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.

Mathematical Origins of Life

The purpose of this post is to demonstrate some very beautiful (I think!) mathematics that arises form Darwinian evolutionary theory. It is a real shame that most courses and discussions dealing with evolution never introduce any type of mathematical formalism which is very strange, since at the most fundamental levels, evolution must also be governed by quantum mechanics and electromagnetism, from which chemistry and biochemistry arise via top-down and bottom-up causation. See this article by George Ellis for more on the role of top-down causation in the universe and the hierarchy of physical matter. Indeed, my personal belief is that if some biologists and evolutionary biologists like Dawkins, Coyne, and others took the time to explain evolution with some modicum of mathematical formalism to properly describe the underlying mechanics instead of using it as an opportunity to attack religious people, the world would be a much better place, and the dialogue between science and religion would be much more smooth and intelligible.

In this post today, I will describe some formalism behind the phenomena of prebiotic evolution. It turns out that there has been a very good book by Claudius Gros and understanding evolution as a complex dynamical system (dynamical systems theory is my main area of research), and the interested reader should check out his book for more details on what follows below.

We can for simplicity consider a quasispecies as a system of macromolecules that have the ability to carry information, and consider the dynamics of the concentrations of the constituent molecules as the following dynamical system:

$\boxed{\dot{x}_{i} = W_{ii}x_{i} + \sum_{j \neq i}W_{ij}x_{j} - x_{i} \phi(t)}$,

where $x_{i}$ are the concentrations of $N$ molecules, $W_{ii}$ is the autocatalytic self-replication rate, and $W_{ij}$ are mutation rates.

From this, we can consider the following catalytic reaction equations:

$\boxed{\dot{x}_i = x_{i} \left(\lambda_{i} + k_i^j x_j - \phi \right)}$,

$\boxed{\phi = x^{k}\left(\lambda_{k} + \kappa_k^j x_j\right) }$,

$x_i$ are the concentrations, $\lambda_i$ are the autocatalytic growth rates, and $\kappa_{ij}$ are the transmolecular catalytic rates. We choose $\phi$ such that

$\boxed{\dot{C} = \sum_i \dot{x}_i = \sum_i x_i \left(\lambda_i + \sum_j \kappa_{ij}x_{j} \right) - C \phi = (1-C)\phi}$.

Clearly:

$\lim_{C \to 1} (1-C)\phi = 0$,

that is, this quick calculation shows that the total concentration $C$ remains constant.

Let us consider now the case of homogeneous interactions such that

$\kappa_{i \neq j} = \kappa$, $\kappa_{ii} = 0$, $\lambda_i = \alpha i$,

$\boxed{\dot{x}_{i} = x_{i} \left(\lambda_i + \kappa \sum_{j \neq i} x_{j} - \phi \right)}$,

which becomes

$\boxed{\dot{x}_i = x_i \left(\lambda_i + \kappa - \kappa x_i - \phi\right)}$.

This is a one-dimensional ODE with the following invariant submanifolds:

$\boxed{x_{i}^* = \frac{\lambda_i + \kappa - \phi}{\kappa}}$,

$\boxed{x_i^* = 0, \quad \lambda_i = N \alpha}$.

With homogeneous interactions, the concentrations with the largest growth rates will dominate, so there exists a $N^*$ such that $1 \leq N^* \leq N$ where

$\boxed{x_i^* = \frac{\lambda_i + \kappa - \phi}{\kappa}, \quad N^* \leq i \leq N}$,

$\boxed{0, \quad 1 \leq i < N^*}$.

The quantities $N^*$ and $\phi$ are determined via normalization conditions that give us a system of equations:

$\boxed{1 = \frac{\alpha}{2\kappa} \left[N(N+1) - N^*(N^* - 1)\right] + \left[\frac{\kappa - \phi}{\kappa}\right] \left(N + 1 - N^*\right)}$,

$\boxed{0 = \frac{\lambda_{N^*-1} + \kappa - \phi}{\kappa} = \frac{\alpha(N^* - 1)}{\kappa} + \frac{\kappa - \phi}{\kappa} }$.

For large $N, N^*$, we obtain the approximation

$\boxed{N - N^* \approx \sqrt{\frac{2 \kappa}{\alpha}}}$,

which is the number of surviving species.

Clearly, this is non-zero for a finite catalytic rate $\kappa$. This shows the formation of a hypercycle of molecules/quasispecies.

These computations clearly should be taken with a grain of salt. As pointed out in several sources, hypercycles describe closed systems, but, life exists in an open system driven by an energy flux. But, the interesting thing is, despite this, the very last calculation shows that there is clear division between molecules $i = N^*, \ldots N$ which can be considered as a type of primordial life-form separated by these molecules belonging to the environment.

Black Holes, Black Holes Everywhere

Nowadays, one cannot watch a popular science tv show, read a popular science book, take an astrophysics class without hearing about black holes. The problem is that very few people discuss this topic appropriately. This is further evidenced that these same people also claim that the universe’s expansion is governed by the Friedmann equation as applied to a Friedmann-Lemaitre-Robertson-Walker (FLRW) universe.

The fact is that black holes despite what is widely claimed, are not astrophysical phenomena, they are a phenomena that arise from mathematical general relativity. That is, we postulate their existence from mathematical general relativity, in particular, Birkhoff’s theorem, which states the following (Hawking and Ellis, 1973):

Any $C^2$ solution of Einstein’s vacuum equations which is spherically symmetric in some open set V is locally equivalent to part of the maximally extended Schwarzschild solution in V.

In other words, if a spacetime contains a region which is spherically symmetric, asymptotically flat/static, and empty such that $T_{ab} = 0$, then the metric inn this region is described by the Schwarzschild metric:

$\boxed{ds^2 = -\left(1 - \frac{2M}{r}\right)dt^2 + \frac{dr^2}{1-\frac{2M}{r}} + r^2\left(d\theta^2 + \sin^2 \theta d\phi^2\right)}$

The concept of a black hole then occurs because of the $r = 0$ singularity that occurs in this metric.

The problem then arises in most discussions nowadays, because the very same astrophysicists that claim that black holes exist, also claim that the universe is expanding according to the Einstein field equations as applied to a FLRW metric, which are frequently written nowadays as:

The Raychaudhuri equation:

$\boxed{\dot{H} = -H^2 - \frac{1}{6} \left(\mu + 3p\right)}$,

(where $H$ is the Hubble parameter)

The Friedmann equation:

$\boxed{\mu = 3H^2 + \frac{1}{2} ^{3}R}$,

(where $\mu$ is the energy density of the dominant matter in the universe and $^{3}R$ is the Ricci 3-scalar of the particular FLRW model),

and

The Energy Conservation equation:

$\boxed{\dot{\mu} = -3H \left(\mu + p\right)}$.

The point is that one cannot have it both ways! One cannot claim on one hand that black holes exist in the universe, while also claiming that the universe is FLRW! Since, by Birkhoff’s theorem, external to the black hole source must be a spherically symmetric and static spacetime, for which a FLRW is not static nor asymptotically flat, because of a lack of global timelike Killing vector.

I therefore believe that models of the universe that incorporate both black holes and large-scale spatial homogeneity and isotropy should be much more widely introduced and discussed in the mainstream cosmology community. One such example are the Swiss-Cheese universe models. These models assume a FLRW spacetime with patches “cut out” in such a way to allow for Schwarzschild solutions to simultaneously exist. Swiss-Cheese universes actually have a tremendous amount of explanatory power. One of the mysteries of current cosmology is the origin of the existence of dark energy. The beautiful thing about Swiss-Cheese universes is that one is not required to postulate the existence of hypothetical dark energy to account for the accelerated expansion of the universe. This interesting article from New Scientist from a few years ago explains some of this.

Also, the original Swiss-Cheese universe model in its simplest foundational form was actually proposed by Einstein and Strauss in 1945.

The basic idea is as follows, and is based on Israel’s junction formalism (See Hervik and Gron’s book, and Israel’s original paper for further details. I will just describe the basic idea in what follows). Let us take a spacetime and partition it into two:

$\boxed{M = M^{+} \cup M^{-}}$

with a boundary

$\boxed{\Sigma \equiv \partial M^{+} \cap \partial M^{-}}$.

Now, within these regions we assume that the Einstein Field equations are satisfied, such that:

$\boxed{\left[R_{uv} - \frac{1}{2}R g_{uv}\right]^{\pm} = \kappa T_{uv}^{\pm}}$,

where we also induce a metric on $\Sigma$ as:

$\boxed{d\sigma^2 = h_{ij}dx^{i} dx^{j}}$.

The trick with Israel’s method is understanding is understanding how $\Sigma$ is embedded in $M^{\pm}$.  This can be quantified by the covariant derivative on some basis vector of $\Sigma$:

$\boxed{K_{uv}^{\pm} = \epsilon n_{a} \Gamma^{a}_{uv}}$.

The projections of the Einstein tensor is then given by Gauss’ theorem and the Codazzi equation:

$\boxed{\left[E_{uv}n^{u}n^{v}\right]^{\pm} = -\frac{1}{2}\epsilon ^{3}R + \frac{1}{2}\left(K^2 - K_{ab}K^{ab}\right)^{\pm}}$,

$\boxed{\left[E_{uv}h^{u}_{a} n^{v}\right]^{\pm} = -\left(^{3}\nabla_{u}K^{u}_{a} - ^{3}\nabla_{a}K\right)^{\pm}}$,

$\boxed{\left[E_{uv}h^{u}_{a}h^{v}_{b}\right]^{\pm} = ^{(3)}E_{ab} + \epsilon n^{u} \nabla_{u} \left(K_{ab} - h_{ab}K\right)^{\pm} - 3 \left[\epsilon K_{ab}K\right]^{\pm} + 2 \epsilon \left[K^{u}_{a} K_{ub}\right]^{\pm} + \frac{1}{2}\epsilon h_{ab} \left(K^2 + K^{uv}K_{uv}\right)^{\pm}}$

Defining the operation $[T] \equiv T^{+} - T^{-}$, the Einstein field equations are given by the Lanczos equation:

$\boxed{\left[K_{ij}\right] - h_{ij} \left[K\right] = \epsilon \kappa S_{ij}}$,

where $S_{ij}$ results from defining an energy-momentum tensor across the boundary, and computing

$\boxed{S_{ij} = \lim_{\tau \to 0} \int^{\tau/2}_{-\tau/2} T_{ij} dy}$.

The remaining dynamical equations are then given by

$\boxed{^{3}\nabla_{j}S^{j}_{i} + \left[T_{in}\right] = 0}$,

and

$\boxed{S_{ij} \left\{K^{ij}\right\} + \left[T_{nn}\right] = 0}$,

with the constraints:

$\boxed{^{3}R - \left\{K\right\}^2 + \left\{K_{ij}\right\} \left\{K^{ij}\right\} = -\frac{\kappa^2}{4} \left(S_{ij}S^{ij} - \frac{1}{2}S^2\right) - 2 \kappa \left\{T_{nn}\right\}}$.

$\boxed{\left\{^{3} \nabla_{j}K^{j}_{i} \right\} - \left\{^{3}\nabla_{i}K\right\} = -\kappa \left\{T_{in}\right\}}$.

Therefore:

1. If black holes exist, then by Birkhoff’s theorem, the spacetime external to the black hole source must be spherically symmetric and static, and cannot represent our universe.
2. Perhaps, a more viable model for our universe is then a spatially inhomogeneous universe on the level of Lemaitre-Tolman-Bondi, Swiss-Cheese, the set of $G_{2}$ cosmologies, etc… The advantage of these models, particular in the case of Swiss-Cheese universes is that one does not need to postulate a hypothetical dark energy to explain the accelerated expansion of the universe, this naturally comes out out of such models.

Under a more general inhomogeneous cosmology, the Einstein field equations now take the form:

Raychauhduri’s Equation:

$\boxed{\dot{H} = -H^2 + \frac{1}{3} \left(h^{a}_{b} \nabla_{a}\dot{u}^{b} + \dot{u}_{a}\dot{u}^{a} - 2\sigma^2 + 2 \omega^2\right) - \frac{1}{6}\left(\mu + 3p\right)}$

Shear Propagation Equation:

$\boxed{h^{a}_{c}h^{b}_{d} \dot{\sigma}^{cd} = -2H\sigma^{ab} + h^{(a}_ch^{b)}_{d}\nabla^{c}\dot{u}^{d} + \dot{u}^{a}\dot{u}^{b} - \sigma^{a}_{c} \sigma^{bc} - \omega^{a}\omega^{b} - \frac{1}{3}\left(h^{c}_{d}\nabla_{c}\dot{u}^{d} + \dot{u}_{c}\dot{u}^{c} - 2\sigma^2 - \omega^2\right)h^{ab} - \left(E^{ab} - \frac{1}{2}\pi^{ab}\right)}$

Vorticity Propagation Equation:

$\boxed{h^{a}_{b}\dot{\omega}^{b} = -2H\omega^{a} + \sigma^{a}_{b}\omega^{b} - \frac{1}{2}\eta^{abcd}\left(\nabla_{b} \omega_{c} + 2\dot{u}_{b}\omega_{c}\right)u_{d} + q^{a}}$

Constraint Equations:

$\boxed{h^{a}_{c} h^{c}_{d} \nabla_{b} \sigma^{cd} - 2h^{a}_{b}\nabla^{b}H - \eta^{abcd}\left(\nabla_{b}\omega_{c} + 2 \dot{u}_{b} \omega_{c}\right)u_{d} + q^{a} = 0}$,

$\boxed{h^{a}_{b} \nabla_{a}\omega^{b} - \dot{u}_{a}\omega^{a} = 0}$,

$\boxed{H_{ab} - 2\dot{u}_{(a}\omega_{b)} - h^{c}_{(a}h^{d}_{b)}\nabla_{c} \omega_{d} + \frac{1}{3} \left(2\dot{u}_{c}\omega_{c} + h^{c}_{d} \nabla_{c}\omega^{d}\right)h_{ab} - h^{c}_{(a}h^{d}_{b)} \eta_{cefg}\left(\nabla^{e}\sigma^{f}_{d}\right)u^{g}=0}$.

Matter Evolution Equations through the Bianchi identities:

$\boxed{\dot{\mu} = -3H\left(\mu + p\right) - h^{a}_{b}\nabla_{a}q^{b} - 2\dot{u}_{a}q^{a} - \sigma^{a}_{b}\pi^{b}_{a}}$,

$\boxed{h^{a}_{b}\dot{q}^{b} = -4Hq^{a} - h^{a}_{b}\nabla^{b}p - \left(\mu + p\right)\dot{u}^{a} - h^{a}_{c}h^{b}_{d}\nabla_{b} \pi^{cd} - \dot{u}_{b}\pi^{ab} - \sigma^{a}_{b}q^{b} + \eta^{abcd}\omega_{b}q_{c}u_{d}}$.

One also has evolution equations for the Weyl curvature tensors $E_{ab}$ and $H_{ab}$, these can be found in Ellis’ Cargese Lectures.

Despite the fact that these modifications are absolutely necessary if one is to take seriously the notion that our universe has black holes in it, most astronomers and indeed most astrophysics courses continue to use the simpler versions assuming that the universe is spatially homogeneous and isotropic, which contradicts by definition the notion of black holes existing in our universe.

On the Importance of Solar Eclipses

As has been well documented over recent days, there has been great excitement over the recent activity of full and partial solar eclipses, with students, astronomy enthusiasts, etc… all over showing great enthusiasm. However, a vast majority seem to be completely unaware as to why these solar eclipses are so important. Their most important purpose is that one can directly confirm as Eddington did in 1919 the validity of Einstein’s theory of General Relativity. When the sun is eclipsed, one can directly observe starlight from behind the sun being bent as should happen according to Einstein’s theory, namely that light bends according to the curvature of spacetime. I document here a brief calculation that demonstrates this:

We will assume that the spacetime under consideration is spherically symmetric and static, and so by Birkhoff’s theorem, outside the spherically symmetric body, the solution to Einstein’s equations is the well-known Schwarzschild metric:

To determine an equation for the path that light should follow in this spacetime, one writes the associated Lagrangian of geodesics as:

Applying the Euler-Lagrange equations and exploiting the fact that the spacetime is spherically symmetric and static, one obtains the orbit equation for light as:

More on this calculation can be found in my handwritten notes by clicking the image below:

For the Sun, the deflection angle turns out to be approximately

That is, the curvature of spacetime induced by the Sun causes starlight from behind the sun to be “bent”/deflected by 1.75”. This is precisely what Eddington’s team observed in their 1919 expedition. This test was what finally confirmed Einstein’s theory of General Relativity, and is the true reason why solar eclipses are so important!