Pi Day and Einstein 

  

A Series of Lectures on Fine-Tuning in Biology

A recent lecture and a series of interviews has been posted online where cosmologist George F.R. Ellis discusses the issue of fine-tuning in biology at considerable length and in considerable detail. Of course, the larger theme here is that to discuss and understand things like Darwinian evolution properly, one needs to have an understanding of the underlying physics, as it is laws of physics that allow life to emerge and for Darwinian evolution to occur in the first place. Here are the lectures:

 

 

 

 

 

Notes on Dynamical Systems

A big part of my research involves dynamical systems theory. A lot of people don’t know what this is, at least, they don’t have a very good idea. It has not helped that the vast majority of Canadian university physics programs have deemphasized classical mechanics and differential equations, but that is an another story! 

Anyways, here are some notes describing what they are and how they work. 

Click the image for the full notes in PDF form.
Click the image above for the rest of the notes.

On Gravitational Waves

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). 

   
    
 

Equations Published in a Cosmology Textbook

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:

image1

 

New Paper on Stochastic Eternal Inflation

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.

prdabstract

The preprint can be read on the arXiv here.

 

On The Acausality of Heat Propagation

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:

\boxed{u_t = \alpha u_{xx}},

where \alpha is the thermal diffusivity and is material dependent. Note also, we are considering the one-dimensional case for simplicity.

Now, let f(x-at) be a solution to this problem, which represents a wave travelling at speed a. We get that

\boxed{-a f' = \alpha f''}.

This implies that

\boxed{u(x,t) = -\frac{\alpha c_1}{a} \exp\left[-\frac{a (x-at)}{\alpha}\right] + c_{2}},

where c_{1}, c_{2} are constants determined by appropriate boundary conditions. We can see that as a \to \infty, u(x,t) < \infty! 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:

\boxed{u_{tt} = c^2 u_{xx}}.

Now, consider a travelling wave solution as before f(x-at). Substituting this into this wave equation, we obtain that

\boxed{a^2 f'' = c^2 f'' \Rightarrow a^2 = c^2 \Rightarrow a = \pm c}.

That is, all solutions to the wave equation travel at the speed of light, i.e., a = c! 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.