Hi I want to get to the bottom of some thaughts I have and for that I need an answer to a – I have to admit – highly constructed example.
Lets assume – against all possibility – we find a strange device fixed earth an on alpha centauri and we could communicate nearly instantly with alpha centauri through this device in both direction. Nobody knows why and how, but it just works.
Those devices can not be moved, and it is not allowed to change the trajectory of either earth or alpha centauri by much (in relativistic scale), or the communication feature would stop working.
Could this situation, even though we don't understand how this device works, be used to break causality, for example by sending messages back in time? And if, how?
UPDATE: I think I might have found the answer myself, also thanks to @Willo for the final push.
If one such device can exist, so can ofcourse multiple such devices exist.
So this is how one could violate causality using this:
Earth tells Alpha Centauri via this communication channel to press a button, which is what they do.
A third observer moving at relativistic speeds compared to Earth and Alpha Centauri observes this, but in reverse order. First Alpha Centauri receives a message and presses the button, and only long AFTER that earth sends the signal to press the button.
So in theory, if they know about the communication channel, they could try to contact earth and prevent the message being sent, AFTER they have witnessed that the message was received.
Problem remains, how they do that. I don't think they can move to earth themselve, because I guess calculations will show that they can't make it in time. But if there was another Planet with a communication channel from where the Ship currently is, to earth, they could stop by there to send the message in time.
My problem is I never understood how accelleration/deaccelleration in relativistic terms can modify expierienced time and if this "Stopping" at that other planet is possible without missing that moment as well. But it seems to me this can be achieved. So we have a paradox and my question is answered by: Yes
Update 2: The more I think,the more I come to the conclusion that deaccellerating into the innertial frame of this thrid planet, earth and alpha centauri will make the ship miss the opportunity to stop earth from sending that message, after all… So I am stuck with a NO again and that seems wrong.
Best Answer
You already seem to have your accepted answer, but your question edits indicate some uncertainty. Let me see if I can offer an easier way to see it.
First off, what is the question? As I read it, your question is essentially,
And the answer is no. (It can be trivially "yes" if they are two-way portals that are not all simultaneous; it remains "no" if they are two-way portals that are all simultaneous.)
Light bubbles and the two light cones of an event
To understand why, we simply look at the light cones.
An event, say a supernova, emits a lot of light. To anything which hasn't seen this light, that event has no causal power yet. We describe this as a "future" light cone which is an expanding "bubble" of photons in spacetime that announces to the universe that the event has happened. Since it expands at constant speed, if you project-out one of your dimensions of space and use that dimension instead to visualize time, these photons describe a cone as the circle expands uniformly in time: hence the name, "light cone". Now every point in spacetime "within" that cone is in the event's objective future: those points in space have observed the event, so there is a definite time-order. There is no definite distinction between where these events are spatially because a spaceship, bound to stay slower than $c$, could have had just the right sort of trajectory to happen to be at the locations of both events when they occurred; its reference frame locates both coordinates "right here", hence they may have happened at the same or different places.
The event also has a "past" light cone that you can get by extending all of those light rays backward through the event to the time before the event. This has an intuitive meaning too: the points inside this "past-pointing" light cone are objectively in the supernova's past: the star, pre-supernova, has observed all of these events by the time that the supernova has occurred. Then the space between these light cones is a sort of "relativistic present" for the event: these points in spacetime are objectively not at the same place as the event, but they may be at the same time or not depending on your frame of reference. The "local presents" for the supernova are hyperplanes in spacetime which pass through the supernova and live between these two light cones; given any such plane, there exists some reference frame which thinks that these spacetime points in the hyperplane are simultaneous; the hyperplane is a "present", but localized only to that reference frame.
If you finally understand these definitions and you want to kill your grandfather or otherwise send a signal to yourself before you send it, the perspective in terms of light cones becomes very simple: To get a grandfather paradox, you have to get information from an event into the past-pointing light cone of an event.
Why the answer is "no".
Relativity says that every reference frame is correct for analyzing the happenings in spacetime and that they all agree on the general topology of these light cones etc.
In particular the privileged frame above is always a correct way to view the whole spacetime. In this privileged frame there is a monotonic-invariant: no matter how information passes through these portals, the local time coordinate in this reference frame starts positive and keeps increasing. But the past-pointing light cone in this reference frame only has negative time-coordinates. So it's really simple: you cannot get into the past light cone using these portals. Some reference frames see this communication as the future having affected the past, but it is not a robust enough effect to actually generate a paradox.
With that said, other reference frames will still have paradoxes to deal with. The simplest one is temporary violations of conservation of energy: you throw a ball through the portal and some reference frame sees a time where two balls coexist briefly.
But it matters that it's all simultaneous in the same reference frame.
Now let's do the reverse. Suppose you have just two portals; they transmit information faster-than-light in two different reference frames. Then some of these configurations allow a grandfather paradox. The exact calculation of when this occurs is a little tedious; let me phrase it like this: in the simplest case of co-inertial portals, each portal consists of two worldlines which can be written $r_{0,1}^\alpha(s) = c_{0,1}^\alpha + s~t^\alpha $ connecting points at the same $s$; hence $r_1^\alpha(s) - r_0^\alpha(s) = c_1^\alpha - c_0^\alpha$ is always a spacelike vector of constant size $c$.
With no loss of generality we can do a bunch of stuff with this. Choose some reference frame and determine which FTL portal is "more" future-pointing in that frame, then boost into the coordinates where its $c_0$ is at the origin and its $c_1$ is simultaneous, at some point $q^\alpha = (0, c, 0, 0).$ This means we only need to mess with one $r_{0,1}^\alpha$ line and its corresponding $s$.
Now in this reference frame, the time-coordinate of $r_1^\alpha(s) - r_0^\alpha(s)$ is negative; then we solve $|r_0^\alpha(s) - q^\alpha| = 0$ for $s$ to find the "closest" portal $r_0(s)$ we can enter from $q.$ We then get to violate causality if the corresponding vector $r_1^\alpha(s)$ is timelike past-pointing; the portal must emerge into the past light cone of the origin. That's the essential criterion for having a causal loop with two fixed portals.
If you can just teleport into a simultaneous moment elsewhere in your own "present" coordinates, then (perhaps with some restrictions on your acceleration and teleport-distance) you can cause grandfather paradoxes willy-nilly: you teleport, accelerate towards where you came from, "tilting" your local-present hyperplane underneath the point in spacetime you teleported from, then you can teleport into your own relativistic past.