I have a decent understanding of Einstein’s concept of spacetime, and understand how time is considered a dimension under that concept. My question is, does that concept apply properly to traditional geometry?
If I go through the progression of line, square, cube, would the next extension of that progression be “cube X time”? Would other dimensions come between the third and time? Or is time not a dimension at all in the geometric sense?
Sua
My WAG is that it is a dimension in a mathmatical sense, but not a geometric sense. For example, I can have a square that is 1m x 1m. I can view that square on an x y plane. I can turn it and view it on a y z plane. I cannot rotate that same suare to produce a “square” that exists as a line in the x plane for t seconds. At least not in any way that my primitive brain can comprehend.
All that the dimensionality of an object tells you is the number of coordinates necessary to specify a point in it. You can interpret one of those dimensions as time if it’s convenient to do so.
The next object in the sequence line, square, cube is the hypercube.
[QUOTE=SuaSponte]
I have a decent understanding of Einstein’s concept of spacetime, and understand how time is considered a dimension under that concept. My question is, does that concept apply properly to traditional geometry?
If I go through the progression of line, square, cube, would the next extension of that progression be “cube X time”? Would other dimensions come between the third and time? Or is time not a dimension at all in the geometric sense?
Sua
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It’s a bit hard to grasp, since most of the geometry stuff is written, really, for spacial dimensions and not for temporal ones.
A square that comes into existence as a particular moment, remains motionless, then is abolished, would look a little bit like a cube twisted into four-dimensional spacetime. (Assuming that the length of time it existed is comparable to its size.) A cube that exists for a finite period of time would look like a four-dimensional hypercube.
There isn’t much application for four-dimensional space-time geometry, AFAIK, especially since our brains can’t easily grok it. Three-dimensional or two-dimensional geometry with time as one of the dimensions can be handy for demonstrating some things, though I can’t remember just what at this moment.
[QUOTE=SuaSponte]
I have a decent understanding of Einstein’s concept of spacetime, and understand how time is considered a dimension under that concept. My question is, does that concept apply properly to traditional geometry?
If I go through the progression of line, square, cube, would the next extension of that progression be “cube X time”? Would other dimensions come between the third and time? Or is time not a dimension at all in the geometric sense?
Sua
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Time may be thought of as a change in the relative distribution of matter and energy in space.
Geometric dimensions are requisite to it but as a concept it is not a geometric dimension.
If one expands the notion of “dimension” to those concepts which embrace the fundamental framework inside of which we exist, sure–it’s a dimension.
There is no “the” fourth dimension. Time is viewable as a fourth dimension, and there’s a lot of treatment of it as “ict” as a fourth dimension of spacetime in undergrad texts*, but you can also have a fourth spatial dimension that’s pretty much like the other three (see Abbott’s “Flatland”, or any of the sequels and imitators that treat higher dimensions). There’s no reason to stop there – you can add fifth, sixth, seventh, etc. spatial dimensions, and do calculations in them and imagine properies of figures drawn in such hyperspaces. You can imagine axes that quantify other properties unrelated to space or time – charge, for instance, and use those as dimensions in some imagined space. Scientist, engineers, economists, and others do this all the time.
*This really bugs some physicists. In the classic text Gravitation by Misner, Thorne, and Wheeler there’s a section called “Farewell to ict”.
[QUOTE=CalMeacham]
…but you can also have a fourth spatial dimension that’s pretty much like the other three (see Abbott’s “Flatland”, or any of the sequels and imitators that treat higher dimensions). There’s no reason to stop there – you can add fifth, sixth, seventh, etc. spatial dimensions, and do calculations in them and imagine properies of figures drawn in such hyperspaces.
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Which makes sense mathmatically, but is difficult to comprehend in reality (much as the sphere was uncomprehensible to the 2d triangles in Flatland).
[QUOTE=ultrafilter]
All that the dimensionality of an object tells you is the number of coordinates necessary to specify a point in it. You can interpret one of those dimensions as time if it’s convenient to do so.
The next object in the sequence line, square, cube is the hypercube.
[/QUOTE]
Isn’t the next object in sequence the Tesseract?
That is, they are all hypercubes, but the specific name for the R[sup]4[/sup] figure is tesseract?
[QUOTE=DSYoungEsq]
Isn’t the next object in sequence the Tesseract?
That is, they are all hypercubes, but the specific name for the R[sup]4[/sup] figure is tesseract?
[/QUOTE]
Hypercube means any cube of four or more dimensions; a tesseract is the hypercube with exactly four dimensions. But when the number of dimensions if left unspecified, a “hypercube” is assumed to be a tesseract.
[QUOTE=CalMeacham]
. . . and there’s a lot of treatment of it as “ict” as a fourth dimension of spacetime in undergrad texts . . .
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Lord knows I’m no scientist, but usually I can at least recognize a scientific term. Not so, however, with “ict”.
Googling terms up references to Information & Communication Technology and the International Campaign for Tibet, neither of which seem quite on point. What the heck is “ict”?
[QUOTE=SuaSponte]
I have a decent understanding of Einstein’s concept of spacetime, and understand how time is considered a dimension under that concept. My question is, does that concept apply properly to traditional geometry?
If I go through the progression of line, square, cube, would the next extension of that progression be “cube X time”? Would other dimensions come between the third and time? Or is time not a dimension at all in the geometric sense?
[/QUOTE]
The four dimensional analog of a cube would be a tesseract or a octachoron, with time being the “out” dimension.
Not only can time be considered a legimate geometric dimension, i.e. you can measure a position in space with respect to any two dimensions and the time it takes light to get there (see light cone) but under general relativity you can actually have closed spacelike paths through time; i.e. you can come back to your same point in time by travelling along a measured path. Considering time just another dimension is actually critical for formulating relativity in Minkoski vector space, and is necessary (for symmetry) when using quaternions to model mechanical behavior.
In most areas of quantum mechanics, and special relativity (not sure about GR) time is not invariant in direction; that is to say, there is no reason that it has to do one direction rather than the other, and in fact there are certain assumptions in QM that require noninvariance. As a practical matter in the real world and per the laws of thermodynamics as we udnerstand them, however, time definitely has a direction and there’s no stepping backwards. As noted above there are solutions in relativity that would permit closed timelike curves, but it seems unlikely that you could “get off” anywhere off the curve prior (in time) to your start point, which protects global causality, i.e. that things have a cause-and-effect relationship. Most people, including the majority of physicists and cosmologists, are offended by the notion that causality could be violated and have come up with various rationales, generally lumped under the Law of Cosmic Censorship, as to why any apparent scheme for going backward in time is paradoxical and unrealizable.
So time is, or at least can be treated like, a real dimension, but like being in line at the drive-thru, you can only go forward, not back. Check out Brian Greene’s very accessible The Fabric of the Cosmos (I forget the chapter, but he makes a bunch of Simpson’s references in it, which gives you an idea of the level at which he communicates these ideas) on the topic of dimensional time.
Stranger
Sorry. It’s “i” (the square root of minus one) times “c” (the speed of light) time “t” (time). It’s a sneaky way of making time look like a linear dimension. You put the “i” in there so that if you compute the “length” of a vector in 4-space it comes out as x^2 + y^2 +z^2 - (ct)^2, which is of constant length in spacetime. But thinking in those terms, say Misner, Thorne, and Wheeler, confuses you to the essential reality of things, blurs the distinction between covariant and contravariant vectors, and causes bad breath in dogs.
[QUOTE=chrisk]
It’s a bit hard to grasp, since most of the geometry stuff is written, really, for spacial dimensions and not for temporal ones.
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Really? Wow. What have Minsky and Kleiner and Perelman and that lot been doing across the hall during their seminars then?
Four-dimensional (or as we like to say, (3+1)-dimensional) geometry is Einstein’s relativity theory. No application, huh? So what have those physicists been doing with all their grant money?
[QUOTE=Freddy the Pig]
Lord knows I’m no scientist, but usually I can at least recognize a scientific term. Not so, however, with “ict”.
Googling terms up references to Information & Communication Technology and the International Campaign for Tibet, neither of which seem quite on point. What the heck is “ict”?
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The imaginary unit i times the speed of light c times the time coordinate t. The imaginary unit’s in there because writing a negative sign in a bilinear form scares the pants off of some people. They prefer to write:
d[sup]2[/sup]=x[sup]2[/sup]+y[sup]2[/sup]+z[sup]2[/sup]+(ict)[sup]2[/sup]
[QUOTE=Stranger On A Train]
In most areas of quantum mechanics, and special relativity (not sure about GR) time is not invariant in direction
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Great stuff, just wanted to pick this nit. Quantum mechanics proper does require a fixed direction of time. It’s when you put it with special relativity to get quantum field theory that you’re thinking of. And since general relativity is just a manifold with special relativity on the tangent spaces, you’ve got no fixed direction there either.
And it is possible to rotate an object through spacetime. The only catch is, you can’t rotate it all the way. When one object is moving relative to another object, it’s actually rotated through spacetime relative to that other object. All of the distortions of lengths and durations you find in relativity are really just projection effects from things being rotated relative to each other.
Also, on the matter of terminology, it doesn’t really matter how you number the dimensions. There’s nothing special about time in particular that makes it the fourth one. In fact, the usual practice among physicists is to label time as the zeroth dimension. Again, there’s no fundamental significance to this; it’s just easier to do physics if everyone agrees on what labelling to use, and that’s the labelling that happend to have caught on.
[QUOTE=Mathochist]
Great stuff, just wanted to pick this nit. Quantum mechanics proper does require a fixed direction of time. It’s when you put it with special relativity to get quantum field theory that you’re thinking of.
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QFT and specifically QED is by itself time symmetric, of course. For my own elucidation, can you go into more detail on your nitpick?
Stranger
OK, I’m more confused than when I started. But confused in a good way. I’ll mull.
Sua
All you smart guys have more typing skills than I do. (How do you do a superscript?) In any case, time is certainly a dimension. Number it as you will.
Superscripts are done with the tag <sup> </sup>, except with square brackets instead of the HTML type. Subscripts, of course, are done with <sub> </sub>.