**Contents**

Basic principles

Appendix 1

Energy in relativistic damping

Appendix 2

The nature of spacetime

Basic principles

This is a brief exposition of some basic notions concerning the principle of wave-particle duality, and the possibility of faster than light travel. The title of this document refers to the damping factor which appears in the related formulas, and by which a correction to the corresponding relativistic expressions can be made.

Classically the momentum, which we may call

*p*, of a material object of mass_{m}*m*, moving at a speed*v*is
On the other hand, the momentum, which we may call

*p*, of a wave of wavelength_{μ}*λ*is given by de Broglie’s formula
The quantity

*μ*describes the mass of the wave, not the inertial mass*m*of the material object. This wave is associated with the oscillations of the medium (spacetime) in which the object moves. Therefore spacetime is not empty, but it contains some mass*μ*.This can also be seen if we use the formula for Planck energy,

The last term gives Einstein’s formula of mass-energy equivalence. Here however the important distinction is made between the mass

*m*of an object and the mass*μ*of spacetime.
Assuming that the total momentum

*p*of the wave- particle (or spacetime- material object) system is conserved, we have_{0}
The related forces are taken if we differentiate the momenta with respect to time. Here however a distinction will be made between the coordinate time

*t*of an observer on the moving object, and the period*T*of the wave on which the object moves.
Incidentally a straightforward relationship between the coordinate displacement

*x*of the object and the displacement*λ*of the wave can be given by the following formulas
While

*a*is the common acceleration of an object of mass*m*(the inertial mass is always a constant),*β*is the damping constant (*μ*is not a constant), and*c*is the speed of the wave (presumably the speed of light, which is constant). The quantity*ρ*is the density of the wave.It is illustrative to make a comparison with Newton’s second law

The problem here is that the mass

*m*is treated both as a constant and as a changing quantity. This problem is solved by separating the inertial mass*m*of the object from the mass*μ*of spacetime. The mass*μ*can also be seen as the mass of the fuel of the object. But if we imagine that the object (e.g. a spaceship) can collect its fuel directly from space, instead of storing them, then the fuel will be all the available energy (or the mass*μ*equivalent) stored in a region of spacetime.The next to last equation of the forces has the following simple solutions

*β*is in fact negative because the mass

*μ*of the wave decreases).

From the previous equation we can take an expression for the damping factor

*γ*(while

*β*will be the damping constant)

The quantities

*μ*and_{0}*T*refer to the total (initial) mass and period of the wave, while_{0}*v*refers to the total (final) speed of the object._{0}What is significant is that the mass

*μ*of spacetime is not a constant. Thus we can write

Therefore it is supposed that the mass of spacetime reduces exponentially with time (as it is utilized by the moving object).

The damping factor

*γ*plays a central role in the whole analysis

The energy

*E*≡_{m}*E*of the object is the common kinetic energy. The energy_{k}*E*≡_{μ}*E*is the energy of the wave, and can be also called damping energy._{d}We may note that the formulas

A final step is to find the rate of change of the energies with respect to the time

*t*(the rate of change of the damping energy*E*can be expressed either with respect to the coordinate time_{d}*t*or with respect to the period*T*of the wave)
This is the energy of the system. The damping energy

*E*of the wave of spacetime reduces exponentially, as it is transformed into the kinetic energy_{d}*E*of the object moving in spacetime._{k}
A fundamental consequence of this analysis and the related equations is that the object can move faster than light. From the total energy

*E*of the system we have_{0}
Therefore the final speed v

*of the object can be greater than the speed of light*_{0}*c*, as long as its mass*m*is smaller than the mass*μ*which is stored in a region of spacetime. If the object reaches the speed of light, its kinetic energy_{0}*E*will simply be_{k}*1/2mc*. But the total energy^{2}*E*of ‘fuel’ (of the region of spacetime where the object is moving) will still be_{0}*μ*_{0}*c*.^{2}Energy in relativistic damping

Comparison between the relativistic expression for the kinetic energy (red line), and the correction function (green line) for the kinetic energy where damping is included.

The functions depicted in the previous graph (red and green lines) are respectively

As a first note, a comparison with the energy in relativity will be made. In relativity the energy is given by the formula

Here

*m*is the relativistic mass, while*m*is the rest mass. Correspondingly,_{0}*E*is the rest energy. The quantity_{0}*γ*is the Lorentz factor. The relativistic kinetic energy is given as the difference_{L}
If the speed of the moving object is sufficiently small then the relativistic kinetic energy is reduced to the classical expression for the kinetic energy

However if the speed of the object approaches the speed of light, the relativistic kinetic energy goes to infinity

It is assumed that as the speed of the moving object approaches the speed of light, its mass becomes infinite, so that the object cannot exceed the speed of light. However as the mass of the object becomes infinite, its kinetic energy will also be infinite. This is because the Lorentz factor goes to infinity at the speed of light. Therefore the relativistic energy cannot be defined at speeds equal to or greater than the speed of light.

Now in order to make the comparison between the kinetic energy in relativity and the kinetic energy in relativistic damping, we have to express the later energy as a function of the speed

*v*
where we have explicitly noted the damping factor

*γ*as*γ*, so that it is not confused with the Lorentz factor_{d}*γ*._{L}In order to express this energy as a function of the speed

*v*, we set

The meaning of the previous substitution is that at some time interval

*Δt*=*t*equal to the difference*ΔT*=*T*-_{0}*T*, where*T*is the initial period and_{0}*T*is the final period of the wave. If the object has reached the speed of light,*v*=*c*, then the kinetic energy of the object will have increased by a factor of*e**E*of spacetime will all be transformed into the kinetic energy of the moving object only if the speed of the object is much greater than the speed of light,

_{0}
The previous equations are in fact a first order approximation. This can be seen from the pairs

Taking the second order approximation (using the second of the previous pairs), we have

This function for the kinetic energy

*E*_{k}*(v)*is depicted in the previous graph (green curve).
If the speed of the moving object is sufficiently smaller than the speed of light, we take back the relativistic expression for the kinetic energy (red curve in the previous graph)

If the moving object approaches the speed of light, its kinetic energy increases by a factor of

*e-*^{1/2}
But the kinetic energy in relativistic damping can still be defined if the speed of the object is greater than the speed of light

In such a way we can have speeds (infinitely) greater than the speed of light, even if the best energy efficiency is at the speed of light (the derivative of the kinetic energy is zero at the speed of light), while the total energy is always bounded (theoretically all the energy available in the universe). This is possible because spacetime is treated as a real entity with some total mass

*μ*, different from the inertial mass_{0}*m*of any object moving in spacetime.The nature of spacetime

As a second note, the ultimate question is about the nature of spacetime and its oscillations. However the notion of Planck energy may refer to any kind of medium. If we know the wavelength

*λ*of the oscillations of the medium, we can also estimate its mass or energy
This is in fact the energy per wavelength

*λ*of the oscillations. For a macroscopic distance*L*to be traveled, if this distance is composed of*N*wavelengths*λ*, the total energy will be
Simpler, if we divide the distance

*L*into Planck lengths*l*(assuming that this is the smallest possible wavelength), so that the distance_{P}*L*is composed of*n*Planck lengths, then the energy will be
Ιt doesn’t really matter if we call the total energy

*E*‘Planck energy’ or ‘gravitational energy,’ or whether we call space ‘quantum vacuum,’ or ‘spacetime,’ as long as we know how much the total energy is.
The equivalence between the various forms of energy, which goes beyond the principle of wave- particle duality, can be shown for example if we include the mass

*m*and wavelength_{g}*λ*of the graviton in the previous energy equation_{g}
Considering these parameters of the problem, some interesting coincidences of universal nature arise. If

*M*and_{U}*R*is the mass and radius of the observable universe respectively, and we call_{U}*ρ*the linear density of the observable universe, then we have_{U}
The last equation relates the ratios of the various fundamental parameters (presumably constants) of the problem to each other, and expresses mathematically the numerical coincidence between the same parameters. In fact the mass

*M*of the observable universe was estimated here so that the ratio of the mass_{U}*M*over the radius_{U}*R*of the observable universe coincides with the ratio of Planck mass_{U}*m*over Planck radius_{P }*r*(as was previously defined in relation to Planck length_{P}*l*). We also see that the product of the linear density_{P}*ρ*of the observable universe by Newton’s gravitational constant_{U}*G*is equal to the speed of light*c*squared.Based on such coincidences, we can also consider another one

The acceleration ℊ which appears in the last equation is a uniform acceleration throughout the universe, and if we accept the numerical coincidence between the graviton’s mass

*m*and Planck’s constant_{g}*h*(transformed into light units), so that the graviton’s wavelength*λ*is exactly_{g}*1ly*, then the value of ℊ is indeed*1ly/y*.^{2}
The previous calculations were made in brief, but the replacements are straightforward. The numerical coincidences involved portray the equivalence between the various forms of energy which may appear in the problem, so that we can ultimately link them to the kinetic energy of a (material) object moving in space (the medium)

The previous equations refer to total energies, thus also to the total (final) speed

*v*. According to these equations, in order for an object (e.g. a spaceship), traveling in spacetime along a distance*L*, to reach the speed of light, it is sufficient to consume a mass*M*, found along the same distance*L*, equal to ½ its own inertial mass*m*.
Assuming for the spaceship a mass

*m*equal to that of a modern supercarrier, if the distance*L*to be traveled is the radius*R*of the whole observable universe, then the spaceship’s final speed will be_{U}
For a more detailed analysis of this subject, on can see another document of mine, ‘Crossing the brachistochrone,’ on the following link:

[https://archive.org/details/CrossingTheBrachistochrone_201804]

© 2018 Chris C. Tselentis

Last updated: 4/18/2018

Mailto: christselentis@gmail.com

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