It has been theorized that black holes are surrounded by firewalls [1–7], although there is not universal agreement concerning this [1–7]. There is a vast literature exploring this topic, of which we have cited only a tiny sample. But the many works cited, and discussed, in our cited Refs. [1–7] could provide at least the beginning of a thorough literature search [1–7].

In Section 2, we review basic concepts pertaining to Schwarzschild black holes and Hawking radiation. In Section 3, we discuss the anticipation of Hawking radiation—albeit from non-black holes—initially by R. C. Tolman and shortly thereafter with P. Ehrenfest, and Tolman’s [8,9] proof, bolstered with Ehrenfest [10], that any gravitator—black hole or non-black hole—must radiate and hence cannot be in thermodynamic equilibrium with a surrounding vacuum at (or at least sufficiently close to) absolute zero ($0 \mathrm{K}$), but must completely evaporate into that vacuum within a finite time. The times required for evaporation of black holes and non-black holes are compared. (See also Garrod [11].) This has been corroborated by recent research [12]. In Section 4, we show that (i) if firewalls exist, they can originate via Hawking radiation at the minimum possible ruler distance [13] (the Planck length [14–16]) beyond the Schwarzschild horizon, where it has not suffered any gravitational redshift [17,18], or, alternatively, suffered maximal gravitational blueshift [17,18] and (ii) the firewall temperature is on the order of the Planck temperature [19], independently of the mass and hence also of the Schwarzschild radius of a Schwarzschild black hole. We emphasize and focus on ruler distance [13] because, unlike other distance measures in General Relativity [13], ruler distance—uniquely! [13]—is actual physical distance: the distance between two points as measured by rulers laid upon the shortest possible spatial path separating them and hence the physical distance separating them [13]. (We employ other distance measures where they are more applicable.) In Section 5, we explain the exponential nature of the gravitational frequency shift as a function of the gravitational potential. In Section 6, we consider the firewall-mass problem [3], and provide an at least prima facie tentative resolution thereto based on: (i) the mass of a firewall being canceled by the negative gravitational mass [17,18] $=($negative gravitational energy$)/c^2$ [17,18] accompanying its formation, (ii) the unchanged observations of a distant observer upon formation of a firewall, and (iii) Birkhoff’s Theorem [20–31]—actually first discovered by Jørg Tofte Jebsen [25–28]. We show that the mass of a firewall is exactly counterbalanced by the (negative) gravitational mass-energy accompanying its formation. Perhaps this may complement other lines of reasoning [4] disputing massiveness [3] of firewalls. (There is a caveat [28–31]${ }^1$ with respect to Birkhoff’s Theorem [20–31], but it [28–31]${ }^1$ is not relevant with respect to our considerations.${ }^1$) In Section 7, we consider one aspect of thermodynamics in gravitational fields, showing that equilibrium relativistic gravitational temperature gradients cannot be exploited to violate the Second Law of Thermodynamics. A concluding synopsis is provided in Section 8. Auxiliary topics are discussed in the Notes.

The Schwarzschild metric of a Schwarzschild black hole of mass M and Schwarzschild radius $r_S=2GM/c^2$ is [32]

Schwarzschild-coordinate radial distance r is not radial ruler distance but is radial area distance and also radial distance from apparent size [13]. By contrast, in accordance with the Euclidean form of the angular term of the Schwarzschild metric [the last term in all three lines of Equation (1)], a spherical shell at $r_{shell}$ has ruler-distance circumference $C_{shell}=2\pi r_{shell}$ and ruler-distance surface area $A_{shell}=4\pi r_{shell}^2$ [13]. At $r\ge r_S$, l—notr—is radial ruler distance, t is Schwarzschild-coordinate time (proper time measured by a clock at rest at $r\to \infty$) [17,18], and $\tau$ is proper time measured by a clock at rest at any given $r\ge r_S$ [17,18]. [A clock at rest at $r=r_S$—if such a clock can exist—must be constructed entirely of photons (and/or other zero-rest-mass particles)!]

Although not necessary for our derivations, it may be helpful, as an aside, to briefly remark on the following three features of the Schwarzschild metric [Equation (1)]: (i) Setting $d{s}^2=0$ in Equation (1) shows that the physical radial velocity of light $V_{phys,light}=dl/d\tau =c$ at all$r\ge r_S$ [32]; but, by contrast, the Schwarzschild-coordinate radial velocity of light $V_{coor,light}=dr/dt=c\left[1-\left(r_S/r\right)\right]$ decreases monotonically with decreasing r from c at $r\to \infty$ to zero at $r=r_S$ [32]. (ii) We focus on distance [13], especially on ruler distance [13], and most especially on radial ruler distance [13], beyond the Schwarzschild horizon $r_S$ in Schwarzschild spacetime [32]. But it may be interesting to note that, in accordance with the angular term of the Schwarzschild metric [the last term in all three lines of Equation (1)] being of the identical Euclidean form at all $r\ge 0$ [32],${ }^2$ even within $r_S$,${ }^2$ where a spherical shell cannot be at rest but must be collapsing, while falling through a given $r_{shell}<r_S$ it has ruler-distance circumference $C_{shell}=2\pi r_{shell}$ and ruler-distance surface area $A_{shell}=4\pi r_{shell}^2$ [32].${ }^2$Even within$r_S$,${ }^2$ where r becomes timelike, r does not become time itself:${ }^2$unlike time itself r still retains these spatial geometrical attributes [32].${ }^2$ Time itself has no spatial geometrical attributes. (iii) Because the gravitational field of a Schwarzschild black hole is purely radial, it seems intuitive that this gravitational field’s stretching [33,34]${ }^3$ of space from the Euclidean [33,34]${ }^3$ is purely in the vertical${ }^3$ radial r direction, and not at all in the horizontal angular $\theta$ and $\varphi$ directions: thus the identical Euclidean form of the angular term of the Schwarzschild metric [the last term in all three lines of Equation (1)] at all $r\ge 0$ [32]. Indeed, more generally, intuition suggests that stretching [33,34]${ }^3$ of space from the Euclidean by any gravitational field is purely in the vertical direction [33,34],${ }^3$ and not at all in any horizontal direction [33,34].${ }^3$ This intuition augments the immediately preceding Item (ii), and is perhaps most clear in relation to Sakharov’s elastic-strain theory of gravity [35–40].${ }^{4,5}$ (Thus, might space be the ether [41,42]?${ }^{4,5}$)

We will be concerned only with radial motions of photons in the gravitational fields of Schwarzschild black holes, because only radial motions can result in gravitational frequency shifts. In this regard we will be concerned only with the temporal-radial part of the Schwarzschild metric [Equation (1)], hence ignoring the angular term thereof (the last term in all three lines thereof).

Hawking radiation, the (at least essentially) blackbody radiation from a Schwarzschild black hole, is most typically construed to have the temperature [16,43–49] where k is Boltzmann’s constant, M is the mass of the black hole, and $r_S=2GM/c^2$ is its Schwarzschild radius [16,43–49]. Note that $T_{H,r\to \infty }$ given by Equation (2) is the temperature of Hawking radiation at a great distance from a Schwarzschild black hole, i.e., at $r\to \infty$, hence after Hawking radiation having suffered the maximum possible gravitational redshift [16–18,43–49]. For all non-primordial black holes, which are all of stellar mass or larger, $T_{H,r\to \infty }$ is extremely low compared to the current temperature $T_{CBR}=2.725 \mathrm{K}$ [50] of the cosmic background radiation [50]. For sufficiently small primordial black holes [51–61] ($M<\frac{\hslash c^3}{8\pi Gk{T}_{CBR}}=4.50\times {10}^{22}kg\iff r_S<\frac{\hslash c}{4\pi k{T}_{CBR}}=6.69\times {10}^{-5}m$), $T_{H,r\to \infty }>T_{CBR}=2.725 \mathrm{K}$ obtains. But as of this writing, to the best knowledge of the author, no such sufficiently small primordial black holes—indeed, no primordial black holes at all—have been discovered [51–61]. Moreover, while there are rationales according to which primordial black holes might contribute, perhaps significantly, to cold dark matter, there also are both theoretical and observational upper limits on their abundance [51–61] and therefore also on their actual contribution to cold dark matter [51–61]. Hence, while it is possible that they could contribute, perhaps significantly, to cold dark matter, as of this writing, to the best knowledge of the author, it is uncertain whether or not they actually exist [51–61].

Thus far we have considered $T_{H,r\to \infty }$. But $T_{H,r}$ at smaller values of r ($r_S\le r<\infty$) has been discussed as well [49]. Closer to $r_S$ (at $r_S\le r<\infty$) [49] than at $r\to \infty$, Hawking radiation has suffered less [17,18,49] gravitational redshift [17,18] and hence has a higher [49] temperature [16,43–49]

It is extremely important to note that Equation (3) is a special case of the more general result derived by Tolman [8,9], bolstered with Ehrenfest [10]. The last two terms of Tolman’s Equation (128.6) and the entirety of Tolman’s Equation (129.10) in Ref. [9] read: where (i)$C_T$ is Tolman’s constant in Equations (128.6) and (129.10) of Ref. [9], and (ii)$T_T$ is the Tolman gravitational thermodynamic-equilibrium temperature as measured by a local observer, written more explicitly as $T_T\left(r,\theta ,\varphi \right)$, at a specified location [specified radial area distance and also radial distance from apparent size [13] r and specified direction $\left(\theta ,\varphi \right)$] from the center of mass of a gravitator (spherically symmetrical or otherwise) where the time-time component of the metric has the value $g_{tt}\left(r,\theta ,\varphi \right)$. [As per the first term of Equation (128.6) in Ref. [9] (this notation is also employed in Refs [8] and [10]), Tolman (with Ehrenfest in Ref. [10]) sets $e^{\nu }=g_{tt}$, but for uniformity in notation we employ only the $g_{tt}$ symbolization. Tolman [8,9] (with Ehrenfest in Ref. [10]) employs the symbols $T_0$ instead of $T_T$ (with the subscript ${ }_0$ designating measurement by a local observer: see p. 489 of Ref. [9]), and C instead of $C_T$. (In Tolman’s paper with Ehrenfest [10] const. is employed instead of C.) We employ the subscript ${ }_T$ to refer to Tolman’s quantities. We take $T_T$ to be the Tolman gravitational thermodynamic-equilibrium temperature as measured by a local observer (the subscript ${ }_0$ omitted for brevity).]

We can choose Tolman’s constant $C_T$ for a given gravitator to be equal to $C_{T,r\to \infty }=$$\underset{r\to \infty }{lim}{T}_T\left(r,\theta ,\varphi \right){\left[g_{tt}\left(r,\theta ,\varphi \right)\right]}^{1/2}=T_{T,r\to \infty }$, the Tolman gravitational thermodynamic-equilibrium temperature measured by a local observer at radial area distance and also radial distance from apparent size [13] $r\to \infty$ in any specified direction $\left(\theta ,\varphi \right)$ from the center of mass of any gravitator (spherically symmetrical or otherwise), because spacetime approaches the Minkowskian and hence $g_{tt,r\to \infty }=\underset{r\to \infty }{lim}{g}_{tt}\left(r,\theta ,\varphi \right)=1$ at radial area distance and also radial distance from apparent size [13] $r\to \infty$ in any specified direction $\left(\theta ,\varphi \right)$ from the center of mass of any gravitator (spherically symmetrical or otherwise):

Hence Equation (4) can be rewritten as:

Tolman presents this general result not only via the last two terms of Equation (128.6) and the entirety of Equation (129.10) in Ref. [9], but also in Ref. [8] via the second equation in the Abstract and Equations (27), (28), (42), (53), and (54). It is also presented in Ref. [10] with Ehrenfest via Equations (2), (3), and (30), and discussed in some detail in Section 7 thereof. [In the weak-field limit ($M\ll r_S{c}^2/2G\iff r\gg r_S=2GM/c^2$ given spherical symmetry) this result is presented via Equations (8) and (29) and the last (unnumbered) equation in Section 8 of Ref. [8], and also via Equations (128.4), (128.5), and (128.10) in Ref. [9].] Tolman also evaluates the gradient $d\left\{ln\left[T_T\left(r\right)/T_{T,r\to \infty }\right]\right\}/dr$ at and near Earth’s surface in the last (unnumbered) equation in Section 2 of Ref. [8] and in Equation (128.7) of Ref. [9]. (In Earth’s weak gravitational field, this gradient can of course with at most negligible error be construed as being either with respect to radial ruler distance [13] or with respect to radial area distance and/or radial distance from apparent size [13].) We focus on the last two terms of Equation (128.6) in Ref. [9] and of Equation (3) in Ref. [10], on Equation (129.10) in Ref. [9], and on Equation (30) and Section 7 in Ref. [10]. (See also Garrod [11].)

For Schwarzschild black holes, we should expect that $T_T=T_H$. Indeed, letting $T_T\to T_H$, the second line of Equation (6) is identical to Equation (3), taking $T_{T,r\to \infty }=T_{H,r\to \infty }$ as per Equation (2). For $T_{T,r\to \infty }$ of Schwarzschild (spherically-symmetrical, non-rotating) non-black holes, we will give a plausibility argument (albeit not a proof) for a conjecture concerning the value of $T_{T,r\to \infty }$.

We define a Schwarzschild non-black hole (of mass M) as a spherically-symmetrical, non-rotating gravitator whose radius $r_{NBH}$ (as per radial area distance/radial distance from apparent size [13]) exceeds the Schwarzschild radius $r_S=2GM/c^2$. (The subscript ${ }_{NBH}$ may be omitted for brevity when that will not result in confusion.)

The apparent gravitational acceleration ${\mathcal{G}}_{BH}\left(r\right)$ towards a Schwarzschild black hole of mass $M_{BH}$, as measured by dangling a unit mass m at $r_S$ from a higher altitude $r>r_S$ (with a massless string), and the limit thereof as $r\to \infty$, are [62]:

The limiting value of ${\mathcal{G}}_{BH}\left(r\right)$ as $r\to \infty$—as per the second line of Equation (7)—is sometimes called the surface gravity of a Schwarzschild black hole [62]. Fortuitously, as if Newtonian theory was adequate for Schwarzschild black holes [62]! Fortuitously despite Newtonian theory taking all distance measures to be equivalent—not distinguishing, for example, between ruler distance [13] on the one hand, and area distance and/or distance from apparent size [13] on the other. Fortuitously because enormous values of ${\mathcal{G}}_{BH}\left(r\right)$ if r is only slightly greater than $r_S$ suffer gravitational redshift down to Newtonian values in the limit $r\to \infty$. (Note: We employ g to denote components—we focus on the time-time component—of the spacetime metric, G to denote the universal gravitational constant, and $\mathcal{G}$ to denote acceleration due to gravity.) For a Schwarzschild (spherically-symmetrical non-rotating) non-black hole of mass $M_{NBH}$ in the weak-field limit ($M\ll r_S{c}^2/2G\iff r\gg r_S=2GM/c^2$)

Based on the fortuitous Newtonian-equivalence of the forms of the second line of Equation (7) on the one hand and Equation (8) on the other, and applying Equation (2), we prima facie suggest the following conjecture for $T_{T,r\to \infty }$ of a Schwarzschild non-black hole of the same mass M as a Schwarzschild black hole ($M_{NBH}=M_{BH}=M$) but with radius (radial area distance and also radial distance from apparent size [13]) from the center to the surface of the Schwarzschild non-black hole) $r_{NBH}>r_S=2GM/c^2$:

We recognize that while the conjecture given by Equation (9) may, at least prima facie, seem plausible, we have not proven it. Nonetheless at least prima facie it seems a reasonable conjecture, especially given that, since at least it is consistent with Equation (2). However, irrespective of the validity (or lack thereof) of this conjecture, two paragraphs hence we will prove the extremely important point that $T_{T,r\to \infty }$ must be finitely greater than absolute zero ($0 \mathrm{K}$) for all non-black holes [as $T_{H,r\to \infty }$ is finitely greater than absolute zero ($0 \mathrm{K}$) for all black holes].

In Ref. [8] and in Sections 128 and 129 of Ref. [9], Tolman implies that our Equations (4)–(6) are valid in any static spacetime [63]. Together with Ehrenfest [10] this is also implied in Ref. [10]. But given rotation at constant angular velocity, a time-independent centrifugal potential can be incorporated into the time-independent gravitational potential that obtains in static spacetime, i.e., into that which obtains neglecting the rotation [63]. This time-independent gravitational-centrifugal potential would then of course be a function of $\theta$ as well as of r, but at a given$\theta$ it can still be expressed as a function of r alone. Thus we can construe Equations (4)–(6) to be valid in any static or stationary spacetime [63]. Hence these results proven by Tolman [8,9], bolstered with Ehrenfest [10], and summarized via our Equations (4)–(6) and the associated discussions, imply that at thermodynamic equilibrium temperature increases downwards${ }^6$ in any static or stationary gravitational field (the centrifugal contribution in a stationary field construed as incorporated therein given rotation at constant angular velocity) [63]. But for simplicity and definiteness we focus on the static spacetimes at $r\ge r_S$ of Schwarzschild, i.e., spherically-symmetrical non-rotating (black hole and non-black-hole) gravitators.

Even more importantly, Tolman [8,9], bolstered with Ehrenfest [10], furthermore implies more than that, as summarized via our Equations (4)–(6): it is furthermore implied [8–10] that $T_{T,r\to \infty }$ must be finitely higher than absolute zero ($0 \mathrm{K}$) for allnon-black holes, as $T_{H,r\to \infty }$ isfinitely higher than absolute zero ($0 \mathrm{K}$) for all black holes. As per Equations (4)–(6) [most explicitly as per Equation (6)], incorrectly assuming that $T_{T,r\to \infty }=0 \mathrm{K}$incorrectly implies that $T_T\left(r,\theta ,\varphi \right)=0 \mathrm{K}$ obtains everywhere—at least, everywhere that $g_{tt}\left(r,\theta ,\varphi \right)>0$ or equivalently that ${\left[g_{tt}\left(r,\theta ,\varphi \right)\right]}^{-1/2}<\infty$. In the Schwarzschild (spherically-symmetrical non-rotating) special case, wherein $g_{tt}\left(r\right)=1-\frac{r_S}{r}\iff {\left[g_{tt}\left(r\right)\right]}^{-1/2}={\left(1-\frac{r_S}{r}\right)}^{-1/2}$, for a black hole this incorrect implication would pertain everywhere in the region $r\ge r_S$ except at exactly $r=r_S$; and for a non-black hole, whose radius exceeds $r_S$, this incorrect implication would pertain everywhere without exception. Thus the correct implication is that any gravitator—black hole or non-black hole—must radiate: a black hole surrounded by a vacuum colder than $T_{H,r\to \infty }$ and a non-black hole surrounded by a vacuum colder than $T_{T,r\to \infty }$ cannot be in thermodynamic equilibrium with that vacuum, but must radiate into that vacuum and completely evaporate into that vacuum within a finite time! Thus at least the qualitative fact that Hawking (Tolman!) radiation emanates from all gravitators—not only from black holes but also from non-black holes—(even if not also quantitative values of $T_{T,r\to \infty }$ and $T_{H,r\to \infty }$ [16,43–49]) was discovered by Tolman [8,9], bolstered with Ehrenfest [10], at least as early as 1930! Any gravitator—black hole or non-black hole—surrounded by a $0 \mathrm{K}$ vacuum cannot be at thermodynamic equilibrium unless it is enclosed within an opaque thermally insulating shell [64,65]${ }^7$ and thereby insulated from that vacuum: otherwise it will completely Hawking- (Tolman!-) evaporate into that vacuum within a finite time! This has been corroborated by recent research [12].

Black holes evaporate ever more rapidly and get hotter as they lose mass, hence completely evaporating into a vacuum at absolute zero ($0 \mathrm{K}$) within a finite time $\Delta t_{BH,evap}$. For evaporation of a Schwarzschild black hole into a $0 \mathrm{K}$ vacuum [43–49]: where the minus signs account for the black hole’s mass M decreasing during evaporation, $A=4\pi r_S^2$ is its (decreasing) surface area, $\sigma =5.670374419\times {10}^{-8}W/m^2 {\mathrm{K}}^4$ is the Stefan-Boltzmann constant [66], and $M_{\odot }\doteq 1.9885\times {10}^{30}kg$ is the mass of the Sun [67]. The dot-equal sign (≐) means very nearly equal to.

By contrast, non-black holes evaporate ever more slowly and get cooler as they lose mass. But the time rate of this slowdown is itself sufficiently slow—they get cooler sufficiently slowly—that they, too, completely evaporate into a vacuum at absolute zero ($0 \mathrm{K}$) within a finite time $\Delta t_{NBH,evap}$. For a weak-field ($M_{initial}\ll r_S{c}^2/2G\iff r_{initial}\gg r_S=2GM/c^2$) Schwarzschild (spherically-symmetrical non-rotating) non-black hole, $g_{tt}\left(r\right)$ and therefore also ${\left[g_{tt}\left(r\right)\right]}^{-1/2}$ is essentially constant at unity, and hence also by Equations (4)–(6) the Tolman [8–10] temperature $T_T\left(r\right)$ is essentially constant at $T_{T,r\to \infty }$, as M decreases from $M_{initial}$ to $M_{final}=0$ and r decreases from $r_{initial}$ to $r_{final}=0$ during the entire evaporation process. Moreover, $A=4\pi r^2\iff r={\left(A/\pi \right)}^{1/2}/2$, and, assuming uniform density $\rho$ for simplicity (justified in the weak-field ($r_{NBH}\gg r_S$) limit because gravity is too weak to significantly compress material with depth), also $M=4\pi \rho r^3/3=\rho rA/3=\rho A^{3/2}/6{\pi }^{1/2}\iff A={\pi }^{1/3}{\left(6M/\rho \right)}^{2/3}$. Hence in the weak-field ($r_{NBH}\gg r_S$) limit for evaporation of a Schwarzschild (spherically-symmetrical non-rotating) uniform-density non-black hole into a $0 \mathrm{K}$ vacuum: where the minus signs account for the non-black hole’s mass M decreasing during evaporation, and

$\Delta t_{NBH,evap}$ is finite because although $dM/dt$ decreases with decreasing M, it does so only proportionately to $M^{2/3}$. (In order to render $\Delta t_{NBH,evap}$ infinite, $dM/dt$ would have to decrease with decreasing M at least proportionately to M itself.) That $\Delta t_{NBH,evap}$ is finite is corroborated by recent research [12].

Equations (11) and (12) yield an exact numerical value for $C_1$. By contrast (even though the exact numerical values of all factors in $C_2$ except $T_{T,r\to \infty }$ are known) Equations (4)–(6), (13), and (14) do not yield enough information to provide an exact (or even less-than-exact) numerical value for $C_2$ and hence also for $T_{T,r\to \infty }$—albeit, as we showed in the eighth paragraph of this Section 3, they do yield enough information to prove that $T_{T,r\to \infty }$ must be finitely higher than absolute zero ($0 \mathrm{K}$). However, if our conjecture as per Equations (7)–(10) and the associated discussions is correct, then applying our result for $T_{T,r\to \infty }$ in Equation (9) into Equation (13) does yield, at least for weak-field ($r_{NBH}\gg r_S$) uniform-density Schwarzschild (spherically-symmetrical non-rotating) non-black holes, exact numerical values for $C_2$, hence also for $T_{T,r\to \infty }$, and thence also for $\Delta t_{NBH,evap}$ as per:

Comparing Equations (11) and (13) with the same $M_{initial}$ for both a Schwarzschild black hole and a weak-field ($r_{NBH}\gg r_S$) uniform-density Schwarzschild non-black hole:

Hence, if our conjecture as per Equations (7)–(10) and (15), and the associated discussions, is correct, then applying Equations (7)–(9), for weak-field ($r_{NBH}\gg r_S$) uniform-density Schwarzschild non-black holes $T_{T,r\to \infty }=T_{H,r\to \infty }{\left(r_{NBH}/r_S\right)}^2$, thereby yielding, at least for weak-field ($r_{NBH}\gg r_S$) uniform-density Schwarzschild non-black holes, an exact numerical value for $C_2$ and thence also for $C_3$: and

Whether or not our conjecture is correct, with respect to Equation (16), this qualitative evaluation is valid: If $M_{initial}<C_3^{3/8}$, $\Delta t_{NBH,evap}>\Delta t_{BH,evap}$; if $M_{initial}=C_3^{3/8}$, $\Delta t_{NBH,evap}=\Delta t_{BH,evap}$; if $M_{initial}>C_3^{3/8}$, $\Delta t_{NBH,evap}<\Delta t_{BH,evap}$.

Thus all gravitators—black holes and non-black holes—are enveloped by atmospheres of equilibrium blackbody radiation. Because both $T_{H,r\to \infty }>0 \mathrm{K}$ [16,43–49] and $T_{T,r\to \infty }>0 \mathrm{K}$ [8–10], neither a black hole nor a non-black hole can be in thermodynamic equilibrium with a surrounding vacuum at (or at least sufficiently close to) absolute zero ($0 \mathrm{K}$), but must radiate into that vacuum and completely evaporate into that vacuum within a finite time, unless shielded from that vacuum by enclosure within an opaque thermally-insulating shell [64,65].${ }^7$

The Tolman-Hawking evaporation of a Schwarzschild black hole into a vacuum colder than $T_{H,r\to \infty }$ and of a non-black hole into a vacuum colder than $T_{T,r\to \infty }$ is in accordance with the Second Law of Thermodynamics. The entropy of a black hole is large [43–49], but the entropy of the radiation dispersed into a vacuum colder than $T_{H,r\to \infty }$ by its Hawking-evaporation is even larger. The entropy of a non-black hole is not as large as that of a black hole of the same mass, affording even more scope for entropy to increase as it Tolman-evaporates into a vacuum colder than $T_{T,r\to \infty }$.

Tolman was aware of the concept of black holes (even if not of the moniker “black hole”): see, for example, the last paragraph of Section 96 of Ref. [9]. Yet nowhere does this enter into Tolman’s [8,9] derivations, bolstered with Ehrenfest [10], that at thermodynamic equilibrium temperature increases downwards${ }^6$ in any static, or even stationary, gravitational field [63]. Indeed, despite early contemplations of the concept of black holes [68–72], this concept [68–72] (and the moniker “black hole” [68–72]) was not mainstream until the 1960s [68–72]. Hence if Tolman’s [8,9] discovery, bolstered with Ehrenfest [10], had borne fruit circa 1930 (or shortly thereafter), it would have (i) initially been construed with respect to non-black holes and (ii) dubbed Tolman radiation rather than Hawking radiation: Hawking radiation would then initially have been construed as emanating from non-black holes—and dubbed Tolman radiation rather than Hawking radiation!

A caveat pertaining to Hawking-Tolman evaporation in general and to Tolman evaporation of Schwarzschild non-black holes in particular: Our results in this paper in general and in this Section 3 in particular relate, respectively, to gravitators in general and Schwarzschild non-black holes in particular that are bound solely by their own gravity. With respect to Schwarzschild non-black holes: any additional, non-gravitational, e.g., chemical, bonding, is neglected. Any additional, non-gravitational, e.g., chemical, bonding abetting gravitational bonding must necessarily increase $\Delta t_{NBH,evap}$, perhaps even rendering $\Delta t_{NBH,evap}=\infty$ if the temperature is absolute zero ($0 \mathrm{K}$). But this is a peripheral issue, not the central issue. The central issue relates only to gravitational bonding: to gravitators in general, and Schwarzschild non-black holes in particular, that are bound solely by their own gravity.

At this point, it is worthwhile to note the similarities${ }^8$—owing to the equivalence principle [73,74]${ }^8$—between Hawking (Tolman) radiation and Unruh radiation; but also a caveat.${ }^8$

These topics, and related ones, will be further discussed in Sections 4, 5, 6, and 7.

In Section 4, it may be helpful to envision a Schwarzschild black hole enclosed concentrically within an opaque thermally-insulating spherical shell at $r_{shell}\to \infty$ [64,65].${ }^7$ Hawking radiation at temperature $T_{H,r\to \infty }$ as per Equation (2) is reradiated and/or reflected downwards${ }^6$ from the inner surface of this spherical shell, suffering increasing gravitational blueshift with decreasing r in accordance with Equations (3)–(6) [8–11,16–18,43–49,64,65]. Since thermodynamic equilibrium obtains perfectly within the shell [64,65],${ }^7$ the caveat “(at least essentially)” can be deleted from the sentence containing Equation (2): radiation within the shell is exactly blackbody [64,65]. Indeed, enclosure of any radiation—whether emanating from a source or freely existing in space${ }^9$—within an opaque thermally-insulating shell ensures perfect thermodynamic equilibrium and hence an exactly Planckian blackbody spectrum [64,65] (even if not immediately upon enclosure, then after the relaxation time). For example, if the Sun was so enclosed, the currently approximately blackbody radiation [67,70–78] at its photosphere would become exactly blackbody [64,65]. Without enclosure within an opaque thermally-insulating shell, radiation can be exactly blackbody; with enclosure, it must be exactly blackbody [64,65]. (Of course, exactly blackbody radiation incorporates the cutoff of the Planckian blackbody spectrum for wavelengths exceeding the size of an enclosure or cavity [79,80]. But this is not a consideration for our spherical shell, because it is at $r_{shell}\to \infty$ [64,65].${ }^7$) Moreover, it should be noted that the Planckian form of any exactly-blackbody spectrum, and thus its having an exactly well-defined temperature, survives gravitational frequency shifting [81]—and also motional Doppler frequency shifting [81], cosmological frequency shifting [81], and any combination of any two or all three types of frequency shifting [81].

Prima facie, by Equation (3), it might seem that arbitrarily close to the Schwarzschild radius $r_S$ (but still at $r>r_S$) $T_{H,r\to r_S}\to \infty$. But this is not so. Thus far, we have not taken into account that, if at $r>r_S$, it is not possible, even in principle (let alone in practice) to be arbitrarily close to $r_S$, because owing to quantum fluctuations spacetime breaks down as ruler distance [13] on the order of the Planck length [14–16] is approached. (The standard uncertainty is $0.000018\times {10}^{-35}m$ [16].) Thus, even in principle (let alone in practice), it is not possible, if at $r>r_S$, to be any closer to $r_S$ than at minimum radial ruler distance [14–16] beyond $r_S$.

We now derive $T_{H,r}$ as a function of radial ruler distance [13] $\delta l$ beyond $r_S$, which we denote as $T_{H,r_S+\delta l}$. We focus on regions just barely beyond $r_S$, i.e., where $\delta l\ll r_S$. Also, let $\delta r=r-r_S$ be Schwarzschild-coordinate radial distance [13] (which is also radial area distance and radial distance from apparent size [13]) beyond $r_S$. We focus on regions just barely beyond $r_S$, i.e., where, also, $\delta r\ll r_S$. Obviously

Applying Equations (1) and (21) [13],

The last step of Equation (21) and the second-to-last step of Equation (22) are justified because we focus on regions just barely beyond $r_S$, where $\delta r\ll r_S$. Applying Equation (22), if $\delta r\ll r_S$:

Hence, applying Equations (2)–(6), (21), (22), and (23), if $\delta r\ll \delta l\ll r_S$:

As noted in the paragraph containing Equations (19) and (20), even in principle (let alone in practice), $\delta l$ can be no smaller than ${\left(\delta l\right)}_{min}=l_{Planck}$ [14–16]. Thus, minimizing $\delta l$ at ${\left(\delta l\right)}_{min}=l_{Planck}$, by Equations (19), (20), and (24) we obtain where is the Planck temperature [19]. (The standard uncertainty is $0.000016\times {10}^{32} \mathrm{K}$ [19].)

This result is independent of the mass M and hence also of the Schwarzschild radius $r_S=2GM/c^2$ of a Schwarzschild black hole. As M and hence also $r_S$ increases, by Equation (2) $T_{H,r\to \infty }$ decreases in inverse proportion. But $r_S/\delta l$ for any given $\delta l\ll r_S$ in general and hence $r_S/{\left(\delta l\right)}_{min}=r_S/l_{Planck}$ in particular increases in direct proportion. Hence in accordance with Equations (2) and (24)–(26) these two opposing factors cancel out. Because of quantum fluctuations in the metric at length scales of $l_{Planck}$ [14–16], Equation (25) may be pushing the limit of accuracy of Equation (24), but we should expect Equation (25) to be valid at least in some average sense. Accordingly, perhaps we should not be too adamant about the small numerical factor of $1/2\pi$ in Equation (25), and hence recapitulate Equation (25) as

By Equation (24), recapitulated with the help of as

$T_{H,r_S+\delta l}$ still has high values in the region $l_{Planck}\ll \delta l\ll r_S$, hence with quantum fluctuations in the metric of Equation (1) being negligible [14–16]. For example, the temperature of the Sun’s core, $1.571\times {10}^7 \mathrm{K}$ [67], is equaled at $\delta l=2.3198\times {10}^{-11}m$, i.e., only slightly less than typical atomic dimensions; the (effective [67,75–78]) temperature of the Sun’s photosphere, $5772 \mathrm{K}$ [67], is equaled at $\delta l=6.3140\times {10}^{-8}m$, i.e., the dimensions of small microbes; and room temperature, $300 \mathrm{K}$, is equaled at $\delta l=1.2148\times {10}^{-6}m$, less than two orders of magnitude below the limit of naked-eye visibility $\approx {10}^{-4}m$.

Now let us consider the ruler-distance [13] wavelength of Hawking radiation in the region only slightly beyond $r_S$, i.e., where $\delta r\ll \delta l\ll r_S$. The ruler-distance [13] wavelength ${\lambda }_{r_S+\delta l}^{Wien,max}$ of blackbody radiation in general and of Hawking radiation in particular at the Wien’s-Displacement-Law maximum with respect to wavelength [66,82] corresponding to temperature T is [66,82]

(Since we are focusing on wavelength, we employ the Wien’s-Displacement-Law maximum with respect to wavelength [66,82] as opposed to that with respect to frequency [66,82].) Hence by Equations (28) and (30) [66,82]:

The numerical factor $1.265466416\times 2\pi =7.951159991$ is dimensionless and hence is valid in any self-consistent system of units. In the third line of Equation (31) we applied the second line of Equation (24). In accordance with the reasoning concerning quantum fluctuations in the metric in the paragraph ending with Equation (27) [14–16], perhaps we should not be too adamant about the small numerical factor of $1.265466416\times 2\pi =7.951159991$ in the last term of Equation (31), and hence recapitulate Equation (31) as

Thus the ruler-distance [13] wavelength ${\lambda }_{r_S+\delta l}^{Wien,max}$ of Hawking radiation in the region $\delta r\ll \delta l\ll r_S$ is on the order of the ruler distance [13] $\delta l$ itself. In particular, at ${\left(\delta l\right)}_{min}=l_{Planck}$ [14–16]

Hawking-radiation photons for which Equations (25)–(27) and (33) apply, and consequently for which $T_{firewall}\approx T_{Planck}$ and thus $E=h\nu \approx k{T}_{firewall}\approx k{T}_{Planck}\approx E_{Planck}=m_{Planck}{c}^2$ [14–16,83–86], are themselves Planck-mass black holes [14–16,87–89], specifically, Planck-mass geons [87–89], and thereby themselves contribute to the breakdown of spacetime as the Planck scale is approached, i.e., as $\delta l\to {\left(\delta l\right)}_{min}=l_{Planck}$ [14–16,87–89].

In accordance with the three immediately preceding paragraphs, and for consistency with Equation (31) keeping the numerical factor $7.951159991$ [66,82], by Equations (30) and (31) [66,82]

Thus Hawking radiation with ${\lambda }^{Wien,max}$ corresponding to values of T that are still high occurs in the region $l_{Planck}\ll \delta l\ll r_S$, hence with quantum fluctuations in the metric of Equation (1) being negligible [14–16]. For example, ${\lambda }^{Wien,max}$ corresponding to the temperature of the Sun’s core, $1.571\times {10}^7 \mathrm{K}$ [67], is equaled at $\delta l=2.3198\times {10}^{-11}m$, i.e., only slightly less than typical atomic dimensions; ${\lambda }^{Wien,max}$ corresponding to the (effective [67,75–78]) temperature of the Sun’s photosphere, $5772 \mathrm{K}$ [67], is equaled at $\delta l=6.3140\times {10}^{-8}m$, i.e., the dimensions of small microbes; and ${\lambda }^{Wien,max}$ corresponding to room temperature, $300 \mathrm{K}$, is equaled at $\delta l=1.2148\times {10}^{-6}m$, less than two orders of magnitude below the limit of naked-eye visibility $\approx {10}^{-4}m$.

Of course, the last two lines of Equation (31), and Equations (32) and (34) [let alone Equation (33)], do not apply in the region $r\gg r_S$. For, as $r\to \infty$, $\delta l\to r-r_S+\left(r_S/2\right)ln\left(r/r_S\right)$ [90], whilst applying Equation (2) and the first two lines of Equation (31) [66,82]:

The numerical factor $1.265466416\times 4\pi =15.90231998$ is dimensionless and hence is valid in any self-consistent system of units.

We have considered Schwarzschild black holes whose only energy source is their own Hawking radiation. This may eventually be the case for actual black holes if the Universe expands forever. But in the current Universe, black holes are bathed by photons emanating from $r\gg r_S$—effectively from $r\to \infty$—far more energetic than thermal photons at temperature $T_{H,r\to \infty }$ as per Equation (2): photons from the $T_{CBR}=2.725 \mathrm{K}$ [50] cosmic background radiation [50], from starlight, etc. [50]. Radiation comprised of these far more energetic photons will be blueshifted to $T_{firewall}\approx T_{Planck}$ as given by Equations (25)–(27) at $r_S+\delta l$ with $\delta l\gg l_{Planck}$. But photons corresponding to $T_{firewall}\approx T_{Planck}$, i.e., for which $E=h\nu \approx k{T}_{firewall}\approx k{T}_{Planck}\approx E_{Planck}=m_{Planck}{c}^2$ [14–16,87–89], are themselves Planck-mass black holes [14–16,87–89], specifically, Planck-mass geons [87–89], and thereby themselves might contribute to the breakdown of spacetime at this $r_S+\delta l$, i.e., at this $\delta l\gg l_{Planck}$, well before ${\left(\delta l\right)}_{min}=l_{Planck}$ is approached [14–16,87–89]. Hence in the current Universe we should consider at least the possibility of the breakdown of spacetime at this $r_S+\delta l$, i.e., at this $\delta l\gg l_{Planck}$, well before ${\left(\delta l\right)}_{min}=l_{Planck}$ is approached [14–16,87–89]. But this is not what we mean by a Schwarzschild black hole’s firewall. By a Schwarzschild black hole’s firewall we mean that which is intrinsic to the black hole itself, i.e., owing solely to its own Hawking radiation.

In Section 5, as in Section 4, it may be helpful to envision a Schwarzschild black hole enclosed concentrically within an opaque thermally-insulating spherical shell at $r_{shell}\to \infty$ [64,65].${ }^7$ Hawking radiation at temperature $T_{H,r\to \infty }$ as per Equation (2) is reradiated and/or reflected downwards${ }^6$ from the inner surface of this spherical shell, suffering increasing gravitational blueshift with decreasing r in accordance with Equations (3)–(6) [8–11,16–18,43–49,64,65].

Expressed in terms of r, at $r\ge r_S$ the relativistic gravitational scalar potential $\Phi$ of a Schwarzschild black hole and its magnitude $\left|\Phi \right|$ are [13,17,18]

Applying Equations (2), (3), (21), (22), and (23) [especially Equation (21) and the last two lines of Equation (23)], if $\delta r\ll \delta l\ll r_S$, expressing $\Phi$ and $\left|\Phi \right|$ in terms of $\delta r$ and $\delta l$ [13,17,18]:

It may be interesting to note that corresponding to minimum-definable ruler distance [13] $\delta l_{min}=l_{Planck}$ [14–16] beyond $r_S$ where A is the surface area of a black hole and S is its entropy [43–49].

We re-emphasize that a relativistic gravitational scalar potential $\Phi$ and hence also its magnitude $\left|\Phi \right|$ [17,18], and the relation thereof to gravitational potential energy [17,18], are valid concepts for all static, and even stationary, spacetimes [63] (not just Schwarzschild spacetime [91–93]). And that the spacetime engendered at $r\ge r_S$ by any Schwarzschild black hole and at all $r\ge 0$ by any Schwarzschild non-black hole (we focus on Schwarzschild metrics) is static [91–93], not merely stationary [63].

The blueshift of any photon (Hawking/Tolman-radiation photon or otherwise) whose frequency, energy, and mass [83–86] at $r\to \infty$ are ${\nu }_{r\to \infty }$, $E_{{ }_{r\to \infty }}=h{\nu }_{r\to \infty }$, and $m_{{ }_{r\to \infty }}=E_{{ }_{r\to \infty }}/c^2=h{\nu }_{r\to \infty }/c^2$ [83–86], respectively, upon falling radially inwards from $r\to \infty$, increases exponentially rather than merely linearly with decreasing $\Phi$ (or, equivalently, with increasing $\left|\Phi \right|$) [17,18], in accordance with [17,18]

This obtains because as a photon falls and gets blueshifted its mass [83–86] $m=E/c^2=h\nu /c^2$ [which of course is solely its (kinetic energy)$/c^2$ [83–86], because a photon’s rest mass is zero [83–86]] increases: the photon gets more massive as it falls. Thus as a photon falls through successive ruler-distance [13] increments $dl$, a Schwarzschild black hole’s gravitational field at $r\ge r_S$, a Schwarzschild non-black hole’s gravitational field at all $r\ge 0$—indeed, the gravitational field $\mathcal{G}=-d\Phi /dl$ in any static, or even stationary, spacetime [63,91–93]—does successive increments of (positive) work [92,93]

not on a fixed mass m but on an ever-increasing mass m. [The minus sign in $\mathcal{G}=-d\Phi /dl$ obtains because $\mathcal{G}$ acts downwards,${ }^6$ i.e., in the direction of decreasing l. $dW$ in Equation (40) is positive, because $\mathcal{G}$ is negative, and both $d\Phi$ and $dl$ are negative during infall.] $dW/d\Phi$ and thus the rate of increase of $m=E/c^2=h\nu /c^2$ with decreasing $\Phi$ is proportional to $m=E/c^2=h\nu /c^2$ itself: consequently the exponential form of Equation (39).

Hence also, in accordance with Equations (3), (24), (25), (36), (37), (39), and (40), the temperature T of any Planckian blackbody distribution of photons increases exponentially rather than merely linearly with decreasing $\Phi$ (or, equivalently, with increasing $\left|\Phi \right|$) [16–18,43–49,64,65,81,83–86].

Of course, the same reasoning also applies in reverse: as a photon rises a Schwarzschild black hole’s gravitational field at $r>r_S$—indeed, the gravitational field $\mathcal{G}=-d\Phi /dl$ in any static, or even stationary, spacetime [63,91–93]—does negative work on, or equivalently receives positive work from, not a fixed mass m but an ever-decreasing mass m. $dW/d\Phi$ and thus the rate of decrease of $m=E/c^2=h\nu /c^2$ is proportional to $m=E/c^2=h\nu /c^2$ itself: consequently as per Equations (39) and (40) a rising photon’s mass [83–86] $m=E/c^2=h\nu /c^2$ decreases exponentially rather than merely linearly with increasing $\Phi$ (or, equivalently, with decreasing $\left|\Phi \right|$) [16–18,43–49,64,65,81,83–86]. Hence also, in accordance with Equations (3), (24), (25), (36), (37), (39), and (40), the temperature T of any Planckian blackbody distribution of photons decreases exponentially rather than merely linearly with increasing $\Phi$ (or, equivalently, with decreasing $\left|\Phi \right|$) [16–18,43–49,64,65,81,83–86].