Is Time Truly a Fourth Dimension in Physics?

[Dale]

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

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.

I think the extraneous ideas of "purity" and "independent entity" are the source of the confusion.

 

[PhilDSP]

Isn't a Galilean transform between frames that are rotated relative to each other also mixing or cross-contaminating the spatial dimensions?

I can see where a person might think so, because values are moved from one spatial dimension to another. But I'd have to say no. In rotating, you're applying a fixed spatial offset to all 3 dimensions which is linear and produces no spillover into time (unless of course your model includes a time offset also which is decoupled from direct dependence on the space factors)

 

[TGlad]

'.. produces no spillover into time ..'
That wasn't the question. Is not a rotation a 'cross-contamination' of the x dimension and the y dimension?

Also, note that the transforms scale, rotate, translate and boost are all conformal transforms. And none of them stop the parameters being dimensions even though all of them cause mixing.

 

[PhilDSP]

That wasn't the question. Is not a rotation a 'cross-contamination' of the x dimension and the y dimension?

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.

 

[PhilDSP]

Isn't what you are calling "distance" in relativity what everyone else calls "Spacetime Interval"?

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?)

 

[danmay]

I think there is some confusion between what is a "dimension" and what is a "degree of freedom". "Dimension" usually means a degree of freedom that has a linear structure to it, while a "degree of freedom" is a more general notion, indicating only that one thing is independent of another.

In my opinion time is the only dimension. It is the progression of events and is closely related to the concept of mass, energy, causation, inertia, etc. It's arithmetics.

Space is more like the character of events, the attributes and possibilities that permeate each instance of time. We usually find 3 independent spatial attributes, but without time, they are simply degrees of freedom without any intrinsic order. With time, however, these 3 spatial attributes give meaning to force and momentum, velocity and position, as opposed to simply distance, speed, and energy. Sometimes we pay more attention to spatial relations than we do to temporal ones, as in the analysis of conservative fields. It's just a shift of emphasis, however, for making calculations and real-life application easier.

When so-called "mass-energy" is present, it must be conserved and cannot exceed the speed c as observed locally. If there were no mass-energy involved, however, then conservation, inertia, c, etc. all go out of the window. In fact, even relations such as position, velocity, acceleration, ... and time itself no longer mean anything without mass-energy. It makes reality different from imagination.

 

[ghwellsjr]

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)

You are one of the few here who seems to understand this matter.

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.

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?

Isn't what you are calling "distance" in relativity what everyone else calls "Spacetime Interval"?

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?)

Remember, he made these statements:

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.

...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.

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?

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.

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?

 

[PhilDSP]

Passionflower believed you understood and agreed with what he was talking about. Do you agree with his assessment?

I couldn't parse out a technical observation from the text you provided, so it's not possible to agree or disagree.

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?

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.

 

[PhilDSP]

One further observation: It's not just the coordinates that are mixed with the LT (except possibly in LET). In SR, measurements and determinations of numerous physical parameters such as the E, B, D and H fields have embedded time and length dimensions that are required to be re-scaled between frames. The dependencies between various physical parameters require that for them to be consistent with each other. I suppose this means that we need to assign an additional quality to real or pure dimensions, that they operate as a basis element between parameters or observables.

The traditional dimensions (as basis elements) are:
Q charge, M mass, T time, L length

I'm sure missing some others. It's interesting to note 3 space is an extrapolation of the length basis element. L^3 or volume assumes that. Presumably you could create and use mathematical spaces like that using the time dimension or other dimension as well. Langevin's concept of SR does that with mass that has perpendicular and parallel directions.

 

[nitsuj]

Isn't what you are calling "distance" in relativity what everyone else calls "Spacetime Interval"?

That's exactly the sense I'm getting.

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/defined.

 

[ghwellsjr]

Passionflower believed you understood and agreed with what he was talking about. Do you agree with his assessment?

I couldn't parse out a technical observation from the text you provided, so it's not possible to agree or disagree.

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?

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?

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.

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?

 

[ghwellsjr]

Isn't what you are calling "distance" in relativity what everyone else calls "Spacetime Interval"?

That's exactly the sense I'm getting.

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.

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.

But I don't know if that is in any way related to what Passionflower was talking about because he has gone mute.

 

[PhilDSP]

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?

In post #28 Passionflower is apparently talking metaphysically and I wouldn't claim to understand what was meant by it. Not much to say about that other than it's not relevant to the topic.

In regard to the other posts I don't find the wording "Galilean spacetime" useful or valid since space and time are fully independent or decoupled in the Galilean-Newtonian way of doing physics (except of course where the dependence is specified according to the particular application).

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?

Okay. My post was a distraction there and had no relation to what Passionflower posted in regard to your question. I merely found your question interesting enough to posit further questions and look at related detail.

 

[ghwellsjr]

PhilDSP, thanks for clarifying.

 

[Dale]

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.

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.

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.

 

[PhilDSP]

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.

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. All of the root dimensions Q, I, M, T and L are undirected. I think that means that a rotation (of everything around a point center) is purely a coordinate transformation.

A boost is a rotation in, for lack of better words, boost-space, isn't it? Boost space operates on t (associated directly to T within SR), on the vector <x, y, z> and implicitly on L. There seem to be 2 factors that set a boost apart from a Euclidean rotation. The first is that T and L are melded together and are refactored together. There might possibly be some theoretical inconsistency with the melding in that t is an undirected scalar while L is vectorized into x, y and z components.

The other factor is that the boost parameter v is an invariant within the boost. In addition to c it specifies an L/T ratio. Proposals for 2 studies immediate come to mind that could have interesting or illuminating results. One is to decompose any occurrence of the t or t' variables into vector components. The other is to decompose the boost parameter \mathsf v_{inv} into \Delta \mathsf t_{inv} and \Delta \mathsf l_{inv} components and then eliminate one of them using algebraic reduction.

 

[Dale]

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.

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]

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.

Okay, he he. fair enough. It seems that Minkowski space differs fundamentally from a classically conceived space. That we knew all along anyway...

 

[DrGreg]

It might help to note that a rotation through angle \phi in 2D Euclidean space is given by\begin{align}<br /> x&#039; &amp;= x \, \cos \phi- y \, \sin \phi\\<br /> y&#039; &amp;= x \, \sin \phi+ y \, \cos \phi<br /> \end{align}which can be rewritten as\begin{align}<br /> x&#039; &amp;= \gamma ( x - \beta y ) \\<br /> y&#039; &amp;= \gamma ( y + \beta x )<br /> \end{align}where\begin{align}<br /> \beta &amp;= \tan \phi\\<br /> \gamma &amp;= \cos \phi= \frac{1}{\sqrt{1 + \beta^2}}<br /> \end{align}Compare that with the Lorentz boost for velocity c\beta:\begin{align}<br /> ct&#039; &amp;= \gamma ( ct - \beta x ) \\<br /> x&#039; &amp;= \gamma ( x - \beta ct ) \\<br /> \gamma &amp;= \frac{1}{\sqrt{1 - \beta^2}}<br /> \end{align}which can be written as\begin{align}<br /> ct&#039; &amp;= ct\, \cosh \phi - x \, \sinh \phi \\<br /> x&#039; &amp;= -ct \, \sinh \phi + x \, \cosh \phi <br /> \end{align}where\begin{align}<br /> \beta &amp;= \tanh \phi \\<br /> \gamma &amp;= \cosh \phi<br /> \end{align}