- #1

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(1) [tex]S_{BH} = {k c^3 \over 4G \hbar} A[/tex]

where A is the surface area of the black hole event horizon. All the other factors on the right hand side of the equation are constants.

From the point of view of an observer moving relative to the black hole the spherical event horizon appears to be an oblate spheriod with its shotest radius parallel to the relative motion. The surface area of an oblate spheriod varies in a non linear way with respect to the contraction of one of the radii and it is certainly not invariant under Lorentz tranformation. This contradicts almost text on relativistic thermodynamics that almost universally accept that entropy is a Lorentz invariant.

Perhaps the situation can be rectified by substituting the the equation for the volume (V)of a sphere into the equation?

The equation would then be:

(2) [tex]S_{BH} = {k c^3 \over 4G \hbar} \times {3 \over R} \times V = {k c^3 \over 4G \hbar} \times {3 \over R} \times {4 \pi R_x R_y R_z \over 3}[/tex]

By assuming that the relative motion is along the x-axis and by assuming the undefined R is the radius parallel to the relative motion the equation becomes:

(3) [tex]S_{BH} = {\pi k c^3 \over G \hbar} \times { R_y R_z } = {A_{tranverse} \over 2L_p^2}[/tex]

where Ax is the transverse cross sectional area of the event volume and Lp is the Planck Area. This formulation is invariant under Lorentz transformation as all length measurements are transverse.

Since the Radii along the y and z axes remain unaltered under transformation we can substitute the values for the Schwarzschild radii :

(4) [tex]S_{BH} = {\pi k c^3 \over G \hbar} \times { 4 G^2 M^2 \over c^4 }[/tex]

which simplifies to

(5) [tex]S_{BH} = {4 G \pi k \over \hbar c} \times {M}^2 = 4 \pi k {M^2 \over M_p^2}[/tex]

where M is the mass of the black hole and Mp is the Planck mass.

Equation (5) appears not to be invariant although it can be noted that equation (1) can be expressed in similar way using Planck units as:

(6) [tex]S_{BH} = {k \over 4 } {A \over L_p^2} [/tex]

so maybe we can assume invariance is assured by the Planck constants transforming in the same way as area and mass respectively? Was the assumption that the "constants" in equation (1) really are constants, misplaced?