Hilbert Book Model Project/Compartments

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Splitting the universe

Compartments

The extended Stokes theorem suggests that it makes sense to divide the universe in compartments that encapsulate discrepant regions.

• Floating platforms that own a private parameter space constitute a discrepant region.
• Modules consist of conglomerates of such platforms that move as one unit.
• Black holes represent spatially more extended discrepant regions that barriers, which block the flow of information encapsulate.

At every progression instant, compartments that encapsulate a subset of such discrepant regions divide the universe.

Black holes

An event horizon characterizes the black hole. That horizon corresponds to the boundary where the escape speed exceeds the speed of warps.

The speed of warps equals the extension speed of clamps.

Thus neither the warps nor the border of the clamps can pass the event horizon.

The escape velocity $v_{e}$ at distance $r$ from the center of mass $M$ equals

$v_{e}={\sqrt {\frac {2GM}{r}}}$

$G$ is the gravitational constant. For spherical regions, the Schwarzschild radius $R_{S}$ defines the radius at which the escape velocity equals the speed of warps.

$R_{S}={\frac {2GM}{c^{2}}}$

Each clamp carries a standard bit of mass and each warp carries a standard bit of energy. Also, every clamp can in principle turn into a warp or vice versa.

Each warp carries a bit of information.

Thus, the Schwarzschild radius is proportional to the number of encapsulated clamps and is proportional to the equivalent of the encapsulated information.

Bekenstein discovered a bound to the amount of entropy $S$ that an energy containing sphere can contain.

$S\leq {\frac {2\pi kRE}{\hbar c}}$

Here $k$ is the Boltzmann constant. $R$ is the radius of the sphere and $E$ equals the energy equivalent of the sum of the clamps and the warps that are contained in the sphere.

The equal sign holds for the event horizon of a spherical black hole. This means that inside the event horizon no warps occur. The spherical black hole represents the most efficient way of packaging entropy.

For the Schwartzschild black hole M replaces E,

$S_{BH}={\frac {2\pi kR_{S}M}{\hbar c}}={\frac {4\pi kGM^{2}}{\hbar c^{3}}}\propto R_{s}^{2}$

Thus, the entropy of a black hole appears to be proportional to the surface area of the event horizon. For a spherical region the surface area equals $2\pi R^{2}$.

With other words the entropy of the spherical black hole is proportional to the square of the number of encapsulated clamps. With an evenly distribution of clamps the entropy per unit of volume is proportional to the square of the number of clamps per unit of volume. The volume-surface integral tells that each fraction of the volume maps on a similar fraction of the surface area.

If surpasses the radius of the event horizon this relation stays the same. This forms the background of the Holographic Principle.

Inside the compartments the entropy per unit of volume reaches a maximum inside the event horizons of the contained black holes.

Warp inversion radius

At $R_{c}=1.5R_{S}$the warps that approach the black hole along a tangent of a concentric circle with radius $R$ encounter a clipping condition at $R=R_{c}$.

At $R>R_{c}$ the warp deflects away from the black hole.

At $R<R_{c}$ the warp deflects towards the black hole.

Black body

Black holes are black body radiators.

Hawking and Bekenstein showed that the black hole owns a temperature that corresponds to its entropy. Therefore it acts as a black body.

The emitted radiation is the Hawking radiation.