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If one is able to travel into the past but at a spatial distance that puts him outside of his own past light cone would this be considered a causality violating trip? Looking at a Minkoski diagram, it would seem that one ought to be able to travel to a spatially displaced past without producing causality violations. In fact, I'm not sure you could even say whether this was the past of not for sure. Generic Minkowski diagram

For instance: If you found a one way wormhole, that took you outside of your own lightcone but into the past, could you say for sure that you traveled through time? Would this be a form of timetravel that did not violate causality? Note; I'm not necessarily suggesting that it is an FTL trip, it may be considered to be instantaneous teleportation, but in the reverse direction of x'.

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Outside the light-cone, distances are spacelike and not timelike.

This means that you can always find a frame of reference such that your "spacially displaced past" is in the future for some other inertial frame of reference.

By definition, a spacelike distance is neither in the past nor the future. By itself, this would not create any causality violations.

Since wormholes (as shown in other answers) could set up a scenario where you do violate causality, I think you will find you cannot use wormholes to travel outside of your light-cone.

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I think this best answers my question but I'm not sure about the purely spacelike aspect outside one's lightcone. Look at the minkowski diagram in my question. X' appears to be a vector in a timelike/space like direction right outside of the light cone. Are you sure there is no timelike component? – WillMcLeod May 28 '14 at 14:05
Again it depends on your frame of reference. In one frame of reference, X' is purely spacial. This means that events that occur along X' will appear simultaneous in your frame of reference. There are other frames of reference where events that occur along X' do not appear simultaneous. So in that sense, those events have a time component to them (either into the future or into the past). A "timelike" vector on the other hand has no frame of reference in which events that occur along X' would appear simultaneous. – chowey May 28 '14 at 18:09

The problem is that if you had two such wormholes, you could still end up in your own past light cone. For example: suppose I took a wormhole trip to Alpha Centauri, leaving in the year 2050 and arriving in 2049 (according to some fixed inertial reference frame). Since Alpha Centauri is 4 light years away, this doesn't take me into my past light cone. But suppose I then find another wormhole on Alpha Centauri that takes me back to Earth in the year 2048. This is also a "faster than light but backward in time" trip that doesn't take me into my own past light cone. But now I'm back on Earth two years before I left, and I can wreak havoc on the space-time continuum by murdering my former self, etc.

Now, there are ways you could still get around this. Basically, you have to come up with a reason why the reverse trip I've described isn't possible after all. You could say that there is only one wormhole of this kind in the whole universe, for example. But then you'd have to explain why, of course; as far as I know there isn't any actual theory that would predict something like this.

In general, if you want to keep Einstein's postulate (for which there is a huge amount of empirical evidence) that the laws of physics are the same in every reference frame, it's going to be very difficult to come up with a theory that allows me to travel this way in one direction (to Alpha Centauri) but not the other (back to Earth again). This is why travelling to any region of space-time other than your future light-cone is generally considered a causality violation.

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My reason for there only being one is that if there were two, it would violate causality. You don't have to kill your grandfather to violate causality. The simple existence of a round trip wormhole is a causality violation but a single one could exist without doing that I think. That's why I'm asking. – WillMcLeod May 26 '14 at 7:21
@WillMcLeod I do understand that, and I covered it in the answer. If you have a theory where there can only ever be one such wormhole then you are right, it would not violate causality. But it is very difficult to have such a theory without violating Einstein's postulate. You can do it, for example by inventing a special preferred reference frame that breaks Lorentz invariance (this is how it's supposed to work in Star Trek, where the special frame is called "subspace"), but it tends to result in something inelegant for which there is no empirical evidence. – Nathaniel May 26 '14 at 7:26
it's hard to phrase this correctly but that is my theory. There can only be one wormhole per light cone because 2 would violate causality. The physical mechanism is that it would collapse if a round trip was attempted (even by radiation) and so they all collapse. However, one could exist on its own. – WillMcLeod May 26 '14 at 7:32

Adding to Nathaniels answer...

If you have a theorie that only one such wormhole can exist at a time, it implies space time would have to "keep book" about your original light cone, and your new one. Because you could travel there without any causality violation, stay there for a year, the old wormhole long collapsed and gone and somehow find a new one, somewhere else which will then collide with your original past... And it's not even about any particle travelling back which has to be prevented. What about information? It would be enough if someone else would travel the other way and you would tell him to murder your former self. So noone who can at any way interact with you could travel the opposite direction, or any direction from which he could than travel to your past...

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Hi Falco. In a Swartzchild wormhole, travel is one directional and overlapping light cones collapse in plank time. This means that they are essentially forbidden. I guess the real question is whether one can say they travelled through time if they travel to outside their own lightcone. – WillMcLeod May 26 '14 at 15:56
You said a round-trip is forbidden and the wormhole would collapse, if something (even radiation) would attempt it. But how does the wormhole "know" that a certain particle would cause a causality violation? If it is just information, the Particle never did a round trip... one goes one way, the other one through a different wormhole the other way... – Falco May 27 '14 at 8:31
I'm sorry. I didn't mean to imply that radiation caused the collapse. Swartzchild wormholes collapse before any radiation enters them. I don't know the math begin why. I'm really asking a broader question about how we think about timelike dimensions outside of our own light cone. For the sake of argument we can assume FTL travel is used using a method that forbids a return trip. – WillMcLeod May 28 '14 at 14:02

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