Crossing the brachistochrone- Lorentz transformation

Here we can show that Doppler shift for the frequency (thus also for the period) can be derived from Lorentz transformation between two reference frames. Lorentz (relativistic) transformation is an expansion of Galileo’s (non- relativistic) transformation, according to the following picture:

[https://www.slideshare.net/HemBhattarai2/relativity-57673106]

In this picture motion occurs on the horizontal axes x and x΄. Two observers are located at points O and O΄ of their respective reference frames K and K΄. An event (red dot in the previous picture) takes place at some distance x΄ from point O΄. The first reference frame K can be treated as stationary, while the second reference frame K΄ will be moving at the relative speed v. Initially the two reference frames may coincide, so that after some time t according to the observer on the first reference frame K (at point O), the distance between the two reference frames K and K΄ will be OO΄=vt. If x is the distance (the coordinates) of the event from point O, and x΄ is its distance from point O΄, then the Galilean transformation between the two reference frames is

The main difference between the Galilean transformation and Lorentz transformation in relativity, is that the times at which the same event takes place are different for the reference frames K and K΄.

This happens because in relativity distances are measured with photons, whose speed c (the speed of light) is constant. The positions x and x΄ of the event with respect to the two different reference frames K and K΄, respectively, will be

Thus the time t΄ according to the observer on the second reference frame K΄, will be different from the time t according to the observer on the first reference frame K, as a consequence of the constant speed of light c.

Now a way to derive Lorentz transformation is to use the following linear combination,

where the Lorentz factor γ_L is added to the transformation as a correction factor.

Replacing in the previous formulas

we take

so that for the Lorentz factor γ_L we have

This is the value of Lorentz factor γ_L.

Intriguingly enough, the previous result relating the two coordinate times t and t΄ (the time as measured by the clocks of two observers on the different reference frames K and K΄),

is identical to the one we earlier took for Doppler shift, which relates the frequencies (thus also the periods) of a light wave, according to two observers (one at rest, and another one moving at a relative speed v). That formula was given as

where here we have preferred a combination of signs implying that the observer (the receiver) approaches the source (the emitter), so that the frequency f_R he/she receives increases.

If instead of the frequencies f_R and f_E we use the corresponding periods T_R and T_E,

and make the appropriate replacements,

we can identify the periods T_R and T_E with the coordinate times t΄ and t, respectively,

The times T_R and T_E (or T΄ and T respectively) refer to the period of photons, while the times t΄ and t refer to the clocks (measuring devices) of the observers. But if the times t΄ and t can be identified with the periods T΄and T, respectively, then there must be an intricate relationship between the properties of the photons and the notion of time as we know it.

The deeper aspect of such a relationship, as we have already proposed, is that the photons with the altered frequency (as disturbances of spacetime) are produced by the motion of the observer in spacetime, in a cause- and- effect relationship.

But according to this description, the oscillating medium (spacetime) cannot be ignored. If we ignore the medium, we may end up with wrong results, or we may not be able to expand the result, in order to find more general formulas.

An example of such a generalization was given in the next to last section. There we derived a formula of the form

where the periods T₂ and T₁ (thus also the oscillations of spacetime measured in the form of photons) are related to the speed v of the observer, and according to his/her own clock Δt.

Here we can show the following thing. Replacing the periods T₂ and T₁, by T΄ and T respectively, and setting

where λ and λ΄ represent the corresponding wavelengths of the oscillations (the photons), then we have that

Synchronizing now the clock Δt of the moving observer with the initial period T (the period of photons he/she measures before he/she begins to move), we take for the periods,

or, equivalently, for the wavelengths,

Therefore by assuming that motion occurs in a medium, that this motion affects the medium by changing the period of its oscillations, and that we can measure this effect in the form of photons, we retrieve the previous relativistic expressions for length contraction (or time dilation), and for Lorentz factor γ_L, essentially from first principles, and directly from a quadratic equation of the general form

In the next section we will see that the same equation is valid in any occasion, even if we move at non- relativistic (much smaller than light) speeds.

Notes:

With respect to the general formula

or equivalently

where

solving for the periods T, or T΄, we take

If instead of

we set

assuming thus a position vector of the form

then we alternatively take

This expression, compared to the expression for Doppler shift,

has the advantage of being more symmetrical, as the speed v increases.

This can be seen in the following graph:

The graph compares the function for the relativistic Doppler shift (blue line), to the function we previously mentioned (orange line). The black line is the difference between these two functions. As the graph suggests, this difference is bigger at about v=0.5c.

Such a difference can be measurable, so that it can be tested if the second formula corresponds better to reality.

More importantly, however, we may plot the following graph:

This graph compares the function (blue line)

to the common relativistic expression (red line)

where

The fundamental difference between the two functions is that while in the case of the classical relativistic expression the period T΄ (thus also the corresponding time t΄) will become zero for the moving observer if his/her speed v becomes equal to the speed of light c, in the case of the expression with the positive Lorentz factor γ_L⁺ the same period T΄ will become zero for the moving observer if his/her speed v becomes infinite.

Such an aspect is shown in the graph, as the red line becomes zero at v=c (c=1 in the graph), while the blue line becomes zero if v→∞. This is an illustrative way to show, what we have already mathematically derived, the possibility of faster than light travel.