Crossing the brachistochrone- The photon and the graviton

We have already discussed, in the section about quantum entanglement, a possible model for the interaction between a brachiston of mass m, which travels on the brachistochrone S, and the reference photon of wavelength λ, which propagates on the linear axis L.

Here the result can be specialized, so that the object moving on the brachistochrone is a graviton, while the photon is related to the graviton either directly (the photon is emitted by the graviton), or indirectly (the photon is produced as a disturbance in spacetime, due to the graviton’s motion).

The relationship between the graviton and the reference photon can be described from the energy equation,

The same equation can also be written in the variable form,

where the index ‘B’ stands for ‘Brachistochrone’, while the quantities E_B, g, M_B, v, λ are variables (the rest of the quantities being constant) with respect to the harmonic, or state, n.

If we replace in this energy equation the mass m of any object by the mass m_g of a graviton, then the same equation will describe the interaction between a graviton of mass m_g, and a photon of wavelength λ. Thus we have,

Comparing the different pairs in this energy equation, we can take a relationship between the wavelength λ of the reference photon, and the wavelength λ_g, or the mass m_g, of the graviton,

Supposing, for example, that the graviton travels at the speed of light, we have

More generally, the relationship between the mass m_g of the graviton, and the mass Μ_Β of the brachistochrone, will be

As far as the final speed of the graviton is concerned, we have that

Therefore, we see that the speed v of the graviton will be √N΄ times greater than the speed of light c, where N΄ is the number of gravitons of mass m_g which compose the brachistochrone of mass Μ_Β.

As we shall see later on, even greater speeds can be attained by a smaller mass (that of a brachiston lighter than the graviton).

Notes:

If we equate in the previous relationships the mass Μ_Β and the length L_Β of the brachistochrone to the mass m_g and the wavelength λ_g of the graviton, respectively, we take that

so that the energy equation is reduced into the following one,

However, comparing the first and the last of the energy terms, we have that

Therefore such substitutions turn the brachistochrone into a micro- black hole (a Planck particle), whose length L_Β will be equal to Planck length l_P, and whose mass Μ_Β will be equal to Planck mass m_P.

In what follows we shall see that particles even smaller than a Planck particle can be obtained.