Crossing the brachistochrone- Energy states

Up till now we have used the terms ‘harmonic’ (referring to the harmonic n of the reference photon), and ‘state’ (referring to the brachistochrone) indiscriminately. For example, in the previous section, it was supposed that the number of photons which fit in the length of the brachistochrone L₀, at some harmonic n, is equal to that number n. However this is not necessarily true. More generally, if N₀ is the number of photons of wavelength λ₀ which fit in the length L₀ of the brachistochrone at the first harmonic, n=1, and N_n is the number of photons of wavelength λ_n which fit in the length L₀ of the brachistochrone at some harmonic, n, then it will be

This helps us distinguish between the energy E_n, at some state n, and the energy per wavelength λ_n, which we may call ε_n, at the same state,

Defining now an energy ε₀ per photon of wavelength λ₀, then for the first harmonic, or state, n=1, it will be

and the total energy at the first state can also be written as,

Similarly, for any state n, the energy ε_n per photon of wavelength λ_n will be

so that the total energy at the same state n will be,

Therefore the energy equations can be written in the following general form,

Here we should mention that the definitions,

bring us back to the principle of wave-particle duality, which we earlier saw in this document, in the form of the following equations,

where m is the mass of a material object, and μ is the mass of the wave ‘accompanying’ that object.

But here the notion of the brachistochrone of mass M_n, (at some state n), and the mass of the brachistochrone μ_n per wavelength λ_n of the reference photon, makes such a principle much clearer: The wave of mass μ_n which accompanies the object of mass m₀ is associated with the wavelength λ_n of the reference photon (thus the mass μ_n refers to the mass of the wave, and not to the inertial mass m₀ of the object), while the same mass μ_n, as a sub-division of the total mass M_n of the brachistochrone, is not the mass of a photon.

Now a similar division of the brachistochrone can be made with respect to Planck length l_P, instead of the wavelength λ of the reference photon, if Planck length l_P is used as a reference unit of length.

Thus in addition to the formulas

which give the number of photons N_n which fit in the length of the brachistochrone L₀ at some state or harmonic n, we may suppose that the distance L₀ is composed of a number N_P of Planck lengths l_P. Since the distance L₀ is supposed to be fixed, the number N_P will be constant, independently of the state n, since Planck length is fixed. Thus, we have

This way we can add another term in the energy equation, referring to the number N_P, as follows. At the first state, n=1, it will be

where m_P is Planck mass.

Thus, at any state n, it will be

This way we have a term in the energy equation referring to Planck length.

Before writing down the full energy equation, including all terms, we will make a third step, in order to include in the energy equation a term which will refer to the graviton mass, which we may call m_g. As we previously did with the numbers N and N_P, referring to the number of photons, or Planck lengths, respectively, which fit in the distance L₀, we may introduce a number N΄, which will refer to the number of gravitons of mass m_g which compose the mass M_n of the brachistochrone, at some state n.

Thus if N₀΄ is the number of gravitons which compose the mass M₀ of the brachistochrone at the first state,

then, if N_n΄ is the number of gravitons which compose the mass M_n of the brachistochrone at some state n, it will be

This way we have the following additional energy term, with respect to the number N΄ of gravitons of mass m_g,

where λ_g is the graviton’s wavelength.

Gathering now together all the terms, we have for the numbers n, N, N_P, and N΄, that

so that the energy E₀ at the first state will be

and the energy E_n, at any state n, will be

while the corresponding energy per wavelength will be,

Here are some relationships which we can derive from the energy equation of the brachistochrone,

with respect to the masses m₀, μ_n, and M_n,

or with respect to the speeds v_n and c,

Thus we see, among other things, that the speed v_n of the object and the speed of light c can be more generally defined by the previous formulas.

Another observation is that the state of the brachistochrone (referring to the number N of photons) is directly related to the harmonic n (referring to the excited state of the photon),

Also, the energy of the brachistochrone E_n at any harmonic n, is a multiple squared of the energy E₀ at the first harmonic,

This gives us the freedom to use the terms ‘state,’ or ‘harmonic,’ of the brachistochrone interchangeably, keeping in mind the distinction between the numbers n and N, wherever this is necessary.

Notes:

We have already used the assumption that if the speed v₀ of the object at the first state, n=1, is set equal to the speed of light c, then from the formulas,

we take the simpler result,

Another possible way to simplify the energy equation is to assume that as the harmonic n increases, the number of photons N_n which compose the length L₀ approaches the number n,

By equating the length L₀ of the brachistochrone to the wavelength λ₀ of the reference photon, the number of photons N will always be equal to their harmonic n, so that we take simpler relationships, such as

If we additionally equate the speed v₀ to the speed of light c,

then we take even simpler expressions,

However, by doing this, from the energy equation we also take that,

so that we take as a result that the wavelength λ₀ of the reference photon will be equal to Planck length l_P, while the mass m₀ of the moving object will be equal to Planck mass m_P.

Thus, although by equating different parameters we may take simple results, oversimplifications may result in our losing the physical meaning of the problem.

Finally we may also note some aspects with respect to the term in the energy equation associated with Planck energy. By dividing the brachistochrone into Planck lengths l_P (instead of wavelengths λ₀ of some photon of reference), we take for the energy per wavelength, that

so that the energy ε₀ per wavelength λ₀ will be identical to Planck energy ε_P.

In that sense the energy E₀ may be considered the total energy (for all Planck lengths l_P),

where, among others, we have the following relationships,

An application of such a substitution will be further explored later on.

Still, if the total energy E_n, for all harmonics n, is n² times greater than the energy E₀ at the first state, n=1,

then we may assume photon wavelengths λ_n much shorter than Planck length l_P.

Therefore, in addition to the meaning of photons with macroscopic wavelengths, as we earlier saw, λ₀≈L₀, λ_n<<L₀, we have also to consider the consequences of having photons with wavelengths smaller than Planck length, λ₀≈l_P, λ_n<<l_P.

However, in such cases, the problem may not lie on the theoretically infinite possibilities, but on the practical limits of measurement.