Crossing the brachistochrone- From the Earth to Alpha Centauri

Figure: Journey to Alpha Centauri

Here is an application of the notion of the brachistochrone. For example, a spaceship travels to Alpha Centauri (the nearest star system to the Sun). The distance between the Sun and Alpha Centauri (Δr in the previous image) is estimated at 4.37 light years:

Alpha Centauri is the closest star system to the Solar System, being 4.37 light-years from the Sun.
[https://en.wikipedia.org/wiki/Alpha_Centauri]

This is the distance which a photon travels ‘on a straight line.’ The route of the spaceship, on the other hand, will be a brachistochrone, because the spaceship will have to accelerate to reach a certain speed.

Here we will suppose that the journey involves the total route ΔS from the one end of the brachistochrone (the Sun), to the other end (Alpha Centauri).

Thus if Δr is the linear distance a photon travels, at some time Δt, and at the speed of light c,

then the distance ΔS the spaceship travels, at a speed v, and at some time ΔT, will be

where ΔT is the time of the brachistochrone

and R is the radius of the brachistochrone.

The point here is that we use the photon to estimate the distance, on one hand, and the time of the brachistochrone to estimate when the spaceship arrives at Alpha Centauri, on the other hand.

Now we will suppose that the spaceship has an average acceleration equal to the acceleration of gravity on the surface of the Earth, g=9.807m/s²:

The average value (of gravity) at the Earth’s surface is, by definition, 9.80665 m/s².
[https://en.wikipedia.org/wiki/Gravity_of_Earth]

We should note that normally the acceleration g refers to the acceleration of gravity, thus it should refer to the average gravitational field between the Sun and Alpha Centauri. However, due to the equivalence between gravitational charge and inertial mass (thus also the equivalence between gravitational and inertial acceleration), the acceleration g can also be seen as the average acceleration produced by the spaceship’s engine.

Incidentally, if we change the units of g from meters per second squared, into light years per year squared, we find out (remarkably enough) that g≈1ly/y²:

This transformation simplifies the calculation, because, in our case, the distance is in light years.

The radius R of the brachistochrone, with respect to the distance Δr, which we may also call ΔL, is given as

so that, given the acceleration g=1.031ly/y², the time of the spaceship’s arrival will be

while the spaceship will have traveled a total distance

at an average speed equal to

If the acceleration doubles, then the time ΔT of the brachistochrone will be √2 times less, so that, for the given distance ΔS, the speed v of the spaceship will be √2 times as much.

So what happens to the relativistic equations, which imply that nothing can travel faster than light? In fact, we have already seen some formulas which can be reduced to the relativistic formulas if the speed of the moving object is less than the speed of light, while they are they still valid if the speed of the object is greater than light.

The key is that by using the notion of the brachistochrone, the general problem is divided into smaller scales. For example, we have previously supposed that the distance to be traveled can be equated to the wavelength of the reference (the ‘emitted’) photon. By doing this, we can also identify the speed of the object with the speed of light ‘per wavelength.’ But if the wavelength is contracted, because of the moving object, then the total speed of the object will be so many times greater than the speed of light, as many contracted wavelengths compose the total distance. Thus we may say that the relativistic expression will be valid as long as we are confined within one wave of spacetime (a wavelength), while a more general formula will be applied when the object exceeds the speed of light, thus ‘jumps’ to the next wave.

Such ideas, and the corresponding formulas, will become clearer after we introduce the energy of the brachistochrone.