Modeling the Interior of Black Holes Utilizing a 4-D Spatial Blackbody Radiation Model with an Exponential Distribution

This is a second follow up paper on a model, which treats the black hole as a 4-D spatial ball filled with blackbody radiation. For the interior radiative mass distribution, we employ a new type of truncated probability distribution function, the exponential distribution. We find that this distribution comes closest to reproducing a singularity at the center, and yet it is finite at 4-D radius, 0 r = . This distribution will give a constant gravitational acceleration for a test particle throughout the black hole, irrespective of radius. The 4-D gravitational acceleration is given by the expression, R is the radius of the black hole, is its mass, and λ is the exponential shape parameter, which depends only on the mass, or radius, of the black hole. We calculate the gravitational force, and the entropy within the black hole interior, as well as on its surface, identified as the event horizon, which separates 3-D from 4-D space. Similar to a truncated Gaussian distribution, the gravitational force increases discontinuously, and dramatically, upon entry into the 4-D black hole from the 3-D side. It is also radius dependent within the 4-D black hole. Moreover, the total entropy is shown to be much less than the Bekenstein result, similar to the truncated Gaussian. For the gravitational force, we obtain, sun, and 10 6 solar masses. The corresponding λ parameters are found to equal, ( ) ( ) 8 , , 0.039569,0.0037996,3.01229 10 A B C λ λ λ − = × , respectively. We compare these results to the truncated Gaussian distribution, which were worked out in another paper.

r = . This distribution will give a constant gravitational acceleration for a test particle throughout the black hole, irrespective of radius. The 4-D gravitational acceleration is given by the expression, ( ) R is the radius of the black hole, R M is its mass, and λ is the exponential shape parameter, which depends only on the mass, or radius, of the black hole. We calculate the gravitational force, and the entropy within the black hole interior, as well as on its surface, identified as the event horizon, which separates 3-D from 4-D space. Similar to a truncated Gaussian distribution, the gravitational force increases discontinuously, and dramatically, upon entry into the 4-D black hole from the 3-D side. It is also radius dependent within the 4-D black hole. Moreover, the total entropy is shown to be much less than the Bekenstein result, similar to the truncated Gaussian. For the gravitational force, we obtain, ( ) contained within that radius, with 0.1λ being the constant of proportionality. For the entropy at radius, r, and on the surface, we obtain an expression which is order of magnitude comparable to the truncated Normal distribution. Tables are presented for three black holes, one having a mass equal to that of the sun. The other two have masses, which are ten times that of the

Introduction
This is a follow up paper, where we consider another probability distribution function (pdf), the exponential distribution, and apply it to modeling the radiative mass concentration within the interior of a black hole. We assume that the black hole is a 4-D spatial ball embedded in 3-D space, and that it is filled (packed) with blackbody photons. Moreover, the radiative energy, in all its forms, internal, pressure, and heat, makes up the radiative mass within the black hole. See references [1] [2] for a detailed description of the model, which we will not reproduce here. We will only give the briefest of sketches, and then proceed to model the interior distribution of radiative mass, employing the truncated exponential function.
In paper [1], we considered the event horizon. We conjectured/hypothesized that a rip or tear in the spatial continuum occurs there, and not at the black hole center. For it is there that we transition from 3-D space to 4-D space. A four dimensional black hole filled with blackbody radiation has many advantages over a 3-D counterpart. First, we can pack an enormous amount of mass, i.e., radiative energy within a, from our perspective, relatively small 3-D volume. Second, employing such a model, the temperature decreases precipitously when we enter the black hole. This follows from a generalized version of the Stefan-Boltzmann law, which was derived for radiative transfers between different adjoining spatial dimensions. Third, the energy density in all its forms, internal, heat, and pressure reduce significantly upon entering the 4-D space. Fourth, we have a natural barrier which prevents wholesale entry of CMB photons, and potentially other forms of radiation, such as dark energy. Fifth, there can only be zero, or net positive radiative inflow within this model, as outside net outflow is not allowed, except through evaporative processes such as Hawking radiation. Sixth, we have an inherent positive radiative surface tension, which allows for inherent black hole formation, and keeps the event horizon positively curved and stable. The Young-Laplace relations were also considered, and generalized, within a 3-D/4-D context. All these factors/advantages were shown and derived in the first work.
For a detailed description we refer the reader to reference [1].
In the 2 nd paper, reference [2], we focused on what the interior of a black hole C. Pilot Journal of High Energy Physics, Gravitation and Cosmology might consist of. We modeled the radiative mass (energy) distribution as a probability distribution function. The specific choice was a truncated Gaussian, i.e., truncated Normal distribution. It was shown that a uniform truncated distribution, in contrast, within this space cannot accommodate gravitational forces within its interior. A temperature gradient is needed for that, which a Normal distribution can satisfy. Moreover, this gradient increases as the 4-D radius decreases. Thus the internal energy density, the heat density, the radiative pressure, the entropy density, etc. all increased, as the radius decreased. For a 4-D spatial black body, the internal energy density is proportional to the fifth power of temperature, 5 r T , where r T is defined as the temperature in an infinitesimal layer at radius, r. The radiative pressure and radiative heat are likewise proportional to, 5 r T , whereas the entropy density is proportional to 4 r T .
The entropy density is treated as a state variable, and it has a well-defined value within a specific layer. Utilizing our truncated Gaussian distribution, we were able to derive localized, as well as other non-localized attributes/characteristics, within the black hole. The localized quantities held within a particular layer, whereas the other properties were the cumulative effects up to, and enclosing, a specific radius. Examples of localized attributes are the mass density, the radiative pressure, the radiative force, the internal energy density, the heat density, the entropy density, etc. These hold within a specific layer at radius, r.
Examples of non-localized quantities are enclosed mass, gravitational force, enclosed entropy, etc., all defined within an enclosing volume of 4-D radius, r.
Our results were listed in table form. We considered three black holes, which we labeled A, B and, C. Black hole A has a mass equal to that of the sun, whereas black holes, B and C had masses, 10 times, and 10 6 times, the mass of the sun, respectively. In this way comparisons between various massive black holes could be made. The localized quantities were given in Tables 1-3, re-spectively, as a function of radius. The non-localized characteristics were tabulated under Tables 4-6, respectively for the three black holes under consideration.
Of particular interest in the second paper was the probability distribution function [3]- [9]. All results relating to the interior of the black hole depended on this choice of function. Quite simply, the pdf told us how the black hole is packed, radiative energy (mass) wise. This determined the internal forces, both radiative and gravitational, the pressures and the densities within the black hole.
We chose a truncated Gaussian distribution. In this paper we will select another pdf, the truncated exponential, [ ] exp ; 0, R λ . The question is how will the results change qualitatively and quantitatively with this choice? Are there specific peculiarities associated with this new choice? To this we now turn. We keep in mind that the truncated exponential distribution is particularly simple in that it is determined by one, and only one, parameter, λ , called the shape parameter. This parameter is defined over the range and has support,     Before we start, we point out some notable differences between our model and the conventional picture. A good survey article on the thermodynamics of black holes is given in reference [10]. The deviations between our model and the standard picture will be given in bullet form, and the list is not all-inclusive. However, these are among the most important differences, which will be seen as we move through the text. It is best to highlight them now such that misconceptions will be avoided. We note that in our model: 2) There is no information loss (only information gain) upon transitioning from 3-D space (the surroundings) to 4-D space, the black hole, if only blackbody radiation is considered. Our black hole can be thought of as a special, exotic type of 4-D spatial capacitor, where pure states, in the form of blackbody radiation obeying Bose-Einstein statistics, are possible within specific layers. Not only is radiative mass stored within the 4-D black hole, but in addition, entropy and many other attributes. Entropy will increase upon massive and massless inflow.
Of all isothermal radiating bodies, a black body is that unique body which produces the largest amount of entropy for a given energy. See, for example, reference [11].
3) Reduced temperatures just inside the event horizon of a black hole is a direct consequence of a spatial transition from 3-D space to 4-D space, as was shown explicitly in reference [1]. Upon transitioning from 3-D to 4-D space, surface area increases dramatically, and as a consequence, temperature decreases discontinuously, and dramatically, according to a generalized version of the Stefan-Boltzmann law. The temperature within the black hole in our model is not due to evaporative processes, such as in Hawking radiation.

4)
Because the temperature changes abruptly, all blackbody attributes which depend exclusively on temperature, such as internal energy density, radiative pressures, entropy density, etc. also change abruptly at the 3-D/4-D interface, identified as the event horizon. The temperature, however, will increase as one penetrates to smaller radii within the 4-D black hole. Thus, all these quantities which can be defined within specific layers will also increase. Ultimately, this will make up the mass of the black hole within this very simplified model.

5)
Layer by layer, within the black hole, outward expansion due to radiative pressures and forces are held in check by gravitational forces pulling in. Hydrostatic equations apply as was shown explicitly in reference [2]. At the surface, we also have gravitational forces, and radiative pressure. Black holes are in thermal Journal of High Energy Physics, Gravitation and Cosmology equilibrium with the surrounding CMB temperature when there is no net inflow.
Radiative forces are such that there will always be a net positive curvature, both within, and on the surface of the black hole. This is not guaranteed on the surface for a 3-D/3-D spatial transition.
6) The gravitational force increases dramatically and abruptly as one enters the spatial 4-D black hole. This is a consequence of the spatial dimension changing. The probability distribution function chosen will determine how the radiative mass is packed or stored within the 4-D space. Typically, in 4-D space, the gravitational constant is a function of radius.
8) By and large, we ignore Hawking radiation and other evaporative processes.
This has to be worked out in another series of papers. Our temperatures just inside the event horizon depend on the size, or mass, of the black hole. They typically are much higher than the corresponding 3-D Hawking temperature, although much lower than the surrounding CMB temperature. The temperature is not constant throughout the black hole in our model, nor is it due to evaporative processes.
Clearly, these assumptions go contrary to much of what is believed to hold true for black holes. Our premise is an unconventional one, as we are dealing with a 4-D spatial object.
The general outline for this paper follows that of reference [2], although in less detail. In Section 2, we introduce the exponential pdf, and determine the shape parameters needed for each of the three black holes under consideration.
We analyze the same three black holes as was done in reference [2], black holes A, B, and, C, such that direct comparisons can be made. In this section, we use the temperature just inside the event horizon, within the 4-D black hole to find the associated shape parameters, ( ) , , With these values, we will be in a position to map out the interior. Since we can find the various densities, the radiative pressures, and the radiative forces at the surface just inside the event horizon, we can determine these values within the interior for any arbitrary radius knowing the various shape parameters. In Section 3, we consider specifically the localized variables which hold within a particular layer, within the interior of the 4-D black hole. We are considering the internal energy density, the radiative pressure, the entropy density, the radiative force, the temperature, and the heat capacity, among other variables. The localized properties are tabulated under Tables 1-3.

C. Pilot Journal of High Energy Physics, Gravitation and Cosmology
In Section 4, we derive and calculate certain cumulative properties. These are cumulative quantities, which hold within an enclosing radius. In this section, we focus on radiative mass, the gravitational force, the gravitational acceleration, and the entropy, all defined within an enclosing radius. The cumulative characteristics are listed under Tables 4-6. The key formulae are given in the text. Finally in Section 5, we discuss our results, and remark upon the similarities, as well as deviations, from the corresponding Normal distribution counterparts.
We shall see that particular pdf's lead to particular ramifications.

The Exponential Distribution
We first derive a general relation which all pdf's have to obey within our model [1]. On the inside surface of a black hole, just inside the event horizon, the radiative force is given by We have assumed that the outside temperature is the CMB temperature and If it is obvious that we are in 4 spatial dimensions, we will dispense with the superscript, such as in Sections 3, 4, and 5.
We next recognize that, according to our model, the radiative pressure just inside the 4-D black hole surface, is related to the radiative mass density, ( ) 4 1 ρ , at that radius, by means of In this relation, the radiative energy density, ( ) 4 1 e , is defined at temperature, 1 T , which is the temperature just inside the event horizon. The total 4-D energy density, which consists of internal energy density, radiative pressure, and heat density, is assumed to contribute to the radiative mass. When multiplied by a 4-D volume and/or 4-D surface area, ( ) The heat density at radius, r, is designated by, ( ) 4 r q . As shown in reference [1], this can be rewritten a variety of ways, This is due to the particular coefficients, which hold only in 4-D space, In another spatial dimension, these factors would assume different values.
Equation, (2)(3)(4)(5), gives us the relative contributions towards total energy density within any layer at radius r, or at a particular radius r. Those contributions are, According to our model, this is precisely equal to, where, R f , is the value of the probability distribution function at r R = . The R is the full radius of the black hole, related to its mass via the Schwarzschild relation, A quick proof follows.
We introduce, quite generally, a pdf, r f , which satisfies, i.e., 100% probable. The cumulative distribution function (cdf), CDF r F , is defined as, The cdf is the sum of all probabilities starting from 0 r = up to, and including, radius, r. The radiative mass within the black hole is assumed to obey such a function. In other words, , within a layer between r and d r r + , we see that, Comparing the last two equations, we see that  -7), is a special case of relation, (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13).

Radiative Pressure, Internal Energy Density, Heat Density, and Other Quantities within Specific Layers
Next we construct tables for black holes A,B and C, where we calculate important attributes as a function of radii. This will give us a snapshot of how the black hole is structured internally. According to our exponential distribution, we will have a very specific profile for all the key attributes within the black hole. We consider in this section, the radiative mass density, ( )  Tables 1-3 will hold for black holes A, B and C, respectively. We go through the same steps and follow the same format, as was done in reference [2], for a truncated Gaussian distribution. We extend the analysis to the exponential distribution.

C. Pilot Journal of High Energy Physics, Gravitation and Cosmology
As was shown in reference [2], the temperature just inside the black hole is determined by the black hole mass, or equivalently, by its radius, because of the Schwarzschild proportionality. The result was, where, 1 T , is the temperature just inside the black hole event horizon, at radius, R. Using this expression, the temperatures for the three black holes under consideration are, These are the radiative mass densities just inside the event horizon, defined in 4-D space. We note that the units in all the above quantities conform to 4-D space. The subscript "1" in all these quantities refers to the temperature, 1 T , which is the temperature just inside the event horizon, within the black hole.
This subscript could just as well be replaced by "R", as this temperature holds at radius, R. In other words, 1, , etc.. The subscripts, A, B, C, refer to black holes, A, B, and, C, respectively.
Once r ρ is specified, we can easily evaluate the internal energy density, r u , the radiative pressure, r p , the heat density, r q , and the temperature, r T .
When comparing these values with those in reference [2], which holds for a truncated Gaussian, we notice a difference. The black holes are the same, and we had the same temperatures, densities, and radiative pressures at the surface.
However the distribution is now different as we are using a truncated exponential, and therefore, the radiative mass density, the energy densities, and the pressures are "packed" differently within the black hole. Hence, we have the differences in the tabulated entries. We will come back to this point when we make formal comparisons between the two distributions (pdf's) in Section 5.
Equation, (3)(4)(5)(6)(7)(8)(9)(10)(11), makes explicit use of the shape parameter, full radius, and considered radius, r. The final expression is quite different than the one obtained for a truncated Gaussian (TG) distribution, obtained in reference [2]. There we had, Again, a different pdf leads to a different distribution of radiative mass, and other quantities within the black hole for the same size, or same total mass, black hole.
The next column entry in the tables is column 9, which relates to radiative force. By definition the radiative force is the radiative pressure at a particular value, multiplied by the surface area at the same radius, r. When looked at from the interior, this is a force acting inwards. When looked at from the outside, it is a force directed outwards. Per definition, For a particular value of r, we can evaluate the right hand side recognizing that both λ and R are known for the three black holes being analyzed. We can also easily evaluate R F on the left hand side, as this is just, . The radii to be employed are as listed in Equations, (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16). The radiative pressure at the surface, just inside the black hole, is specified by Equations, (3)(4). Thus it is possible for us to determine, R F , and using the right hand side of Equation, (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16), the values for r F , as well. These are the values worked out in column 9.
We give three further columns in Tables We This was worked out in reference [2]. Upon comparing the two results, we notice that the second terms on the right hand side are equal. The first term differs due to the differing shape parameters.
For the 11 th column, we want to evaluate the quantity, d d

Radiative Mass, Gravitational Forces, and Entropy as Functions of Radius
We have focused on radiative mass density, internal energy density, radiative pressure, heat density, entropy density, temperature, and radiative forces within the black hole. These held within specific layers inside the black hole. It is now time to consider cumulative effects. What is the total mass contained within an enclosing radius, r? What is the total internal energy, the total heat, the total gravitational force, the total entropy, etc. enclosed within radius, ? And what are the values at the surface, where r R = ? It is now time to consider these questions. We gave an answer to these questions in a previous paper but these held for a truncated Gaussian distribution. We wish to formulate answers but ones, which now hold for a truncated exponential. The goal is to compare results, and see what the similarities and differences are.
We will follow the same format as in reference [2]. We investigate the same variables, and give similar tables, which will summarize our results. In the next section we will compare what we discover here with what was determined in paper [2], where we had a different pdf. Just like in the previous paper, we construct three Tables 4-6, at the end of the references. Table 4 holds for black hole A, which has one solar mass. Table 5 and Table 6 refer to black holes Table 2 and Table 3, respectively. They have masses 10 times, and 10 6 times, the mass of the sun.
The radiative energy at radius, r, is also quite simple. We know that the radiative energy at radius r, between r and d r r + , must equal The total radiative energy is not entered under any specific column. Rather, we focus on the constituent parts, which are the internal energy, r U , the heat energy, r Q , and the work done against radiative pressure, r W , which has to be contained by gravity. The sum of these three contributions will give us, The factors of 4 10 , 5 10 , and, 1 10 , in all these relations remain the same, whether we consider, a layer, an enclosing volume, or the total volume. The factors are imposed upon us by the dimensionality of space and the fact that we are dealing with blackbody radiation. What is obvious with these relations is the fact that the internal energy makes up 40% of the total energy, the trapped heat represents 50% of the total energy, and the work done against pressure makes up 10% of the total. In 3-D space, the proportions would be different.
The net radiative pressure pushing out is balanced by the gravitational force We next focus on the gravitational force. Equation, (4-3), will be our starting point. We know from Equations, (3)(4)(5)(6)(7)(8)(9)(10), and, (3)(4)(5)(6)(7)(8), that We take the derivative of the first and last part of this equation with respect to radius, and find We next divide the left hand side of Equation, (4)(5), by the left hand side of Equation, . We do the same for the right hand side. This allows us to write, ( ) There will be a sharp discontinuity in the gravitational constant from its 3-D value, G, which is Newton's constant, to a new value, ( ) 4 R G , upon entering 4-D space. It will turn out that, 2 0.2 1 R λ ≠ . From the tables, it is seen that, under column 8. This drastic and abrupt increase in gravitational coupling "constant" when entering 4-D space has to do with the discontinuity of space itself. Even though the radius does not change, the surface area, and volume do. Therefore many quantities upon entry into the 4-D black hole experience a sharp discontinuity. The temperature decreases abruptly. So do the internal energy density, the radiative pressure, the heat density, etc. We referred to this as our waterfall model, and the details are explained in reference [1].
We next derive a surprising result. As shown in reference [2], utilizing a 4-D version of Gauss' law, we saw, quite generally, that the gravitational acceleration within the black hole, reduced to, A test particle within the 4-D black hole will thus experience the same gravitational force, irrespective of radius! Moreover, the exponential shape parameter, which depends only on the mass, or size, of the black hole, will determine its numerical strength. This holds true only for the truncated exponential distribution. Even though the acceleration due to gravity is a constant for the truncated exponential distribution (TE), we have listed them under column 9 for each black hole being analyzed in Tables 4-6. This is to emphasize a point, but also to show the differences between the various masses.
For the truncated Gaussian distribution, we had quite another result. There, we obtained, which is proportional to, r, for a given size (mass) black hole. The gravitational acceleration increased linearly with radius in 4-D space, even though the radiative mass density was not a constant. What we see with these two examples is that the pdf chosen will determine the gravitational "packing" within the black hole, i.e., how radiative mass is stored, and contained within the black hole.
We have already mentioned that at the 3-D/4-D interface, which is at the surface, or event horizon, of the black hole in our model, we can expect discontinuities. This holds true for the gravitational acceleration as well. We consider the gravitational acceleration at the event horizon, where r R = . Using Equations, (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16), and, (4-17), we notice that These values can be compared to the acceleration due to gravity on the 3-D side. There we would obtain, ( ) We next turn to the gravitational potential, r ϕ . This is a four dimensional quantity which depends on radius, r. The defining equation is The constant of integration will be fixed by setting the gravitational potential in 4-D space equal to that in 3-D space at the event horizon. In this way we can guarantee for the same equipotential surface in both spaces.
Equation, , can be compared to the corresponding truncated Gaussian Journal of High Energy Physics, Gravitation and Cosmology result, worked out in reference [2]. There we found that ( ) ( ) Upon comparing results, there is a certain similarity in that solutions match at the 3-D/4-D interface, and the gravitational potential becomes more and more negative as the radius decreases. But there the similarity ends as we have, in Equation, (4-24), an entirely different function.
We have two more column entries, columns 11 and 12. Under column 11 we calculate the gravitational force at particular radii, and under column 12, we give the entropy, also as a function of radius. We first consider the 4-D gravitational force. Within a layer between, r, and, d r r + , we have a well-defined gravitational force acting on that layer, Equation, (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16), was used. To find the total gravitational force at a specific radius, we integrate this expression, and find, 2 2 , From this equation, we see that the gravitational force at radius, r, is simply proportional to the radiative energy enclosed within that radius, 2 r M c . The shape parameter determines the constant of proportionality. That the gravitational force should be proportional to r M is not surprising, but that it depends on little else is. In fact, a force which is proportional to stored energy is very unusual. The gravitational force is listed under column 11 in Tables 4-6. We only need the r M values, given under column 2, and the shape parameters, specified by Equations, (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16).
For a truncated Gaussian or Normal distribution, the corresponding expression for the gravitational force was somewhat different. There we needed the 4-D radiative force at the surface, R F . The expression for the gravitational force was [2], C. Pilot Journal of High Energy Physics, Gravitation and Cosmology As stated, , is the 4-D radiative force just inside the black hole, at radius, R. This radiative force can be calculated. We find, using the arguments of reference [2], For the latter, we find, We evaluate numerically both sets of values for each of the black holes under consideration. We have the radii specified in Equations, (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) . The λ values are indicated in Equations, . We also have the σ values specified for each of the black holes in Equations, (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14). Numerically, the results are, Upon comparing these two sets of results, we notice that they are comparable in value.
We have listed the r S entries under column 12 in Tables 4-6, for the specific radii indicated under column 1.
We can specialize to, r R = , and compare these values for entropy. We find, upon referring to the last row in Tables  [17], just as in the truncated Gaussian case. This would seem to suggest that, if this model is correct, the black hole is a highly ordered state. This is very contrary to what is commonly thought.
We have calculated the entropy using our truncated Gaussian distribution. As seen in reference [2], we obtained, , , or without.

Discussion of Results
Now that the values have been evaluated, and entered into the tables, it is time to compare results. We start with Tables 1-3. We consider values at 0.01 r R = , and compare those values to the corresponding quantities at the surface. The ra- , is close to the core, and it will be insightful to compare those values to surface values. We will also give, right next to the truncated exponential values, the corresponding truncated Gaussian values. This will enable us to make further comparisons.
We start with temperature. Ultimately, for a black body, everything depends on temperature. This includes the internal energy density, the radiative pressure, the radiative forces, the heat density, the entropy density, etc.. These quantities are, by and large, some factor times the temperature taken to the fifth power.
Entropy density is an exception, as well as radiative force. The temperature, We see very clearly that the smaller the black hole (lessor mass), the higher the core temperature. Equations, (5-1), can be compared to the corresponding values for a truncated Normal distribution. These values were worked out in reference [2], and referring to those tables, we had The surface temperatures are the same as in Equations, . The core temperatures are comparable, but less than in relations, . The least massive black holes have the highest core temperatures.
Another quantity we focus on as it relates to Tables 1-3 All forces are measured in Newtons. As is plain to see, the radiative forces are close to zero at the surface but quite substantial at the core. In fact, at the core, they have almost identical values.
Equations, (5-3), can be compared to the corresponding values for a truncated Normal distribution. Those values are found in the same manner, and referring to the tables in reference [2], we obtain, The surface radiative forces are the same as in Equations, . The core values are likewise very close to each other, irrespective of the size (mass) of the black hole. The core radiative forces in Equations, , are less than those indicated in Equations, (5-3).
We next proceed to Tables 4-6. These quantities are cumulative in nature, i.e., they hold from, 0 r = , up to, and including, some finite value for r. We start with radiative mass. Using the same conventions as before, and referring to the Tables 4-6 These masses are in, kg. Upon comparing the core values to the surface values, we notice something surprising. The core values are exceedingly large in relation to the total mass. Within one percent of the radius, we already have the following proportions of the entire (total) mass.
We first give the gravitational coupling in 4-D space using our truncated exponential distribution. Using the same notation as before, and referring to the Tables 4-6, we find, The units are different than their three dimensional counterpart, For black holes with larger mass, the gravitational acceleration does decrease.
However, for a specific mass, this value is uniform. We compare this to the corresponding values if one were to use a truncated Normal distribution instead.
We focus on two more sets of entries. The first is the gravitational force, These are the same percentages as in Equations, (5)(6), which hold for the radiative mass. That the two are proportional is not surprising. That the core values are a high percentage of the total is. The gravitational force at the surface is a major contribution to the surface tension, as shown in reference [1].

Conclusion
Summarizing, by studying various pdf's, we have a rich structure by which to analyze and explore various scenarios for the internal structure of a black hole. If we accept this model of a 4-D spatial ball for a black hole filled with black body radiation (and potentially other forms of radiation), we can investigate various unique characteristics within the black hole. These characteristics within the black hole will lead to specific surface conditions which can be measured from our 3-D perspective. We are thinking of surface forces, surface entropy, surface acceleration, etc., attributes, which will define which pdf is best suited for modeling the interior. In this way valuable insights can be gained, and perhaps, finally, a complete picture as to what is happening within the black hole, and with space itself. This concludes our discussion.