Our new article was recently published in The Journal of Geometry and Physics. It is shown that under certain conditions, The Einstein Field Equations have the same form as a fold bifurcation seen in Dynamical Systems theory, showing even a deeper connection between General Relativity and Dynamical Systems theory! (You can click the image below to be taken to the article):
My new paper has now been published in Annalen der Physik, which is a great honour, because 100 years ago, Einstein’s General Theory of Relativity was also published in the same journal.
This paper describes a method by which one is able to determine whether a given spatially flat cosmological model produces finite-time singularities, and also gives some examples of interesting cosmological model configurations.
The paper can be accessed by clicking the image below:
The preprint can be accessed here on the arXiv.
What goes into making a cosmological model? Here is a presentation (that was part of my Ph.D. dissertation) that I have reproduced and embedded here to describe what actually goes into the making of a cosmological model. After describing some general properties, I describe specifically a early-universe model that contains a viscous fluid and a magnetic field.
The background mathematics can be found in this old presentation of mine here:
Since I expect the concept of gravitational waves to once become very popular in the next few days, I wrote some quick notes on them, I.e., where they come from. They are handwritten, as I didn’t have time to LaTeX them, but, hopefully, they’ll be useful to interested readers!
Also note that, gravitational waves are not necessarily evidence of inflation. I wrote a paper a few years ago, describing a anisotropic early universe that had an epoch of plane waves that isotropized to our present-day universe. It can be seen here. It was subsequently published in Physical Review D.
Anyways, here are the notes (Interested readers should see the classic texts by Misner, Thorne, Wheeler, Landau and Lifshitz, or Stephani for more details).
One of my earliest works was deriving equations which themselves were forms of Einstein’s field equations that described the state of the early universe, which may have had dominant viscous effects. I was delighted to learn that these equations were published in Springer’s Handbook of Spacetime Cosmology textbook.
Here is a snapshot of the textbook page citing these equations:
Our new paper was accepted for publication in Physical Review D. The goal of the paper was to calculate the probability that a multiverse could emerge from a more general background spacetime, in this case, Bianchi Type I coupled to a chaotic inflaton potential. Basically, we found that a multiverse being generated from such a scenario has a small probability of occurring. Further, the fine-tuning problem that the multiverse / eternal inflation is supposed to solve doesn’t actually occur, because fine-tuning is still required of the geometry of the background spacetime, the initial conditions, and most importantly, the amount of anisotropy.
The preprint can be read on the arXiv here.
In many physics and chemistry courses, one is typically taught that heat propagates according to the heat equation, which is a parabolic partial differential equation:
where is the thermal diffusivity and is material dependent. Note also, we are considering the one-dimensional case for simplicity.
Now, let be a solution to this problem, which represents a wave travelling at speed . We get that
This implies that
where are constants determined by appropriate boundary conditions. We can see that as , ! That is, that even under an infinite propagation speed (greater than the speed of light), the solution to the heat equation remains bounded. PDE folks will also say that solutions to the heat equation have characteristics that propagate at an infinite speed. Thus, the heat equation is fundamentally acausal, indeed, all such distribution propagations from Brownian motions to simple diffusions are fundamentally acausal, and violate relativity theory.
Some efforts have been made, and it is still an active area of mathematical physics research to form a relativistic heat conduction theory, see here, for more information.
What we really need are hyperbolic partial differential equations to maintain causality. That is why, Einstein’s field equations, Maxwell’s equations, and the Schrodinger equation are hyperbolic partial differential equations, to maintain causality. This can be seen by considering an analogous methodology to the wave equation in 1-D:
Now, consider a travelling wave solution as before . Substituting this into this wave equation, we obtain that
That is, all solutions to the wave equation travel at the speed of light, i.e., ! Therefore, wave equations are fundamentally causal, and all dynamical laws of nature, must be given in terms of hyperbolic partial differential equations, as to be consistent with Relativity theory.