Here we will make a comparison between the formulas for the amplitude and the energy of the driven and damped harmonic oscillator, and the respective formulas in relativity.
One may wonder what the relationship could be between an object moving in spacetime, and an object oscillating ‘on a spring.’ Nevertheless, we have already seen in previous examples (the ship- in- the- sea, and the rocket- in- space, or even the spaceship- on- the- brachistochrone) that if spacetime is an oscillating medium, then the motion of an object can be approximated by the equations of an oscillator. The connection between the oscillator and the brachistochrone will be established later on.
The new element here is that, besides the harmonic and the damping force of the damped harmonic oscillator, we also have the driving force. This force can either be seen as independent from the other forces, imposed by an external source onto the oscillator, or, better, as a force intimately related to the other forces, caused by the moving object itself (by the engine of a spaceship for example). Consequently, in the latter sense, the motion of the object can be described by the equation of the driven and damped harmonic oscillator.
As far as the displacement of the moving object is concerned, instead of regarding a vertical displacement y(t), we may equivalently consider a horizontal displacement x(t). But the description of the problem will be the same. As long as the moving object can be seen as a mass ‘hanging from a spring’ (where in this case the spring is a wave in spacetime), the amplitude of the oscillation y(t)=y0(ω)cos(ωt) will give the displacement of the object.
Now will make some considerations with respect to the speed of the object. The speed is taken from the equation of the oscillator, and its steady state solution,
where
is the amplitude of the speed v(t).
The speed v(t) takes its maximum value, at ω=ω0, and at some time t=T0. To see this, using the boundary condition
and the values for the amplitude y0(ω) at ω=0, and ω=ω0,
where
we have that,
Thus the speed
can be no greater than the speed of light c (presumably the speed of the propagating wave).
Alternatively, we can define a speed of the following form,
The perspective of this speed is that it can be greater than the speed of light c,
The number n which appears here refers in fact to the harmonics of the oscillator,
Thus the speed
can be directly linked to all our previous considerations about motion in spacetime, and motion on the brachistochrone. Such motion will be further described later on.
A comparison between the speed vn, and the speed amplitude v0(ω), is the following one,
where, setting n=1, vn=v0, we have
so that, specifically, for the speed vn we have that,
After defining such a speed, in order to make the comparison with the relativistic expressions for the displacement and the energy, we need first to define the damping factor γ(ω) with respect to the speed vn, where
To do this, we have to manipulate the fraction, with the angular frequencies ω and ω0, which appears in the square root of the previous expression.
Introducing the wavenumber,
we may express the angular frequencies ω and ω0 with respect to the speeds v and c, respectively, as follows,
Thus the quantity which appears in the expression for the damping factor γ(ω), takes the form,
where here the speed v refers to the speed vn, and the damping factor γ takes the form
Now, in order to find a corresponding expression for the amplitude of the displacement y0(ω), with respect to the speed v, from the following formulas
and substituting
we have
Before plotting the graph for the amplitude of the displacement y0(v), we will also find the corresponding expression for the energy amplitude with respect to the speed v, and then make any assumptions necessary in order to simplify the expressions.
The energy amplitude Em(ω), or simply E(ω), is given, as we have already seen, by the following formula,
The same amplitude can be expressed with respect to the speed v, as follows
and in an expanded form as
Now we are able to plot the following couple of graphs. The first one refers to the amplitude of the displacement y0(v):
The function depicted is,
where
The previous graph is similar to the one we plotted earlier for the amplitude y0(ω), where here the orange line corresponds to a small damping factor (γ0=0.1), the green line corresponds to critical damping (γ0=2ω0, ω0=1), while the red line corresponds to a damping factor equal to one (γ0=1).
The difference here is that we have included the relativistic expressions for the displacement of the moving object (black dotted lines),
or
where
is the relativistic Lorentz factor.
The reason why I have included both relativistic expressions, is because it is rather ambiguous which distance is ‘contracted’ or ‘expanded’ in relativity.
For example, supposing that the displacement (the amplitude) is expanded with respect to a moving observer, then we have to compare the upward black dotted line (corresponding to γL) to the orange line in the previous graph (corresponding to a small damping factor, γ0=0.1, γ02≈0). But, as the speed of the moving object v reaches the speed c of the oscillator, while the amplitude according to relativity will go to infinity, the amplitude can still be defined according to the orange line, even if v>c.
If we compare the upward black dotted line, to the red line (corresponding to a damping factor equal to one (γ0≈ω0≈1), then these two lines begin to diverge somewhere at v=0.5c. But, while again the amplitude for the moving observer, according to relativity, will go to infinity, if v=c, the amplitude y0(v) will be equal to the original amplitude y0, at v=c, according to the red line, while it can still be defined, if v>c.
On the other hand, if we suppose that the distance is contracted with respect to a moving observer, then we may compare the downward black dotted line (corresponding to 1/γL), to the green line in the previous graph (corresponding to critical damping, γ0=2ω0≈2, ω0=1). These two lines begin to diverge somewhere at v=0.7c. But, while, in this case, the displacement according to relativity will go to zero, as the speed of the moving object v reaches the speed of light c, the displacement can still be defined according to the green line, even if v>c.
Similar considerations can be made with respect to the amplitude of the energy E(v), according to the following second graph:
The function depicted is
where, again,
The red, orange, and green lines correspond to a damping factor γ0=1, γ0=0.1 (γ02≈0), and γ0=2ω0=2, ω0=1, respectively.
The relativistic expression for the energy is given by the following formula,
This notation suggests that the total energy is EEin, because the Lorentz factor γL increases as the speed v of the object increases, while the total energy will go to infinity if the speed v of the object reaches the speed of light c,
(black dotted lines in the previous graph).
On the other hand, we have the energy amplitude E0(ω), or E0(v), as it was defined here,
Supposing that the speed of the moving object v is larger than the speed of light c, we have that
so that the energy (which in fact at large speeds is purely kinetic energy) of the object can be defined for all speeds, v>>c, as it goes to zero, E(v)→0, only if the speed of the object goes to infinity, v→∞.
On the other hand, at small speeds, v<<c, the energy amplitude E(v) will approximately be equal to the initial energy E0 (which is mostly elastic energy at small speeds).
If the speed is close to the speed of light, v≈c, which is the case of critical damping, γ0=2ω0=2κc, as we have already seen it will be,
Thus at the speed of light the moving object utilizes half the available energy.
If we assume a small damping factor, then the energy amplitude will go to infinity, if the speed of the object approaches the speed of light. This would be the consequence of ignoring the medium (spacetime), in which objects move. By acknowledging the existence of the medium, and supposing that it oscillates, then we may use the equations of an oscillator in order to describe the motion of the object. Such equations give us for the displacement and the energy of the object formulas which can be defined even if the speed of the object exceeds the speed of light. Later on we will establish the connection between these equations and motion on the brachistochrone.
One may wonder what the relationship could be between an object moving in spacetime, and an object oscillating ‘on a spring.’ Nevertheless, we have already seen in previous examples (the ship- in- the- sea, and the rocket- in- space, or even the spaceship- on- the- brachistochrone) that if spacetime is an oscillating medium, then the motion of an object can be approximated by the equations of an oscillator. The connection between the oscillator and the brachistochrone will be established later on.
The new element here is that, besides the harmonic and the damping force of the damped harmonic oscillator, we also have the driving force. This force can either be seen as independent from the other forces, imposed by an external source onto the oscillator, or, better, as a force intimately related to the other forces, caused by the moving object itself (by the engine of a spaceship for example). Consequently, in the latter sense, the motion of the object can be described by the equation of the driven and damped harmonic oscillator.
As far as the displacement of the moving object is concerned, instead of regarding a vertical displacement y(t), we may equivalently consider a horizontal displacement x(t). But the description of the problem will be the same. As long as the moving object can be seen as a mass ‘hanging from a spring’ (where in this case the spring is a wave in spacetime), the amplitude of the oscillation y(t)=y0(ω)cos(ωt) will give the displacement of the object.
Now will make some considerations with respect to the speed of the object. The speed is taken from the equation of the oscillator, and its steady state solution,
The speed v(t) takes its maximum value, at ω=ω0, and at some time t=T0. To see this, using the boundary condition
Alternatively, we can define a speed of the following form,
A comparison between the speed vn, and the speed amplitude v0(ω), is the following one,
Introducing the wavenumber,
The energy amplitude Em(ω), or simply E(ω), is given, as we have already seen, by the following formula,
The displacement y0(v) of an object as a function of its speed v
The function depicted is,
The difference here is that we have included the relativistic expressions for the displacement of the moving object (black dotted lines),
The reason why I have included both relativistic expressions, is because it is rather ambiguous which distance is ‘contracted’ or ‘expanded’ in relativity.
For example, supposing that the displacement (the amplitude) is expanded with respect to a moving observer, then we have to compare the upward black dotted line (corresponding to γL) to the orange line in the previous graph (corresponding to a small damping factor, γ0=0.1, γ02≈0). But, as the speed of the moving object v reaches the speed c of the oscillator, while the amplitude according to relativity will go to infinity, the amplitude can still be defined according to the orange line, even if v>c.
If we compare the upward black dotted line, to the red line (corresponding to a damping factor equal to one (γ0≈ω0≈1), then these two lines begin to diverge somewhere at v=0.5c. But, while again the amplitude for the moving observer, according to relativity, will go to infinity, if v=c, the amplitude y0(v) will be equal to the original amplitude y0, at v=c, according to the red line, while it can still be defined, if v>c.
On the other hand, if we suppose that the distance is contracted with respect to a moving observer, then we may compare the downward black dotted line (corresponding to 1/γL), to the green line in the previous graph (corresponding to critical damping, γ0=2ω0≈2, ω0=1). These two lines begin to diverge somewhere at v=0.7c. But, while, in this case, the displacement according to relativity will go to zero, as the speed of the moving object v reaches the speed of light c, the displacement can still be defined according to the green line, even if v>c.
Similar considerations can be made with respect to the amplitude of the energy E(v), according to the following second graph:
The energy amplitude E(v) as a function of the speed v
The function depicted is
The relativistic expression for the energy is given by the following formula,
On the other hand, we have the energy amplitude E0(ω), or E0(v), as it was defined here,
On the other hand, at small speeds, v<<c, the energy amplitude E(v) will approximately be equal to the initial energy E0 (which is mostly elastic energy at small speeds).
If the speed is close to the speed of light, v≈c, which is the case of critical damping, γ0=2ω0=2κc, as we have already seen it will be,
If we assume a small damping factor, then the energy amplitude will go to infinity, if the speed of the object approaches the speed of light. This would be the consequence of ignoring the medium (spacetime), in which objects move. By acknowledging the existence of the medium, and supposing that it oscillates, then we may use the equations of an oscillator in order to describe the motion of the object. Such equations give us for the displacement and the energy of the object formulas which can be defined even if the speed of the object exceeds the speed of light. Later on we will establish the connection between these equations and motion on the brachistochrone.
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