To the Land of Dreams: Relativistic Damping

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_m, of a material object of mass m, moving at a speed v is

On the other hand, the momentum, which we may call p_μ, of a wave of wavelength λ is given by de Broglie’s formula

Combining this formula with that of Compton’s wavelength

we take

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 p₀ of the wave- particle (or spacetime- material object) system is conserved, we have

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

Thus we have

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

where we took the measure of these quantities, ignoring thus any negative sign (the rate 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 μ₀ and T₀ refer to the total (initial) mass and period of the wave, while v₀ refers to the total (final) speed of the object.

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 next step is to define the energy equation of the wave-object system

The energy E_m≡E_k of the object is the common kinetic energy. The energy E_μ≡E_d is the energy of the wave, and can be also called damping energy.

We may note that the formulas

give us respectively a first and second order relationship between the masses and the speeds.

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_d can be expressed either with respect to the coordinate time t or with respect to the period T of the wave)

This is the energy of the system. The damping energy E_d of the wave of spacetime reduces exponentially, as it is transformed into the kinetic energy E_k of the object moving in spacetime.

A fundamental consequence of this analysis and the related equations is that the object can move faster than light. From the total energy E₀ of the system we have

Therefore the final speed v₀ of the object can be greater than the speed of light c, as long as its mass m is smaller than the mass μ₀ which is stored in a region of spacetime. If the object reaches the speed of light, its kinetic energy E_k will simply be 1/2mc². But the total energy E₀ of ‘fuel’ (of the region of spacetime where the object is moving) will still be μ₀c².

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 m₀ is the rest mass. Correspondingly, E₀ is the rest energy. The quantity γ_L is the Lorentz factor. The relativistic kinetic energy is given as the difference

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_d, so that it is not confused with the Lorentz factor γ_L.

In order to express this energy as a function of the speed v, we set

This substitution is justified by the expression for the damping factor

The meaning of the previous substitution is that at some time interval Δt=t equal to the difference ΔT=T₀-T, where T₀ is the initial period and 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

But the total energy E₀ 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,

If we compare the previous energy to the relativistic kinetic energy, we have

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 μ₀, different from the inertial mass 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_P (assuming that this is the smallest possible wavelength), so that the distance 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_g and wavelength λ_g of the graviton in the previous energy equation

Considering these parameters of the problem, some interesting coincidences of universal nature arise. If M_U and R_U is the mass and radius of the observable universe respectively, and we call ρ_U the linear density of the observable universe, then we have

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_U of the observable universe was estimated here so that the ratio of the mass M_U over the radius R_U of the observable universe coincides with the ratio of Planck mass m_Pover Planck radius r_P (as was previously defined in relation to Planck length l_P). We also see that the product of the linear density ρ_U of the observable universe by Newton’s gravitational constant 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_g and Planck’s constant h (transformed into light units), so that the graviton’s wavelength λ_g is exactly 1ly, then the value of ℊ is indeed 1ly/y².

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_U of the whole observable universe, then the spaceship’s final speed will be