Crossing the brachistochrone- The universe as a brachistochrone

Apart from the numerical coincidences which we saw in the previous section, if in the energy equation of the brachistochrone we replace the mass M_B of the brachistochrone with the mass M_U of the observable universe, and the radius R_B of the brachistochrone with the radius R_U of the observable universe, then we take

where the ratio ρ, giving the linear density of the observable universe, was identified in the previous section.

Thus we take as result that the wavelength λ of the reference photon is reduced into Planck length l_P.

This result is not obvious, but it is a consequence of the notion of the brachistochrone, and of the energy equation which brings together a pair of phenomenally unrelated quantities, the ratio of the mass M_U of the observable universe to Planck mass m_P, or, equivalently, the ratio of the radius R_U of the observable universe to Planck length l_P,

The number N_P is a fixed quantity, which we have already calculated in the previous section.

But if we ignore the distinction between the different masses or lengths which appear in the latter formula, then the energy equation reduces to the following undifferentiated form,

and loses its meaning.

Therefore the hypothesis of the brachistochrone is significant in order to take more general results.

In fact such results may offer us a better estimation for the mass M_U of the observable universe. The current estimate for this mass is

[https://en.wikipedia.org/wiki/Observable_universe]

But here we have that

If we accept this result as more accurate, then the brachistochrone hypothesis can be used in order to give physical meaning to the numerical coincidence of the previous ratio.

Another consequence of such a coincidence is the following one. We have already talked about Schwarzschild radius, which is given by the formula

where R_S is the radius of a black hole, and M_S is its mass.

This formula can be derived from the mechanical energy of a system, if this energy is gravitational and kinetic energy. Supposing, for example, that an object of mass m leaves the surface of the Earth with a speed v, and is set in orbit around the Earth, then if R is the radius of the orbit, and M_E is the mass of the Earth, the conservation of the mechanical energy gives

The speed v is the escape speed of the object from Earth’s gravity. If this speed was equal to the speed of light c, then the Earth would turn into a black hole, and its radius would be given by Schwarzschild’s formula, replacing v by c in the previous equation,

This would be about 10 millimeters.

Now we will see what happens if we replace in Schwarzschild’s formula the mass of the gravitating body with the mass M_U of the observable universe.

It would also be helpful to change the units of Newton’s gravitational constant G into light units, as follows

Replacing these values into Schwarzschild’s formula (omitting the factor of 2), we take

Remarkably enough, the value of Schwarzschild radius R_S in this case is identical to the radius R_U of the observable universe. But since the Schwarzschild radius R_S of the universe is (exactly or about) equal to the radius R_U of the observable universe, then we must conclude that the universe is a black hole!

An explanation of this coincidence could be that the linear density ρ of the observable universe, which we may also call ρ_U, is constant,

An observation here is that the (inverse) linear density of the universe ρ_U numerically coincides with the gravitational constant G (in units of light),

Calculating the product of these two quantities, we have

Therefore, returning to Schwarzschild radius, we take that

Now another interesting aspect arises if we take the following two ratios, which we also saw in the previous section,

If the wavelength λ_g of the graviton numerically coincides with the speed of light c, then we can write that

Presumably, the quantity ℊ will be constant throughout the universe, since all the related quantities are constants, and will be equal to 1ly/y².

But the coincidences do not end here. Another observation we can make is that that acceleration of gravity, on the Earth’s surface, which we may call g_E, is approximately equal to 1ly/y2, if we change the units into light units as follows,

Another way to express this coincidence is with respect to the linear density ρ or area density σ of the Earth and of the universe. Given the values for the mass M_E and radius R_E of the Earth, and for the mass M_U and radius R_U of the observable universe, respectively,

for the linear density ρ_E and area density σ_E of the Earth we have,

while for the linear density ρ_U and area densit σ_U of the observable universe we have

We can also calculate the corresponding accelerations for those densities,

so that if we identify the acceleration on the Earth’s surface g_E with the acceleration ℊ, it will be

Incidentally, the ratio

which has units of angular frequency, gives us Hubble constant,

Hubble constant is estimated to be

[https://map.gsfc.nasa.gov/universe/uni_expansion.html]

where

so that,

Therefore the value of

can be seen as a more accurate estimation of Hubble constant.

Conclusively, from the equation for the acceleration ℊ,

we have the equivalent expressions,

We should mention here that in the calculation of the area density σ we have assumed that the Earth was flat, or perhaps that the Earth’s mass was spread across the surface of a sphere of radius equal to the Earth’s radius (while we have ignored the radian factor π for simplicity). Such assumptions can be based on the holographic principle, which we will explore later on.

The same assumptions can also be based on the brachistochrone hypothesis. As the universe, so the Earth can be seen as a brachistochrone whose mass is either spread across the curve of the brachistochrone, or concentrated on the focus of the brachistochrone. This is as far as the linear density is concerned.

With respect to the area density, we have to suppose that the mass of the brachistochrone is spread within the area of the brachistochrone, thus the region enclosed by the rim L and the curve S of the brachistochrone. The area A of the brachistochrone is in fact A=3πR², thus analogous to the radius of the brachistochrone squared.

Therefore, ultimately, approximating either the Earth or the universe with a brachistochrone, we take meaningful results which, besides their implication with respect to the appearance of life on Earth (considering for example that the acceleration of 1ly/y² is ideal for the appearance of life as we know it), apply everywhere.

Incidentally, the acceleration ℊ, which is constant throughout the universe, and (as has been treated here) equal to 1ly/y², could be identified with a field, yet unknown.