I've been going through my somewhat rusty understanding of relativity again given the occasion and started to draw Minkowski diagrams to get my thoughts in order. Since I think they are actually quite helpful in understanding what "time travel" in the context of relativity actually means, I jotted them down digitally and added a bit of an introduction for those unfamiliar with them:
Minkowski diagrams are a standard way of depicting what happens in special relativity. "Special" meaning we're not dealing with the crazy 
of curved space yet, just the speed of light as a finite upper limit (including for the transmission of information).
This is a basic Minkowski diagram, as of yet unpopulated:
In order to make things representable we restrict ourselves to one space dimension, so what you see before you is not a spatial plane, but a one dimensional space (a line) and a time axis forming a flat 2D spacetime. Usually one chooses the units on the two axes such that the speed of light corresponds to one space unit per time unit, i.e. light emitted from the origin of this diagram moves on the yellow dotted lines towards the future.
Since the yellow dotted lines represent the speed of light (remember, this is a space-by-time diagram), they form two cones with respect to the origin. The upper one is called the future cone, containing all events (points in spacetime) that you could causally influence from the origin. The lower one is called the past cone, containing all events that could causally influence the observer sitting in the origin.
The past and future cones together only make up half of the diagram. Why? The rest is called the elsewhere (incidentally also the place where Khajiit live), and consists of those space-time coordinates that you can never reach and that can not influence you currently: You might be able to reach the space-coordinate at some time, but not at the time corresponding to these specific points in spacetime.
Now, we populate this flat 1Dx1D world. Here is an unmoving observer sitting in the origin:
The arrow does not indicate movement in space in this case (the space coordinate does not change), but simply the passage of time. These curves showing which point in space is passed by an observer at which point in time are called worldlines.
If you start moving around, they will look like this:
The observer in our example is moving at half the speed of light, they are making one unit of space in two units of time. This example also allows us to demonstrate the concept of time dilation: Say you left a clock sitting at rest at the origin and then take off at half the speed of light. After you have travelled one space unit (taking two units of your Eigenzeit) you take a look back at the clock sitting in the origin. Naively (or newtonially) one would depict "looking back at the origin" by the black dotted line. However, since "looking" means you have to receive light rays reflected of the clock in your eyes, we may not draw a horizontal line from you to the vertical line representing the origin in the passage of time, we have to draw a 45° angled light ray back to the origin. Now this light ray meets the origin at the event (0 space units, 1 time unit), which means the time you are seeing on that clock in the origin is 1, not 2 as on your own watch. That means that from your perspective time at the origin is advancing more slowly than with you.
This is the first effect on the passage of time one has to consider: Since measurement of time at different spacetime points involves the transmission of information, which happens at most at the speed of light, movement changes what time coordinates we measure for distant points in space.
Now a common talking point is that going faster than light means going back in time. Let us make that precise, as it will turn out that what you get from superluminal speeds is pretty far from a naive "turning up in your own past" DeLorean-style. Here is the worldline of a hypothetical observer that somehow managed to go at double the speed of light:
If we apply the same principle as above for reading off the time at the origin (we left a clock there again), we see that the time at the origin seems to be -1, so indeed it seems like we travelled back in time with respect to the origin. However, your Eigenzeit ("own-time") still progressed by one unit to 1. Furthermore, since we left the origin, we cannot influence its past in any way anyhow (our own future light cone still only contains future points of the origin). The only thing we have achieved so far is that while looking back at the origing it looks like time at the origin is going backwards (i.e. we receive earlier and earlier light signals emitted by the resting clock).
Frustrated by this ineffective method of "time travel" we decide to turn around and travel back to the origin at double the speed of light, which looks like this:
Now something funny happens: Suppose we left a farewell/welcoming party committee at the origin and they are looking at us to see when we turn around, so they can prepare the party. They are in for a surprise: We arrive back at the origin way before they even see us turning around. Why? The event of us turning around (a) emits a ray of light travelling towards the origin at the speed of light. But since we travel at double that speed the events of us passing Alpha Centauri (b) and the Kuyper Belt (c) emit rays of light that arrive even earlier at the origin. In toto, seen from the origin our journey back seems to be entirely backwards: First we arrive completely unforeseen (literally) at time 2, then they see us moving backwards via c, b, a to the point of turning around on our superluminal voyage - all the while we're standing next to them on earth! The only "causal" stunt you could pull here is shooting a laser cannon at one of your friends from (a), then travel back to earth, and comfortably wait until it arrives one time unit later to either heroically save your friend or murder them with the perfect alibi - but that's just (slightly psychopathic) cosmic one-man-runaround ping pong, not actual time travel.
So, superluminal speeds make the succession of time measured between observers moving relative to each other somewhat awkward, but doesn't really create time travel opportunities.
There is one interesting effect of superluminal speed though: It changes what events are contained in your past and future cones in a weird way. Here's ordinary, sublight travel:
You move at half the speed of light from the origin along the arrow representing your worldline. When you arrive, the green dotted lines represent your new future and past cones. Notice that the new future cones is always contained in any older future cone (e.g. the yellow one from sitting in the origin earlier), and that the new past cone containes all older past cones. In other words, the passage of time narrows down what you can potentially influence in the future, while it adds new events that can potentially influence you.
Now this only holds for sublight speeds. Here's the same diagram for travelling at twice the speed of light:
Notice how the green future cone is no longer contained in the yellow one, and the green past cone no longer contains the entirety of its yellow counterpart. In particular the blue area used to be part of your past, but is no longer, while the orange area used to be in your "elsewhere" and has now become part of your future. Caution, though: "Past" and "Future" do not mean sections of your worldline here, but the cones of events that can potentially influence you/that you can potentially influence.
So far for special relativity in a flat spacetime. Of course things are more complicated in general relativity (of which special relativity is a linear approximation for small areas of low energy/mass if you will), and spacetime itself being curved could, at least, theoretically create settings where you can actually have loops in your worldline (vulgo: arrive in your own, actual, past, and say hello to yourself like Spock). This can only happen, though, if spacetime is topologically non-trivial, i.e. if it "has holes" like a donut:
In the example above our 2D spacetime is curved and sports a kind of handle-like structure in addition to the mostly flat part. If you ignore the "entrance" or "exit" of the handle you can just move around ordinarily in spacetime (1), but if you "travel" through the handle (remember, one of the dimensions is time!) you could end up with a loop in your worldline (2). However, this kind of topological structure is highly unlikely and/or requires immense amounts of mass/energy to maintain. Some used to posit that black holes could create the ruptures in spacetime that are the "entrance" and "exit" in the image above (and then only need to "link up" to create such a "wormhole"), but Hawking and Penrose have shown that Quantum effects prohibit black holes from actually reaching infinite curvature at the centre, so it is unlikely spacetime actually "rips".
In toto, this is crazy, but doesn't really provide us with means for time travelling in the more common sense. For anyone interested in further reading I can recommend Richard Gott, "Time travel in Einstein' universe".
Anyone here know anything about timelines?
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Originally Posted by Douchebag
