In the previous sections (the ship- in- the- sea, and the rocket- in- space), we saw the intrinsic relationship between spacetime and the motion of an object in spacetime. In both cases the object is able to move by reducing the energy of the wave of spacetime, gaining thus kinetic energy.
Such an intimate relationship between objects and spacetime, which led Einstein to the mass- energy equivalence (expressed as E=mc2), is described in physics through the notion of wave-particle duality, and the uncertainty principle (which can also be called the complementarity principle).
The momentum p of a wave is expressed by de Broglie’s formula
where h is Planck constant, and λ is the wavelength associated with the wave-like aspects of a material object.
The momentum of a material object (or particle) is given by the common expression
where m is the inertial mass of the object, and v is its speed.
Combining the two previous formulas for the momentum, we take
This expression relates the properties of a wave (the wavelength λ), to those of a particle (the mass m, and speed v), or of any material object. The constant of proportionality h (Planck’s constant), connects quantum physics with classical physics.
The same formula also expresses the momentum-position uncertainty principle. Such an uncertainty can be stated as
where ħ is called reduced Planck constant.
The uncertainty in fact has to do with the aspect of complementarity between the wave-like properties of spacetime, and the material-like properties of an object moving in spacetime.
Another fundamental expression in quantum physics, closely related to de Broglie’s formula, is Planck energy
where the index ‘P’ stands for ‘Planck.’ The second formula expresses the speed of the wave c as the product of its frequency f by its wavelength λ.
From the formula of Planck energy we can take the equivalent expression for the energy-time uncertainty,
If we combine Planck energy with de Broglie momentum, we take that
Strictly speaking, the momentum p=h/λ refers to that of a wave, while the momentum p=mv refers to that of an object. But if we literary equate the two expressions, we have
Setting v=c in the last expression, we take
It has been such a substitution which led to the confusion between the energy of a wave and the energy of a material object, because the mass m does not refer to a wave, while the energy EP does not refer to a particle (a material object). Thus the aspect that material objects have wavelike properties was a conclusion based on this misidentification.
This is what made me suspect that in the last expression m does not refer to the mass of the object (the constant of inertia) but to the mass of the wave.
Thus here we can make the following distinction between the two momenta. Calling pm the momentum of the object, and pμ the momentum of the wave, we have
where m will be the mass of the object, while μ will be the mass of the wave.
If we compare these expressions to Newton’s second law of motion,
where the index ‘N’ in FN stands for ‘Newton,’ we see that we can identify two forces, a force Fm, related to the material object, and another force Fμ, related to the wave,
where m is the inertial mass of the object, and c is the speed of the wave, so that
Comparing the expressions for the forces Fm and Fμ, to Newton’s formula for the force FN, we see that
This formula can also be written as follows. Substituting,
we take
where ρ is the density of the wave, and b is the damping constant, which we have already seen.
The formula bv=ρc2 gives us the equivalence between the properties of the wave (μ,c), and those of a material object (m,v), moving in the wave.
Thus the equation of motion of the moving object can be written in two equivalent ways,
where
From the equation of motion, we can also derive an equation for the energy of the system. Having,
we take that,
where μ0c2 will be the total energy stored in (a region of) spacetime.
This equation for the energy of the system is equivalent to the one we previously saw for the rocket- in- space, where
To see this, we may set
so that
The difference in this energy will be the kinetic energy Em of the object,
so that for the total energy we have the equivalent terms,
The factor of ½, which was omitted in the previous expression, is not significant and may disappear if we include a harmonic term in the equation of motion, as we will do later on.
The basic aspect here is that the energy Eμ=μc2 can be identified with the damping energy of the system, (so that the associated force Fμ can also be identified with the damping force). This is the energy stored in the wave of spacetime, and it is transformed by the moving object into its own kinetic energy Ek,
Therefore the wave-like properties do not refer to the moving material object, but to spacetime. Such a conclusion can only be drawn if we make the distinction between the two masses, the inertial mass m of the object moving in spacetime, and the mass μ of the wave associated with the oscillations of spacetime.
Another remark we can make, is that the expression for the damping energy stored in the wave
is handled much easier than if we tried to calculate the damping energy directly from the following integral,
where the index ‘d’ stands for ‘damping.’
This is also true for the rate of change of the energy
Furthermore, comparing the previous expression to the one we have already seen,
we take that
This is a second formula of equivalence, additional to the earlier formula
Α more accurate description can be made if we separate the coordinate time t, related to a clock on the moving object, from the period T, referring to the oscillations of the wave. These two times need not be the same, if the object is moving at a speed v different from the speed of the wave (presumably the speed of light c).
A direct comparison between the times t and T can be made if we set,
so that
The coordinate y measures the displacement of the moving object on a linear axis y, while λ is the displacement of the wave on its curved path.
If the displacement y is identified with the amplitude of the wave, then it will simply be λ=2πy. In this case, it will also be
so that from the first formula of equivalence bv=ρc2,
we take the second formula of equivalence.
The terms in the latter expression may be accompanied by a negative sign, since the energy on the right side of the equation (referring to the energy stored in a region of spacetime) is transformed into the energy on the left side (referring to the kinetic energy of the object moving in the same region).
Significantly, if the speed v of the object approaches the speed of light c, then the rate at which the wave loses mass μ, will be equal to the rate at which the object loses mass m (or burns fuel),
so that we may suggest that at the speed of light the object will have the best energy efficiency.
Notes:
The envelope (the green line) travels at the group velocity. The carrier wave (the blue line) travels at the phase velocity. Group velocity and phase velocity are not necessarily the same.
[http://physics.gmu.edu/~dmaria/590%20Web%20Page/public_html/qm_topics/phase_vel/phase.html]
This is an example of how the introduction of the mass μ of the wave helps us solve the problem of phase velocity. The previous picture illustrates a wave-packet which travels in space. In the context of wave-particle duality, an object of mass m travelling at a speed v will have a momentum p=mv, and it will be accompanied by a ‘guiding’ wave, whose velocity will be the phase velocity vp=fλ, f being the frequency, and λ the wavelength of the wave. If we use the formulas
for the momentum and the energy, respectively, of a material particle (as if it behaved like a wave), we take that
This creates the paradox that the phase velocity can be greater than the speed of light. In quantum physics the paradox is resolved by using the notion of group velocity, which never exceeds the speed of light. Yet the problem of the phase velocity remains, if we confuse the properties of the moving object (m,v) with those of the guiding wave (μ,c). But by acknowledging this distinction, we have that
so that the phase velocity will simply be
where the previous formulas for the momentum and the energy strictly refer to a wave.
Thus while the energy Eμ=μc2 refers to the energy of a wave (whose speed c will be that of light), the energy Em=mc2 refers to the kinetic energy of an object of mass m which, in this case, travels at the speed of light, v=c.
Such an intimate relationship between objects and spacetime, which led Einstein to the mass- energy equivalence (expressed as E=mc2), is described in physics through the notion of wave-particle duality, and the uncertainty principle (which can also be called the complementarity principle).
The momentum p of a wave is expressed by de Broglie’s formula
The momentum of a material object (or particle) is given by the common expression
Combining the two previous formulas for the momentum, we take
The same formula also expresses the momentum-position uncertainty principle. Such an uncertainty can be stated as
The uncertainty in fact has to do with the aspect of complementarity between the wave-like properties of spacetime, and the material-like properties of an object moving in spacetime.
Another fundamental expression in quantum physics, closely related to de Broglie’s formula, is Planck energy
From the formula of Planck energy we can take the equivalent expression for the energy-time uncertainty,
This is what made me suspect that in the last expression m does not refer to the mass of the object (the constant of inertia) but to the mass of the wave.
Thus here we can make the following distinction between the two momenta. Calling pm the momentum of the object, and pμ the momentum of the wave, we have
If we compare these expressions to Newton’s second law of motion,
The formula bv=ρc2 gives us the equivalence between the properties of the wave (μ,c), and those of a material object (m,v), moving in the wave.
Thus the equation of motion of the moving object can be written in two equivalent ways,
so that
This equation for the energy of the system is equivalent to the one we previously saw for the rocket- in- space, where
The basic aspect here is that the energy Eμ=μc2 can be identified with the damping energy of the system, (so that the associated force Fμ can also be identified with the damping force). This is the energy stored in the wave of spacetime, and it is transformed by the moving object into its own kinetic energy Ek,
Another remark we can make, is that the expression for the damping energy stored in the wave
This is also true for the rate of change of the energy
A direct comparison between the times t and T can be made if we set,
If the displacement y is identified with the amplitude of the wave, then it will simply be λ=2πy. In this case, it will also be
The terms in the latter expression may be accompanied by a negative sign, since the energy on the right side of the equation (referring to the energy stored in a region of spacetime) is transformed into the energy on the left side (referring to the kinetic energy of the object moving in the same region).
Significantly, if the speed v of the object approaches the speed of light c, then the rate at which the wave loses mass μ, will be equal to the rate at which the object loses mass m (or burns fuel),
Notes:
[http://physics.gmu.edu/~dmaria/590%20Web%20Page/public_html/qm_topics/phase_vel/phase.html]
This is an example of how the introduction of the mass μ of the wave helps us solve the problem of phase velocity. The previous picture illustrates a wave-packet which travels in space. In the context of wave-particle duality, an object of mass m travelling at a speed v will have a momentum p=mv, and it will be accompanied by a ‘guiding’ wave, whose velocity will be the phase velocity vp=fλ, f being the frequency, and λ the wavelength of the wave. If we use the formulas
Thus while the energy Eμ=μc2 refers to the energy of a wave (whose speed c will be that of light), the energy Em=mc2 refers to the kinetic energy of an object of mass m which, in this case, travels at the speed of light, v=c.
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