Propagation Issues

The last couple of discussions have focused on the issue of causality in respect to how, and possibly why, energy propagates through the universe. As a broad generalisation, a wave is an understood mechanism which can dissipate a localised excess energy, such that the wave media can return to a point of equilibrium associated with the lowest energy state possible for the system. We might attempt to initially visualise this model as being similar to the effect of dropping a pebble into a pond, where the initial central displacement is dissipated through a series of expanding 2D surface waves, which actually needs to be translated into a more complex 3D model. While it is a general assumption that energy is proportional to the square of the wave amplitude [A²], this is not always true. For there is a point, where the wave has zero amplitude, although within the confines of the wave model being described, we might realise that the wave amplitude [A], only reflects potential energy. However, when the amplitude is zero, this coincides with the maximum, and equivalent, amount of kinetic energy [mv²] associated with the point in the wave media in motion. In a lossless wave system, the total wave system always has a constant amount of energy, although when distributed across an expanding 3D surface, energy at any given point will be subject to the inverse square law. While this model might serve as an interesting analogy, we still need to question whether it viable as a causal mechanism.

How might energy actually propagate within a physical 3D universe?

Of course, in practice, the idea of energy is complicated by a multitude of different descriptions, e.g. rest-mass, thermal, mechanical, electrical, electrostatic, magnetic, electromagnetic, gravitational, chemical, radiation, nuclear, ionization, chromo-dynamic etc. However, we might have to question whether any of these forms is truly reflective of energy at the most fundamental level of the universe. In this respect, it might be argued that some potential offset from a point of equilibrium is not an unreasonable starting assumption, which if taking place within a wave packet of some description might be equated to kinetic energy.

So, what model might better explain causality?

We might attempt to differentiate the causal probability of a particle model based on its ability to explain how energy [E] is physically propagated from [A] to [B]. Without being too rigorous, the particle model might first be considered in terms of Newton’s first law, where an object either remains at rest or continues to move at a constant velocity [v], unless acted upon by a force. However, as argued elsewhere, the particle model also depends on the idea of a physical reality associated with rest mass [kg], which is clearly problematic in terms of a photon or EM wave, where neither have rest mass. While supporting the idea that a field can contain energy, it has been argued that it does not provide any mechanism that explains how energy physically propagates through the field. As outlined, we might consider a wave model, where energy can be propagated based on known wave mechanisms, although it is far from clear that these mechanisms can simply be assumed to work in the quantum domain.

Note: Most wave models are invariably described in terms of mathematically abstracted formulation of perfect sinusoidal waves, restricted to one or two dimensions, propagating through a supportive wave media. In contrast, the physical universe, as described by the quantum model, is apparently a seething and chaotic field of energy. If so, we might question how any wave could propagate through this model, although there is considerable empirical evidence that waves of various description do so, possibly because wave-particle wavelength may be 20 orders of magnitude greater than the quantum Planck scale.

Therefore, in the case of an EM wave, we possibly need to be more specific as to how the electric and magnetic field components propagate with velocity [c] in a given direction. As the details can be referenced in EM Propagation and EM Energy, this discussion will only present some of the key arguments based on Maxwell’s 3^rd and 4^th equations shown in [1].

[1]  

In [1], we see only the time-dependent Maxwell equations in the form of the cross products in [xyz] space, which can be expanded based on the following generic formulation shown in [2].

[2]   

However, an EM plane wave solution can simplify the 3D form implied in [1] and [2] by proceeding on the basis that [E] and [B] fields are perpendicular to each other in just one dimension, as suggested in [3].

[3]     

We can now solve for the [E] and [B] fields based on the assumptions in [3] starting with the electric field [E].

[4]     

We can now do the same for the magnetic field [B].

[5]     

Substituting the solutions in [4] and [5] back into [1] we get [6]

[6]     

Based on [6], we see that any value of [E] or [B] depends on only [x] and [t]. However, we might pursue a 2nd derivative solution of [6], as shown in [7], based on Maxwell’s 3^rd and 4^th equations shown in [1].

[7]     

If we reverse the order in [7], we arrive at something that has the form of a wave equation, as shown in [8], along with a similar result for the magnetic field [B].

[8]     

So, based on the equations in [6] and [8], we see a change [d/dt] in [E] or [B] drives the change [d/dx] and vice versa, such that it might be described as self-propagating, such that it might avoid the issues of a chaotic quantum media. However, we will first consider an alternative solution that is rooted in wave physics, where the change in the [E] or [B] field strength is based on a sine wave function, as shown in [9].

[9]     

Based on [9], we need to determine the 1^st derivatives in-line with Maxwell’s 3^rd equation, as per [7]:

[10]   

However, based on [6], we know the expressions in [10] are equal, as shown in [11], where the cosine functions cancel.

[11]   

Again, we are led towards an expression of velocity [v], not only in terms of [ω,κ], but also in terms of [E,B] as shown in [12], where the value of [c] is also shown in terms of the permittivity [ε₀] and permeability [μ₀] constants, as required by Maxwell’s equations in the SI system.

[12]   

So, is it more logical to describe light-energy as an EM wave or as a photon?

Let us assume that the laws of nature do not require a duality. If so, what might we say about ‘light’ as observed from our own sun, and the stars beyond, which suggest an ability to transport energy through the near vacuum of space and over vast distances.

Note: This observational model requires the sun to radiate energy in all directions, such that energy is subject to an inverse square law with distance. In order to address the inverse square law, a wave model must decrease the E-M field strength as a function of radial distance [r] or the photon model must reduce its number density aligned to the inverse relationship.

Based on Maxwell’s equations, which we might assume is essentially a wave model, the electric and magnetic fields propagate in-phase, such that there is a point in time and space, where these fields are both zero. However, in a self-propagating model, without a physical media, there is no obvious equivalent mechanism where the potential energy associated with the wave amplitude [E] or [B] is converted into kinetic energy within the structure of the propagating media. Therefore, we might question the implications of the relationship [E/B=c], which remains constant at all values except when [E=B=0]. As indicated, within the basic wave model, total energy is conserved within the conversion between potential and kinetic energy, although still subject to the inverse square law at any given point. In the classical model of an EM wave, energy is often described as propagating outwards from a source charge oscillating under acceleration in space, as illustrated in the animation.

However, this model requires some clarifications as outlined in the note below.

Note: In practice, man-made sources of EM waves are produced in oscillating dipole antenna, where an alternating voltage can make millions of charged particles within the length of the antenna, analogous to the red line in the diagram, oscillate in synchronisation. As such, the strength of the electric [E] and magnetic [B] reflects the sum total of charged particles undergoing acceleration, not just one in isolation. In this context, it might be recognised that the antenna is amplifying the energy of the aggregated EM wave being produced, which in-turn is a function of the electrical power being used to generated the alternating voltage at a specific frequency.

As outlined in the note above, the EM wave energy is proportional to the electric field driving the oscillation of the charges, not the individual charges in isolation. Likewise, the rate of oscillation is also driven by the frequency of the alternating voltage, such that it defines the frequency of the EM radiation being emitted. However, as previously discussed, see EM Wave Issues for details, it is unclear how this continuous model fits with the energy quanta model of a photon

Note: It is highlighted that the EM model appears to be very different from the model of a photon produced by an orbital transition within an atom. For this process appears to define a finite transit time, such that a photon might be better modelled as a wave-pulse. Likewise, the photon is assumed to be a quantised unit of energy, as the electron orbital is quantised to the orbital circumference aligning to an integer wavelength. In contrast, the EM model in the diagram provides no causal explanation of energy quantisation, if the oscillation frequency of the charge has no obvious quantisation restrictions - see Photon Issues for more details.

So, what are Maxwell’s time-dependent equations really telling us?

Let us consider the units of the electric [E] and magnetic [B] fields along with permittivity [ε₀] and permeability [μ₀] constants, as required by the SI system:

[13]   

At face value, these SI units do not seem to convey anything meaningful, which we might associate with the propagation of energy within an EM wave. However, we might consider another form of [13] related to Gaussian units as shown in [14].

[14]   

In order to justify the squaring of [E²] and [B²], it might be highlighted that the energy of a classical wave corresponds to the square of its amplitude [A²], such that [14] might suggest that [E] and [B] do indeed correspond to the amplitudes within the EM wave. If we pursue this idea, then the units of the composite terms [ε₀E²] and [B²/μ₀] equate to the units of an energy-density. If course, if we were to drop the SI requirement for [ε₀] and [μ₀], then we might recognise that [E] and [B] have the same units in the Gaussian system and the original equations in [6] might be shown in a more symmetrical form, as per [15].

[15]   

It seems curious that by simply changing the units by which the electric [E] and magnetic [B] fields are defined, we get a different perspective of these quantities and the necessity for the constants [ε₀] and [μ₀] ] used normalise the equations in SI units. However, we might extend this ambiguity to the question of energy in general, i.e. what is it? At one level, we accept that macroscopic objects have energy by virtue of the rest-mass and kinetic energy linked to the concept of mass and velocity plus potential energy based on a position within a field, such as gravitation. However, such concepts are not so tangible when it comes to radiation, which has no rest-mass, but still retains the concept of kinetic energy linked to momentum, which if translated into kinetic mass might allow radiation to also be affected by a potential energy field. In the context of [14], we appear to have a format where the electric [E] and magnetic [B] field strengths have the units of an energy-density, such that we might table another question.

What is energy-density?

At first, we might attempt to answer this question in terms of the energy contained within some defined volume, which in terms of a mass-particle might be equated to some physical radius. Of course, this certainty has to be questioned in the quantum model, if the substance of mass [kg] is also questioned in preference of some form of energy-density, which possibly takes the form of the anecdotal description of ‘a ripple in a quantum field’. However, as this discussion has attempted to highlight, energy as a scalar quantity needs a causal mechanism to explain its propagation through space as a function of time. In this respect, the idea of an EM wave, at least, provides an empirically tested model for wave propagation, which appears absent in the description of a photon being a particle. We might characterise some of the ambiguity surrounding the causal mechanisms associated with the photon concept in the following question and answer exchange taken from an on-line forum.

Question: How can light travel forever and why don’t photons lose energy and stop?

Answer: This is two questions in one. Addressing the first part, the term ‘forever’ is a relativistic term, where humans only ‘sense’ forever because their lifetimes are finite. Not so with photons, which travel at the speed of light [c], where relativity in the form of time dilation, at the speed of light, causes time to stop. So, a photon does not travel forever because photons do not experience the passage of time. Regarding the second part, photons are packets of quantised energy defined by its frequency [E=hf]. It would be erroneous to consider a photon having kinetic energy, which is a mechanical form of energy. Photons have quantum energy, which is exchanged in an ‘all-or-nothing’ process.

While the answer submitted by a professor of physics has been heavily edited, it is not clear that the unedited version provides any useful insight to causality. We might therefore consider another description, this time in terms of electromagnetic waves by somebody holding a Master’s degree in science offering on-line tuition. 

Electromagnetic waves propagate energy forward, but do not use a medium. This is why light can reach Earth from the Sun. If electromagnetic waves were mechanical, there would be no light, heat or life on Earth!

Clearly, this description is making indirect reference to the idea that EM waves self-propagate and therefore have no requirement for a wave propagation media. While it is understandable that such a statement would be forwarded in this type of on-line forum, there is no attempt to question whether EM waves are the preferred explanation of energy transfer rather than photons. Of course, if the description of a photon is adopted, the statement does not address how photons propagate with velocity [c] unless we turn to the ambiguity in the first answer. The equation in [16] simply quantifies, as a point of reference, how much energy is received on Earth from the Sun per square metre as a normalised average.

[16]   

While not directly pertinent to this discussion, [16] shows the conformance to the inverse square law of radiated energy being distributed over a spherical surface, defined by 4πr², where [r] is the radius of Earth’s average orbital distance from the Sun. However, what is not made clear in [16] is the make-up of the energy, as considerable energy is emitted by the Sun as charged particles – see Solar Wind for details. In essence, the flow of charged particle would constitute an electric current, which we might assume would create both electric and magnetic fields in space. Of course, if these particles were subject to an acceleration, we might assume that some form of EM wave might be the result.

Note: One of the issues this discussion is trying to highlight is the apparent multitude of different ways energy is said to exist. For example, we quantify energy in terms of rest-mass, thermal, mechanical, electrical, electrostatic, magnetic, electromagnetic, gravitational, chemical, radiation, nuclear, ionization etc. However, while speculative, the general principle of Occum’s Razor might question whether all these different forms are simply manifestations of the potential and kinetic energy being propagated by different wave structure, where the question of a propagation media is left as an open issue.