Crossing the brachistochrone- Black holes

Image: Illustration of a black hole as a gravitational well of brachistochrones

Another application of the brachistochrone, besides quantum entanglement (although here it is at the macroscopic level), is black holes.

The previous graph is similar to the next to last graph, describing quantum entanglement, with one exception. Here the edges of the successive brachistochrones S_n are repositioned at the edges of the first brachistochrone S₀, corresponding to the points (-x₁,x₁) in the next to last graph).

Although such brachistochrones (with the exception of the first one S₀) are not ‘true’ brachistochrones (their equations do not correspond to that of a cycloid), each one respectively has the same total length L_n (thus also the same radius R_n) with the ‘true’ brachistochrones in the next to last graph.

This way, even if the shape of the brachistochrone is ‘deformed,’ we can treat the deformed curves as new brachistochrones S_n at a higher harmonic n. Thus the last graph may illustrate the black hole as a ‘gravitational well’ of brachistochrones.

Beyond this, the considerations made in the previous section, about quantum entanglement, including the related formulas, can also be applied in the case of black holes.

An additional element is that the mass M_B (or M_n at some state n) is assumed to be concentrated on the focus (the center of the generating circle) of the respective brachistochrone S_n. Thus the mass M_n represents the mass of the black hole or radius R_n.

As far as far the acceleration g_n is concerned, we should mention that, in the original problem of the brachistochrone, this acceleration is the acceleration of gravity. However, because of the equivalence between the gravitational charge and the inertial mass of an object (that an accelerating object creates in spacetime a field equivalent to a gravitational field), we may treat the gravitational acceleration as equivalent to the acceleration generated, or gained, by the moving object, during its journey on the brachistochrone.

Therefore, if M_n is the mass of the black hole, and R_n is its radius, at some state n, then for the acceleration g_n we may write that,

where here we have used the index ‘0’ for the state n=1.

The proposal that the mass of the black hole M_B increases by a factor of n², as the state n increases, can be based on the following general energy equation,

which can also be expressed in relation to the state n,

so that

The energy equation will be explored in more detail in the next section.

The significance of this result is that the speed of an object moving across the black hole will be,

Since, presumably, the inertial mass m₀ of an object will be much less than the mass M_B of the black hole, then the final speed v₀ of the object will be much greater than the speed of light c.

Such a result is possible only if we distinguish between the two masses m₀ and M_B, where the latter mass refers to the properties of spacetime (the black hole in this case), while the former mass refers to the properties of an object (moving across the same region).

The time it takes for the object to move across the black hole, will be the time T_B of the brachistochrone,

Using now the equations for the energy and for the gravitational acceleration, we can take an expression for the radius R_B of the black hole (presumably the radius of its event horizon),

Here we may compare the previous formula to that which gives the Schwarzschild radius of a black hole,

We see that, while Schwarzschild formula does not depend on the speed of the object, the formula for R_B tells us that the event horizon of the black hole shrinks if the object moves faster than light.

Furthermore, the expression

is also interesting in the sense that it reveals a relationship between the inertial mass m₀ of the object, and the radius R_B of the black hole.

The dependence of the properties of the black hole, to those of a material object moving with respect to the black hole, can also be seen if we use a photon of reference, as we have already done in quantum entanglement. If λ is the photon’s wavelength, then, we have

Among other things, the previous formula shows the dependence of the wavelength of the reference photon on the speed of the object.

The reference photons can either be seen as photons emitted by the moving object, or as fluctuations on the event horizon L_B (the rim) of the black hole of radius R_B, perceived in the form of photons.

Incidentally, we should mention here that in the formula for the wavelength λ,

if we equate the speed of the object v to the speed of light c, we take

where l_P stands for Planck length, and r_P stands for Planck radius.

The last relationship implies that an observer on board an object approaching, or crossing, the event horizon of a black hole at the speed of light, v=c, will measure the wavelength λ of the oscillating event horizon equal to Planck length l_P. If his/her speed is greater than the speed of light, the same wavelength will be even shorter than Planck length, as the black hole, according to him/her, will shrink to a singularity of zero dimension, if his/her speed becomes infinite.

Notes:

Here is a way to estimate the wavelength λ decrease (or frequency f increase) of the reference photon, with respect to the speed v of an object of mass m. From the energy equation we have,

Noting λ₀ and λ΄ the initial and final wavelength respectively (or f₀ and f΄ the frequency), and E₀ and E΄ the corresponding energy, we take

where

For the total energy we have

Thus, for the speed v of the object we take

and for the final frequency f΄ of the reference photon we take

where

The quantity f_m has units of frequency, and it can be associated with the motion of the object.

If we set

then the object will be motionless, and there will not be any change in the frequency f΄.

On the other hand, setting

then the frequency f_m of the moving object will be equal to the initial frequency f₀ of the photon, and the frequency f΄ of the fluctuations of spacetime will seize.

The point is that since the reference photons are fluctuations (oscillations) of the medium (spacetime), and because the energy of spacetime is proportional to the frequency of those fluctuations, as the moving object transforms this energy into its own kinetic energy, then the frequency of the fluctuations will decrease, as long as the speed of the object increases.

This is analogous to the common experience we have when moving in the sea with a ship. If the speed of the ship increases, we perceive that the frequency of the waves increases (we hear the waves hitting the ship more often). But this is the frequency related to our motion. In contrast, an external observer may expect that the frequency of the waves decreases, although such a change is miniscule compared to the vast amount of energy stored in the waves.

The frequency f΄ corresponds to the reduced frequency of the oscillations within a harmonic n. A simpler formula can be taken assuming the motion at any harmonic n, as follows.

Solving the previous formula which gives the frequency f΄, for the initial frequency f₀,

and supposing that the speed of the object is large enough, so that

then we may identify the frequency f₀ with the frequency of the oscillations at the first state, n=1, so that, with respect to the harmonics n of the fluctuations, the speed v_n and the frequency f_n, at any state n, will be

This can also be seen from the energy E_n, at any state n,

Setting now

we take

The last frequency, which, incidentally, will be equal to Planck frequency (in analogy to the wavelength λ₀ we earlier saw),

can be seen as the frequency of the oscillations on the event horizon of the black hole, as the object of mass m crosses the event horizon at the speed of light.

In any case, the notion of the brachistochrone can be used to represent a two- dimensional model of a black hole, where the successive excited states n of the black hole depend on the speed of an object approaching or crossing the black hole at a corresponding speed v_n, so that, according to an observer on this object, both the radius R_n of the event horizon of the black hole, and the wavelength λ_n of its oscillations, will shrink.