From Wikipedia:
"The relativistic principle of causality says that the cause must precede its effect according to all inertial observers"
What exactly does this mean?
Also, is it an assumption or is proven?
I was looking for something in Physics which establishes the correlation between cause and effect, and either proves or assumes that cause precedes the effects.
The notion of "Causality" is notoriously subtle, something philosophers still struggle with and is probably impossible to define rigorously. A search of something like the Stanford Encyclopedia of Philosophy will turn up a dizzying array of articles that all show this notion tests the intellect of very many highly capable and insightful people.
But from the point of view of physics, one can say that it's simply an inductive inference that because, experimentally, certain physical processes are always observed to happen in the same order as time measured by our clock advances, we hypothesize that they will always happen in this order. I've never eaten boiled eggs in my 52 years of existence without my having to boil them first, for example. Notice how this notion depends on known, experimental physics: we have to make observations for each new process to establish whether there is a "preferred" order and what that order is, although this preferred order in physical processes is all around us from the time we are born. There'd barely be a human alive who could not tell when a movie of a glass shattering on the floor is being played backwards.
So to answer one of your basic questions, no, causality is certainly not proven: it's simply an hypothesis begotten of inductive inference. An experimental result.
"Relativistic Causality" is simply the hypothesis that observed preferred orders of physical processes are not disturbed by transformation between inertial frames. There's no reason to believe that well established orders of processes are observer dependent, so we make the hypothesis that they aren't. Not the least because a theory wherein they were would be greatly more complicated!
This is a particularly important point in special relativity, where the notion of simultaneity is observer dependent. The Lorentz transformation, with its conserved signatured metric, gives us a way to make sure that established orders of physical processes are not observer dependent, even though simultaneity is. And that is if make the hypothesis that no two observers can move at speed greater than $c$ relative to one another. Supraluminal boosts do change the time-order of events. So if we make the hypothesis that certain processes happen in a given order and that that this order is never observer dependent, then we are forced to make this no signalling-faster-than light inference; to do otherwise would be to allow the possibilities that gainsay our initial hypothesis. Interestingly, the only other ways that a time and a space dimension can be mixed through linear transformations is by rotation and this is ruled out because one could then always find a boost that would reverse the order of any set of events.
In closing, notice in physics that there is a great deal that doesn't easily fit into this "causality" framework: for very simple systems - like systems of currents and the system of electromagnetic fields that they "source" described by Maxwell's equations - it is very hard if not impossible to ascribe the categories of "causative agent" and "effect" to the system components: current and field are intimately mixed by coupled equations. This is not surprising, because Maxwell's equations by themselves are acausal: any solution can be turned into an equally valid one by inverting the time co-ordinate. Causality is imposed on Maxwellian electrodynamics by hand simply by discarding the so called advanced wave potentials and keeping only the retarded potentials. This situation is true for most of the fundamental laws of physics. However our everyday World and its processes are complicated enough that thermodynamics establishes the preferred orders which we ground our notions of causality on.
Causality in physics is not defined in a satisfactory way. The meaning of cause and effect has been debated for centuries. In the end, at the microscopic scale (with the exception of a few processes) the laws of physics are reversible, and the best you can do is define if two events could have influence on each other. Everything are particles that collide, split or merge, and each of these processes can happen in reverse. Thus there is no much sense of talking of cause and effect at the microscopic level. It does at the macroscopic level though, and it is related with the perceived arrow of time, which it is also debated on how it is originated but is most likely a result of the second law of thermodynamics. It is a statistical law and thus causality is easier to define, and very useful as a concept in many contexts. But while in some situations the interpretation is clear cut (such as the car killed the boy), in others is less clear (the butterfly effect), and in others still less clear (like in a closed time loop, for instance, you are a scientist that travels to the past and clone yourself, so you are born not from parents but from yourself. This last example does not involve any paradox and does not yet contradict any laws of physics. Just my two cents, not a thorough answer.
Causality and chronological order of the events are very interesting notions - especially when considered under Relativity. I don't yet know much about GR rigorously, so I will not touch that part. (But I have heard that things become exceedingly interesting in that regime!)
It can be shown with the help of Lorentz Transformations that if the square of the spacetime interval between two events ( i.e. $-c^2t^2+x^2+y^2+z^2$) is negative then there doesn't exist any frame of reference in which they can be simultaneous. Such events are called Timelike separated. But if the square of the spacetime interval is positive then there certainly exist some frames in which the events are simultaneous. Such events are called Spacelike separated.
Now the most intuitive definition of 'causality' (whatever the precise definition may be) must imply that for an event to cause another event there must be some carrier of information from one event to the other. We will keep in mind that Special Relativity forbids any information to travel at a speed that is greater than the speed of light.
Now, if the spacelike separated events are causally connected (i.e. one of them causes the other) then in at least some frames (the frames in which these events are simultaneous) there has to be some transfer of information at an infinite speed - which is just impossible. So no spacelike separated events can be causally connected.
On the other hand, in any frame, the ratio of the spatial interval to the temporal interval between any timelike separated events will be lesser than the speed of light. Thus, at least some carrier of information must have reached from one event to the other. So they must be causally connected.
Actually, we can reverse the choice of forward time and the chronological order of causally connected events will also reverse. But remarkably, one can say for sure that if two observers agree on the chronological order of two different pairs of causally connected events then they will agree on the same of all the rest.
I think what this means is that:
Note I have not defined what I mean by 'future-directed' but this can be made precise (see below for a slightly rough hack at this).
And no, this is not proven: it's possible to construct spacetimes containing closed timelike curves for which at least (2) fails. Whether such spacetimes are physically plausible is a different question: I personally suspect strongly that they are not, but I think that is also an open question.
Future- and past-directed curves. Consider a point of spacetime $P$ and its tangent space $T_PM$ ($M$ is spacetime here) and let the metric have signature $-2$ (equivalently, $(+,-,-,-)$). Consider timelike vectors $\in T_PM$: for such a vector $\vec{v}$, $g(\vec{v},\vec{v}) > 0$ (this is the definition of being timelike), and equivalently for some basis $\{\vec{e}_i\}$, $v^iv^jg_{ij} > 0$. Now consider a pair of timelike vectors, $\vec{v}, \vec{u}$, and consider $g(\vec{v},\vec{u}) = v^iu^jg_{ij}$.
Now it turns out that this can have either sign. But (this is slightly more subtle), for any three vectors $\vec{v}, \vec{u}, \vec{w}$, if $g(\vec{v},\vec{u}) > 0$ and $g(\vec{u},\vec{w}) > 0$ then $g(\vec{v},\vec{w}) > 0$. This means that vectors form equivalence classes, and we can say that $\vec{v} \sim \vec{u}$ if $g(\vec{v},\vec{u}) > 0$.
And now it turns out that there are just two such equivalence classes (I have not shown this, but it's not that hard). We call one class future-directed, and one class past-directed.
Now a timelike curve in spacetime is future-directed (respectively past-directed) if its tangent vector is everywhere future-directed (respectively past-directed).
Note I have talked about timelike vectors and curves and not for timelike or null vectors and curves, while above I made a claim about timelike or null curves. I think the generalisation is fairly straightforward: you have to change $>$ to $\ge$, and then you need to add some conditions about the vectors not being zero (because clearly $g(\vec{0},\vec{0}) = 0$). However I am not confident enough about this to claim it without looking things up, and I am a bit lazy.
Note: although I once understood this stuff my memory of it is now quite distant: I'd welcome correction.
Two things, both simple. It is not that complicated,physics assumes causality and insures that all its laws obey it. Anything else at this point is speculation.
1) Special relativity was built based in the speed of light being invariant in all inertial coordinate frames. Events inside the null cones of each other (one past, one future, maybe both at apex) are causal - they can be causally connected. If they do or not depends on their interactions but it is causally possible. With one inside and one outside one can not cause the other. If on light comes it's the same as inside for causality purposes. Simple. Special Relativity has it that These relationships are independent of the coordinates. That answers most of the question. In General Relativity it is the same results for any two coordinate frames.
There is a well known general technical treatment of causality in general relativity called The Causal Structure of Spacetime.
2) Maxwell and other equations having solutions with time going forward, also have the same solutions with time going backwards and charge going to anti charge. That is a symmetry of nature. When we use solutions we pick the ones going forward. Why time goes forward instead of backwards is not known but various things tell us to pick one. Either way causality would be obeyed just reverse the symbols. Since we are not too stupid we do
There are speculative theories about it. One less speculative is that it is basically thermodynamic time, and isolated systems time forward is where entropy increases.
Maxwell and other physics equations are totally compare to with causality as in Special Relativity. Maxwell's generalized are also consistent with general relativity. All quantum field theory is consistent with causality as is special relativity. Eg, probabilities of a state going to another is zero if the latter is outside the light cone of the other.
There is no, at a standard level, problems with causality in physics
