Is Time Truly a Fourth Dimension in Physics?

[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}