- #36
yuiop
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Would anyone care to explain to me the difference between coordinate axes and dimensions. I am certain they are different entities, but I am not clear on the definitions and distinctions here.
I don't know how Passionflower is using the term, but the usual meaning is a property of a (pseudo-) Riemannian manifold or a vector space.yuiop said:Would anyone care to explain to me the difference between coordinate axes and dimensions. I am certain they are different entities, but I am not clear on the definitions and distinctions here.
Passionflower said:So you think that if a particular observer travels between event A and B the amount of time passed is not the length of this path?
I think your comparison is flawed for two reasons:yuiop said:More casually, when we say city A is "one hour away" from city B, the expression is meaningless unless we specify traveling by jet, car, or on foot, whereas if we specify that the distance between the two cities is 300 km the expression is less ambiguous. Time and distance are not exactly the same thing.
Passionflower said:But none of those dimensions is pure space or time.
It seems many people have trouble distinguishing between a chart of spacetime and spacetime itself.
Isn't what you are calling "distance" in relativity what everyone else calls "Spacetime Interval"?Passionflower said:In relativity the path length between event A and B also depends on how one travels and the distance between A and B is also the maximum path length.
The property of dimensionality is more fundamental than the concept of coordinates. I.e. you can define the dimensions of a manifold or a vector space without ever defining sets of N numbers and mapping them to the space. See my post #37.Passionflower said:Sapcetime is clearly 4-dimensional as it takes four numbers to identify an event uniquely. But none of those dimensions is pure space or time.
I would be happy with an answer to my question:Passionflower said:Let's just let you folks settle with "in relativity time is a dimension of spacetime" and we part ways, looks like the majority would be happy with that!
ghwellsjr said:Isn't what you are calling "distance" in relativity what everyone else calls "Spacetime Interval"?
Passionflower said:Let's just let you folks settle with "in relativity time is a dimension of spacetime" and we part ways, looks like the majority would be happy with that!
bobc2 said:Passionflower, I get the impression that folks here did not give careful thought to one of your most instructive comments--I think it kind of went over their heads and they responded after jumping to all of the wrong conclusions about the point you were making. For the benefit of those who may wish to attempt to try again to understand your earlier comment, here it is again:
Dimensions are independent entities, however in relativity space and time are mere shadows.
As Minkowski wrote more than 100 years ago:
The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.
The first part of the comment is wrong, there is nothing about a dimension being an "independent entity" in the definition. And the Minkowski quote doesn't contradict the claim that time is a dimension of spacetime.bobc2 said:Passionflower, I get the impression that folks here did not give careful thought to one of your most instructive comments--I think it kind of went over their heads
A.T. said:Isn't a Galilean transform between frames that are rotated relative to each other also mixing or cross-contaminating the spatial dimensions?
TGlad said:That wasn't the question. Is not a rotation a 'cross-contamination' of the x dimension and the y dimension?
ghwellsjr said:Isn't what you are calling "distance" in relativity what everyone else calls "Spacetime Interval"?
Passionflower believed you understood and agreed with what he was talking about. Do you agree with his assessment?Passionflower said:You are one of the few here who seems to understand this matter.PhilDSP said:In Galilean terms, time and the 3 space units are dimensions and fully independent (based on the rules of Euclidean geometry). In SR, they are dimensions and independent within an inertial frame but not outside a single one, right? Frame to frame translations do not follow the rules of Euclidean geometry and therefore mix or cross-contaminate the one time "dimensions". In Minkowski space, the concept of independence of dimensions loses its traditional meaning entirely so that the relationships between "dimensions" must be re-defined (especially the inner product)
Others, keep singing the mantra that time is the fourth dimension in relativity, they should know better but they hate to change the words of an old song even when they know the words are wrong.
Remember, he made these statements:PhilDSP said:Isn't what we call "distance" actually the "metric"? The metric in Galilean reckoning is merely the spatial distance while in SR and Minkowski space it includes the differential in time between space-time points or "Spacetime Interval". (Maybe the term "metric" more properly means the proscription for determining the distance?)ghwellsjr said:Isn't what you are calling "distance" in relativity what everyone else calls "Spacetime Interval"?
Passionflower said:In Galilean spacetime time is the "distance" traveled in the time dimension between two events.
In Minkowski and Lorentzian spacetimes time is the path length between two events.
Passionflower said:...time in Galilean spacetime is the difference between the time coordinates of the two events...
However in case of Minkowskian or Lorentzian spacetime the path length determines the time between two events.
Passionflower said:So you think that if a particular observer travels between event A and B the amount of time passed is not the length of this path?
Do you understand what he is talking about when he says that in relativity the pathlength between two events is the time it takes a traveler to travel between between those two events?Passionflower said:A and B in my example where events yours are not.
You use a distance and I use a path length.
In Euclidean geometry the path length between city A and B really depends on how one travels while the distance between A and B is also the minimum path length. In relativity the path length between event A and B also depends on how one travels and the distance between A and B is also the maximum path length.
ghwellsjr said:Passionflower believed you understood and agreed with what he was talking about. Do you agree with his assessment?
If so, then are you saying that he (and you) would have answered "yes" to my question?
Do you understand what he is talking about when he says that in relativity the pathlength between two events is the time it takes a traveler to travel between between those two events?
ghwellsjr said:Isn't what you are calling "distance" in relativity what everyone else calls "Spacetime Interval"?
I provided links so that you could go back to the original posts. Please look at your post #26 and then look at Passionflower's post #28, both on page 2. Don't you have any reaction to his statement that you understand what he is talking about and the implication that you agree with him?PhilDSP said:I couldn't parse out a technical observation from the text you provided, so it's not possible to agree or disagree.ghwellsjr said:Passionflower believed you understood and agreed with what he was talking about. Do you agree with his assessment?
The question I was referring to was the one you quoted in post #55. It was a simple "yes" or "no" question but instead of providing a "yes" or "no", you asked more questions. If you don't agree with Passionflower and/or you don't know what he is talking about, then I would have expected you to have clearly stated that before attempting to comment further about my question. So that's the first question: going back to the posts on page 2, do you understand and/or agree with Passionflower?PhilDSP said:I can't say that I quite follow what he or she might mean by that. And sorry, I haven't paid enough attention to the thread to know which question of yours you're referring to above.ghwellsjr said:If so, then are you saying that he (and you) would have answered "yes" to my question?
Do you understand what he is talking about when he says that in relativity the pathlength between two events is the time it takes a traveler to travel between between those two events?
The one-way measure of c is not just difficult, it's impossible. The whole point of the Spacetime Interval is that it doesn't require any postulate or knowledge regarding the propagation of light. It doesn't require any theory. It doesn't require the establishment of any frame. It doesn't require any measurement of both time and length. For the case that Passionflower stated, that of a traveler going between events A and B, it only requires an inertial clock traveling between A and B. The time accumulated on that clock is the Spacetime Interval.nitsuj said:That's exactly the sense I'm getting.ghwellsjr said:Isn't what you are calling "distance" in relativity what everyone else calls "Spacetime Interval"?
The difficulty with the one-way measure of c shows pretty clearly the "relation" between the measure of time & length. One of many "things" that show this "relation", which couldn't be more implied with the way time & length are measured.
ghwellsjr said:I provided links so that you could go back to the original posts. Please look at your post #26 and then look at Passionflower's post #28, both on page 2. Don't you have any reaction to his statement that you understand what he is talking about and the implication that you agree with him?
ghwellsjr said:The question I was referring to was the one you quoted in post #55. It was a simple "yes" or "no" question but instead of providing a "yes" or "no", you asked more questions. If you don't agree with Passionflower and/or you don't know what he is talking about, then I would have expected you to have clearly stated that before attempting to comment further about my question. So that's the first question: going back to the posts on page 2, do you understand and/or agree with Passionflower?
I disagree with this. A rotation is a linear transformation which preserves the origin and preserves distances. A boost is a linear transformation which preserves the origin and preserves intervals. They are mathematically the same from a symmetry perspective and from an operation perspective, only the signature of the metric is different.PhilDSP said:Again, no. For the rotation you're adding some external offset or value that has no dependency on the present position in the x, y and z dimensions. In spherical coordinates, for example, a rotation can be performed merely by changing one of the angular coordinates. The position of every object in the space then transforms as a result.
With the Lorentz transformation, the time position is defined on the basis of spatial position while the spatial position is defined on the basis of the time position. In effect, the cross-definition locks us out of being able to make a definite determination of their values outside of a local domain.
DaleSpam said:Pretty much everything you say about a rotation you can say about a boost and vice versa. The very few distinctions are due to the facts that the signature is different and there are 3 dimensions of space and only 1 of time, neither of which disqualifies time from being a dimension.
A boost in Minkowski spacetime leaves the root dimension s invariant between points because s is in principle a scalar and not directed in physical spacetime. ... I think that means that a boost is purely a coordinate transformation.PhilDSP said:The situation between the two is analogous, but not quite parallel I think. A rotation in Euclidean space leaves the root dimension L invariant between any points because L is in principle a scalar and not directed in physical space. ... I think that means that a rotation (of everything around a point center) is purely a coordinate transformation.
DaleSpam said:A boost in Minkowski spacetime leaves the root dimension s invariant between points because s is in principle a scalar and not directed in physical spacetime. ... I think that means that a boost is purely a coordinate transformation.