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In a lot of laymen explanations of general relativity it is implied that the four dimensions of the space-time are equivalent, and we perceive time as different only because it is embedded in our human perception to do so.

My question is: is that really how general relativity treats the 4 dimensions?

If so - what are the implications (if any) this has on causality?

If no - can the theory support more than one time dimension?

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As pointed out in Moshe's excellent answer, the key observation is the the "inner product" between two 4-vectors is not positive definite, since there are space like and time like intervals. – Matt Calhoun Mar 4 '11 at 15:09
up vote 7 down vote accepted

That statement, that space and time are equivalent, is not really correct. In special relativity there is a distinction between spacelike and timelike events, those are events that cannot or can (respectively) be causally connected. This replaces the notion of "simultaneous" and "before or after" to something all inertial observers can agree on. In general relativity, this distinction is made locally - causal influence propagates locally in speeds less than c, but that constraints changes from point to point according to the local metric (which encodes the force of gravity).

All of this is encoded in the statement that the metric is indefinite, with a specific signature that supports one time direction. While the mathematics of GR can be generalized to other signatures, the physics cannot - the Lorentzian signature is essential in correctly interpreting the theory. Anyone can play all kinds of mathematical games, but those are meaningless unless you are very clear on how the calculations you perform are related (at least in principle) to physically observable quantities.

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The formal statement would be that there is no one notion of time, and that one persons definition of time may be intermixed with another's definition of space. What is not correct is that time and space are wholly interchangeable, as Moshe says--there is a distinction between events separated in time and events separated in space. Causality theory is constructed around this notion--spacetimes that fail to keep the distinct separation between spacelike and timelike separated events will inevitably violate rules of cause and effect. Also, all observers will agree on which events are timelike and spacelike.

Theoretically, it is possible to define a theory with two linearly independent timelike directions. The physics in such a theory would be extremely novel (for example, if I time evolved in direction one, and then time evolved in direction two, I wouldn't necessarily end up at the same event if I reveresed this order--what would 'time' even end up meaning in such a theory?), and I don't see a particular reason to do so. But there certainly is existent math to describe a theory with two time directions.

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I think I can answer this in more laymen's terms. In relativity theory, space and time are very closely related. For example, when you pass by me, I measure that your wristwatch runs slower than mine, and I measure that you are shorter in your direction of motion than you would be if you stopped to talk to me. Even at walking speed this is true, if only undetectably. From your perspective your wristwatch always runs at its normal rate, and your length in your direction of motion isn't affected by your motion relative to me.

In relativity theory an observed distortion of time always goes hand-in-hand with an observed distortion of space (length), and by the exact same percentage, so it became convenient to consider that those two concepts are two sides of a coin, so to speak, hence the term spacetime. It also became convenient to think of observed objects and events in terms of 4 dimensions, the 3 of space and one of time.

Of course that's a rough way to put it, but I think it's not much more involved than that. So no, the theory doesn't support more than one time dimension.

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The only time which has any special status is proper time. This is the length of a geodesic or path in space. The interval $ds^2~=~g_{ab}dx^adx^b$ is a measure of proper time. This is invariant under all Lorentz transformations. For $U$ an element of the Lorentz group the interval length $s~=~x^T\cdot g\cdot x$ these element transform as $x’~=~Ux$, $x’^T~=~x^TU^{-1}$ and $g’~=~UgU^{-1}$ and it is trivial to see $$ s’~=~x^TU^{-1} UgU^{-1}Ux~=~x^Tgx~=~s. $$ So the proper time is invariant. The coordinate time, usually written as $t$ is transform dependent. This is a manifestation of the coordinate system you chose to work in.

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The interval is a measure of proper distance for a spacelike interval, so I don't think this answer addresses the question. – Ben Crowell Sep 6 '13 at 21:20

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