# Causality and Determinism

If one has a deterministic model of physics, why is causality so important?

Let's work in a fixed frame. Suppose that event A in the future causes event B, which happens before event A. Now, given the conditions at some time $t_0$ before both events (or at the time of event B), one can predict what will happen in the future, hence one already knows that event A will happen.

Therefore, there's no event that happens after B that causes B and that we didn't already know about at the time of event B. Rather, we could say that because event A is caused by the conditions at time $t_0$, event B is in essence caused by the conditions at time $t_0$ and not by event A (in the sense that it can be explained completely in terms of what happens at time $t_0$). Therefore, if we have a theory that says that event A can cause event B, we could re-explain the exact same theory by considering the mechanism that allows A to cause B as simply part of the laws which allow us to determine all future events from the initial conditions at time $t_0$. Maybe, there's some mathematical aspect of those laws, some term in the equations, which we would philosophically like to think of as 'event A causing event B.' But physically, causality is already in this sense satisfied under a deterministic model.

I came to wonder about this as a result of classical electrodynamics (without any other forces or gravity). If we know the electromagnetic field at a given point in time in addition to its time derivative, as well as the position and velocity of all charge and mass, we can use Maxwell's equations and the Lorentz force law to find out what will happen at all future events.

The implication of this is that in special relativity, we shouldn't really have to worry so much about information traveling faster than the speed of light (something we impose on a system because otherwise it would violate the law of causality after applying the Lorentz transformations). It does so happen that the current classical special theory of relativity works nicely mathematically and also satisfies that principle, so we have no reason to reject it, but I don't see why it's so important for information not to travel faster than the speed of light.

One further idea: We are assuming that we can choose some time $t_0$ before all events caused by A, but what if A causes events arbitrarily far into the past. One idea is that we could in some sense consider the limit of the state of the universe as $t$ approaches $-\infty$ as our initial conditions.

Maybe the independence of causality and determinism could make sense in a universe with 'divine intervention' of sorts, where we could suddenly decide to intervene at any point and break the deterministic model. For example, suppose we could suddenly give an electron in the universe a mysterious push, or we could even magically make an electron appear or disappear. Then if the laws of physics allowed that divine push to influence what happened in the past, that would violate causality. But that situation would not be deterministic.

Causality means that for any two events A,B, there has to exist an ordering that says whether A can influence B or B can influence A, and in the normal examples with "time", the ordering is the condition $$t(A) < t(B).$$ If the condition above is satisfied (i.e. if A precedes B), then A may influence B.

An ordering - a transitive relation - has to exist in order to avoid logical contradictions. If the relationship were not transitive, for example, it would be possible to find triplets of events A,B,C such that A influences B, B influences C, C influences A. That would be a "closed time-like curve" and it would lead to logical inconsistencies because in general, there would be no way to choose the outcomes of the events A,B,C so that all the three implications are preserved.

Those contradictions are avoided in any causal theory because the outcome of event A (in a causal and deterministic theory, to be specific) is never calculated from conditions at event B if A is the cause of B (if A precedes B). It's the other way around. Causality makes it clear which data are "inputs" and which data are their "outputs", so because of this orientation, there can't be any contradiction.

In a geometric setup, the comparison of a coordinate associated with the events is the only way how to produce ordering as a relationship. We call this coordinate "time".

In special (and similarly general) relativity, the condition for A to be able to influence B becomes sharper - $t(A)<t(B)$ has to hold in all reference frames which means that B has to belong to the future light cone of A.

You want to discard the idea of "causality". At the same time you are talking about "past" and "future", a time $t_0$ which is earlier than A and B etc. etc. which is funny since the very concept of past and future depends crucially on the concept of causality!

It is the causality which enables you to decide which event has taken place earlier and which latter. So all the logic of your arguments falls flatly.

As already pointed out that any kind of "retro-causality" will bring inconsistency in the logical structure itself. In special relativity the concept of causality is even more forceful. Without that it can't work.

• Actually, note that I work in a fixed frame. What I mean by past and future is time before and after in that particular coordinate system – Davidac897 Apr 27 '11 at 1:16
• @Davida897: It doesn't matter. In a particular frame also the concept of past and future (which event has taken place earlier and which latter) depends upon the causality. – user1355 Apr 27 '11 at 1:45

"we shouldn't really have to worry so much about information traveling faster than the speed of light (something we impose on a system... I don't see why it's so important for information not to travel faster than the speed of light."

The fact that nothing can move with a speed greater than the speed of light is not an "assumption" that ensures causality. Rather, it is a consequence of the special theory of relativity as follows: the energy of a massive body of mass $m$ moving with a speed $v$ is $\gamma m c^2$, where $\gamma$ is the Lorentz factor. Thus, for a body to be accelerated to the speed $c$ you will require an infinite amount of energy! This makes $c$ a natural upper bound to the velocity a massive body can attain.

Causality is a logical principle that is appealed to when you want to explain why $c$ is the upper bound for velocities using the relativity of simultaneity. Like Lubos said, you run into all sorts of contradictions if superluminal velocities are possible. I don't see how causality has anything to do with determinism, though.

• "The fact that nothing can move with a speed greater than the speed of light is not an "assumption" that ensures causality" - No. This is exactly the basic postulate behind the SR, from which the Lorentz transformations are derived. – Anixx Apr 24 '11 at 13:29
• @Anixx - I was under the impression that the two postulates said the following: (1) The laws of physics are the same for all inertial observers and (2) the speed of light as measured by all observers is always $c$, irrespective of their state of motion, or the state of motion of the emitter. That nothing can move faster than $c$ is then a consequence, isn't it? Please clarify. – Madhusudhan Raman Apr 24 '11 at 13:47
• You are absolutely right regarding the above comment. – user1355 Apr 24 '11 at 14:36
• SR is only geometry and mathematics. The only what You need for SR is as follows: (1) The laws of physics are the same for all inertial observers, (2) coordinates of different inertial observers transform in linear (affine) way. From that two postulates You will give Lorenz group of symmetry, with one constant - velocity of unknown value which is "largest possible velocity inertial observer may notice". Then You may turn on some physics. You may identify this velocity value with the speed of light by comparison with Maxwell equations which obeys Lorentzian symmetry. – kakaz Apr 25 '11 at 19:19
• So please remember - the maximality of seed of light is kind of experimental results ( from Maxwell theory of electrodynamics) whilst existence of maximal speed as a constant of nature is a effects of Galilean principle ( all inertial observers are equivalent) and certain mathematical assumption about symmetry ( linearity or affinity of coordinate transformations among them) – kakaz Apr 25 '11 at 19:21

Good question. Yes, it is possible to travel faster than light (and faster than infinite speed, i.e backwards in time) in a relativistic theory that is symmetric against the direction of time, i.e. reversible.

The problem is that classical physics is not time-reversible due to the second law of thermodynamics. The second law of thermodynamic in turn originates in quantum theory, namely in the postulate of the wave function collapse. The collapse cannot be reversed, so the casualty means that the result of a completed measurement cannot be affected by any future measurements.

But if we do not make any measurements, the system undergoes unitary, time-reversible evolution. This means that faster-than-light travel is actually possible in quantum mechanics in well isolated quantum system. And this is supported by experimental evidence (for example, photons can travel faster than light due to uncertainty principle). One just cannot transfer classical information this way because such transfer unavoidably includes measurements.

• ""The second law of thermodynamic in turn originates in quantum theory,"" -1 for such nonsense – Georg Apr 24 '11 at 14:10
• The answer is full of nonsense! – user1355 Apr 24 '11 at 14:38