The mass event horizon is defined in “Gravity and Space-Time” as the radial distance from the center of a body of mass M at which the orbiting velocity gradient generated by the body is equal to the speed of light:
Rs = M G / C^2 (1)
This equation is obtained directly from the two classical equations defining gravitational and centripetal accelerations.
The first one is Newton’s inverse square relation:
g = M G / d^2 (2)
where: g is the acceleration due to gravity M is the mass of the body to which gravity is attributed G is the universal gravitational constant 6.67E-8 [cm3 g-1sec-2] d is the distance from the body
The second one is the equation relating the centripetal acceleration of a body rotating at a constant velocity around a center:
a = V^2 / d (3)
where: a is the centripetal acceleration directed toward the center V is the linear velocity of the rotating body d is the distance of the body from the center
The velocity of a body orbiting another body of mass M at a distance d is given by equating gravitational force to centripetal force:
V = sqr (M G / d) (4)
Velocity V will be equal to the speed of light C at the distance:
d = M G / C^2 (5)
This distance defines the radius Rs of the event horizon of mass contracted into a spherical volume of that radius.
The mass event horizon distance Rs is not the Schwarzschild radius (which is twice the Rs distance): the event horizon is based on orbital velocity while the Schwarzschild radius is based on escape velocity.
If a body of mass M is contracted into a spherical volume of radius Rs (the event) then the horizon of that event has the following properties: orbital velocity gradient equal to the speed of light; light deflection angle of 90 degrees; space-time warp angle of 90 degrees; and total energy content of mass orbiting at the horizon equal to zero. Sub- event worlds are closed systems where energy can neither get in nor get out: they are invariant.
If a Planck mass is contracted to its event horizon then the radius of that horizon is equal to the Plank length:
Pm = sqr (hbar C / G) (6)
Rs = Pm G / C^2 = sqr (hbar G / C^3) = Pl (7)
We can also calculate an orbital rotational frequency equal to:
f = C / (2pi Rs) (8)
Using Planck’s equation:
m = f h / C^2 (9)
where: m is mass f is frequency h is the Planck constant C is the speed of light we have:
m = C h / (2pi Rs C^2) = hbar / (Rs C) (10)
Independently of the mass of the rotating body, m is the equivalent mass of the frequency of rotation at velocity C around a body of mass M at a distance equal to the event horizon of that body.
At the event horizon of Planck mass (equal to the Planck length) the equivalent mass of the frequency of rotation is equal to the Planck mass:
From equations (6) and (10), setting Rs = Pl:
m = M = Pm (11)
For any mass M expressed in terms of number of Planck masses, the relation is:
M = n Pm (12)
m = M / n^2 = Pm / n (13)
Using equation (9):
f = F / n (14)
where: f is the orbital rotational frequency F is the equivalent frequency of Planck mass n is the number of Planck masses
The mass event horizon is the locus at which the orbital velocity gradient is equal to the speed of light and the total energy of a point mass orbiting at the horizon is equal to zero.
The radius of the event horizon of Planck mass is equal to the Planck length. At the event horizon of Planck mass the orbital rotational frequency and the equivalent frequency of Planck mass are synchronous. The Cosmological Model
The total energy E of a body in motion is equal to the sum of its energy at rest Eo plus kinetic energy Ke and potential energy Kp:
E = Eo + Ke + Kp (15)
The energy at rest is given by
Eo = mo C^2 (16)
The kinetic energy Ke is the difference in energy levels between the body in motion (mass m) and the body at rest (mass mo), or:
Ke = m C^2 – mo C^2 (17)
The potential energy is given by:
Kp = – m M G / d (18)
where d is the distance of the body in motion (mass m) to the center of the body at rest (mass M).
From equations (15), (16), (17), and (18) we have:
E = mo C^2 + (m C^2 – mo C^2) – m M G / d = m C^2 – m M G / d (19)
At the event horizon of mass M the kinetic potential C^2 and the gravitational potential M G / Rs are equal and equation (19) becomes:
E = m C^2 – m M G / Rs = 0 (20)
If we define the universe as an event then the event horizon of the universe is given by equation (1) where M is the mass of the universe. This event horizon has the same properties as all mass event horizons and the equality of potentials has to be true for the life of the universe. We can then use equation (20) as a cosmological model:
M C^2 – M^2 G / r = 0 (21)
where M is the mass of the universe and r is its “radius” equal to:
r = M G / C^2 (22)
Equation (22) is identical to equation (1) that defines the radius Rs at the event horizon of mass M. However, the Rs definition in equation (1) is based on a point mass orbiting at the horizon with a velocity equal to the speed of light while equation (22) is for a universe of a radius r that is expanding at the speed of light. This introduces a time parameter that requires either or both mass and speed of light to be time dependent. Furthermore, equation (1) defines a system when viewed from above its mass event horizon; equation (22), instead, defines a system when viewed from below the horizon.
According to this model the mass of the universe and the speed of light are functions of time and the relations between their current values and those at any epoch of the universe are:
M = Mo (t / to)^(2/5) (23) C = Co (to / t)^(1/5) (24)
Where: Mo is the mass of the universe (in grams) at the current epoch Co is the current speed of light of 2.998E10 [cm sec-1] to is the current epoch of 4.37E17 [sec] (or 13.85 billions years) t is any epoch of the universe M is the mass of the universe at time t C is the speed of light at time t
In the same reference as above, the current values of to and Mo have been defined as:
to = 1 / Ho = 4.37E17 [sec] (25) Mo = (5/4) Co^3 to / G = 2.2E56 [g] (26)
Where Ho is the current value of the Hubble constant of 2.29E-18 [sec-1]
Applying Equations (23) and (24) to the definitions of the mass event horizon, Planck mass, and Planck length gives:
Rs = Rso (t / to)^(4/5) (27) Rso = Mo G / Co^2 (28)
If at any time a Planck mass defined by Equation (29) is contracted to its event horizon defined by Equation (27) then the radius of that horizon is equal to the Plank length defined by Equation (31):
Rs = Pm G / C^2 = sqr (hbar G / Co^3) (t / to)^(3/10) = Pl (33)
The observation that has been made about the synchronicity of the orbital rotational frequency at the event horizon and the equivalent frequency of Planck mass contracted to that horizon is valid.