Crossing the brachistochrone- The many-photons example

In previous sections we made some preliminary remarks about how the properties (period, wavelength, etc.) of a photon change if we move with respect to the photon. Now we will focus attention on the motion of the photon itself.

This is a related picture:

The many-photons example

Background picture: Shorter wavelength (λ₁) compared to longer wavelength (λ₂)

[https://perg.phys.ksu.edu/vqm/laserweb/Ch-1/F1s1t1p3.htm]

This is a picture of two photons with different wavelengths, λ₁ and λ₂, but the example can be generalized to include many photons with different wavelengths λ_n, where n will be the number of the harmonic.

The problem is the following one:

An observer located at point O uses photons to measure the distance OO΄=Δx. The speed c of the photon is considered (also experimentally verified) fixed, so that if Δt is the time (as recorded on the clock of the observer) it takes the photon to travel the distance from O to O΄, and back to O, then the observer at point O can determine the distance OO΄, as

For simplicity we will remove the ½ factor, assuming that the time Δt refers to the time it takes the photon to travel the distance OO΄ (without returning to point O). Still this time interval refers to the clock of the observer. As a consequence the photons (of different wavelengths) do not travel the same distance at the same time, according to their frame of reference. This is because, assuming that the photon is a point traveling on the wave, the path the photon follows is the curved path (on the sinusoidal wave), not the linear path (on the straight line) Δx.

But if the time Δt is fixed (all photons arrive at point O΄ from point O, thus cover the distance Δx, at the same time Δt), then they cannot have the same speed (with respect to their own reference frame on the curved path), since the distances they travel on their curved paths will be different (the smaller the wavelength the longer the distance they travel till they reach point O΄).

This is a more thorough analysis of the problem. Firstly, the fixed linear distance is

Now if λ_n is the wavelength of the n-th photon (assuming many photons of different wavelengths covering the same distance Δx), the distance this photon will have traveled on its curved path across the wave will be

where n represents the harmonic, or the number of wavelengths λ_n which fit in the distance Δx for the n-th photon. (It may be assumed that also half-wavelengths can fit into the distance, but here we will consider integer wavelengths for simplicity).

The number n of wavelengths will be different for photons of different wavelength λ_n. But we can use a photon of certain wavelength as reference. Thus we will regard a photon of just one wavelength λ₁ (at the first harmonic, n=1) as the reference photon, so that the distance this photon travels will approximately be equal to the distance Δx,

The rest of the photons will have smaller wavelengths (thus also smaller periods, or bigger frequencies), thus they will travel longer distances,

Accordingly, the period, which we may call τ_n, of a photon with wavelength λ_n will be n times shorter than the period τ₁ of the reference photon with wavelength λ₁, where by definition,

The time τ₁, which we may also call T₁, can be identified with the time Δt (as we have identified the distance λ₁≡L₁ with the linear path Δx).

The revealing aspect here is that any photon with wavelength λ_n (n times shorter than the wavelength λ₁ of the photon at the first harmonic, n=1), in order to cover the same distance L₁≡Δx at the fixed time T₁≡Δt, will have to travel n times faster than the speed c (i.e. the speed of light), so that if we call v_n the speed of the photon at some harmonic n, it will be:

Still, the speed of light will be constant,

Thus for the ‘external’ observer the speed of any photon will never exceed the speed of light c.

As we shall see later on, the speed of light can be strictly reserved for the photons (travelling the linear distance Δx on the rim of the brachistochrone), while the speed v will be associated with another particle, connecting the photons, and travelling on the (curved path of) brachistochrone.

Notes:

The time T₁ which we previously mentioned, where

can in fact be identified with the time of the brachistochrone.

To see this, we may define a time T_n, using the formula for the speed v_n,

where

This expression can be justified as follows. Taking the formula for the time of the brachistochrone

(where here we wrote 2π, instead of π, considering the total journey, from top to top, on the brachistochrone,)

and setting

we have that

The form of the acceleration g_n will be further explored later on.

Taking here the equivalent expressions for the distance L₁, we have that

so that

The previous two formulas represent conditions of the tautochrone (of simultaneity), and relate the time T of (an object travelling on) the brachistochrone to the period τ of the photon.

Conclusively, we may write down the following two basic expressions. On one hand, we have the constant time of the brachistochrone T₁,

On the other hand, we have the constant speed of light c,

Although these two formulas are interrelated, their origin and implications are fundamentally different. In the later, keeping the speed of the photon constant, both the wavelength and period of the photon (thus the spatial and temporal aspects of spacetime) have to change. In the former, keeping the time of the brachistochrone constant, the speed of the photon has to change, if the distance it travels changes.

Later on this speed will be dissociated from the photon, and will be attributed to- let me call it- the brachiston.