- #1

johne1618

- 371

- 0

Imagine we are at the center of a sphere of radius R containing a mass M.

The Beckenstein bound states that the entropy inside that sphere, S, must be given by the inequality:

[itex] \large S \leq \frac{2 \pi k c R M}{\hbar}. [/itex]

In order to maximise the entropy we need to fill the sphere of radius R with as much mass M as possible. The limit is reached when we have created a black hole. This occurs when the following relationship holds:

[itex] \large \frac{G M}{R} = \frac{c^2}{2}. \ \ \ \ \ \ \ \ \ \ (1)[/itex]

Now in the case of a black hole the event horizon is at a constant radius R and we are sitting on the singularity at the center.

But instead let us assume that the sphere is expanding.

Let us also assume that Equation (1) always holds so that the expanding sphere always has a maximum entropy. This of course means that the mass M must increase with radius R. Thus matter is being continuously created.

If we assume flat space then Equation (1) implies that the mass density, [itex]\rho[/itex], is given by

[itex] \large \rho = \frac{3 c^2}{8 \pi G R^2}. \ \ \ \ \ \ \ \ (2) [/itex]

If we assume that the radius [itex]R(t)[/itex] is expanding with the Universal scale factor [itex]a(t)[/itex] then we can say:

[itex] R(t) = R_0 a(t) \ \ \ \ \ \ \ \ (3)[/itex]

where t is the cosmological time, [itex]R_0[/itex] is the radius at the present time [itex]t_0[/itex] and [itex] a(t_0) = 1 [/itex].

Now let us consider the Friedmann equation for flat space with no cosmological constant:

[itex] \large (\frac{\dot{a}}{a})^2 = \frac{8 \pi G}{3} \rho \ \ \ \ \ (4) [/itex]

Substituting Equations (2) and (3) into Equation (4) we obtain

[itex] \large (\frac{\dot{a}}{a})^2 = \frac{c^2}{R_0^2 a^2}. [/itex]

As the Hubble parameter at the present time, [itex]H_0[/itex], is given by

[itex] H_0 = c / R_0,[/itex]

we finally arrive at

[itex] \large (\frac{\dot{a}}{a})^2 = \frac{H_0^2}{a^2}, [/itex]

which has the simple linear solution

[itex] a = H_0 t. [/itex]

This solution to the Friedmann equation is thus the maximum entropy solution.

It is very close to what is observed.

The Beckenstein bound states that the entropy inside that sphere, S, must be given by the inequality:

[itex] \large S \leq \frac{2 \pi k c R M}{\hbar}. [/itex]

In order to maximise the entropy we need to fill the sphere of radius R with as much mass M as possible. The limit is reached when we have created a black hole. This occurs when the following relationship holds:

[itex] \large \frac{G M}{R} = \frac{c^2}{2}. \ \ \ \ \ \ \ \ \ \ (1)[/itex]

Now in the case of a black hole the event horizon is at a constant radius R and we are sitting on the singularity at the center.

But instead let us assume that the sphere is expanding.

Let us also assume that Equation (1) always holds so that the expanding sphere always has a maximum entropy. This of course means that the mass M must increase with radius R. Thus matter is being continuously created.

If we assume flat space then Equation (1) implies that the mass density, [itex]\rho[/itex], is given by

[itex] \large \rho = \frac{3 c^2}{8 \pi G R^2}. \ \ \ \ \ \ \ \ (2) [/itex]

If we assume that the radius [itex]R(t)[/itex] is expanding with the Universal scale factor [itex]a(t)[/itex] then we can say:

[itex] R(t) = R_0 a(t) \ \ \ \ \ \ \ \ (3)[/itex]

where t is the cosmological time, [itex]R_0[/itex] is the radius at the present time [itex]t_0[/itex] and [itex] a(t_0) = 1 [/itex].

Now let us consider the Friedmann equation for flat space with no cosmological constant:

[itex] \large (\frac{\dot{a}}{a})^2 = \frac{8 \pi G}{3} \rho \ \ \ \ \ (4) [/itex]

Substituting Equations (2) and (3) into Equation (4) we obtain

[itex] \large (\frac{\dot{a}}{a})^2 = \frac{c^2}{R_0^2 a^2}. [/itex]

As the Hubble parameter at the present time, [itex]H_0[/itex], is given by

[itex] H_0 = c / R_0,[/itex]

we finally arrive at

[itex] \large (\frac{\dot{a}}{a})^2 = \frac{H_0^2}{a^2}, [/itex]

which has the simple linear solution

[itex] a = H_0 t. [/itex]

This solution to the Friedmann equation is thus the maximum entropy solution.

It is very close to what is observed.

Last edited: