# Causality principle and Entropy (Second Law)

I was reading about the light cone in relativity and I got to the point where in order to avoid paradoxes one can introduce the causality principle:

Causality Principle: For every inertial observer , there is no event whatsoever that can affect past events . Only past events can affect future events .

This principle states that time has a direction. That is related to the second law fo thermodynamics,

Second law of thermodynamics: the entropy of an isolated system can increase, but not decrease. Hence, from one perspective, entropy measurement is a way of distinguishing the past from the future.

This again states that time has a direction.

Whats the relation between both statements ?, Does one imply another ?

• Where did you read the "causality principle"? It seems to me that one would need to provide some formal definition of what it means for one event to "affect" another one. Some physicists describe the time-symmetry or time/charge/parity symmetry of all known physical laws as meaning it makes just as much sense to consider later events as "causes" and earlier events as "effects" as the more usual reversed way of speaking of cause and effect. Apr 14, 2015 at 10:09
• For example, in the book Hawking on the Big Bang and Black Holes, p. 266, Stephen Hawking says "Suppose state A evolved into state B. Then one could say that A caused B. But one could equally well look at it in the other direction of time, and say that B caused A. So causality does not define a direction of time." So he'd disagree with your "causality principle" as stated, although as I said it really depends on defining words like "affect" and "cause" that don't really have a pre-existing technical definition in physics. Apr 14, 2015 at 10:11
• "there is no event whatsoever that can affect past events" that seems like a tautology; past events are, by definition, those that already happened and thus, by definition, they cannot change. Apr 16, 2015 at 22:45
• "Second law of thermodynamics: the entropy of an isolated system can increase, but not decrease." That is a corollary of 2nd law of thermodynamics (the law itself is more general, involving transfer or energy via heat). "Hence, from one perspective, entropy measurement is a way of distinguishing the past from the future." In principle change in entropy could be used to infer some time has elapsed. But it is a ridiculous idea - measuring changes of entropy is much more difficult than measuring time directly with clock. Apr 16, 2015 at 22:51

There are several possible approaches to this question, but I've always been a fan of the one taken by Edwin Jaynes in his 1965 paper Gibbs vs Boltzmann Entropies. (See sections V and VI for the discussion, which I think can be read in isolation from the rest of the paper.) Here he derives the second law from the empirical fact that we as scientists and engineers are able to manipulate the initial conditions of an experiment, but we can't directly manipulate the final conditions. (The only way we can affect the final conditions of an experiment is via the initial conditions and the boundary conditions.) Together with Liouville's theorem this is enough to derive the second law.

Jaynes doesn't mention relativity explicitly, but if we take his view then the causality principle can be seen as a common assumption in both relativity and thermodynamics.

Of course, this doesn't address the other side of this issue, which is why there would be a causality principle at all, given that the microscopic laws of physics are reversible in time. This is known as Loschmidt's paradox, which itself has many possible resolutions.

IMHO it's important to look hard at the ontology of what's actually there and take care to distinguish between reality and abstraction. For example:

Relativity is just about the best-tested theory we've got. I "root for relativity". But I will say this: a light cone is an abstract thing. You cannot point up to the clear night sky and say, "Look, there's a light cone". The future light cone models an expanding sphere of light. A past light cone models light coming at you from all directions. And that's just about it.

I got to the point where in order to avoid paradoxes one can introduce the causality principle

That principle is a pat statement that doesn't actually explain anything. IMHO you'd be better off thinking about a light beam moving from A to B to C. When the light reaches B, that's an event, and there's nothing that can undo the motion of that light from A to B. There is no such thing as negative motion. That's why there are no paradoxes.

This principle states that time has a direction.

That's what people say, but look closely. Your light beam has to move from A to B to be able to move from B to C. You could order your events A B C and say this denotes the direction of time, but there is no actual time flowing or moving in any direction. All that's there is light, moving. Likewise there is no time flowing in an optical clock. Or in any other clock. A quartz wristwatch "clocks up" the piezo-electric vibrations of crystal and shows you a cumulative display called the time. A grandfather clock "clocks up" the swings of a pendulum, and so on. A clock is not some cosmic gas-meter gizmo with time flowing through it. Motion is motion whichever way it goes, and more motion means more time, because the big hand moves, and the little hand moves, and the gizzards of a clock are called a movement. All this might sound alien to you, but check out A World Without Time: The Forgotten Legacy of Gödel and Einstein. It's just something you don't hear much about, that's all.

That is related to the second law fo thermodynamics... the entropy of an isolated system can increase, but not decrease. Hence, from one perspective, entropy measurement is a way of distinguishing the past from the future. This again states that time has a direction.

Again, that's what people say, but this direction is an abstract thing. You can't literally point towards the future. It's merely some abstract "direction" associated with... more motion.

Whats the relation between both statements? Does one imply another?

In a way, but IMHO neither get to the heart of the matter. IMHO what does is that relativity works, but spacetime is an abstract mathematical space which represents space at all times. Because of this there is no motion in spacetime. You can draw worldlines in it to represent motion through space over time, the time being some cumulative display of the regular cyclical motion inside a clock. But like lightcones, these worldlines do not actually exist in this real world, and nor does the literal flow of time. Things move, things like light, and piezo-electric crystals, and pendulums, and hearts and blood and electrochemical signals, and cars and stars. Through space. We live in a world of space and motion, and the map is not the territory.

You asked: "Does one imply another?"

No. Neither implies the other. However, I think there are benefits to first being clear what the ideas are, particularly since I think each idea actually already assumes an arrow of time.

In the first case, you start with an arrow of time that only earlier times affect later times, and then end up strengthening that to be that causes have to be within the past light cone in order to have causes precede effects in all frames. When they say past, they mean past light cone. There is already an idea of cause and effect with cause coming before effect.

In the second cause, again when they say that entropy increases, they already have an idea of earlier times and later times when they say entropy increases. In this cases they only mention that you can measure this direction by looking at entropy measurements.

So, now, does one imply the other? An easy way to show that A does not imply B is to exhibit an example where A is true but B is not. Sometimes the example really just reveals a third thing that was needed (i.e. if when A is true and B is not you realize that C is not true then you might turn around and state that A&C together jointly imply B).

Neither implies the other because the first is a not really operationally defined in a non-tautological fashion.. So let's look at an example where the first is false and the second is true (this will show the second cannot imply the first). The idea of causality presented in the first idea makes no predictions. When you make a theory, you simply predict that observations are restricted to a smaller collection of possibilities, those selected by your theory. The theory is falsified when observations are observed that do not belong to the limited collection. And making the collection be limited is actually what the theory does. For instance Newtonian Physics is a theory about a correspondence between solutions of second order Ordinary Differential Equations and the observed dynamics of material bodies. The correspondence is what the theory is about. The equations themselves don't identify a preferred direction for time, you can tell stories after the fa t and call some thing causes and other things effects, but it is just a story on top of the actual theory and doesn't carry any scientific wait. You could swap your labels and still have Newtonian Physics. And if you do you can have one where entropy increases (because we didn't change any dynamics) in time, but effects now reliably and consistently precede causes because we changed the labels.

You can now probably see the real reason they don't imply each other, they don't even use the same words. The second one never mentions causes, so it can't possibly tell us anything about causes. But for completeness, let's make an example where the first is true but the second is not.

Here we can label the causes and effects in the usual way (causes before effects). Now we need to look careful at what entropy is. We can imagine a nice classical system with nice time reversible dynamics (this doesn't always hold even for Newtonian Physics, but let's make a theory with very limited Force Laws where it does hold).

And the big thing is that entropy doesn't actually always have to increase even for an isolated system. For terminology let's be clear that a macrostate is an actual state of the system as it really is. And a macrostate is a set of microstates with the same macro description such as volume, pressure, and temperature. Microstates do not have entropy per se, macrostates do, and the entropy of a macrostate goes up when the the size of a macrostate goes up. And the size of a macrostate is nothing more than the number of microstates that are in the collection.

When a system evolves from a low entropy system to a high entropy system what happens is that the actual microstate initially belonged to a macrostate that was, say, large (i.e had many microstates in it) and evolved into a macrostate that was even larger (i.e. had even more microstates in it).

So clearly the actual microstate is one that can come from a macrostate with lower entropy. So you can setup a universe where say, the microstate is the opposite of the current microstate (e.g give every particle the opposite momentum), but all the dynamical laws run backwards (forces that used to be attractive are now repulsive, etc.) Now evolution in the forward time will make it evolve (in the future) to what the earlier time evolution would be in the original universe. So in this new universe we measure causes as before effects, but entropy decreases.

In reality, this is just like the first example where we relabelled the cause and effects, it's just that we also relabelled the direction time goes as well (that made all the momentums be opposite, and made attraction be repulsion and so forth).

So it's the same idea, except I'm explicitly pointing out that entropy can increases in the opposite direction as the time coordinate/parameter used in the laws of physics, as well as the direction that we label as cause and effect.

So what is the relationship?

Both are already implicitly using a preexisting time in their formulation, so they both have that in common. They are unrelated in the sense that it is possible for each to point in opposite directions.

There are other unaddressed issues. For instance to test theories you have to have controls and repetition. In both cases these interact in nontrivial ways with the principles. When you try to make sense of something you are trying to look at the information in terms of the things you can measure, and a small device operating fora limited duration can't know everything about everything. And our sense of control is seeped with particular assumptions that influence what we consider good enough. The sense of control influences what we colloquially call a cause, and affects what we consider spontaneous (which affects practical conceptions of entropy).

I don't think those two statements are the same or related.

1. If you reversed time, the causality principle wouldn't be broken; the two events could still be causally related. In other words, for a future event B in the lightcone of event A, if you reverse time, A would now be in the lightcone of B and could be caused by B.

So if you want a way to check whether time is running forward or backwards, you couldn't do it with special relativity. Specially relativity would work the same in either case. The same is true of Newtonian dynamics; with a set of billiard balls, if you run time backwards, everything would look much the same as with time running forwards, you wouldn't notice any anomalies.

2. However, with the second law of thermodynamics, you could definitely make a test of whether time is running forwards or backwards. If it's running backwards, total entropy would be decreasing. That is a real change or anomaly you could measure.

IMHO, of course Time has singular direction for all real life events. E.g. living bodies, human bodies, etc. age and cannot be still or start moving the opposite direction. (The Curious Case of Benjamin Button is a fiction. Lets not mix facts with fiction here.)

Causality principle is true and so is the second law of thermodynamics. But the former is more general and the latter is more or less thermo/heat based.

But the principle of entropy (2nd Law of Thermodynamics) works best for definite and reasonable intervals of time. I say this because, just after the production of the Universe (Big Bang) the universe was tremendously HOT and would that degree of tremendous hotness be considered extremely high entropy? And following which the entropy decreased as the Universe cooled off? No! This hypothesis is violating the second law of thermodynamics. But the hypothesis is true nevertheless, i.e. Universe was initially HOT and eventually cooled off.

Second Law of Thermodynamics works best for shorter / reasonable intervals of time. Whereas Causality principle can be applied to any intervals of time (large or short) OR to whole of the Time altogether since the inception of the Universe.