_{1}

^{*}

This is the first paper in a two part series on black holes. In this work, we concern ourselves with the event horizon. A second follow-up paper will deal with its internal structure. We hypothesize that black holes are 4-dimensional spatial, steady state, self-contained spheres filled with black-body radiation. As such, the event horizon marks the boundary between two adjacent spaces, 4-D and 3-D, and there, we consider the radiative transfers involving black- body photons. We generalize the Stefan-Boltzmann law assuming that photons can transition between different dimensional spaces, and we can show how for a 3-D/4-D interface, one can
only have zero, or
net positive, transfer of radiative energy into the black hole. We find that we can predict the temperature just inside the event horizon, on the 4-D side, given the mass, or radius, of the black hole. For an isolated black hole with no radiative heat inflow, we will assume that the temperature, on the outside, is the CMB temperature,
T
_{2} = 2.725 K. We take into account the full complement of radiative energy, which for a black body will consist of internal energy density, radiative pressure, and entropy density. It is specifically the entropy density which is responsible for the heat flowing in. We also generalize the Young- Laplace equation for a 4-D/3-D interface. We derive an expression for the surface tension, and prove that it is necessarily positive, and finite, for a 4-D/3-D membrane. This is important as it will lead to an inherently positively curved object, which a black hole is. With this surface tension, we can determine the work needed to expand the black hole. We give two formulations, one involving the surface tension directly, and the other involving the coefficient of surface tension. Because two surfaces are expanding, the 4-D and the 3-D surfaces, there are two radiative contributions to the work done, one positive, which assists expansion. The other is negative, which will resist an increase in volume. The 4-D side promotes expansion whereas the 3-D side hinders it. At the surface itself, we also have gravity, which is the major contribution to the finite surface tension in almost all situations, which we calculate in the second paper. The surface tension depends not only on the size, or mass, of the black hole, but also on the outside surface temperature, quantities which are accessible observationally. Outside surface temperature will also determine inflow. Finally, we develop a “waterfall model” for a black hole, based on what happens at the event horizon. There we find a sharp discontinuity in temperature upon entering the event horizon, from the 3-D side. This is due to the increased surface area in 4-D space,
A_{R}
^{(4)} = 2π
^{2}
R
^{3}, versus the 3-D surface area,
A_{R}
^{(3)} = 4π
R
^{2}. This leads to much reduced radiative pressures, internal energy densities, and total energy densities just inside the event horizon. All quantities are explicitly calculated in terms of the outside surface temperature, and size of a black hole. Any net radiative heat inflow into the black hole, if it is non-zero, is restricted by the condition that, 0<1/
cd
Q/d
t < 4
F_{R}
^{(3)}, where,
F_{R}
^{(3)}, is the 3-D radiative force applied to the event horizon, pushing it in. We argue throughout this paper that a 3-D/3-D interface would not have the same desirable characteristics as a 4-D/3-D interface. This includes allowing for only zero or net positive heat inflow into the black hole, an inherently positive finite radiative surface tension, much reduced temperatures just inside the event horizon, and limits on inflow.

Black holes are mysterious objects about which we still know very little. They have long intrigued physicists and fascinated the general public ever since they were first proposed [

The event horizon, in particular, is a complete mystery. This interface must have certain characteristics, which allow for net radiative heat inflow but zero (or very little) heat outflow. It must be a relatively stable structure because it can withstand tremendous gravitational forces. And yet, it can, and does expand upon massive and massless particle inflow. It seems to function as a sort of membrane, almost bubble like in character. There must be internal forces pushing out, which prevent complete collapse due to gravity. With or without inflow, there is a tremendous mass packed within a very small volume. How is that mass distributed, and what prevents complete implosion? Is there really a rip or tear in the space-time continuum at the center of a black hole, as is commonly thought?

A third central question is why do black holes have a much reduced temperature within their interiors? As such, would they not continuously swallow up the surrounding CMB radiation? There is also dark matter and dark energy. All are in direct contact with the event horizon, and all have been in direct contact since black holes were formed. Why does a black hole not feed upon this continuous and plentiful source of matter/energy? One would think that due to the ambient mass/energy, a black hole would expand continuously. And, that expansion would have been much more pronounced in earlier cosmological epochs, when the CMB temperature was much, much higher, and the universe was much more compact. The standard argument is that this inflow would lead to very small effects, numerically, and therefore does not contribute significantly to expansion. Also, the rip or tear at the center of a black hole would prevent temperature from building up within its interior. We will argue, however, that there is no rip or tear at the center of a black hole. Instead, there must be a barrier of sorts which prevents permanent inflow of CMB photons and other pervasive forms of mass/energy surrounding the black hole. Having very low temperatures on the inside, the standard 3-D Stefan-Boltzmann law would not prevent black holes from permanently absorbing CMB photons, and bringing the black hole temperature up to CMB temperatures.

We can take this a step further. If the CMB temperature is the lowest possible temperature since formation of the universe, and if black holes are in constant thermal contact with the cosmic background, then how could they form, and evolve, at a lower temperature? We believe this is a valid argument against the permanent inflow model, which we will sometimes refer to as the PIM model. The permanent inflow model cannot be disproven, or proven, observationally, because, as stated, the effects are numerically small, at least in the present epoch. Expansion would not be measurable. Logically, however, we think that the PIM model makes little sense. Unless there is a mechanism, which would force the temperature out of the black hole, we believe in a natural barrier to entry. Our thinking is that, black holes, being 4-D spatial objects, have an inherently lower temperature on the inside, at least close to its surface, which is near the event horizon. We also believe that true isolated, static black holes, can exist. In fact this is the basic assumption behind our two papers. From this, we can show that, then, we have either no net radiative heat inflow, d Q / d t = 0 , or, positive net radiative heat inflow, d Q / d t > 0 , but now between 3-D/4-D space. Radiative heat flowing out of the black hole is not possible other than through evaporative processes such as Hawking radiation. In this regard, it can be noted that, observationally, orbiting stars around black hole candidates seem to have stable orbits. Isolated, static black holes would certainly conform to this picture.

We will make two central assumptions in both papers, other than a black hole being a 4-D object. The first is that isolated, static black holes are not only possible, but likely. In fact, we will assume they must exist. And the second assumption will be that the CMB temperature can be used to find the temperature just inside the event horizon, on the 4-D side. This holds true today, as well as in earlier cosmological times. It is interesting to note that positive, net radiative heat outflow out of a black hole will not be possible given our assumptions above. It is something we can show within our model. The fact that black holes are black, observationally, demands a theoretical explanation.

To answer the questions posed above, we will make a leap of faith. We propose that black holes are 4-D spatial objects, spherically symmetric and packed with blackbody radiation, embedded in 3-D space. Their radiative mass distribution is distributed in such a way as to make them appear black. A three dimensional analogy would be liquid droplets in a gas, but here we are dealing with a 4-D droplet, and, furthermore, as it will turn out, not of uniform density. We can imagine black holes to be droplets of 4-D radiation, to be precise, within a greater 3-D universe. The event horizon is the interface between 4-D space and 3-D space. This is where the rip or tear in the space-time continuum occurs, and not at the center of the black hole, as commonly thought. Indeed, as we shall see in the follow up paper, the black hole is well-behaved within its interior and has no singularity at its center. While at first sight, this interpretation may seem fanciful and even far-fetched, we will soon see that certain characteristics emerge within this picture, which seem to make sense. It is the goal of this paper, and the follow-up paper, to show that this hypothesis may have some validity.

We will build our model with two papers. The first work, this paper, deals with the event horizon itself. The second paper, which follows this, will deal almost exclusively with the internal structure of a black hole. Both papers are lengthy and involve a considerable amount of formulae. However we believe this to be necessary in order to make a convincing case, which will support this novel hypothesis.

Interestingly, as we develop our model, we will also show that a black hole cannot be a 3-D ball packed with black-body radiation, or for that matter, a 2-D construct. There are several reasons for this, the most important of which are the following. We list these in bullet form.

1) 2-D and 3-D balls of blackbody radiation cannot pack the requisite amount of radiative mass in such a small volume. The temperatures would have to be incredibly large, even at the surface of a black hole.

2) 3-D objects will not allow for a natural discontinuity at the interface, which is needed to define a radiative surface tension. A finite, positive surface tension is required to define a curved object in space, which a black hole inherently is.

3) A 3-D object cannot guarantee that there is no net heat outflow whereas a 4-D object can.

4) 3-D/3-D radiative transfers of energy cannot allow for substantially lower surface temperatures within the black hole event horizon, which is just underneath the surface.

5) 3-D/3-D transfers of radiative energy will not allow for much reduced radiation pressures, internal energy densities, entropy densities, and total energy (radiative mass) densities, etc. just inside the event horizon. These quantities, incidentally, will all increase dramatically within the black hole itself, as one approaches, r → 0 , in order to pack in the required radiative mass.

6) 3-D/3-D interfaces will not prevent CMB photons, and other pervasive forms of matter/energy surrounding a black hole, from being continuously pulled in. With our 4-D/3-D model we can provide a barrier, or lip, which prevents permanent inflow and expansion of a black hole. In fact, the outside 3-D surface temperature, T 2 = 2.725 K , will serve as an input in order to define an equilibrium temperature for a black hole, on its inside surface, when there is no inflow. This is what we will call an isolated, static black hole. For temperatures, T 2 > 2.725 K , we will have radiative heat inflow, i.e., d Q / d t > 0 , the amount of which will depend on the value of T 2 . We reserve temperature, T 1 , for the temperature just inside the event horizon, on the 4-D side. This will always be substantially lower than the temperature just outside the event horizon, with or without radiative inflow.

There are other reasons for settling on a 4-D/3-D interface, but these will be among the most important.

It has not gone unnoticed that black holes appear, and act very much like balls of blackbody radiation [^{16} K, all particles in the standard model are relativistic, i.e., in the form of radiation. Thirdly, the concept of ball lightning (Kugelblitz in German) has been applied to blackbody radiation and, in particular, to the formation of black holes. The idea is that the black body radiation is so concentrated in intensity that it curves that space-time itself around it, and forms a black hole [

There have been 2-D models proposed for black holes, so-called holographic models [^{37} J/K. This is only about 2 orders of magnitude greater than the entropy of the sun itself, which is approximately, 10^{35} J/K. The Bekenstein model gives the entropy as, S Bekenstein = ( 1 / 4 ) c 3 k B / ( G ℏ ) ( 4 π R 2 ) , which for a black hole of one solar mass gives, 1.50 × 10^{54} J/K. His entropy is much greater than that of the sun by almost nineteen orders of magnitude! Moreover, the Bekenstein entropy scales as, S Bekenstein ~ R 2 . Our entropy also scales appreciably, comparable to, R 2 , but not according to a clear power law. If our model is correct, a black hole is thus a highly-ordered state, contrary to what current consensus claims. The calculation will be presented in the follow-up paper for several black holes, each with a different mass.

In this paper, and the next, we will ignore/discount Hawking radiation [^{nd} order corrections, at best, to the results presented here. The temperatures just inside the event horizon within our model will be shown to be considerably higher than the Hawking temperature, which is given by the formula, T H = ℏ c 3 / ( 8 π G M k B ) . Nevertheless, the inside surface temperatures will still be much less than they are on the outside. Moreover, within the black hole itself, those same 4-D temperatures will increase dramatically as one approaches the interior of the 4-D black hole. At the very center of the black hole, at, r = 0 , we theorize that we will have a maximum but finite radiative energy density for a finite volume.

The outline of the paper is as follows. In Section 2, we consider the 4-D/3-D interface, and generalize the Stefan-Boltzmann law to account for radiative energy transfers between different spaces. We show that radiative heat energy density is not the only component, which transfers. When transferring between spatial dimensions, other forms of radiative energy flow, such as internal energy density and radiative pressure. It is an all or nothing proposition. To maintain the blackbody identity of the photons at a particular temperature, all components, or none, carry over. In this section, we will prove that there can only be positive radiative energy inflow into the black hole once the event horizon is reached, or none. In Section 3, we consider the expansion of a black hole upon net inflow of radiation. We define the surface tension and model the event horizon as an infinitely thin membrane, a bubble of sorts. We derive key relations for the work done, in terms of surface tension and coefficient of surface tension. Because there are two surfaces expanding, the four-dimensional and the three-dimensional, we must take both into account for radiation. Then there is also gravity, which will also want to prevent the 4-D surface from expanding in size. This is included although the specific details will be worked out in the subsequent paper. In this section we generalize the Young-Laplace equations for an interface separating two different spaces, one 3-D and the other 4-D.

In Section 4, we build upon the ideas developed in sections II and III. We show that there is a sharp discontinuity in temperature when crossing the 3-D/4-D threshold. This discontinuity is due to the discontinuity in space itself, because in going from the 3-D world to the 4-D space, the surface area increases abruptly and dramatically, from, A R ( 3 ) = 4 π R 2 , to A R ( 4 ) = 2 π 2 R 3 , for the same radius, R. This discontinuity in surface area leads to a precipitous drop in temperature just inside the event horizon. Moreover, this will translate into decreased internal energy densities, reduced radiative pressures, and much smaller entropy densities just inside the black hole. All these quantities will depend on the outside radiative temperature and the size, or mass, of the black hole. In this section, we present our so-called “waterfall model” for the event horizon of a black hole. The summary and conclusions are highlighted in Section 5, our final section. Finally we have an appendix (Appendix A), where we consider non-spherical symmetry, and the emission of quadrupole gravitational radiation. We show that our model can be extended to this situation, and we work out a few numerical examples.

We start with the radiative flux emitted in N-dimensional space. As is known [

Φ ( N ) ( T ) = d Q / d t 1 / A ( N ) = σ ( N ) T N + 1 (2-1)

In this equation, dQ stands for the amount of radiative heat emitted in time, dt, and the superscript, (N), refers to the number of spatial dimensions. The surface area in N-dimensional space, A ( N ) , is that area through which radiative energy can escape, and σ ( N ) is the Stefan-Boltzmann constant, generalized to N-spatial dimensions. The hyper-surface, A ( N ) , in Equation (2-1), can be calculated using the formula [

A ( N ) = A ( N ) ( R ) = 2 π N / 2 R N − 1 / Γ ( N / 2 ) (2-2)

In Equation, (2-2), R is the radius in N-space, and Γ ( x ) is the gamma function. We assume spherical symmetry for this self-contained ball of blackbody radiation. The generalized Stefan-Boltzmann constant, σ ( N ) , in Equation (2-1), is determined by the following formula [

σ ( N ) = ( 2 / c ) N − 1 ( π ) N − 2 ( k B N + 1 / h N ) N ( N − 1 ) Γ ( N 2 ) ζ ( N + 1 ) (2-3)

In this equation, k B is Boltzmann’s constant, c equals the speed of light, h is Planck’s constant, ζ ( x ) is the zeta function, and Γ ( x ) is, again, the gamma function.

In 4-D space, Equation (2-2) gives, A R ( 4 ) = 2 π 2 R 3 , whereas in 3-D space, we obtain the familiar A R ( 3 ) = 4 π R 2 . We can also determine the values for the Stefan-Boltzmann constants, in 4-dimensional, 3-dimensional and 2-dimensional space. Using Equation, (2-3), we obtain numerically

σ ( 4 ) = 3.021 × 10 − 5 Watts / ( m 3 ⋅ K 5 ) (2-4a)

σ ( 3 ) = 5.670 × 10 − 8 Watts / ( m 2 ⋅ K 4 ) (2-4b)

σ ( 2 ) = 9.614 × 10 − 11 Watts / ( m ⋅ K 3 ) (2-4c)

We will be using MKS units throughout this paper, even when not explicitly written out. The radiative flux, Φ ( N ) , is measured in Watts/m^{N} in Equation, (2-1). The emissivity factor will always be taken as unity as we are assuming a perfect blackbody. All superscripts in parenthesis, next to a quantity, will refer to the number of spatial dimensions over which the quantity is defined.

We use Equation (2-1), as our basic starting point, to find the radiative transfer of heat energy between adjoining spatial dimensions. For a 3-D to 4-D spatial transfer of radiative energy, we may claim that, using this equation,

d Q / d t = d Q ( 3 ) / d t − d Q ( 4 ) / d t = A ( 3 ) σ ( 3 ) T 2 4 − A ( 4 ) σ ( 4 ) T 1 5 (2-5)

In this expression, d Q ( 3 ) / d t , is the radiative heat power exiting the 3-D space and entering the 4-D space. The quantity, d Q ( 4 ) / d t , on the other hand, is the radiative heat power exiting the 4-D space and entering the 3-D space. The temperature, T 2 , is the temperature just outside the black hole, in 3-D space. We define the temperature, T 1 , as the temperature just inside the event horizon, in 4-D space. The respective surface areas are found using Equation, (2-2). Equation, (2-5), is a direct extension of Equation, (2-1), and we call this the first generalized version of the Stefan-Boltzmann equation for radiative transfers between adjoining spatial dimensions. We note that even though surface areas, A ( 3 ) , and, A ( 4 ) , have differing units, Equation, (2-5), is dimensionally consistent. We will assume that the event horizon is infinitesimally thin, and as such, the temperatures, T 1 , and, T 2 , are defined at effectively the same radius, just on different sides of radius, R.

It is obvious from relation, (2-5), that d Q / d t will be positive if the first term on the right hand side exceeds the second. If d Q / d t > 0 , then we will have net inflow. For a black hole, we can prove that this is the only possibility, other than d Q / d t = 0 . We will, however, have to assume a 4-D/3-D interface. We will designate a black hole with no net inflow as an isolated, static black hole (ISBH). If there is net inflow, we call that a dynamic black hole (DBH). We called the temperature just outside the event horizon, on the 3-D side, T 2 . For T 2 , we will assume a temperature of 2.725 K if the black hole is an ISBH. In other cosmological epochs, this temperature would have to be modified. For a DBH, the temperature on the outside will be larger than the CMB temperature. In Equation, (2-5), T 1 , is the temperature just inside the event horizon, within the black hole on the 4-D side. As will be seen shortly, this temperature is determined by the radius of the black hole, or what is equivalent, by its mass. Because of the Schwarzschild condition, R = 2 G M / c 2 , the two quantities are proportional. We will often make use of the Schwarzschild relation throughout this paper without explaining it. In practice, T 2 , can be quite large. Due to friction and superheating of massive and massless inflows, the temperatures can reach X-ray temperatures, 1.16 × 10^{6} K to 1.16 × 10^{9} K for soft and hard X-rays, and higher. These X-ray emissions would correspond to photon energies from a few MeV to a few GeV. Emissions of this type are readily discernable, observationally, if not too far away. Black hole masses, and thus radii, can also be estimated in many instances. This will give us enough information to calculate the specific amount of inflow using Equation, (2-5). For the temperature, T 1 , we assume an equilibrium temperature, calculated as if the black hole had no inflow. This quantity is based strictly on the size, or mass, of the black hole, which at the moment of inflow has a unique value.

Focusing further on Equation, (2-5), we specialize to the case where there is no inflow. In this instance, d Q / d t = 0 , and Equation, (2-5), reduces to

A ( 4 ) σ ( 4 ) T 1 5 = A ( 3 ) σ ( 3 ) T 2 4

2 π 2 R 3 σ ( 4 ) T 1 5 = 4 π R 2 σ ( 3 ) 2.725 4

Solving for T 1 gives

T 1 = 0.581 R − 1 / 5 (2-6)

We have used Equations, (2-4a), and, (2-4b), to obtain this simplified result. This we call the equilibrium temperature, just inside the event horizon. It is determined strictly in terms of radius, or equivalently, mass, for a specified black hole. The black hole can and will expand upon inflow. And inflow will be determined using a different and higher value for T 2 by means of Equation, (2-5), and this T 1 value. However, until such time that the black hole has expanded, the temperature to be used at the event horizon on the inside is T 1 , as determined by this Equation, (2-6). From Equation, (2-6), it is clear that a black hole having greater mass will have a lessor temperature just inside the event horizon.

If we consider a black hole having the mass of the sun (not realistic), then the Hawking temperature would give an inside temperature of about 62 nK. Equation, (2-6), gives a much higher inside surface temperature of 0.117 K, at the event horizon. Clearly Hawking radiation and evaporative processes are second order effects. For black holes having masses of 10 times and 10^{6} times solar mass, we obtain using Equation, (2-6),

Coming back to Equation, (2-5), in time dt, the amount of radiant heat entering the black hole is

In the third line of this equation, we made use of,

In N-dimensional space, a hyper-volume can be defined for a N-dimensional ball. The expression [

The superscript “N” in parenthesis on a physical quantity will always refer to the spatial dimension over which the quantity is defined.

The expression, Equation, (2-3), and the relations, Equations, (2-4), through to, (2-7), are not quite correct. The ^{st} order phase transition at a particular temperature and pressure. The situation here is totally different because, as we shall soon see, there will be an abrupt change in temperature at the event horizon. In the previous work, the temperature remained fixed when the transition occurred. Nevertheless, even though the situation is very different because we are talking about radiative transfer, versus a discontinuous phase transition, maintaining the identity of the photons in their respective spatial dimension requires that all forms of blackbody energy transfer.

To see this more clearly, let

In Equation (2-9),

Equation (2-10), really only makes sense when multiplied by a corresponding volume. We are mixing pressure, having units of N/m^{2}, with energy density, measured in J/m^{3}, on the right hand side of Equation, (2-10), which is really only valid if we form a product with a volume. When forming a product with volume, we obtain energy and stored work, which are both measured in Joules, even though the pressure and energy density have different units. So, when we write an “equation” such as relation, (2-10), this is what is inferred.

In 4-D space we have a similar definition. Let

Moreover, by definition, the total radiative energy “density” is found by using

In these equation,

We next note that

In this equation,

These relations depend only on the dimensionality of space, N, and the temperature, T, because

However, by Equations, (2-14),

This implies that

In Equations, (2-17),

Using Equations, (2-16), it follows that in 3-D space,

In 4-D space, we find, however, that

Notice the different factors in both sets of equations, as they will be important. Moreover, Equations, (2-17), show us that, if

If, however, the number of spatial dimensions equals,

We emphasize, once more, that Equations (2-10), (2-12), (2-16), (2-19) and (2-20) really only make sense when multiplied by a corresponding volume because we are mixing pressure and energy density within the same expression. Technically, these equations are only correct if we form a product with a volume, which is how we use these equations.

Coming back to Equation, (2-18), we can write this equation for the radiative heat transfer a variety of ways. We utilize Equations, (2-21), and, (2-22), and find

Furthermore, we keep in mind that,

In Equations, (2-24), and (2-25), the 3-D surface area equals,

In the last line, we have defined the radiative forces at radius, R, in three and four spatial dimensions as

, (2-27a,b)

We keep in mind that,

Next, we revisit Equations, (2-13), and, (2-3). We notice that

In Equations (2-28a)-(2-28c), the coefficients

We substitute Equations, (2-28a), and, (2-28b), into the second line of Equation, (2-26). We thereby obtain, after multiplying through by the velocity of light, c,

The coefficients,

We now compare Equation, (2-29), to Equation, (2-5). We notice right away that we have a discrepancy. Equation, (2-29), has an extra factor of (16/3) in front of the

With this in mind, let us rewrite Equation, (2-29), as

where, by definition, the primed variables have been recast in terms of the original unprimed variables, as follows,

The unprimed sigma values were given by Equations ((2-4a), (2-4b)). We will be working with this version of the generalized Stefan-Boltzmann law, Equation (2-30), which we call the 2^{nd} generalized version of the Stefan-Boltzmann law. It is our contention that this is correct. This Equation (2-30), is equivalent to Equations, (2-26).

With the new Stefan-Boltzmann constants, defined in Equations ((2-31a), (2-31b)), we can recalculate the equilibrium inside surface temperature of a black hole. By setting,

Solving for

or, what is equivalent,

Upon comparison with Equation, (2-6), we note that, numerically, there is virtually no difference between the two results. As far as this result is concerned, the 2^{nd} generalized Stefan-Boltzmann law gives an almost identical calculation for the equilibrium temperature just inside the event horizon, as our 1^{st} version, Equation, (2-5). Nevertheless, let us calculate the inside surface temperature for three black holes. We focus on three black holes; one having the mass of the sun, another having a mass 10 times the mass of the sun, and for the third black hole, we assume a mass, 10^{6} times the mass of the sun. For these three massive black holes, we calculate the radii using the Schwarzschild relation, and substitute these radii into Equation, (2-33). We thereby obtain,

As stated, these values for

There are many ways of writing our 2^{nd} generalized version of the Stefan-Boltzmann law. Equation (2-30), is one way to express it. Other ways are by means of Equations (2-26). If we use the second line of Equation (2-26), and insert the values for

Again,

The lip, or barrier to entry, is also readily apparent. Utilizing the last line of Equation, (2-35), we see that there can be no net outflow. For that to happen, the external outside temperature would have to be less than the cosmic microwave background, which cannot happen! The cosmic microwave background is what we used to eliminate the 4-D term, the

We close this section by establishing a limit for the amount of inflow. We focus on the last line of Equation, (2-26). If,

If, on the other hand,

It is clear that, in general,

However, the left hand side of this equation is positive definite. Thus, the right hand side must also be. This, in turn, leads to the condition,

This shows that the amount of radiative heat inflow is restricted to be less than,

We emphasize that this limit placed on radiative heat inflow is a direct consequence of having a 4-D/3-D interface for the event horizon. Equation, (2-38), follow from Equations (2-36), and (2-37). These, in turn, depend on the last line of Equation (2-26). The factors of 4 and 5 in Equation (2-26), are due to the dimensionality of space itself. See, for example, Equations (2-19) and (2-20). More specifically, refer to the last equalities in each. Radiative inflow would not be restricted if the black hole were a 3-D object, because those factors sitting out in front of

Also, for a 3-D/3-D boundary, the Stefan-Boltzmann law would not allow us to have a lessor temperature just inside the event horizon. A 4-D/3-D is a requirement for that. The different surface areas between the 3-D and the 4-D space, is what causes the sudden drop in temperature, as will be demonstrated in section IV. See, also, the second line of Equation, (2-35), and specialize to the particular situation where we have no net heat inflow. This will relate the temperature on the inside of the event horizon,

We can evaluate the quantity,

If the outside surface temperature is,

For outside surface temperatures in excess of, 2.725 K, the value for ^{9} K. Because of the ^{36} times higher than what is indicated by Equation (2-40). The inequality (2-38), may also help explain the circulating nature of radiative heat inflow before entering the event horizon of a black hole.

The surface tension will play a key role in our analysis of the black hole event horizon. We first recognize that expanding a black hole event horizon requires work, or input energy. That work can be expressed very simply as

here,

Another formulation for the positive work done in expanding the event horizon is in terms of pressure. Ordinarily, the pressure just inside the membrane has to be larger than the pressure just outside in order to guarantee a positive curvature for the object. However, in the case of a 4-D/3-D membrane, this will not hold true. The surface area changes in this case, and as a consequence, the radiative pressure on the inside will actually be less than that on the outside. Remember that we are going from a surface area of,

For conventional 3-D objects such as liquid droplets or bubble films, it is known that

For a 3-D sphere, the Young-Laplace equation reads

The coefficient of surface tension,

The 4-D coefficient of surface tension, ^{2}, and

To expand a surface in 3-D space, we use Equation, (3-1), and write

where dV is the increase in volume. In 4-D we use the same formula, Equation, (3-1), but now,

here, ^{3} and dV is measured in (meters)^{4}. We see that,

Substituting Equation, (3-7a), into Equation, (3-5), gives

This holds true because,

here,

For a black hole to expand, two surfaces have to expand, the 3-D surface and the 4-D surface. For our 4-D blackbody black hole radiation model, we set

In this equation,

As we have seen in the last section the radiative force,

Therefore, by Equations, (3-10), and, (3-11), another way to write the 4-D/3-D radiative surface tension is

For an isolated static black hole (ISBH),

The full surface tension has to include the gravitational force acting on the event horizon. By Equations, (3-10), and, (3-11), we can write,

We can thus calculate the associated surface tension using Equation, (3-13), if we know the value of the 4-D gravitational force at the event horizon, and

An alternative formulation to Equations, (3-10), and, (3-13), is

This is the infinitesimal positive work done by inflowing matter/radiation in expanding the black hole event horizon in both 3-D and 4-D space. The term,

Equations, (3-1), and, (3-10), led us to conclude that the radiative portion of surface tension is the difference in radiative forces,

Hence, a new way to write this part of the surface tension is

In the second line of Equation, (3-16), we have used the identity,

This is a disguised version of Equation, (2-33). We previously saw that

We next give some numerical values. We first consider an isolated, static, black hole, where,

In the last equality, we have set the temperature,

For the three black hole masses considered in Equations, (2-34), we use Equation (3-18), to calculate the associated radiative surface tensions. Mass first gets converted to radius using the Schwarzschild relation,

All forces are measured in Newtons. We notice that both small and large black holes have hardly any radiative surface tension, given their significant size. The values indicated in Equations, (3-19), are really quite small. We will see, however, that what holds the black hole together isn’t just radiative surface tension. It is gravity! And gravity acts within the black hole, as well as on its surface, as will be shown in the follow up paper. The values indicated in Equation, (3-19), hold for an isolated, static black hole (ISBH), where there is zero inflow.

We should also mention that other forces will invariably come into play beside blackbody radiation and gravity, in stabilizing the event horizon. These will also prevent the black hole from expanding outwards. There is dark energy, and dark matter, and ordinary matter. These components are not taken into account in Equations, (3-10), nor in, (3-14), which is equivalent. CMB photons contribute very little, in the present epoch, to the total energy density of the universe, only about 10^{−5} of the total amount. Also, in those regions of space where matter is prevalent, and where black holes are to be found, CMB photons contribute an even smaller percentage. Equations, (3-19), only considers blackbody photons. Nonetheless, they do indicate that an inherently positive, finite value exists for the radiative surface tension, even if the black hole is pure photonic radiation. This was the case in the very early universe.

We next consider a dynamic black hole, where

We emphasize that a positive surface tension is needed in order to define a positively curved object such as a black hole. Our surface tension is inherently positive definite as seen by Equations, (3-12), or (3-16); it can never be less than

We close this section by mentioning that in the Appendix A, we consider non-spherical symmetry. Surface tension leads to work done, and we consider radiative black body inflow under the assumption that the black hole has the shape of an oblate spheroid. Rotating black holes assume this shape, and if there is net inflow, we can demonstrate that gravitational quadrupole radiation will be emitted. We calculate a few such examples to show the robustness of the model.

We now have the required tools to discuss what happens at the event horizon, given our 4-D radiation model for a black hole. The results will only hold at the 4-D/3-D interface. In a follow-up paper we will discuss the internal structure of a black hole.

For now, however, we will indicate that hydrostatic equilibrium has to be maintained layer by layer within the black hole. We therefore set

(4-1)

here,

We now come back to the event horizon. At the event horizon, we will assume that the 4-D space abruptly changes into 3-D space, at radius,^{nd} paper. The gravitational coupling constant, which is Newton’s constant in 3-dimensions, will increase abruptly upon entry into the 4-D space. The gravitational potential, however, can be chosen to have the same value at,

We first calculate the relevant densities and radiative pressure, just outside the event horizon, on the 3-D side. We assume, for the time being, that we are dealing with an isolated, static black hole. Using our CMB temperature of, 2.725 K, we find using Equations, (2-28b), and, (2-19),

, (ISBH) (4-2a,b)

, (ISBH) (4-2c,d)

Just inside the event horizon, on the 4-D side, we use a different set of equations. We use the temperature, specified by Equation, (2-33). We also employ relations, (2-28a), and, (2-20). The results are now black hole radius dependent,

^{4} (ISBH) (4-3a)

^{3} (ISBH) (4-3b)

^{4} (ISBH) (4-3c)

^{4}∙K) (ISBH) (4-3d)

For the second equality of Equation, (4-3a), we have substituted Equation, (2-33). We notice that the size, or mass, of a black hole will have a direct impact on the various densities and radiative pressure, just inside the surface. In fact, as the radius increases, the densities and pressure will decrease according to Equations, (4-3a,b,c,d). Also note the different units between Equations, (4.2a,b,c,d), and, (4-3a,b,c,d). This makes a difference determination impossible. However, we can set up ratios to see how the 4-D quantities relate to the corresponding 3-D quantities.

For an isolated, static black hole, we know that Equation, (2-36), holds. Therefore, by Equations, (2-27a,b), we must have

We use this to construct the ratio,

Moreover, due to the identities in Equations, (2-19), and (2-20), we can further prove that

In the last equation, we have made use of

See Equation, (2-33). What is important to note in Equations, (4-5), through to, (4-9), is the fact that, due to the presence of R, we have a discontinuous jump in value in all these quantities as one enters the black hole. The decreasing jump in value, or gap, depends on the size (or mass) of a black hole. The 4-D densities, will decrease abruptly from the corresponding 3-D quantities as one breaks through the event horizon envelope. This holds for the radiative pressure and temperature, as well, as seen in Equation, (4-5), and, (4-9). This break in value is what we refer to as our “waterfall model”. Many quantities will drop precipitously as one makes their way into the black hole. Again, this is a direct consequence of the change in dimensionality of space. No such drop in value would occur if the black hole were a 3-D construct.

We have seen that the temperature changes abruptly upon entering the 4-D space through Equation, (2-9). Another way of expressing it is to make use of the second line in relation, (2-35). If

We solve this equation for

Equation, (4-11), is another way to express the discontinuity in temperature, because of the presence of R on the right hand side. The mass, or size, of the black hole will determine the discontinuous jump in temperature. If we substitute

Equations, (4-5), through to, (4-9), are also intriguing from another perspective, namely elementary particles. Attempts have been made in the past, to identify elementary particles as mini-black holes. As with black holes, outward characteristics such as charge, spin and mass are what is observed. The interior seems to defy an explanation. We notice that if the radius is made very, very small, in Equations, (4-5), through to, (4-9), we obtain incredibly large values for 4-D radiative pressures, internal energy densities, heat densities, entropy densities, etc.. This may turn out to be an intriguing way to model elementary particles. These mini-black holes have the opposite behavior than their macroscopic counterparts, in that their temperatures, radiative pressures, internal energy densities, etc. would increase (sky-rocket) within their immediate interior. For macroscopic black holes, we have seen that those values for density and pressure decrease in value. Another compelling argument for this identification with elementary particles is the fact that no gravity is assumed. We would obtain a natural curvature in space due to radiative surface tension, and radiative surface tension only. This, of course, would take us too far afield in this work. But we definitely find this to be an intriguing line of research. The second paper will show us how to go into the specifics about modeling this very possibility.

Our waterfall model also extends to the radiative force,

It is important to realize that, within this model, what causes the photons to disappear is not that the escape velocity exceeds that of light. This is one way of looking at it, and it assumes that there are no other photons pushing back at the event horizon. We would have a situation where the black hole just grows and grows. We do believe that 4-D internal photons do push back at the event horizon. The 4-D photons hold back the floodgates, so to speak. But because they are 4-D, they cannot push back with the same force as the 3-D photons pushing in. Our interpretation is that, as a consequence, the 3-D photons are pulled in and fall into a gravitational hole once they enter this new 4-D space. Those photons have to exceed a certain threshold temperature on the outside, which we chose to equal, 2.725 K. In a sense the photons fall off a cliff upon entering the event horizon, and are never seen again. We can also think of the event horizon as a kind of simple machine. The sudden increase in surface area when entering the 3-D/4-D threshold causes a step-like decrease in radiative pressure, and radiative force. This is a trap from which photons cannot escape. What makes this unique is that it is the change in spatial dimension which causes this increase in surface area, and ultimately, this photon trap.

For a dynamic black hole, we must have the condition,

Substituting,

Upon comparison with Equation, (4-5), we see that this makes sense because the denominator on the left hand side is greater than that in Equation, (4-5). In Equation, (4-5), we assumed an outside temperature of 2.725 K, whereas here,

Using our formulae for surface areas in 3-dimensional and 4-dimensional space, and rearranging, we find that

Comparing with Equation, (4-6), we notice that the inequality is again due to the higher outside temperature,

and,

For the inequality in relation, (4-17), we note that in terms of proportionality,

We next consider net heat inflow. From section III, we know that positive work has to be done by an external agent in order to cause the black hole to expand its volume. That external agent can only be in-falling matter/energy. One way to describe the work done is through Equations, (3-1), and, (3-10). Another formulation is through Equation, (3-14). We focus, on Equation, (3-10). The radiative surface tension is given by the expression, (3-16). Using this expression, we calculate the work done against radiation in increasing the black hole from an initial radius,

In the 3^{rd} line, we assumed that the outside surface temperature is held constant during the expansion process. For the 4^{th} line we used Equations, (2-19), and, (2-28b). We also defined,

Another way to find the work done is through Equation, (3-14), which makes use of the coefficients of surface tension. This will give us the same result as Equation, (4-19), and so we will not reproduce it. Instead we will focus of the coefficients themselves, which will round off the discussion. Knowing the surface areas in 3-D and 4-D space, we can easily demonstrate that,

Upon comparison with Equation, (3-1), it is clear that

Moreover, since,

, (4-22a,b)

The radiative forces at the event horizon can thus be related to the radiative coefficients of surface tension. Taking a final step, we use Equations, (2-27a,b), and substitute these into Equations, (4-22a,b), to eliminate

, (4-23a,b)

These just reproduce Equations, (3-3), and, (3-4), which were put forward previously, more as a claim, than as a fact. The coefficients,

We can rewrite Equations, (4-23a,b), to show this more explicitly. From Equations, (2-19), and (2-20), we know that

We see clearly that

To see just how easy it is to stretch the membrane, the event horizon, against radiative forces, we consider a numerical example. We consider a black hole having a mass, ten times the mass of the sun. We will assume an increase in radius by a factor of 1.1, or, 10%. Thus, the radius will increase from an initial value,

The 3-D radiative pressure at,

We use the expression, (4-19), to find the work done against radiative forces. At a temperature,

This is the work done against radiative, and radiative forces only. Gravity will also have to be considered, which will be the case in a follow up paper. The radiative work done in Equation, (4-29), may seem considerable, but it will be next to nothing when compared to the work done against gravity, for the same situation. We will see that in the next paper, where we will give an expression for the 4-D gravitational force. However a crude order of magnitude estimate shows us that an increase of mass by 10% for the black hole of 10 solar masses, amounts to a mass difference of one solar mass, (1.99 × 10^{30} kg). If we multiply this by,

We close this section by emphasizing, once more, that the waterfall model presented here is a direct consequence of the assumed change in spatial dimension at the event horizon. A 3-D/3-D interface would not allow for a precipitous drop in temperature when crossing the boundary. We would not have much reduced internal energy densities, entropy densities, and radiative pressures, when entering the black hole. Nor would the radiative force due to blackbody photons drop abruptly upon entry into the black hole. Finally, a 3-D/3-D event horizon will not allow for gravity to increase dramatically upon entry into the black hole, as will be shown in a follow-up paper. But then again, a 3-D/3-D interface would also not allow for a rip in the space-time fabric, as there would be no a-priori discontinuity in space at the event horizon.

In this work, we presented a model for a black hole based on a 4-D spatial sphere filled with blackbody radiation. We focused on the event horizon and argued for an interface, which separates 4-D space, the black hole, from 3-D space, the surroundings. For a static black hole with no inflow,

In Section 2, we first generalized the Stefan-Boltzmann law for radiative transfers between three dimensional and four dimensional spaces. The result is Equation (2-26), or Equation (2-35), if we bring the right hand side in terms of 3-D quantities. All superscripts refer to the dimensionality of space over which the physical quantity is defined. In Equation, (2-26),

We argued that the first version of the Stefan-Boltzmann generalization for radiative transfers between spatial dimensions, given by Equations (2-5) and (2-7), is incorrect. These equations do not take into account all forms of blackbody energy, which consists of internal energy density, radiation pressure, and radiative heat energy. These quantities are all defined at a specific temperature, and if the temperature changes, as, for example, upon entering the event horizon, then these quantities must also change collectively as one unit. It is an all or nothing proposition. Moreover, we have to concern ourselves with the different dimensionality of space, when moving from one dimension to the next. Hence, Equations (2-5) and (2-7), have to be modified. The correct expressions for the generalized Stefan-Boltzmann law are Equations (2-26) and (2-35). There, the factors are correct, and we call these relations, the 2^{nd} generalized versions of the Stefan-Boltzmann law. They take the dimensionality of space into account, as well as all forms of radiative energy transfer. See the discussion following Equation (2-29), and Equations, (2-28a,b). Numerically, the differences between the two versions are slight. Nevertheless the distinction is important.

With our generalized Stefan-Boltzmann equations, we can predict the amount of radiative heat inflow, given the temperature just outside the event horizon, and the mass, or radius, of a black hole. These are quantities, which are accessible observationally. If the temperature just outside the event horizon,

In Section 3, we focused on the surface tension. The black hole has positive curvature, and therefore, there must be a positive surface tension associated with the event horizon. The radiative surface tension is the difference in radiative forces between the outside, and the inside, of the event horizon. The formal relation is Equation, (3-10), or, (3-12), where it is seen that

With or without heat inflow, there is a jump or gap in radiative force upon entering the black hole. The radiative surface tension can never equal zero. In fact, Equation, (3-12), shows that at a very minimum, for

In this section, section III, we have calculated the radiative work done in expanding a black hole upon net radiative inflow. The work done is given by expressions, (3-11), or, equivalently, (3-14). To expand the black hole, both the 3-D and the 4-D surface areas have to expand. We have generalized the Young-Laplace relations in Equations, (3-3), and (3-4). Both contribute to the expansion process as shown in Equation, (3-14), where one term assists, and the other term hinders, the expansion process. See Equations, (3-7a,b), for the connection between radiative surface tension and radiative coefficients of surface tension. Both formulations give the same results for the amount of work required if we expand from an initial volume to a final volume, as they must.

In Section 4, we brought the ideas developed in sections II and III together. We proposed a so-called “waterfall model” to describe what happens at the event horizon. Due to the change in spatial dimension, we have a sharp discontinuity at the event horizon. The temperature, the radiative pressure, the internal energy density, the radiative heat density, and the entropy density, all drop precipitously in value at the 3-D/4-D interface upon entering the black hole. We have conditions, (4-5), (4-6), (4-7), (4-8), and (4-9), which hold for an isolated, static, black hole (ISBH). All physical quantities on the inside depend strictly on their counterparts in 3-D space, as well as, on size, or mass, of the black hole. For a dynamic black hole (DBH), we have the inequalities, relations, (4-12), through to, (4-18). To find the values for the inside variables, we resort to relations, (4-12), through to, (4-18). We see that we need not only the size, or mass, of the black hole, but also a specific outside temperature,

If we have an isolated, static black hole, then,

Our waterfall model explains why we have a discontinuity at the event horizon. We cannot see the black hole, because once photons enter, they disappear from view due to the much reduced temperatures, radiative pressures, internal and other energy densities on the 4-D side. But more importantly, the radiative forces are such that, once entered, there is no escape. Due to the sudden drop in radiative force, they have entered a radiative trap. They have reached the point of no return, because of the inherent drop in radiative force. It is as if we have a waterfall which drops abruptly, and enters a region which opens up into a much wider valley (space). The water cannot jump back up into the original space. All radiative quantities take a sharp drop in value, and the ultimate flow depends on the size, or mass, of the black hole, as well as on the outside surface temperature.

Some numerical values for an isolated, static black hole (ISBH) are indicated by Equations, (4-2a,b,c,d), and (4-3a,b,c,d). Equations (4-2a,b,c,d) hold just outside the event horizon, while Equations, (4-3a,b,c,d), hold just inside the event horizon. Just inside the event horizon, the values are quite low as can be appreciated by comparing values relative to the outside. The Young-Laplace coefficients associated with radiative surface tension are given by Equations, (4-24a), and (4-26). We see that

We have emphasized throughout this paper why a black hole cannot be identified with a 3-D blackbody ball of radiation. First, no intrinsic, finite positive radiative surface tension can be defined for such a situation. The positive-definite radiative surface tension is due precisely to the change in spatial dimension. Second, our waterfall model goes a long way towards explaining why photons are pulled into the black hole once the event horizon is reached. We have a sudden drop in key thermodynamic variables at the interface, including temperature. All quantities drop abruptly because of the change in spatial dimension. Third, if black holes were three dimensional, the Stefan-Boltzmann law would not prevent continuous inflow of CMB photons. The cosmic microwave background has been in thermal contact with the event horizon since cosmological time, and, as such, a black hole should have fed indiscriminately on such CMB photons, as well as on dark matter and dark energy. One would imagine that the same equilibrium temperature would be reached in short order, on both sides of the event horizon, due to the conventional Stefan-Boltzmann law. With a 4-D black hole, we have an internal mechanism to prevent indiscriminate expansion. We have a natural barrier, or lip, for entry. Fourth, for a 4-D black hole, radiative inflow is restricted by Equation, (2-38). This factors significantly if the outside surface temperature,

We have spent a considerable amount of time discussing the positive aspects of this model. However, much work needs to be done, as we have only scratched the surface. One task would be to catalogue black holes according to their mass and check if the relations for heat inflow, specified here, make sense, given the outside surface temperatures. Our generalized Stefan-Boltzmann equations give clear predictions for

Another line of inquiry would be to consider other forms of radiation, including fermionic components. How would their incorporation affect the results presented here? What about expansion? Equations, (3-10), and, (3-14), could be modified to include other surface tension contributions, such as dark matter, ordinary matter, and dark energy. They may increase, or potentially even decrease (in the case of dark energy), the total work done when a black hole expands in volume. We can define surface tension coefficients for such additional contributions, and evaluate the work done upon expansion in these instances.

An additional area of research would be to consider the symmetries of the metric within our 4-dimensional black hole. Which symmetries are allowed, i.e., are exhibited, if we have a 4-D spatial black hole? At first sight, we can expect that the metric associated with this higher dimensional space will be invariant under specific general coordinate transformations, but this has to be investigated, and worked out in detail. A systematic and straightforward way to find symmetries of the metric in higher curved spaces is to use the Killing equation [

Upon entering the 4-D black hole, filled with black body radiation, the energy-momentum stress tensor defined for a black body [

We can also consider the formation of black holes as a function of cosmological time. The CMB temperature has evolved as a function of time. Thus, the black hole should also evolve. Does the distribution of black holes throughout the universe match the predictions of Equation, (2-35)? According to this equation, there should be reduced heat inflow at higher CMB temperatures for a specific outside temperature and radius. Did the black hole radius increase much less in the distant past for the same external conditions? Moreover, in regards to evolution, one might think in terms of formation. How do black holes form? For a water droplet to form, it takes nucleation, such a dust, about which the water molecules can coalesce. Does something similar happen here with black hole formation? What would be the conditions under which they would form, and could that be tied into the WMAP and Planck CMB temperature data? The most massive black holes are found at the centers of most galaxies. This might not be an accident as these are natural nucleation sites about which stars and other black holes could aggregate.

An even wilder conjecture might be to consider black holes from the perspective of spatial dimension of the universe as a whole, and fragmentation. Are black holes dinosaurs, left over from a bygone era, when space itself might have been 4-D? Perhaps they are the remnants of a time when the universe was four dimensional, and upon cooling, underwent a phase transition into three dimensions. See references, [

We close by remarking that there is a follow-up paper, which builds upon the ideas presented here. In the follow-up paper, we develop a model to help explain the internal structure of a black hole. We focus on how the radiation is layered within the black hole, and calculate important quantities such as total mass, entropy, gravitational force, gravitational potential, etc. There, we will obtain some results, which are even more surprising. Without further elaboration, we encourage the reader to view this work as well.

With these two papers, the author would like to thank his children, Kira, Juergen, and Henry, for their love and support. They have been a constant source of encouragement, inspiration, joy and goodwill, throughout my entire life. They are, first and foremost, my proudest achievement, and will always remain so.

The author declares no conflicts of interest regarding the publication of this paper.

Pilot, C. (2019) Are Black Holes 4-D Spatial Balls Filled with Black Body Radiation? Generalization of the Stefan-Boltzmann Law and Young-Laplace Relation for Spatial Radiative Transfers. Journal of High Energy Physics, Gravitation and Cosmology, 5, 638-682. https://doi.org/10.4236/jhepgc.2019.53036

In this section, we consider our generalized Stefan-Boltzmann law in light of a non-spherically symmetric black hole. We show that this leads to gravitational quadrupole radiation.

Gravitational quadrupole radiation is given by the formula [A1]

where,

The dot over a quantity denotes a derivative with respect to time. For gravitational quadrupole radiation we recognize that, 1) a non-spherically symmetric distribution of mass is needed, and 2) a time dependent quadrupole moment is necessary with non-vanishing

A simple extension of spherical symmetry is to assume an oblate spheroid for the externally viewed 3-D black hole, where the equatorial axis is larger than the polar axis. A rotating body assumes this shape, and we will consider black body radiative inflow into this type of black hole. The surface is characterized by the equation,

where the polar radius, b, is less than the equatorial radius, a. The 3-D volume is,

In this equation, the eccentricity is defined as

To keep the discussion simple, we will assume a mass distribution which has constant density. In actual fact, in our model, the radiative mass density,

For a constant mass density, it can be shown that [A3],

In the last line, we notice that,

In this equation, we have used our definitions for eccentricity,

We will assume that as the black hole expands, the shape stays constant. The black hole is pulling in black body radiation uniformly from all sides, and at an equal rate, surface area wise. Therefore, when we take the time derivative of Equation, (A-6), with respect to time, we obtain,

Taking further time derivatives allows us to write, after some simplification,

We are interested in a linear approximation. Therefore, we set,

We substitute our linearized version of Equation, (A-8), into Equation, (A-4), and obtain

Our goal is to provide an estimate for this expression. This will demonstrate that black body radiative inflow can, and will, generate gravitational waves if the black hole is, a-priori, non-spherically symmetric.

We first focus on the

With this assignment, we are assuming very little deviation from a perfect spherically symmetric black hole. Second, let us assume that the equatorial radius, a, assumes the value,

This is close, but not equal to the mass of the sun,

Substituting this into Equation, (A-9), renders

The emitted gravitational radiation is measured in Watts since we are using MKS units throughout this paper for numerical evaluations.

We next focus on the

In this equation,

This is the work required per unit time to expand the black hole assuming spherical symmetry.

Our generalized Stefan-Boltzmann law reads (see section II),

where,

In this equation,

We next relax this requirement. If the surface is not spherically symmetric, as in Equation, (A-4), which holds for an oblate spheroid, then we must substitute (A-4), in place of

This is to be compared to,

We also note that with our new shape, Equation, (A-10), should be rewritten as,

where,

As mentioned,

This amounts to an increase of only 2.86 meters per year, if the outside blackbody temperature remains steady. Clearly electromagnetic radiation, on its own, will not cause dramatic increases in size.

Moreover, we can now evaluate the gravitational radiation thrown off by this slightly non-spherically symmetric black hole. Using Equation, (A-13), we find,

This is a very, very small amount of gravitational radiation, and yet, it is non-zero. We have thus shown that quadrupole gravitational radiation does exist within our model if we have a dynamic inflow situation, and the mass is not distributed in a perfectly spherically symmetric manner within the black hole.

If we carry out a similar analysis for a much more massive black hole, having mass,

This is greater than what we obtained in Equation, (A-22), but still relatively unimpressive given the size of the object under consideration, the eccentricity, and the external temperature. This gravitational luminosity is approximately equal to 10,000 times the luminosity of the sun.

In our model, instreaming black body radiation will increase the radiative mass of the black hole if the outside black body temperature exceeds the CMB temperature. Even for substantial outside temperatures, we have seen that the rate of increase is not large. Under extreme conditions, such as coalescence, black hole mergers, in-fall of a star, etc., the black hole mass will increase much more dramatically. And as a consequence, the gravitational radiative bursts will be much more impressive. These are also dynamical and non-spherically symmetric situations, but ones where black body radiation will, almost certainly, play a secondary role. It is difficult to imagine how instreaming radiative inflows can compete with these other inflows, under such circumstances.

We close by remarking that almost all of the inflowing radiative black body radiation will increase the size of the black hole. In a spherically symmetric situation, it appears that all of it will be used up to increase the binding energy of the black hole. It is just that, if inflow is asymmetric, a tiny amount will be siphoned off as gravitational radiation. This will escape to the outside world. The asymmetry does not seem to allow for a perfect 100% utilization, i.e., a perfect 100% increase in radiative mass.