Wave Properties of Matter
By: Nicolus Rotich
Tags: Science, Education, Physics, Mathematics, General

It has always been known that there is an elegant relationship between matter and waves, albeit in elementary particles. This was proposed way back in 1924 by Louis de Broglie — a French physicist¹. The hypothesis was not well-received then, and even to date unfortunately it is still a subject that is mainly introduced much later in physics education. Many previous attempts to extend these connections to classical mechanics, let a lone general relativistic studies have failed miserably. The vice versa has also been the case, when transiting from studies of objects of larger extent to that of the minute — the quantum mechanics. That is to imply, we cannot look at a kilogram mass and tell straight away what it’s wave equivalent properties are. There is one missing detail that could be used to elucidate this, and is the subject of this essay.

Unification of general relativistic studies to the calculations involved in the quantum level today represents an active area of research. We will take a step in this thought-experiment to abstract a simple concept of classical mechanics as well as attempt to broaden it to fit our daily understanding of both worlds.

To construct the scenario, a piece of stone with a known dimensional extent (radius), r meters is elevated to a height of H meters above the ground with an intention to drop and measure the instantaneous positions and velocities under free-fall. Sir Isaac Newton predicted that these heights would be modeled as H= 0.5gt². With the very intention to introduce the wave properties of matter, we will suppose that the initial position H was a radius of a sinusoidal wave of wavelength λ, such that:

Eq. (1)

Read up to the end to know why Eq. (1) is true. Since the resulting wavelength may also be represented as a function of another (smaller) radius, this process can be conducted iteratively with each successful cycle corresponding to the instantaneous position (and time, t) of the free-falling object. Rather than tracking the velocity of the stone v=gt as usual, in this case we will be recording the properties of the wave generated i.e. wave’s velocity — renamed u, and the resulting wave frequency, f. We would then be able to draw some conclusions and attempt to generalize the phenomenon based on the thought experiment.

It turns out that the relationship between such wavelength and the wave’s speed can be used to predict the Newtonian gravitational potential, and free fall times in a similar way we do for Newtonian mechanics to yield the same results in the following manner:

Eq. (2)

The resulting form(s) in Eq. (2) is no surprise. However, replacing the earlier form of the wavelength λ in Eq.(2), we find that the wave’s linear, and angular speed ω, and thus frequencies, and period may then be estimated through the following methods.

Eq. (3)

We know that for classical waves, the associated frequency can be treated as a stand alone entity, since it can be measured independently as a vibration. What is more intriguing is that multiplying the frequency, f with the wave’s speed predicts an exact value of Newtonian gravity g in that g=uf.

Extending the analogy to general relativity

While the methodology described may seem over simplified, it has far reaching implications upon up-scaling to the general relativistic context. The wave properties instantly transforms into electromagnetic entities. For instance, we know that planets are in constant “falling” around their respective stars, and thus governed by Newton’s universal laws of gravity. An equivalent wavelength in that case would be given as a function of Schwarzschild radius as follows:

Eq. (4)

Where r is the radius of the object under consideration (of the earth in this case ≃ 6375087.54925801 m) as it is free-falling under the gravitational field of the sun². G is Newton’s universal gravitational constant, M is the mass of the object, and c is the speed of light in a vacuum. The rest of the calculations, namely frequency and velocity are of pure electromagnetic nature, and thus related by the generic formula λf= c. Most importantly, unlike in the thought-experiment, the retarded time of the wave t₀ becomes a known constant such that:

Eq. (5)

Where T is the period of the wave, and fᵣ is a known set of constant ultra low frequencies (ULF) for each celestial body large enough to exert its own gravity. A more detailed description on these frequencies was presented earlier on extending Einstein’s field equations. The physical implication of the frequencies is presumed to be position dependent, rather than belonging to the matter. This is the case even though both matter and the frequencies occupy the same spot in Einsteins coordinates of x, y, z, and t in spacetime.

Once more, as the frequencies described in the outgoing paragraph are independently measurable entities, this would imply that pointing a radar to the position where a massive body is located should read the scalar vector frequencies which when multiplied by the speed of light in a vacuum yields the gravitational acceleration component of the body as depicted in Eq. (5).

For the sake of generalization, a quick test on our nearby planetary bodies would indicate that Jupiter has the highest radio frequency of 77.05 nHz, while Pluto has the lowest of only 2.335 nHz. For planet Earth, this value stands at 32.71 nano Hertz. Taking a multiplication of the above frequencies with c must indeed yield the respective values as g=23.1, 0.7, and 9.80665 m/s² for Jupiter, Pluto, and Earth respectively.

Characterizing black holes is another benefit of understanding this methodology. For those objects characterized as non-rotating black holes, their radii are known to be equal to the Schwarzschild radius by definition. Meaning that r=r_s in Eq.(4), and thus we can state with confidence that the equivalent wavelength for a Schwarzschild black hole is simply given as 2r_s or twice the radius.

A metric tensor solution that fully characterizes the nature of an object in a gravitational field based on this method can be found by letting the wavelength recede by a factor of Δh= λ-r. When the λ from Eq.(4) is replaced in this difference, the solution to the quadratic equation formed yields a metric solution to Einstein’s field equations as follows:

Eq. (6)

Eq.(6) implies that an object will only be characterized as a black hole if and only if Δ h ≤ rₛ. The mass of the resulting black hole if and when formed, for instance due to gravitational collapse due to its on gravity³, will be given as follows:

Eq. (7)

The classical question as to what exactly is waving does not arise here since the understanding is that matter itself is a wave, in the same way elementary particles are characterized from de Broglie-Compton relations in the subatomic level.

NB: To prove Eq.(1) is true, imagine you were to calculate the gravitational potential energy having been told to assume H=R, the radius of a traveling sinusoidal wave. You would then use E=mgR, in place of E=mgH. These two are equal for the simple fact that R=H=λ/2π. You could also conclude that when no loses occur, gravitational potential energy must transform fully to kinetic energy since:

Eq. (8)

With the fact that the object’s speed v = gt, and thus K = mv²/2 as we know from classical mechanics.

Conclusion

We have therefore transformed a simple abstraction of equations of motion to give a full description of how matter and waves interact in a gravitational field. We then extended the generalization to solve for previously counterintuitive subjects in both classical mechanics and general relativity. In general, matter tend to (broad)cast a wave around them, that is much larger (in extent) compared to their own radii and can be analysed in every way like classical waves. The essay thus presents an easier means for new physics readers to look at matter as indifferent from waves with non-conflicting mathematical correlations as compared to the very way objects behaves in classical mechanics work.

References

¹ Feynman, R.P., 2006. QED: The strange theory of light and matter. Princeton University Press.

² https://www.britannica.com/science/free-fall-physics

³ Eppelbaum, L., Kutasov, I. and Pilchin, A., 2014. Applied geothermics (pp. 99–149). Springer Berlin Heidelberg.

⁴ https://www.nasa.gov/multimedia/imagegallery/image_feature_532.html