Crossing the brachistochrone- Quantum entanglement

Figure: Two photons are entangled, and travel on the horizontal axis x in opposite directions. The entanglement is achieved by a signal (a brachiston) travelling on the curved line (the brachistochrone) which connects the two photons. The time it takes the two photons to interact is the time of the brachistochrone.

In the previous picture, all the curves S_n (brachistochrones) are also tautochrones (brachistochrone= shortest time, tautochrone= same time). This is not the same brachistochrone, on which different objects end up at the bottom at the same time from different heights, but different brachistochrones, on which the same object reaches the bottom of each brachistochrone, supposedly, at different time.

According to experiments on quantum entanglement, the two entangled particles communicate one with the other ‘instantaneously.’ However the time of the brachistochrone may offer the opportunity to measure such ‘instantaneity’ (supposing that the time can be very small, but not zero).

The photons travel on the linear path Δx_n (the interval [-x_n…x_n] in the previous graph), so that if it takes for a photon a time Δt_n to travel this distance at the speed of light c, and each interval Δx_n is n times bigger than the initial interval Δx₁, then we have that,

where

On the other hand, the brachiston will travel on the curved line S_n, which connects the two ends of the linear distance Δx_n, so that, in general, it will be

where v is the speed of the brachiston, and T is the time of the brachistochrone.

The advantage of having introduced the notion of the brachistochrone, is that each brachistochrone S_n is directly related to the length of the brachistochrone L_n, where the length L_n corresponds to the distance Δx_n by which the two entangled photons are separated at some time interval Δt_n,

For simplicity we will assume that the length of the brachistochrone S_n is analogous to the length of the brachistochrone’s rim L_n, so that

The problem here is that we have to define some initial condition, referring to the separation Δx₁ (the interval [-x₁…x₁] in the previous graph) between the two photons. This can be done if we assume that at the initial separation, L₁=Δx₁=cΔt₁, the brachiston will have to travel as fast as the speed of light, so that the photons are also initially entangled. Thus, it will be

Therefore, due to the initial condition, the time of the brachistochrone T₁, will be equal to the time t₁ of the photon.

As the photons fly apart from each other, the distance L_n will increase,

As a consequence, the speed v_n of the brachiston will be n times greater than the speed of the photons c, as the separation L_n increases by a factor of n.

This can be done without violating causality, as the brachiston (although travelling faster than light) will meet the photon (the carrier of information) at the end of the journey.

Another problem is to define some acceleration g, in relation to the time of the brachistochrone T, which by definition is

where R is the radius of the brachistochrone.

Solving for the initial separation L₁, we have

Defining the speed v_n,

in such a way that the speed of light c stays constant,

we may also solve the time of the brachistochrone T_n for some final separation L_n,

where

so that

The formulas above relate different parameters of the system to each other, taking into account the given initial condition.

Incidentally, the same formula can be derived if we consider an energy equation of the form,

where the factor of ½ in front of the kinetic energy was dropped for simplicity, and

where

Additionally, we have the properties related to the photons, given by the equation

where λ is the photon’s wavelength, and τ is its period.

If, in addition to the initial condition that the initial speed v₁ of the brachiston which entangles the two photons is the speed of light c,

we also assume that the initial separation L₁ of the photons is comparable to their wavelength λ₁, then it will be

so that, for some final separation L_n, we have that

where

and

The last equation represents a condition of simultaneity, or synchronicity.

The aspect that the photons’ period τ_n decreases, while their harmonic n increases, is a consequence of the definitions and assumptions given above.

This can also be seen from the last equation, where the speed of the brachiston v_n is inversely proportional to the period of the photon τ_n, since their product is constant.

A more general relationship between the same two parameters, v_n and τ_n, can be taken if we assume an energy equation of the form,

The last equation shows that the period of the photon τ_n, is inversely proportional to the speed of the brachiston v_n.

If here we set

equating thus, for some reason, the mass m of the brachiston, to the mass-equivalent of a region of spacetime of size λ₁, we take

so that we retrieve the condition of synchronicity we met earlier.

Notes:

The main point is that we cannot define the time of the brachistochrone, thus the time of the entanglement, independently of the photons, that is their wavelength, or period.

This can be seen if we add another energy term of the form Mc², and using the indices ‘B’ for ‘Brachistochrone,’ and ‘p’ for ‘photon,’ so that

where the mass M_B refers to the mass of the brachistochrone (the mass in the region of spacetime where the object of mass m travels).

Replacing

and identifying the acceleration g with the common gravitational acceleration,

we have

Thus the time of the brachistochrone T_B will be

where λ_p and τ_p is respectively the wavelength and period of the reference (the entangled) photons, and ħ is (the reduced) Planck constant.

If in the last relationship we substitute

where t_P is Planck time, we take

This is a simple formula to estimate the time it takes for the entanglement to occur.

This time however cannot be defined independently of the period of the entangled photons. Thus the problem is how one can measure such period, since, presumably, it will be extremely low (even smaller than Planck time), if the time of the entanglement T_B is miniscule.

As far as the aforementioned energy relationships are concerned, they will become clearer after the energy of the brachistochrone is introduced.