Energy in orbit black holes:
There are two ways of defining a black hole and two black hole formulas. The original black hole definition goes back centuries and equates gravitational energy with kinetic energy. It is based on escape velocity from a star exceeding the speed of light. There is more in the appendix.

We have been using a second black hole definition in looking at the cosmos. It equates gravitational energy with rest energy. It is based on the idea that a black hole has enough mass for light and energy to orbit. This may be the first time it has been published. This is a simple and logical classical definition. We will continue using this new definition.
m*vt²/r = G*m*M/r², The centrifugal force equals the gravitational force.
m*c²/r = G*m*M/r², replaced vt with c, now the tangent velocity is c and light can orbit.
m*c² = G*m*M/r, multiplied by r
The orbiting energy equals the gravitational energy. If G and r both increase at the same rate then the orbiting and gravitational energy are constant as the black hole expands. Energy is conserved. The tangent velocity is c. Light orbits at the specific radius r. Can light follow an elliptical orbit which requires changes in velocity? Light might follow a circular orbital path, a spiral path or a bent path as in gravitational lensing. These paths could form a spherical shell or an empty shell since light has energy and consequently mass. Any mass inside the shell would contribute to the total mass, increasing the radius of the shell. An empty shell of a black hole does requires some thought.
c² *r = G*M, multiplied by r/m
c² *r = G*M, or
c² *r /(G*M) = 1
c² *c*age /(c³*age) = 1, substituted the values we calculated in the Introduction for r=c*age and G*M=c³*age.
All the terms cancel, telling us that our cosmos is a black hole.

Expanding black holes:
M/r = c²/G = 1.35E27_kg/m, the current value which decreases with age.
r = M *G /c²
If G = c³*age/Mc, varies with age then, r = vr*age, must also vary with age, if the mass and c are constant. vr, is the constant radial velocity of expansion at its perimeter.
vr *age = M *G /c², substituted vr*age for r
vr *age = M * c³*age/Mc /c², substituted c³*age/Mc for G, age cancels
vr = M *c/Mc, collected terms, since the age cancels, the radial velocity of expansion is constant and proportional to mass. Black holes expand.
vr = M *1.568E-45_m/(s*kg), or
vr/c = M/Mc

For a ten solar mass black hole, 1.99E31_kg, vr = 3.12E-14_m/s
For a billion solar mass black hole, 1.99E39_kg, vr = 3.12E-6_m/s. This expands at 98_m/year .
For a 9.61E22 solar mass black hole, 1.91E53_kg, the mass of the cosmos, Mc, vr=299792458_m/s = c
Black holes expand proportional to their mass. The radius of the cosmos expands at the speed of light c, or 1_light year per year, because it has enough mass for it to expand at c. This expansion is one year in the age of the cosmos or one part in 15 billion per year currently. This is quite small.

Variable G:
You may have noticed that G has a value of one in 15 billion. It is increasing in proportion to the age of the cosmos.
1/(15 billion) - 1/(15 billion and one).
G = c³*age /Mc, so in a year, deltaG = c³*(seconds in a year) /Mc = 4.44E-21_m³/(kg*s²)
The gravitational force was stronger in the past. How do we detect such a small ongoing increase in the gravitational constant?
c²*r = G*M, after canceling the m’s from above
c²*c*age = G*M, substitute c*age for r.
c³*age = G*M, collect the c's.
We take c and M to be constants. If r is a variable and increases with age then G is a variable and must increase with age. This can be written G = gk*age, G increases with time.
gk = G/age = G*Ho = 1.41E­28_m³/ (kg*s³), gk appears to be a universal constant in contrast to Ho, Hubble's variable constant and G, the variable gravitational constant which we see is a variable.
c³*age = gk*age*M, Both r and G increase with age,
c³ = gk*M, we can cancel the age.
c³/gk = Mc = 1.91E53_kg = The mass of the cosmos.

G increases with time while gravitational force decreases with time:
force = G*m*M/r², gravitational force
force = gk*age*m*M/(c*age)², substituted for G and r the size of the cosmos.
force = gk*age*m*M/(c²*age²),
force = gk*m*Mc/(c²*age), the force decreases with age. The gravitational force was much stronger earlier. Light from a distant earlier time has a greater gravitational redshift. How does this affect distance calculations? Star formation, accretion and fusion rates would have been affected by the stronger gravitation in the past. The standard candle, Type 1A supernovas would have been affected. What does this do to the accelerating universe dark energy theory?
force = gk*m*c³/(gk*c²*age), substituted for Mc = c³/gk.
force = m*c/(c*age) = m/age, force decreases with age or radius of the cosmos for a energy equivalent of a mass at the radius of the universe.
Mc/r = c²/G = 1.35E27_kg/m, The cosmos has always been a blackhole. This mass to radius ratio quantifies an energy in orbit black hole at the present time. If the mass is constant and the radius increases then this ratio must decrease with time.
Mc/(c*age) = c²/(gk*age), expand r and G.
Mc = c³/gk, multiplied by age and collected c's, the mass of the cosmos does not vary with age but its radius does.

Black hole density:
mass = c²*r/G = c²*c*age/(gk*age) = c³/gk
r = mass*G/c² = mass*gk*age/c²
density = mass /volume
c²*r/G /(4*pi*r³/3) =
3*c²/(4*pi*G*r²), using the radius, the density decreases with r² as r increases with age.
3*c²/(4*pi*gk*age*c²*age²), substituted for r=c*age.
3/(4*pi*gk*age³), using the radius, the density decreases with age³.
3*c²/(4*pi*G*r²),
3*c⁶/(4*pi*gk³*age³*mass²), using the mass, the density decreases with with age³.

Three examples of black hole density are a
twenty solar mass = 3.68E17_kg/m³,
ten solar mass = 1.47E18_kg/m³ and
five solar mass = 5.90E18_kg/m³.

These are in the range of the density of a neutron star or nuclear density, inside the nucleus of an atom. The density increases as the mass of the blackhole decreases up to a limit. Less than five solar masses is observable as a neutron star not a black hole. The highest density in energy in orbit black holes is therefore, nuclear density. The density decreases as the square of the radius or mass so a larger radius or mass means a lower density. We have black holes without infinities.

Cosmic density:
The lowest density black hole is our cosmos.
Mc/ (4/3*pi*c³*age³) = 1.6E-26_kg/m³ or 1.6E-29_g/cm³
The mass of a proton or hydrogen atom is 1.67E-27_kg so the average density of the universe is about ten protons per cubic meter and is decreasing with the cube of the age of the cosmos. This is in the range of reported values.

Merging black holes:

Visualize two spheres that overlap, their contents merge, thereafter being enclosed by a single larger sphere, of their combined diameters. The light and energy in orbit around each blackhole, will after merging, and after their travel through space, eventually orbit at this new larger radius.

In eggland, two eggs touch. Their shells merge much like soap bubbles merge. Their contents merge. Where there was two eggs, there is now one larger egg with two merged yokes. Over time, there is a very big egg with many yokes merged together. The yokels, being unaware of the mechanics of merging, make up odd stories of their creation and their importance to the creator.

There are groupings of mass in space so great, that gravity in the age of the cosmos, would be inadequate for their formation from hydrogen gas. These are called large-scale structure. An example is the Sloan Great Wall. Their great mass should have been reflected in the observed inhomogeneities in the CMB, if the conventional theory is right. The merging of black holes, does however, explain these structures. The smaller black hole has a much higher density. The merged contents are enclosed in a much larger volume. From within, one sees only the merged contents. The smaller black hole leaves behind a higher residual mass density, in the stretched out, merged contents, which is the artifact or footprint of their merging.

M/r = mass/radius = c²/G = 1.35E27_kg/m
radius = r = mass *G/c²
surface area = mass² *4*pi* G²/c⁴
volume = mass³ *4*pi*G³/(3*c⁶)
density = mass /volume = 3*c⁶/(mass² *4*pi*G³)
Two times mass equals; two times radius, four times surface area, eight times volume, density divided by four, and two times vr.

Spherical caps of merging spheres: See the figure above.
A spherical cap is a part cut off a sphere. When two spheres merge they create a lense shaped merged region. The volume of the lense shaped merged region includes twice the volume of the spherical caps of each sphere. The volume of a spherical cap is,
1/3*pi*r³*(3-fr)*fr², with fr, being the fraction of r, that is the height of the cap. The volume of a sphere equal to four spherical caps would be
4/3*pi*r³ = 4/3*pi*r³*(3-fr)*fr², or
1 = (3-fr)*fr², or fr = .6527036
When the spherical caps of merging black holes of the same size reach .6527 of their radius, the volume and the mass of the merged portions satisfies the mass/radius formula for a black hole.

When black holes merge they also incorporate a lot of volume and matter which was outside the two original black holes. The material outside the merging volumes would be relatively undisturbed in large low density black holes. This seems a very natural way for black holes to grow and many stars and galaxies would survive as would the life they might harbor. This is another level being added to the ancient idea of panspermia or transfer of life between planets or stars as popularized by Fred Hoyle and others. The viral or bacterial spores in rock can survive for many thousands of years so a super nova, planetary impacts or planetary explosions could spread rock fragments and spores to other planets or star systems. Life could spread across a solar system, galaxy or merging cosmoses.

*Index* *Next, Rotation of the universe*