
#19
Nov3012, 10:01 AM

Physics
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PF Gold
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#20
Nov3012, 11:18 AM

P: 48





#21
Nov3012, 11:22 AM

Physics
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PF Gold
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#22
Nov3012, 11:31 AM

P: 48

The three spatial axes are free to be traversed in any direction at any speed, but the fourth time axis is not. You cannot travel it backwards, you cannot stop. "Something" forces you to travel it in one direction, and your speed is determined by something else than your acceleration (basically mass determines it, rather than anything else, although I may be really wrong here.) So it's both handled as "just another dimensional axis", but it's also quite different from the other three. This is, AFAIK, a fundamental aspect of GR, but I just cannot grasp it. 



#23
Nov3012, 11:36 AM

Physics
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PF Gold
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#24
Nov3012, 12:58 PM

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#25
Nov3012, 02:18 PM

Physics
Sci Advisor
PF Gold
P: 5,503

[tex]d\tau^2 = dt^2  dx^2  dy^2  dz^2[/tex] The [itex]dt^2[/itex] term has opposite sign from the other terms; that means the interval [itex]d\tau^2[/itex] is not positive definite. That is, the interval between two distinct points can be positive, zero, or negative. That's not possible in ordinary Euclidean geometry: there, the distance between two points can only be zero if the points are identical, and it can never be negative. What all this means is that, in spacetime, there is something fundamentally different about a timelike interval with [itex]d\tau^2 > 0[/itex], vs. a spacelike interval with [itex]d\tau^2 < 0[/itex] or a null interval with [itex]d\tau^2 = 0[/itex]. They are three physically distinct kinds of intervals. The same is true in GR; the only difference there is that the line element can look different than the formula above, due to spacetime curvature. But you'll notice that nowhere in any of this did I talk about anything "moving" along a curve, or through an interval. If two points are separated by a timelike interval, that means some timelike curve connects them, so some object's worldline can pass through both points. But that's just a fact about the geometry of spacetime, in the same way that the statement "the Earth's equator passes through Quito, Ecuador and Nairobi, Kenya" is a fact about the geometry of the Earth's surface. (I don't know that that's exactly a fact, btw; those cities are close to the equator but probably not exactly on it. But it illustrates what I'm getting at.) We can describe the geometry of spacetime without talking about anything "moving" in it, just as we can describe the geometry of the Earth without talking about any objects moving on it. 


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