This is a much more dynamic object than a static black hole sitting in an asymptotically flat spacetime. It resembles one interpretation of the McVittie metric ([4]). The case $C=1$ describes a singular black hole horizon that becomes a space-like future event horizon. The solution inevitably develops the ‘firewall’ behavior at the horizon.