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

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 blackbody 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, 2 2.725 K T = . 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 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, ( versus the 3-D surface area, ) 3 . 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, ( ) 3 0 , where, ( ) 3 R F , 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.

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, ( ) 4 2 3 versus the 3-D surface area, ( ) . 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, 3 R Q t > , 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.

C. Pilot Journal of High Energy Physics, Gravitation and Cosmology
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, 0 r → , 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, 2 2.725 K T = , 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, 2 2.725 K T > , we will have radiative heat inflow, i.e., d d 0 Q t > , the amount of which will depend on the value of 2 T . We reserve temperature, 1 T , 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 [6] [7]. Moreover, it is also known that blackbody radiation was the primordial substance in the early universe [8] [9] [10] [11]. It filled essentially all of space, and it has been conjectured that the particles in the standard model "froze-out", each at a particular temperature, as the universe cooled [12] [13] [14]. For energies above, 1 TeV, corresponding to a background temperature in excess of 10 16  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 [15] [16] [17]. A black hole is thus a ball of radiation which gives it its radiant mass. John Wheeler [16], himself, already in 1955, even explored the notion of creating elementary particles in this way. We will also allude to this as a possible mechanism for producing "elementary particles". So the basic ideas presented in this paper have been thought of before. What is new here is the hypothesis that the black hole is, in reality, a 4-D spatial object, filled with blackbody radiation, and possibly other radiations. As such, the temperature does not have to exceed the Planck temperature. Far from it, as we shall see. We will also show in our second, follow-up work how to pack that radiation. This is also novel. We will introduce a probability distribution function to pack the required radiative mass, and still keep the inside surface temperature, just inside the event horizon on the 4-D side, very low. The black hole will therefore not emit radiation, other than through mechanisms such as Hawking radiation.
There have been 2-D models proposed for black holes, so-called holographic models [18] [19] [20]. This ties in to the work done by Bekenstein [21] [22], and others relating to black hole entropy. The entropy is calculated in terms of the 3-D black hole surface area, as multiples of Planck area, so-called Plankions.
Such models predict enormous amounts of entropy associated with a black hole; in fact, using such calculations, most of the entropy in the universe is in the form of black hole entropy [23] [24] [25] [26]. Supermassive black holes contribute, by far, the most entropy. We do not believe entropy to be an intrinsic variable, dependent on surface area. Rather, we think that entropy is an extrinsic variable, dependent on volume. Moreover, it is a 4-D volume we should be considering, and integrating over. In the follow up paper, we calculate the total entropy associated with a black hole. For a black hole having the mass of the sun, the entropy in our model is calculated to be only 1.63 × 10 37 J/K. This is only about 2 orders of magnitude greater than the entropy of the sun itself, which is approximately, 10  In this paper, and the next, we will ignore/discount Hawking radiation [27] [28] [29] [30]. While it may exist, we proceed as if it does not. Other evaporative or leakage processes such as quantum mechanical tunneling [31] [32] [33] through the event horizon will also be ignored. Should they exist, they will be 2 nd C. Pilot Journal of High Energy Physics, Gravitation and Cosmology 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, 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, 0 r = , 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 threedimensional, 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, ( ) , 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.
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Generalization of the Stefan-Boltzmann Law
We start with the radiative flux emitted in N-dimensional space. As is known [34], a blackbody at temperature, T, emits a radiative flux, sometimes called ra- 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 In Equation, (2-2), R is the radius in N-space, and ( ) is the gamma function. We assume spherical symmetry for this self-contained ball of blackbody radiation. The generalized Stefan-Boltzmann constant, ( ) is determined by the following formula [34], In We will be using MKS units throughout this paper, even when not explicitly written out. The radiative flux, ( ) (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, In this expression, t, is the radiative heat power exiting the 3-D space t, on the other hand, is the radiative heat power exiting the 4-D space and entering the 3-D space. The temperature, 2 T , is the temperature just outside the black hole, in 3-D space. We define the temperature, 1 T , as the temperature just inside the event horizon, in 4-D space. The respective surface areas are found using Equation, . Equation, (2)(3)(4)(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, ( ) 3 A , and, ( ) 4 A , have differing units, Equation, (2)(3)(4)(5), is dimensionally consistent. We will assume that the event horizon is infinitesimally thin, and as such, the temperatures, 1 T , and, 2 T , are defined at effectively the same radius, just on different sides of radius, R. It is obvious from relation, (2)(3)(4)(5) , the two quantities are proportional. We will often make use of the Schwarzschild relation throughout this paper without explaining it. In practice, 2 T , 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 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 2 T by means of Equation, (2)(3)(4)(5), In the third line of this equation, we made use of, and a corresponding equation in 4-D, The infinitesimal volume element, In N-dimensional space, a hyper-volume can be defined for a N-dimensional ball. The expression [35] [36] is The superscript "N" in parenthesis on a physical quantity will always refer to the spatial dimension over which the quantity is defined. ( ) is, again, the gamma function, and N equals the number of spatial dimensions. From Equation, (2)(3)(4)(5)(6)(7)(8), it follows that The expression, Equation, (2)(3), and the relations, Equations, (2)(3)(4), through to, (2-7), are not quite correct. The ( ) N σ coefficients are very close to being perfect, but have to be adjusted slightly. This is due to the fact that when blackbody photons transfer between spatial dimensions, it is not just internal energy density or radiative heat density, separately, which transfer. When blackbody photons transfer, the associated internal energy density, plus the radiative pressure, plus the heat density all transfer as one unit. It is an all, or nothing, proposition such that the black body identity of the photons can be maintained in both spaces. All of these quantities depend on temperature, and if temperature changes, which it does, so do all of the above at the same time. This was shown in a previous work [37], where we considered a 1 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.
Technically, these equations are only correct if we form a product with a volume, which is how we use these equations.
We substitute Equations, (2-28a), and, (2-28b), into the second line of Equation, . We thereby obtain, after multiplying through by the velocity of light, c, The coefficients, ( ) Within the black hole, just inside the event horizon, in 4-D space, we have tem- A , are those appropriate for three and four spatial dimensions.
We take half of that for heat transfer to obtain the factor, (15π/8). We call Equation, , the second generalized Stefan-Boltzmann equation for radiative transfers between adjoining spatial dimensions. In our view, it is the correct expression.
With this in mind, let us rewrite Equation, , as

C. Pilot Journal of High Energy Physics, Gravitation and Cosmology
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 , which we call the 2 nd generalized version of the Stefan-Boltzmann law. It is our contention that this is correct. This Equation , is equivalent to Equations, .
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, d d 0 Solving for 1 T gives 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)(3)(4)(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, . We thereby obtain, As stated, these values for 1 T are very close to the values obtained previously.
We also note that these temperatures, just inside the event horizon in 4-D space, are very much higher in value when compared to the Hawking temperature.
There are many ways of writing our 2 nd generalized version of the Ste- It is clear that, in general, . We further note that,   This shows that the amount of radiative heat inflow is restricted to be less than, F , is the 3-D radiative force acting from the outside in. This places limits on the intake of radiative heat into a black hole. This is a specific, and interesting, prediction of our model.
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, , follow from Equations , and . These, in turn, depend on the last line of Equation . The factors of 4 and 5 in Equation , are due to the dimensionality of space itself. See, for example, Equations (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) and (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(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 ( ) would essentially be the same. In other words, we would not be able to distinguish between, 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, , 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, 1 T , defined over 4-D space, to that on the outside of the event horizon, 2 T , defined over 3-D space. We can evaluate the quantity, ( ) 3 4 R F , using the first term on the right hand side of Equation . We obtain For outside surface temperatures in excess of, 2.725 K, the value for ( ) 3 4 R F can be much higher as seen by Equation . For example, hard photonic X-ray emissions just outside a black hole would indicate temperatures in excess of, 10 9 K. Because of the 4 2 T factor in Equation (2-39), we can expect much higher limits placed for radiative heat inflow under these conditions, 10 36 times higher than what is indicated by Equation . The inequality , may also help explain the circulating nature of radiative heat inflow before entering the event horizon of a black hole.

Generalization of the Young-Laplace Equations for Surface Tension at the Event Horizon
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 C. Pilot Journal of High Energy Physics, Gravitation and Cosmology here, ST F is the surface tension, and dR is the increase in radius. The surface tension acts on the 4-D/3-D membrane, identified as the event horizon, and ST F will pull the surface in.
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, ( ) , upon entering the black hole. But we first consider the conventional 3-D case where surface area does not change upon crossing the membrane.
For conventional 3-D objects such as liquid droplets or bubble films, it is gives the outward force, where i p is the inner pressure, o p is the outside pressure, and A is the surface area. For hydrostatic equilibrium, the two forces, the surface tension, and the outward force due to pressure difference, must balance. Thus, For a 3-D sphere, the Young-Laplace equation reads The coefficient of surface tension, ( ) The 4-D coefficient of surface tension, ( ) 4 γ , is measured in Newtons/meter 2 , and , , x y z R R R are the curvatures in the x, y, z sense. Again, we set , because we want spherical symmetry. The cause of surface tension in the case of liquid droplets, gas bubbles, soap bubbles, etc. are intermolecular forces. For the case of a black hole, the surface tension is caused by the difference in radiative force at the interface, and, more importantly, gravity, as we shall see shortly.
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, ( ) 4 p has units of Newtons/meters 3 and dV is measured in (meters) 4 . We This holds true because, Equations, (3)(4)(5)(6)(7)(8), and, (3)(4)(5)(6)(7)(8)(9), give the work done in terms of, ( ) In this equation, ( ) 3 R F , is the 3-D radiative pressure force pushing the event horizon in, and ( ) is the 4-D radiative pressure force pushing the event horizon out. We also have the absolute value of the gravitational force, ( ) which acts on the event horizon, and which wants to pull it in. We only count this once, and because it is derived within the black hole, which is a 4-D construct, the gravitational force, itself, is 4-D. The total contribution to actual work done is given by Equation, (3)(4)(5)(6)(7)(8)(9)(10), which will be positive for an increase in R, from R to R + dR. As mentioned, R, characterizes the event horizon. From Equation, (3)(4)(5)(6)(7)(8)(9)(10), we see that the total work will consist of three separate components, and each goes into defining the surface tension, on the left hand side of this equation.
As we have seen in the last section the radiative force, ( ) 3 R F , is always larger than its 4-D counterpart, ( ) 4 R F . Refer to Equations, , and, . In fact, using these two conditions, and Equation, , we see that Therefore, by Equations, (3)(4)(5)(6)(7)(8)(9)(10), and, (3)(4)(5)(6)(7)(8)(9)(10)(11), another way to write the 4-D/3-D radiative surface tension is  (3)(4)(5)(6)(7)(8)(9)(10)(11)(12). In fact, knowing the outer surface temperature allows us to calculate the amount of heat inflow as shown in the last section. We also have a larger value for ( ) because the outside temperature is now larger. It is interesting to remark that, even without gravity, a ball of radiation is positively curved, if that ball of blackbody radiation has 4 spatial dimensions. This, we believe, is a very significant result, as it may relate to elementary particles, and the formation of black holes. The full surface tension has to include the gravitational force acting on the event horizon. By Equations, (3)(4)(5)(6)(7)(8)(9)(10), and, (3-11), we can write, We can thus calculate the associated surface tension using Equation, (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13), if we know the value of the 4-D gravitational force at the event horizon, and In the next paper, we will calculate an expression for, F . It will turn out to be immense. In fact, the radiative forces are insignificant in comparison, except in the most extreme of circumstances. In expanding the black hole, we will see that the work done is essentially against gravity. Moreover, with, or without, gravity, the surface tension is positive definite, making the black hole a positively curved object in space.
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 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)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(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.

Radiative Waterfall Model for the Event Horizon of a Black Hole
We now have the required tools to discuss what happens at the event horizon, C. Pilot Journal of High Energy Physics, Gravitation and Cosmology 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 here, ( ) 4 r F , is the radiative force pushing the layer out, at radius, r, which acts on a segment of thickness, dr, between radii, r, and, r + dr. The quantity, ( ) 4 d r r F + , is the radiative force pushing the layer in, at radius, r + dr. On the right hand side we have, ( ) 4 , d G r F , which is the gravitational force pulling the layer in. The layer is at radius, r and of thickness, dr. In a follow-up paper, we will give a specific temperature gradient within the black hole. Because it will depend on radius, r, we will write, ( ) r T T r = . It will turn out that,  T T + > , the internal energy density, the radiative pressure, and the radiative heat density, will all decrease as one increases the radius, starting from the center of the black hole. As was seen in section II, the quantities depend on temperature and temperature only. See Equations, (2-28a), and, (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20). One can also claim that these quantities are radius dependent because the temperature inside the black hole is radius dependent. Outside of the black hole we do not have any such layering because the temperature is, by and large, uniform. Also, the blackbody photons there on the outside are "unbounded", i.e., not trapped within a confining volume except that of the entire universe, itself.
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, r R = . In other words, the event horizon has no thickness. Moreover the radiative force, which pushes the event horizon out, is, ( ) On the other hand, the radiative force, which pushes the event horizon in, from the 3-D side, is, p , is defined in terms of 2 T , as can be seen by Equation, (2-28b).
From the second equality in Equations, (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19), we recognize that ( ) ( ) ( ) p , acting from within the black hole and pushing the event horizon out, is at a different temperature, 1 T . The event horizon is assumed to be infinitely thin, i.e., it has no thickness. Gravity will make an abrupt jump in value when entering the black hole, as will be shown in the 2 nd paper. The gravitational C. Pilot Journal of High Energy Physics, Gravitation and Cosmology 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, r R = , in both three and four dimensional space, by choosing the constant of integration appropriately. We will show The difference in radiative force is, of course, the radiative surface tension. The actual expression for surface tension is given by the first line in Equations, (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13), and this net force pushes the event horizon in.
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)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19), (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 See Equation, . What is important to note in Equations, (4)(5), through to, (4)(5)(6)(7)(8)(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)(5)(6)(7)(8)(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)(3)(4)(5)(6)(7)(8)(9). Another way of expressing it is to make use of the second line in relation, . If d d 0 Q t = , then this allows us to write Equation, (4)(5)(6)(7)(8)(9)(10)(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 substi- (4)(5)(6)(7)(8)(9)(10)(11), and solve for 1 T , then we would obtain . There has to be a break in temperature at the event horizon. As far as we know, this abrupt change in temperature upon entering the black hole proper has never been theoretically modeled before. It has been stated more as an accepted observational fact, given that black holes appear black. Here we provide an unequivocal theoretical reason for why this is so. A change in spatial dimension gives a natural explanation for not only this decrease in temperature, but also for the other quantities decreasing abruptly and discontinuously, upon entering a black hole.
Equations, (4)(5), through to, (4)(5)(6)(7)(8)(9), are also intriguing from another perspec- is the radiative force just inside the 4-D space. See Equations, , and, . We have a step-function decrease due to the discontinuity of space, and the different surface areas involved. This is literally what causes the radiative surface tension, defined as, to be greater than zero. The inequality holds for net inflow of radiative heat (and energy), whereas the equality is valid for an isolated, static black hole with no net inflow. In the second paper which follows, we will also see that the gravitational force changes abruptly at the event horizon. Upon crossing the 3-D/4-D threshold, it will suddenly increase in value over its 3-D counterpart. All these facts will support our waterfall hypothesis. We believe that the 3-D photons literally drop out of view upon reaching the event horizon due to the abrupt change in spatial dimension.
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 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.
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, where we have used the Schwarzschild relation to find the radius, given a specific mass. Finally let us assume an outside surface temperature of, 9 2 10 K T = , which is held constant during the expansion process. The 3-D volume increase is, correspondingly, We use the expression, (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19), to find the work done against radiative forces.
At a temperature, 9 2 10 K T = , the second term within the square brackets in Equation, (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19), is truly insignificant, compared to the value calculated in Equation, . Thus, we can ignore it, and the total work done against radiative forces is simply, .014 10 Joules 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, , 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, 2 c , the corresponding increase in energy is,  , is insignificant when compared to this. Larger surrounding temperatures will be considered in our follow-up paper.

C. Pilot Journal of High Energy Physics, Gravitation and Cosmology
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.

Summary and Conclusions
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 sur- T , just outside the event horizon, which is on the 3-D side. Because the event horizon is assumed infinitely thin, both temperatures are effectively at the same radius, R. The event horizon is a membrane separating the 4-D space from our 3-D world, and we argue that there is a sharp discontinuity at the event horizon.
As such we established, in this paper, conditions and equations, which must apply if such a scenario is realized in nature. We focused exclusively on blackbody photonic radiation, although this scheme can be generalized later to include other types of radiation including fermionic components.
In Section 2, we first generalized the Stefan-Boltzmann law for radiative transfers between three dimensional and four dimensional spaces. The result is Equation ( no net inflow/outflow due to the different dimensionality of space. It will turn out that the temperature within the event horizon is always less than that on the outside, even if there is no net radiative flow. We also find that for there to be net outflow from the black hole, the temperature on the outside would have to be less than the CMB temperature, which is not possible. See the last line of Equation , where this is clearly stated. This equation enforces the condition that, d d 0 Q t ≥ . The right hand side of Equation , is either positive for a dynamic black hole with inflow, or zero for an isolated, static black hole with no net heat inflow. We have discounted/ignored Hawking radiation and other forms of evaporative leakages emanating from the black hole, as these will turn out to be second order corrections at best. In Equations  and , , and ( ) We argued that the first version of the Stefan-Boltzmann generalization for radiative transfers between spatial dimensions, given by Equations (2)(3)(4)(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)(3)(4)(5) and ( In this situation, the temperature just inside the event horizon can be determined from Equation . We find that  . We also have an interesting restriction on radiative heat inflow, specified C. Pilot Journal of High Energy Physics, Gravitation and Cosmology by relation, . This restriction is important if outside surface temperatures are close to, 2.725 K; at higher outside temperatures, this lip, or barrier, is inconsequential. This restriction is a by-product of the 4-D/3-D interface. 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)(4)(5)(6)(7)(8)(9)(10), or, (3)(4)(5)(6)(7)(8)(9)(10)(11)(12), where it is seen that as the radiative force acting from the outside in, and is the radiative force acting from the inside out, at radius, R. The radiative pressure, ( ) 3 2 p , is defined exclusively in terms of temperature, 2 T , just outside the event horizon, whereas the radiative pressure, ( ) 4 1 p , is defined exclusively in terms of temperature, 1 T , which is the equilibrium temperature just inside the event horizon. The total expression for surface tension was derived by considering the amount of work done in expanding the black hole. It also includes gravity, which in Equation, (3)(4)(5)(6)(7)(8)(9)(10), is given by the last term on the right hand side. In Equation, (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14), we have, correspondingly, the last term on the right hand side. We will derive an expression for the 4-D gravitational force in our follow up paper. For now, we will just indicate that this is, by far, the major component which will contribute to the total work done upon expanding the black hole, except in the most extreme of situations. It is also to be noticed that radiative heat inflow can be expressed in terms of, ( ) 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)(4)(5)(6)(7)(8)(9)(10)(11)(12), shows that at a very minimum, for d d 0 Q t = , the radiative surface tension equals ( ) 3 0.2 R F . In this situation, Equation, (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18), applies, and we notice that the radiative surface tension is strictly proportional to 2 R .
In Section 4, we brought the ideas developed in sections II and III together.
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 γ , is evaluated using the temperature just inside the event horizon.
We have emphasized throughout this paper why a black hole cannot be iden-  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, [37], and, [45], in this regard. The black holes, representing sufficiently dense pockets of trapped radiation, may have resisted conversion into three-dimensional space. These are all intriguing aspects, which could be studied.
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.
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 . We will also make use of the oblateness, or flattening parameter, f, defined as, For this oblate spheroid, we focus on the 33 I moment, which is the relevant moment for this kind of geometry.
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, Either of these parameters determines the shape of the black hole.
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, 0 a a = = on the right hand side of Equation, (A-8). We consider here a rate of expansion which is assumed a-priori small, and the accelerating components will be ignored. For a burst of quadrupole gravitational radiation, the second and third terms on the right hand side will, undoubtedly, be important, and, in this situation, cannot be ignored.
We 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 2 33 I term. To be specific, let us assume that, 2 0.01 e = .
This implies that With this assignment, we are assuming very little deviation from a perfect spherically symmetric black hole. Second, let us assume that the equatorial ra-C. Pilot Journal of High Energy Physics, Gravitation and Cosmology dius, a, assumes the value, 2954 a = meters. This would correspond to a Schwarzschild radius where the black hole has a mass equal to that of the sun.
Our black hole will have a mass slightly less, due to our slightly reduced volume.
For the eccentricity chosen, 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 ( )   1.097 10 m A R = = × , which holds for a perfect sphere. Our oblate spheroid has a lessor surface area due to its slightly deformed shape.
We also note that with our new shape, Equation, (A-10), should be rewritten T , let us assume a specific value. We will take this temperature to equal, 9 2 10 K T = , a relatively large blackbody temperature, just outside the event horizon. We want substantial radiative heat inflow into our non-spherically symmetric black hole. Evaluating the expansion rate in the equatorial plane us- 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, , with eccentricity, 2 0.75 e = , we obtain a much larger value for the gravitational quadrupole radiation. Here, 0.5 b a = , and we still take the outside black body temperature to equal, 9 2 10 K T = , for the sake of argument. The relevant surface gravity at the equator has also been worked out, and equals, 43 , 6.36 10 G R F = × Newtons. Using these inputs, and following the same steps as before, we find ( ) 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 C. Pilot