Dr. Jonathan Kenigson, FRSA*
Gravitational waves are features of Einstein’s cosmology, General Relativity (GR). Black hole (BH) collisions should produce gravitational radiation in any theory more general than Einstein’s. This is (to the author’s knowledge) a minor new result in mathematics. It bears mention that what appears as a prediction of the theory is actually a consequence of the original analogical reasoning process that suggests that certain BH properties should exhibit a wavelike behavior. Mathematics is merely a precise linguistic formalization of this assumption. The intuition is borrowed from fluid dynamics by considering the waves emitted from two vortices of water falling down drains that are close together and rotate about each other periodically. The waves produced are generally of very low amplitude compared to the amplitude that would be produced in a wave of comparable mass impacting the still surface of the water. Indeed, the reasoning presented herein may hint at a possible unification with Navier-Stokes theory in fluid dynamics. Some mathematicians begin to undertake the study of spacetime as an ideal fluid for similar reasons of pure analogical reasoning. A similar result from pure quantum gravity (String Theory, or ST) is likely available but has not yet been published.
Mathematical practitioners attempting to model this situation could consider the case as an extension of fluid-dynamical thinking again: A tornado that hits nothing, even if it has very high intrinsic energy (e.g. wind speeds), will have less disorder (Entropy) in its vortex than one which attracts a large amount of soil or debris. The larger mass of incident particles increases the entropy of the vortex. A BH should obey the same dynamics if it is rotating. A model that did not include the mass M would thus be highly suspect. Similarly, the charge Q of the BH should influence the entropy, because it would otherwise be possible to assign a unique state (namely the entropy) without taking a mandated “No-Hair’ parameter into consideration. The effects of the charge Q are intuitively less clear. In such cases (which commonly occur), one seeks an explicit solution to ‘tinker’ with and see if a clear, physically motivated pattern emerges. If an intuitive explanation cannot be postulated after some effort, the solution becomes suspect. If other mathematicians arrive at the same solution by other means, the result may be considered novel.
A novel solution does not necessarily mean that the equations have ‘predicted’ a counterintuitive result; rather, such novelty is often taken to mean that the assumptions of the resulting theory are not well understood, and that further clarifications are needed before a consensus on the physical meaning is reached. In the current context, the BH entropy solution arising from a union of integration theory and field theory produces a beautiful “closed form” solution. Mathematics forms a symbolic language for the equation of the entities involved, but none of the entities need to exist abstractly to provide explanatory power to the physical theory. The explanatory power of the relevant mathematics is derived from the physicality, naturalness, and intuitiveness of the assumptions of the underlying BH theory. In addition, mathematical formalism acts as a precise common language for exploration of the underlying phenomenon, facilitating clearer and sharper reasoning from prior assumptions. In this way, mathematics ‘acts like’ logic when applied to the subject matter of cosmology or theoretical physics.
Works Consulted.
Abbott, Benjamin P., et al. “Binary black hole mergers in the first advanced LIGO observing run.” Physical Review X 6.4 (2016): 041015.
Antoniadis, Ignatios, and N. A. Obers. “Plane gravitational waves in string theory.” Nuclear Physics B 423.2-3 (1994): 639-660.
Bekenstein, Jacob D. “Do we understand black hole entropy?.” arXiv preprint gr-qc/9409015 (1994).
Cervantes-Cota, Jorge L., Salvador Galindo-Uribarri, and George F. Smoot. “A brief history of gravitational waves.” Universe 2.3 (2016): 22.
Constantin, Peter, and Ciprian Foias. Navier-stokes equations. University of Chicago Press, 2020.
Durrer, Ruth, and Jasper Hasenkamp. “Testing superstring theories with gravitational waves.” Physical Review D 84.6 (2011): 064027.
Einstein, Albert, and Nathan Rosen. “On gravitational waves.” Journal of the Franklin Institute 223.1 (1937): 43-54.
Lidsey, James E., and David Seery. “Primordial non-Gaussianity and gravitational waves: observational tests of brane inflation in string theory.” Physical Review D 75.4 (2007): 043505.
Raidal, Martti, Ville Vaskonen, and Hardi Veermäe. “Gravitational waves from primordial black hole mergers.” Journal of Cosmology and Astroparticle Physics 2017.09 (2017): 037.
Tsallis, Constantino. “Black hole entropy: a closer look.” Entropy 22.1 (2019): 17.
Zwart, Simon F. Portegies, and Stephen LW McMillan. “Black hole mergers in the universe.” The Astrophysical Journal 528.1 (1999): L17.
