A Prediction on the Graviton
Properties of the graviton to be
We have always wondered how particle physicists are able to predict the properties of a particle even way before its detection is imminent. In deed about half a century ago, Peter Higgs, a British theoretical physicist and now a renowned Nobel Laureate will be forever remembered for his advance knowledge of a particle that previously never exist in our physics textbooks. The prediction of Higgs and his colleagues was confirmed in the early morning of July 4th 2012 by CERN laboratory of particle physics (Fig. 1).
A year later, the outgoing confirmation would earn the team a Nobel prize in physics for conceiving what we now understand as the Higgs Boson. The press described this achievement as an astonishing triumph of the power of mathematics¹. This essay is meant to give a generic introduction on why such achievements are possible, as we intend to follow through with an example of the graviton — a fundamental particle yet to be detected.
Physicists and human minds in general are obsessed with patterns. As an example you might relate to, humans have been known to be good at identifying other humans’ faces. Physicists also search for patterns and symmetries connecting one equation to another. We then hypothesize, or come up with the laws or theories that could characterize future scenarios. In the next sections, we will first discuss what necessitates the notion that a particle such as a graviton would exist, and how one might go about theoretically in an attempt to predict its perceived properties.
Why quantum gravity?
One of the biggest mysteries of the past nearly century, is that there exists a particle that probably every physicist must have a sleepless night over — the graviton. This is the fundamental particle that mediates gravitational force in a quantum mechanical framework. But why do we even think the graviton exists? Well, there are a total of 13 particles predicted by the standard model of particles — a blueprint believed to be crucial in the fundamental description of our universe. In the spirit of finding patterns, scientists have hypothesized that the 13th one must exist since the first 12 of them have been already detected.
Besides, the claim is also pegged on the overall understanding that general relativity however successful it has been, is not fundamental. This is another way to say it doesn’t really get into the nitty gritty of the substance constituting our universe, more like electrons, quarks, photons, and now the Higgs Boson among others does. There are therefore many compelling reasons to believe that the fundamental theory of gravity is quantum in nature². But the main reason for the need for a quantum gravity model can be summarized as follows:
While the other three fundamental forces of physics namely: electromagnetic, the weak, and strong interactions, are founded on quantum field theories, that leaves out gravity for lack of its quantum field counterpart description. It is always argued that it is not sufficient to consistently couple a classical to a quantum system — Whitaker Andrew³
Expected properties of the graviton
The current expectation, is that if graviton exists, it must be massless as it seems to propagate at the speed of light more like photons. However, here is a scenario. Let us suppose that we knew the mass of a graviton. We would naturally be intrigued to express its de Broglie-Compton mass as follows:
Where, h is Planck’s constant, λ its wavelength, and c the speed of light in a vacuum. To get the field force exerted by a single particle on spacetime, we might want to proceed and multiply both sides by Newtonian gravitational acceleration, g. That would yield:
We seem to have characterized the term in bracket to the RHS of Eq. (2), as a constant achievable through some general relativistic calculations. In general, the ratio g/c for an object that is large enough to exert its own gravity may also be represented as follows:
Where, rₛ is the Schwarzschild radius, and r is its radius. Eq. (3) is applicable for every existing object, be it an everyday object we interact with, a planet, a Neutron star, or a black hole, except for the latter r=rₛ. A more detailed explanation on the genesis of Eq. (3) was given in a separate article on extending Einstein’s field equations .What is even more interesting, is that the ratio yields the units of per second, or frequency — we will rename this as fᵣ. Now, rewriting Eq. (2) based on this notion, would result in an energy component as shown below:
The general Planck-Einstein statement concerning photo-electric effect of light-matter interaction is given as E=hf. We can therefore substitute this in Eq. (4) and solve for the graviton wave-energy as E=hfᵣ. Notice that we know all the components of Eq. (3), and thus the particle’s frequency fᵣ, may be easily estimated. The particle’s wavelength λ, may also be drawn from the relations:
Where: c, r, rₛ, and fᵣ takes the earlier defined meanings. For spherical objects, A is the surface area such that A=4π r², while Cₛ is the circumference based off the Schwarzschild radius, i.e. Cₛ=2π rₛ. Even though Eq. (4) applies for all objects, our particular concern is those taken in the context of planet Earth. The next operation is the particle’s wave momentum, given as:
The parameter κ is the wave number — quantifying the spatial frequency, or the spacing of the particles, while the crossed h is the reduced Planck’s constant. Finally, we would want to apply the full version of E=mc² as follows:
Since we now know all the components of Eq. (7) except for the rest mass m₀, we should be able to solve for it in the next steps to completion. You would expect that the rest mass results in a perfect zero, but not. Instead, the results of the discussed analysis suggests that the graviton must be described by a miniature mass-energy of about 1.35364E-22 eV/c², and an even smaller rest mass, m₀ of 4.66E-24 eV/c². As simple as this methodology may seems, it is worthwhile understanding all the core functions derived, as they are well representative and repeatedly verifiable. More importantly, is to bear in mind that the LIGO/Virgo team in 2016 estimated the relativistic mass for gravitational waves observed from a binary black hole merger to be about 1.22E-22 eV/c², citing uncertainties in their measured Compton wavelength, of the wave⁴. We therefore find a close correlation between the obtained quantities herein with those of previous predictions. A further probe to these methods would oversee a development of a more refined methodology that could be representative to the true nature of the particle once successfully detected.
We have highlighted the outstanding problem(s) on the need for a quantum gravitation model, that is currently a focus in many areas of particle and theoretical physics in general. We started by outlining how some previous particles were predicted way before they were detected, as well as touched on some of the patterns that particle physicists look for. We then went ahead to anticipate the expected properties of the graviton once detected. The obtained values seems to be within the range of some of the previously conducted studies. It is important to note that the actual particle — the graviton has not been detected and is less likely within the next few years. However, if past experience has taught us anything, the process is always efficient when we let the practical detection be preceded by theoretical predictions.
² Kiefer, C., 2007. Why quantum gravity?. In Approaches to fundamental physics (pp. 123–130). Springer, Berlin, Heidelberg.
³ Whitaker, A., 2006. Einstein, Bohr and the quantum dilemma: From quantum theory to quantum information. Cambridge University Press.
⁴ Abbott, B.P., Abbott, R., Abbott, T.D., Abernathy, M.R., Acernese, F., Ackley, K., Adams, C., Adams, T., Addesso, P., Adhikari, R.X. and Adya, V.B., 2016. Observation of gravitational waves from a binary black hole merger. Physical review letters, 116(6), p.061102.