摘要
A black hole is treated as a self-contained, steady state, spherically symmetric, 4-dimensional spatial ball filled with blackbody radiation, which is embedded in 3-D space. To model the interior distribution of radiation, we invoke two stellar-like equations, generalized to 4-D space, and a probability distribution function (pdf) for the actual radiative mass distribution within its interior. For our purposes, we choose a truncated Gaussian distribution, although other pdf’s with support, r ∈[0, R], are possible. The variable, r = r(4), refers to the 4-D radius within the black hole. To fix the coefficients, (μ,σ), associated with this distribution, we choose the mode to equal zero, which will give maximum energy density at the center of the black hole. This fixes the parameter, μ = 0. Our black hole does not have a singularity at the center, and, moreover, it is well-behaved within its volume. The rip or tear in the space-time continuum occurs at the event horizon, as shown in a previous work, because it is there that we transition from 3-D space to 4-D space. For the shape parameter, σ , we make use of the temperature just inside the event horizon, which is determined by the mass, or radius, of the black hole. The amount of radiative heat inflow depends on mass, or radius, and temperature, T2 ≥ 2.275K , where, T2, is the temperature just outside the event horizon. Among the interesting consequences of this model is that the entropy, S(4), can be calculated as an extrinsic, versus intrinsic, variable, albeit in 4-D space. It is found that S(4) is much less than the comparable Bekenstein result. It also scales not as, R2 , where R is the radius of the black hole. Rather, it is given by an expression involving the lower incomplete gamma function, γ(s,x), and interestingly, scales with a more complicated function of radius. Thus, within our framework, the black hole is a highly-ordered state, in sharp contrast to current consensus. Moreover, the model-dependent gravitational “constant” in 4-D space, Gr(4), can be determined, and this will depend on radius. For the specific pdf chosen, Gr(4)Mr = 0.1c2(r4/σ2), where Mr is the enclosed radiative mass of the black hole, up to, and including, radius r. At the event horizon, where, r = R, this reduces to GR(4) = 0.2GR3/σ2, due to the Schwarzschild relation between mass and radius. The quantity, G, is Newton’s constant. There is a sharp discontinuity in gravitational strength at the 3-D/4-D interface, identified as the event horizon, which we show. The 3-D and 4-D gravitational potentials, however, can be made to match at the interface. This lines up with previous work done by the author where a discontinuity between 3-D and 4-D quantities is required in order to properly define a positive-definite radiative surface tension at the event horizon. We generalize Gauss’ law in 4-D space as this will enable us to find the strength of gravity at any radius within the spherically symmetric, 4-D black hole. For the pdf chosen, gr(4) = Gr(4)Mr/r3 = 0.1c2r/σ2, a remarkably simple and elegant result. Finally, we show that the work required to assemble the black hole against radiative pressure, which pushes out, is equal to, 0.1MRc2. This factor of 0.1 is specific to 4-D space.
A black hole is treated as a self-contained, steady state, spherically symmetric, 4-dimensional spatial ball filled with blackbody radiation, which is embedded in 3-D space. To model the interior distribution of radiation, we invoke two stellar-like equations, generalized to 4-D space, and a probability distribution function (pdf) for the actual radiative mass distribution within its interior. For our purposes, we choose a truncated Gaussian distribution, although other pdf’s with support, r ∈[0, R], are possible. The variable, r = r(4), refers to the 4-D radius within the black hole. To fix the coefficients, (μ,σ), associated with this distribution, we choose the mode to equal zero, which will give maximum energy density at the center of the black hole. This fixes the parameter, μ = 0. Our black hole does not have a singularity at the center, and, moreover, it is well-behaved within its volume. The rip or tear in the space-time continuum occurs at the event horizon, as shown in a previous work, because it is there that we transition from 3-D space to 4-D space. For the shape parameter, σ , we make use of the temperature just inside the event horizon, which is determined by the mass, or radius, of the black hole. The amount of radiative heat inflow depends on mass, or radius, and temperature, T2 ≥ 2.275K , where, T2, is the temperature just outside the event horizon. Among the interesting consequences of this model is that the entropy, S(4), can be calculated as an extrinsic, versus intrinsic, variable, albeit in 4-D space. It is found that S(4) is much less than the comparable Bekenstein result. It also scales not as, R2 , where R is the radius of the black hole. Rather, it is given by an expression involving the lower incomplete gamma function, γ(s,x), and interestingly, scales with a more complicated function of radius. Thus, within our framework, the black hole is a highly-ordered state, in sharp contrast to current consensus. Moreover, the model-dependent gravitational “constant” in 4-D space, Gr(4), can be determined, and this will depend on radius. For the specific pdf chosen, Gr(4)Mr = 0.1c2(r4/σ2), where Mr is the enclosed radiative mass of the black hole, up to, and including, radius r. At the event horizon, where, r = R, this reduces to GR(4) = 0.2GR3/σ2, due to the Schwarzschild relation between mass and radius. The quantity, G, is Newton’s constant. There is a sharp discontinuity in gravitational strength at the 3-D/4-D interface, identified as the event horizon, which we show. The 3-D and 4-D gravitational potentials, however, can be made to match at the interface. This lines up with previous work done by the author where a discontinuity between 3-D and 4-D quantities is required in order to properly define a positive-definite radiative surface tension at the event horizon. We generalize Gauss’ law in 4-D space as this will enable us to find the strength of gravity at any radius within the spherically symmetric, 4-D black hole. For the pdf chosen, gr(4) = Gr(4)Mr/r3 = 0.1c2r/σ2, a remarkably simple and elegant result. Finally, we show that the work required to assemble the black hole against radiative pressure, which pushes out, is equal to, 0.1MRc2. This factor of 0.1 is specific to 4-D space.
