Can FTL-Communication between two points in the same frame of reference break causality? 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.
 A: 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,

Suppose we take special relativity and add, in one privileged reference frame, $N$ one-way portals which can teleport matter/information to points on corresponding worldlines, so that the 4-displacement from the portal entrance to exit is either locally simultaneous or future-pointing. Can we use these portals to create a grandfather paradox?

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.
A: The problem is that special relativity predicts that the laws of physics are invariant under every reference frame. 
If you modify special relativity, choose a special reference frame, and then say, "instantaneous communication works in this frame and only this frame", then you can make up a consistent theory about the universe where you have faster than light communication*. But this already violates two big parts of physics. Firstly, special relativity postulates that there is no privileged reference frame, so we already have a major contradiction with known laws. Secondly, say the communication takes place through light -- the electromagnetic field. As a quick refresher, part of Maxwell's equations can be written $\varepsilon_0\nabla \cdot {\bf E} =\rho$, where $\rho$ is the charge, $\varepsilon_0$ is a constant, and ${\bf E}$ is the electric field. So let's say that the coupling takes place in $\rho$, and that:
$\varepsilon_0\nabla \cdot {\bf E} =\rho$ everywhere, but that in a certain region, in our privileged reference frame, $$\varepsilon_0\nabla \cdot {\bf E}(\mathrm{Earth},t)=\varepsilon_0\nabla \cdot {\bf E}(\mathrm{Alpha\ Centauri},t)=\rho(\mathrm{Earth},t)+\rho(\mathrm{Alpha\ Centauri},t)$$
But this equation no longer respects Lorentz invariance! That is to say, our modified Maxwell's equations would be different in different reference frames! This is VERY bad, and betrays the entire history of the development of special relativity. So, to summarize:


*

*You would violate the postulates of special relativity by introducing a privileged reference frame. This isn't a nail-in-the-coffin by itself, because -- although the laws of physics are the same -- the rest frame of the cosmic microwave background is a sort-of special frame. 

*You violate Lorentz symmetry, which has never been observed before. ("No Lorentz violations could be measured thus far, and exceptions in which positive results were reported have been refuted or lack further confirmations.")
However, this is all assuming you stick to a single reference frame. If you allow instantaneous communication to occur in just two distinct reference frames, then you do violate causality**.
No comment on viewing it in the framework of general relativity.
*(The reason I say this: I challenge you to draw a Minkowski diagram, using whatever timelike lines you want but only a set of parallel spacelike lines, in which backwards time travel is physically realized. You can't do it!)
**Draw a Minkowski diagram in 1+1 dimensions that contains a parallelogram with all sides spacelike and all sides representing an instantaneous communication between two points. By changing your frame of reference, any one of these four points could have caused the other three. This Minkowski diagram is plotted below. Lets say Earth operates by the rule that it will send a declaration of war if it doesn't hear from the aliens. It doesn't hear it, so it sends a declaration of war. This causes the aliens to rethink things and quickly send a declaration of peace by instantly (in some frame) sending a message to outpost 1, which instantly (in some frame) sends a message back to the aliens (this is ab event labeled "outpost 2" and is at an earlier time), which instantly (in some frame) sends a message to Earth before they send the declaration of war, causing them not to send the declaration of war in the first place. Contradiction!
In the following images, each blue line represents a "person". They are all sitting still with respect to one another. The vertical axis is time, and the horizontal axis is position. The yellow dots are NOT stars! A star would be a line, because it exists over time. The yellow dots could be explosions or supernova -- they are events in spacetime, not objects sitting in space.
The frame with instantaneous Earth-alien and outpost1-outpost2 communication:

The frame with instantaneous alien-outpost1 and outpost2-Earth communication:

Animation of the transformation under which the laws of physics must be invariant (ie, animation of the lorentz transformation):

This is the Mathematica source used to generate the images:
gamma[v_]:=1/Sqrt[1-v*v];
lorentz[v_]:=Function[{x,t},gamma[v]{x-v t,t-v x}];
starpts=Thread[{RandomReal[{-2,6},100],RandomReal[{-4,4},100]}];
starsizes=RandomReal[{0.001,0.01},100];
Diagram[v_]:=Module[{p1,p2,p3,p4,p5,p6,p7,p8,p9,p10},{p1,p2,p3,p4}=lorentz[v]@@@{{0,0},{2,1},{4,0},{2,-1}};
p5={0,-10};p6={0,10};p7={2,-10};p8={2,10};p9={4,-10};p10={4,10};
{p5,p6,p7,p8,p9,p10}=lorentz[v]@@@{p5,p6,p7,p8,p9,p10};
stars=MapThread[{PointSize[#1],Point[#2]}&,{starsizes,lorentz[v]@@@starpts}];
plot1=Graphics[{Red,
Arrow[{{0.2,0-2},{0.6,0-2}}],Text["space",{0.4,0.2-2}],
Green,Arrow[{{0.2,0.4-2},{0.2,1.0-2}}],Text["time",{0.4,0.6-2}],Blue,Line[{p5,p6}],Line[{p7,p8}],Line[{p9,p10}],
White,Arrow[{p1,p2}],Arrow[{p2,p3}],Arrow[{p3,p4}],Arrow[{p4,p1}],Text["Earth",p1+{0,0.13}],Text["Aliens",p2+{0,0.13}],Text["Outpost 1",p3-{0,0.13}],Text["Outpost 2",p4-{0,0.13}]},Background->Black,PlotRange->All];
plot2=Graphics[{Yellow,stars}];
Show[plot1,plot2,PlotRange->{{-0.5,5.1},{-3,3}}]];
Export["EarthToAliens.png",Diagram[0.5]];
Export["AliensToOutpost.png",Diagram[-0.5]];
Export["change.gif",Join[Table[Diagram[v],{v,-0.5,0.5,1/100}],Table[Diagram[v],{v,0.5,-0.5,-1/100}]],"DisplayDurations"->1/30];

A: Yes, such a device could send messages back in time.
It might appear that it can't because you imagine that there is some "now" and that the device merely connects the now at two distant events.
But the reality is that general relativity is local and says that what clocks measure and what people notice is the length of curves on 4d spacetime. So we measure lengths of curves and and the length depends on the path as well as the endpoints.
The physics is local it tells you how long a segment of a path is in a little region. It does not have a built in sense of now and some spacetimes do not have a sense of now that can even be imagined on top.
So what you are really doing is allowing a here-now to interact with a there-then. But time a measurement of length along a curve. It is different than 3d measurements in that there can be a longest path and all the other paths are shorter. So for instance if your friend shoots off super fast and goes to a distant star and back their clock could read a very short length for that path but for some path there is a longest path someone whose clock ticked more than any other clock with the same endpoints.
So we are trying to vet to how the causality violation can occur.  Since time is measured along a curve in 4d we are allowing communications between curves with corresponding points being able to communicate. So imagine we label two points red, one on each curve to indicate they can communicate bilaterally. Then we label two points (one on each curve) to indicate they can communicate bilaterally and so on.
The length along a curve between two communications is either different for one curve than the other or they are the same. And the two devices are either in equally deep gravity wells or else one is in a deeper well.
Case 1 the length along a curve between two communications is different for one curve and the other And the wells are equally deep. Then the one that is measuring a longer length will be able to send messages to their own past. They start telling the other person today's winning lottery numbers. The other person starts sending the messages out at light speed along the normal route between the stars.  Eventually, but after a finite amount of time, the messages start to arrive via the long route. But they are arrive at a rate faster than they are being sent. Just sit back and wait and you'll start to get messages you haven't sent yet. Causality violation.
Case 2, the length along a curve between two communications is the same for one curve and the other. If they are in equally deep gravity wells just take the slightest bit of matter or energy and move it ever so slightly close to the communicate device. Now that one where you moved the matter closer is in a deeper gravity well. Now the person in the deeper well can send messages to their past. How? Same thing, they send messages pit of the well via the device. The one outside the well starts sending the messages out at light speed along the normal route between the stars.  Eventually, but after a finite amount of time, the messages start to arrive via the long route. But they arrive at a rate faster than they are being sent (because of the deeper well). Just sit back and wait and you'll start to get messages you haven't sent yet. Causality violation.
And basically you just tune those rates. The gravity well dictates the rates you send versus receive messages if you just sent them via a normal route. The device has some relative rate of the messages sending and receiving based on its own magic. If they happen to line up then you can adjust the well rate a tiny bit by moving some matter nearer to one end of the device.
Note that it did take time for the off sync to become a causality violation. Some people speculate that the device would self destruct before a causality violation occurred.  That sounds bad either way.
Edit to respond to your edit By the time someone sees either you telling them to push the button or sees them pish the bottom then it is totally too late for them to tell you.
I think your whole notion of before and now are misleading and going to bite you over and over again. You have curves and they have lengths and that is how we make predictions about whether an event is observed at a location where the clocks is between this tick and that tick.  Whether you think a distant event happened before or after another  distant event is an opinion that affects nothing. It's like saying something is above the xy plane. The xy plane does exist it is a computational choice. And so is the notion of before and after. And that is why different observers disagree, but it is merely a mathematical convenience that is natural for a particular observer. And sometimes with gravity it isn't even that natural, so just say no.
So your method is wrong. And you should stop thinking that an observers opinion about distant events is meaningful. If there is a timelike curve from one to the other that is meaningful and so it the length of that curve. If there is a spacelike curve that is meaningful and so is the length. But if there is no timelike curve then before and after is not physical.
A: 
Can FTL communication between two points in the same frame of reference break causality?

No. Time travel is science fiction. See this answer for something about the nature of time. In a nutshell it's a bit like heat in that's an emergent property, related to motion. Time is real like heat is real, in that a hundred years will kill you just as surely as a hundred degrees centigrade. But just as you can't literally climb to a higher temperature, you can't actually travel to another time. Neither forwards nor backwards. There is no way you can move such that everything else not only moves back to where it was, but never moved at all. In similar vein you can't move something else to send it back in time, even when it's just a message or signal. 

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? 

No. What happened happened, and it happened because something moved. And since it happened, light has moved, the Earth has moved, blood in your veins has moved, electrochemical signals in your brain has moved, and so on. That motion cannot be undone by moving something else. Sending some instant "subspace" signal to Alpha Centauri and back doesn't change a thing.  
