And I’ll just add that different fields of physics typically use different sign conventions. Those doing general relativity usually use the convention that space is positive and time is negative, since that leaves the spatial part of the metric identical to the familiar Euclidean metric. Those doing particle physics usually use the space-negative, time-positive convention, since they’re often concerned with times, but very seldom with distances. A good paper in either field really ought to start with a clarification of which of these (and of various other arbitrary conventions) they’re using.
That paper is quite heavy going if you’re not familiar with anything beyond the basic quarternions, but the argument boils down to the maths becoming a bit more elegant in the six dimensional setting, which itself boils down to the asymmetry in the signature of a (3,1)/(1,3) metric. Whilst the question of why there is an asymmetry in the signature of the metric is clearly an interesting one, as a counterpoint to that paper I would say I’m not sure there’s a compelling reason to assume the best resolution is the introduction of two redundant dimensions to create symmetry.
There’s a quote from Tegmark above which I would modify slightly, to say that if you can’t embed a 3+1 dimensional spacetime into a flat (N+1) (where N ≥ 3) pseudo-Riemannian manifold then that spacetime has some undesirable, probably unphysical features. So extra time dimensions must be (largely) redundant even when embedding.
As an aside, I’ve got a pet… probably not even a hypothesis, more like a hunch… that there’s actually only one minus sign in all of physics. The relative minus sign in the metric, if you track it through all of the equations, is the same minus sign as the relative minus sign in Maxwell’s equations, which is of course the same minus sign as the one in Lenz’s Law, which is at least very similar to the minus sign in any frictional law, and so on.
OK, here goes nothing.
I’m going to attempt to give a good mathematical argument why time has to be the odd-dimension-out in the metric tensor, as Chronos has said, and do it using nothing more than undergraduate calculus, a few new axioms (which are over a century old by now…), and probably enough hand-waving to reinterpret Rap God in sign language…
First, the axioms:
[ol]
[li]x is a vector of length 1 pointing in a spatial direction. As it is a physicist’s gauge-given right to orient the spatial coordinate system however they choose, x can point any direction we want. As physicists leave units for engineers, we’ll ignore length entirely.[/li][li]t is the vector of length -1 pointing in the time direction.[/li][li]x[sup]2[/sup] = xx = 1; that is, squaring a vector is taking a dot product of the vector with itself. Therefore, t[sup]2[/sup] = tt = -1.[/li][li]xt = -tx. If you’ve seen a cross product, antisymmetry (or anticommutativity) shouldn’t surprise you. If you haven’t, imagine that the product xy where x and y are vectors in orthogonal spatial directions is an area created by taking x and dragging it along y. This kind of area defined by the product of two orthogonal vectors is called a bivector. (The sign comes from a convention; don’t worry about it.) (Because this is operation produces an area, it is perfectly well-defined in two dimensions, thank you very much.)[/li][li](The geometric product works fine for vectors which are neither colinear nor orthogonal. It just gets involved and we don’t need those results.)[/li][/ol]
OK, that’s all the axioms. Let’s work through some algebra to give a feel for what kind of a world the axioms describe.
First, what’s the square of a bivector in the x-y plane? Well, we can work it out with a pencil (or eat more fiber…) but let’s do it on the screen:
(**xy**)[sup]2[/sup]
**xyxy**
-**xyyx** (Flip the second bivector, flip the sign
due to axiom 4 (antisymmetry of orthogonal product).)
-**xx** (Remove the middle pair of **y** vectors
due to axiom 3 (squaring a vector is dot product).)
-1 (Ditto, in **x**.)
Well, well, well, what do we have here? It seems like the unit bivector in the x-y plane squares to -1! What we had here is a failure to commutate! … which induced the sign flip, which was not undone, because all of the vectors squared to 1, because they all point in spatial directions, so they all have length 1, due to the metric tensor convention I defined above in axioms 1 and 2.
Because it squares to -1, I claim xy can be identified with i, the imaginary unit. You know, the square root of -1. The i in e[sup]x i[/sup] = cos(x) + i sin(x).
How do I justify this? By taking the Taylor series expansion of e[sup]xy[/sup] and seeing what pops out:
e[sup]**xy**[/sup]
= 1 + **xy**/1! + (**xy**)[sup]2[/sup]/2! + (**xy**)[sup]3[/sup]/3! + ...
= 1 + **xy** - 1/2 - **xy**/6 + 1/24 + **xy**/120 - ...
= (1 - 1/2 + 1/24 - 1/720 + ...) + **xy** (1 - 1/6 + 1/120 - ...)
= cos(1) + **xy** sin(1)
If any of that was unclear, it probably deserves a whole other post. The point of that argument is that xy behaves so much like i that it can be used in mathematical descriptions of Euclidean rotation, which is the kind of rotation where things go around in circles and 360 degrees is a full turn and so on. It spins you right round, baby, right round, like the complex unit baby, right round round round…
Now, let’s see what happens in the x-t plane; that is, the plane spanned by one spatial direction and the time direction:
(**xt**)[sup]2[/sup]
**xtxt**
-**xttx**
**xx** (**t**[sup]2[/sup] = -1, which flips the above sign flip.)
1
It seems the fact t squares to -1 means the unit bivector in the x-t plane squares to 1. This seems to make the above argument about rotations go boom.
Let’s try it!
e[sup]**xt**[/sup]
= 1 + **xt**/1! + (**xt**)[sup]2[/sup]/2! + (**xt**)[sup]3[/sup]/3! + ...
= 1 + **xt** + 1/2 + **xt**/6 + 1/24 + **xt**/120 + ...
= (1 + 1/2 + 1/24 + ...) + **xt** (1 + 1/6 + 1/120 + ...)
= cosh(1) + **xt** sinh(1)
Now it’s interesting. cosh and sinh are the hyperbolic equivalents of cos and sin, respectively. It seems that this implies rotation in the x-t plane is hyperbolic; that is, it follows hyperbolae pointing towards the origin, as opposed to Euclidean rotation, which follows circles centered at the origin.
The most salient fact about hyperbolic rotation is that it’s asymptotic: It’s impossible for a line lying on the x-axis, for example, to be rotated a full 45 degrees. It instead follows a line which lengthens it towards infinity, so it gets longer and longer as it infinitesimally approaches that 45-degree angle, but it never reaches it. Now, what behaves like that… What in physics behaves in that fashion…
Special Relativity. Look at the diagrams on this page for enlightenment and notice especially that light moving away from a fixed point moves up the space-time diagram (with x-axis horizontal and t-axis vertical) at that magic, unreachable 45-degree angle. The fact hyperbolic rotation is asymptotic models the fact you can’t accelerate to light speed; you can get arbitrarily close, you can become your own personal Oh My God particle, but you can’t get there from here.
Further, the fact hyperbolic rotation changes length models Lorentz transformations extremely well; in fact, it’s reasonable to say that the simplest way to model all of SR is to think about accelerations as hyperbolic rotations in the x-t plane, where x is the direction of acceleration.
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Now, what happens if we flip the metric tensor around and make x[sup]2[/sup] = -1 and t[sup]2[/sup] = 1? Nothing. Absolutely nothing. I’ll prove it by working the algebra; the calculus follows directly from that anyway:
(**xy**)[sup]2[/sup]
**xyxy**
-**xyyx**
**xx**
-1
See that? The sign flip was done, undone, and redone; first by antisymmetry, then by the dot products of the spatial vectors.
(**xt**)[sup]2[/sup]
**xtxt**
-**xttx**
-**xx**
1
Again, the dot product of a spatial vector undid a sign flip, but as there was only one spatial direction involved, it wasn’t redone, so the result is positive.
So just a bit of a sign flip induces a time slip, and rotation will never be the same… Sorry.
As with many theories in physics that deal with relativity, or unusual geometries, or quantum anything, a lot of this can only be properly understood by studying the math, rather than by any “intuitive” descriptions relating to our everyday life experiences.
Can anyone speculate, in any way that we could comprehend, what life would actually be like in a world with two (or more) time dimensions? There was a few posts above with brief remarks (like Post #2, asking if you could have lunch and then have lunch again), and a few posts with more detailed ideas about a world with a second but compacted time dimension.
What would a world be like with two non-compacted time dimensions, in terms of what life would be like and how we would experience it? Would would it mean to move forward in time(s) along two distinct timelines, independently of one another? Would would it be like for an “event” (like having lunch) to occur at a particular point on one of the time axes but an independently different point along the other time axis? Would a sequence of events in a certain chronological order along one t-axis necessarily occur in the same (or possibly reverse) chronological order along the other t-axis? I assume the duration of an event (or interval between two events) along one t-axis must be independent of the duration or interval along the other t-axis (if the two time dimensions actually run independently of one another). What would that be like?
Is it possible to describe what life would be like in such a world, other than by pure wild speculating? Is it possible to do even that?
And what would it be like to make an appointment with, say, your doctor? Would you make an appointment for a particular time on one or the other other the time axes? Or would you have to specify two particular times, independently of each other, one along each of the t-axes? And what would it be like from the doctor’s point-of-view (or his receptionist, more specifically)? Would the doctor’s office get to specify that their office schedules appointments along the t[sub]1[/sub]-axis or along the t[sub]2[/sub]-axis? Or would the office have to keep two separate appointment books, one for each t-axis, and schedule each appointment in both of them?
Would it be possible to arrive for the appointment at the right t[sub]1[/sub] time but at the wrong t[sub]2[/sub] time? If you arrived t[sub]1[/sub]-on-time but a whole hour t[sub]2[/sub]-early, what would happen? Would that even be possible? Would you have to sit in the waiting room an entire t[sub]2[/sub]-hour until your appointment? And if so, would that make you t[sub]1[/sub]-late? Or would you get to see the doctor immediately in t[sub]1[/sub]-time (since you arrived on-time) and then see him again an hour later in t[sub]2[/sub] time? (This is what ianzin seems to envision in Post #2.) And if so, would that be two audiences with the doctor, or just one (albeit at separate times along t[sub]1[/sub] and t[sub]2[/sub])?
If we are continually moving forward along both time axes (possibly at different rates) in the same way that we currently move forward along our one time axis, does that mean that every point in time in our lives along t[sub]1[/sub] corresponds one-to-one with exactly one point in time in our lives along t[sub]2[/sub]? If that’s so, then in what sense are the two time dimensions independent of each other (even if the times run at different rates)?
Could we define a “moment” in time along one time axis that exists forever (or for some interval) along the other axis? Could we experience moments like that? (This seems to be what Keeve is suggesting in Post #5.)
In some ways this is actually a bad example as the geometric product on the two different signatures is not the same (i.e. not isomorphic), which in itself is a point of interest.
Really? I rather think that’s the point: The mere fact the time dimension is the odd one out in the metric signature is in itself sufficient to derive SR, simply through the geometric product and the usual definition of rotation from undergrad calculus. It means that time has to be special, as if it had the same gorm (dot-product-with-itself) as the rest you’d just get Euclidean rotation.
I guess I’m not fully understanding you, because to me this is a rather simple and elegant justification for the metric signature (or it is once you’ve internalized the idea that a bivector can be a generator of rotations in its plane…).
I was being nit-picking really, but there’s a a genuine point: the geometric product is extra structure here that actually makes the two different signatures inequivalent. You do still end up with the same group for rotations/boosts and in fact the classical theory is insensitive to the difference, but the difference is not unimportant.