One way of asking "what is the current state of the universe?" is to pick a Cauchy surface. This is just a "slice" of the entire universe at a given time. There is a lot of freedom in the choice of slice: In Minkowsky space for example, there are slices corresponding to every choice of rest frame, and many more besides those. We just need to make sure that no points on the surface lie within each-other's light cones ().

The information (about field values & derivatives) lying on any particular Cauchy surface is enough to predict the future and past from that surface. Pick any two Cauchy surfaces, and there's a unitary operator mapping one to the other. This is the relativistic version of a time-evolution operator.

Some Cauchy surfaces are entirely later in time than other Cauchy surfaces. (Though some pairs of Cauchy surfaces are partially later and partially earlier than each other.) We'll say that for Cauchy surfaces $A, B$ , that $A < B$ exactly when for all points $a \in A, b \in B$ , either $a$ is spacelike separated from $b$ or $b$ is in the future lightcone of $a$ .

Let $S ()$ be a function that measures the entropy on a given Cauchy surface. The second law of thermodynamics then says that if $A < B$ then $S (A) \leq S (B)$ .

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[-]James Camacho5h10

I think it's because Schroedinger's equation is

which, breaking $H$ into its energy eigenstates gives

$| ψ (t) ⟩ = \sum p_{n} e^{i E_{n} t} | ψ_{E_{n}} ⟩ .$

In Minkowski space, the time $t$ is literally imaginary. What if we rotated it $90^{\circ}$ in the complex plane so that it's just like the other dimensions? This is called the Wick rotation, $i t = - β$ , and we recover the Boltzmann (Gibbs) distribution

$| ψ (t) ⟩ = \sum a_{n} e^{- β E_{n}} | ψ_{E_{n}} ⟩$

or equivalently

$Pr [ψ (t) = ψ_{E_{n}}] \propto e^{- β E_{n}} .$

$β$ takes on the role of inverse-temperature. Now, what is the entropy-maximizing distribution? Let

$p_{n} = Pr [ψ (t) = ψ_{E_{n}}] .$

We want

$\begin{matrix} max entropy & = - \sum p_{n} ln p_{n}, s.t. \sum p_{n} E_{n} & = constant . \end{matrix}$

Lagrange multipliers give

$ln p_{n} = β E_{n} + constant$

or the same Boltzmann (Gibbs) distribution we saw earlier. So basically, if we switch to a coordinate axis where time is identical to the other distributions, we find that evolution through time is equivalent to a maximum-entopy distribution in this other coordinate axis.

Now, this is almost circular reasoning, because why is Schroedinger's equation the way it is? Basically, if you have some observer $t$ (careful here! we're reusing variable names with different meanings!), and you have some object $ψ$ , and you say that $ψ$ looks the same to the observer as the observer changes, that's written mathematically as

$\frac{\partial ψ}{\partial t} = A ψ$

where $A$ is some matrix/linear transformation. So

$ψ (t) = e^{A t} ψ (0) = e^{scaling    Z t + r o t a t i o n    i H t} ψ (0) .$

After a long enough time, we're really only looking at rotation. So, why is our "observer" the same as the time dimension? Well, it's not. But imagine it's moves mostly in that dimension, maybe around $299, 792, 458$ times faster, with a little mixing in of the other three. Then the mixing in of the other three will make it so the entropy-maximizing distribution we see in 4 space dimensions should also be entropy-maximizing if we Wick rotate any one of those dimensions.

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Arrows of time and space

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The Question

The Basics

Causality in space and time

The arrow of time in Euclidean space

The arrow of time in Minkowski space