Exploring Time as a Dimension

[yuiop]

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.

 

[Dale]

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.

For a vector space, it is the number of independent vectors required to span the space.

For a (pseudo-) Riemannian manifold it is the number of eigenvalues of the metric. The eigenvalues are strictly non-zero, with the positive eigenvalues identifying dimensions of space and the negative eigenvalues identifying dimensions of time. So a pseudo-Riemannian manifold with 3 positive eigenvalues and 1 negative eigenvalue has a metric signature of (3,1) and is used to represent spacetime with 3 dimensions of space and 1 dimension of time.

All of the above is done without defining any coordinates. Of course, you can take the independent vectors and use them to form a coordinate system, but it is not necessary. The dimensionality of a space is more fundamental than any coordinate systems.

 

[Passionflower]

When a finger points to the moon do we want to talk about the finger?

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.

It seems many people have trouble distinguishing between a chart of spacetime and spacetime itself.

 

[yuiop]

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?

Let's say that the proper time that passes for the inertial observer that travels between event A and event B is tau = 0.75 seconds. Is this the length of the path between the two events? I would say no, because the actual length is c*tau = 299792458 meters/second * 0.75 seconds = 224844344 metres which has the correct dimensional units of length whereas tau has dimensional units of time. When we use units such that c=1, tau and c * tau can appear numerically equal, but they have different units and the time passed is not the same as the length of the path, because they have different dimensional units.

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.

 

[dogzer]

We could definitely agree that time is the fourth dimension, if we wanted to. It's only a logical way of seeing it.

After all, time is correlated with space, that's how our brains measure it, and that's how clocks measure time, using motion. In other words, we trust the constant motion of a clock just as we trust a ruler to keep its size constant.

I get confused when I read that time moves slower at higher velocities. Because time is measured with motion, something must move, and it must do so reliably in order to be able to log time.
If a ruler changed its size, does that mean centimeters got bigger? No, it means the ruler got bigger.
If a clock for whatever reason moves slower (gaining mass for some reason, or battery running low) all it means is that its motion is being restricted for some physical reason. Not because its offsetting in time.

I don't see the logic in things offsetting in time, that's not how we agreed to correlate time and space. Either we need a new type of measurement called different than "time", or just agree time and space are correlated directly.

Anyway, just as we use a ruler's length to calculate distance, we use a clock's motion to calculate time. Meaning that's what time is! The motion of things.

We could agree for just a moment that we're going to take temperature as the 1st dimension, and time as the 2nd dimension. We can definitely agree on that, if we wanted to. We'd get a graph of temperature varying over time!

But the difference is that when we agree to measure space in three dimensions, it's a correlation we're making between three dimensions of the same type, which is distance (distance x distance x distance).

Distance alone is a single dimension, right? But if we agree on correlate a single distance dimension with another one, we have two distance dimensions, logically speaking. We can correlate one single value of the first dimension with another value of second dimension, that means we get a point in 2D space.
Our brains naturally respond to this correlation visually, because that's how our sense of sight works. There's no trouble comprehending this logic of 2 dimensions.

Funny thing is, we just correlated two dimensions of the SAME unit type (distance). And we can even do it one more time, adding a 3rd dimension of the same unit type.
The correlation between three dimensions, logically speaking is a representation of space.

But what about a 4th dimension of the same unit type? Can we correlate a point in space, with yet a new dimension of distance?

The truth is that when we use the sense of sight, we're not even perceiving the world as a 3D space. Both our eyes gather 2D information. Little hints like (depth of field blurring, and eye separation), even parallax effects on motion grants our brain enough info to interpret the world in this manner (3D), but it's only after the brain processed it.

To us it wouldn't make much sense to correlate 3D space with yet another distance dimension, it's not the way our brain works!
We are mass, hence we're energy, and so is our brain. And we never chose how our brains would interact with the rest of the universe. In fact the sense that our brain is mass "separated" with the rest of the universe is a fabrication of our thoughts!

In the biological sense, using 4 dimensions is not how our brain works, we don't perceive the world that way. I don't think there's a single living creature on Earth capable of actually working that way.

But humans are logical creatures. Just as we can correlate temperature with time. We can choose to correlate 3D space with time, if we decide to.
And that is in fact useful to our brains and its biology, because that's how the brain works. Constantly receiving information, processing it and more importantly STORING it, this gives us a perception of time.

Just like a clock can tell us the time, our brain, biologically speaking, can tell us the time as well (even if it's not as reliable as a clock).

So yeah, we can perfectly agree that time is an extra dimension. Except it's not a dimension of distance, we use a time dimension.

Meaning we can correlate all space to time. Which gives us a 4D representation, of something our brain is not made to interpret visually, yet we're perfectly able to understand.

Even if we tried to picture 4 dimensions visually, we can only work with the tools our brain gives us.
We naturally interpret it as motion. Or we could picture as a 3D still, of everything overlapping, too!
In fact, the whole universe, since the beginning of time, and to its end, can be interpreted as this big overlapped 3D volume, illustrating everything that happened since the beginning of time in a single shape. Needless to say only god can perceive something so complex!

 

[Passionflower]

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.

I think your comparison is flawed for two reasons:

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.

 

[nitsuj]

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.

If that is you point, it was made in a very "round-about way".

lol, I'm pretty sure the others posting retorts are aware that space exists at the same time as time.

 

[ghwellsjr]

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.

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

 

[Dale]

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.

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.

Also, I don't think that any concept related to "purity" is in any definition of dimension that I am aware of. This is probably the reason that your personal usage is contrary to the mainstream usage.

 

[Passionflower]

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!

 

[Dale]

That works for me.

 

[ghwellsjr]

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!

I would be happy with an answer to my question:

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

 

[bobc2]

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!

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.

 

[elfmotat]

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.

Well, yes. A Lorentz boost jumbles space and time coordinates together into new space and time coordinates. This doesn't say anything about the dimensions themselves. There are still clearly three spatial dimensions and one temporal.

 

[TGlad]

bobc2- Your quote is unrelated to whether time is a dimension.

Minkowski is saying that space and time are part of the same continuum, not separate.

In particular, in Minkowski space you can rotate between space and time (called a boost), just like you can rotate between two space dimensions with a standard rotation to change reference frame. That doesn't stop the space dimensions being dimensions so nor does a boost stop time being a dimension.
Every event in the universe can be given a coordinate in x, y, z and t. The fact that these coordinates change based on your inertial frame doesn't stop them being dimensions. The coordinates change based on your rotation too.

Of course going from special to general relativity complicates things, with wormholes and singularities etc. But on the whole they are independent variables.

 

[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$$\begin{align} x' &= x \, \cos \phi- y \, \sin \phi\\ y' &= x \, \sin \phi+ y \, \cos \phi \end{align}$$which can be rewritten as$$\begin{align} x' &= \gamma ( x - \beta y ) \\ y' &= \gamma ( y + \beta x ) \end{align}$$where$$\begin{align} \beta &= \tan \phi\\ \gamma &= \cos \phi= \frac{1}{\sqrt{1 + \beta^2}} \end{align}$$Compare that with the Lorentz boost for velocity $c\beta$:$$\begin{align} ct' &= \gamma ( ct - \beta x ) \\ x' &= \gamma ( x - \beta ct ) \\ \gamma &= \frac{1}{\sqrt{1 - \beta^2}} \end{align}$$which can be written as$$\begin{align} ct' &= ct\, \cosh \phi - x \, \sinh \phi \\ x' &= -ct \, \sinh \phi + x \, \cosh \phi \end{align}$$where$$\begin{align} \beta &= \tanh \phi \\ \gamma &= \cosh \phi \end{align}$$