Exploring Time as a Dimension

[UberGoober]

I know the topic of time has been brought up on multiple threads, and they are interesting. But I would like to ask the question a bit differently than I've seen it asked.

When I took Physics 1A the instructor basically said that Einstein showed that time was an independent variable, then showed how that impacted a lot of equations.

Over the years, I've enjoyed reading the various laymans physics books like Elegant Universe. In these books I've seen several different definitions of time.

One suggested that time is a physical dimension of space/time, just like the normal 3 physical dimensions. It said that we are traveling through the time dimension at the speed of light, and that whenever we accelerate we are just "redirecting" velocity from the time vector into a spatial vector. It also suggested that matter has a "shape" in the time dimension. It also went on to suggest, based on Einsteins work, that the past, present and future all exist simultaneously and that for some unknown reason we just "experience" the forward arrow of time. This had something to do with relative acceleration causing bodies to experience a "plane" through the time/space continuum like a sideways slice in a loaf of bread.

Others have said that while the passage of time may be relativistically tied to space, that when people refer to it as a "dimension" they simply mean in the mathematical sense (as in an independent variable).

Anyway, I'm sure I've butchered these theories and made myself sound completely ignorant. What I'm curious about is the latest thinking into the physicality of time and if we have any idea why relativistic velocities have an impact on how an object experiences time.

 

[Passionflower]

When I took Physics 1A the instructor basically said that Einstein showed that time was an independent variable, then showed how that impacted a lot of equations.

I know most people think like that but I think they are wrong.

Time is a dimension and independent variable in Galilean spacetime however Einstein showed that in Minkowski and Lorentzian spacetimes time is NOT a dimension and NOT an independent variable.

 

[Bill_K]

I think like that. So I guess I am wrong too.

 

[Dale]

Einstein showed that in Minkowski and Lorentzian spacetimes time is NOT a dimension

I think this is factually incorrect. Can you provide a reference where Einstein said that time is not a dimension?

I suspect that you are confusing dimensions with coordinates or basis vectors.

 

[Mentz114]

Others have said that while the passage of time may be relativistically tied to space, that when people refer to it as a "dimension" they simply mean in the mathematical sense (as in an independent variable).

Anyway, I'm sure I've butchered these theories and made myself sound completely ignorant. What I'm curious about is the latest thinking into the physicality of time and if we have any idea why relativistic velocities have an impact on how an object experiences time.

I think it's important to distinguish the coordinate t, and time, which is the thing measured by clocks.

 

[Passionflower]

I think this is factually incorrect. Can you provide a reference where Einstein said that time is not a dimension?

I suspect that you are confusing dimensions with coordinates or basis vectors.

No I am not.

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.

 

[Bill_K]

What you are confusing, Passionflower, is the distinction between coordinate time and proper time.

 

[Dale]

In Minkowski and Lorentzian spacetimes time is the path length between two events.

That is proper time.

Again, do you have any reference where Einstein explicitly "showed that in Minkowski and Lorentzian spacetimes time is NOT a dimension."?

 

[elfmotat]

I know most people think like that but I think they are wrong.

Time is a dimension and independent variable in Galilean spacetime however Einstein showed that in Minkowski and Lorentzian spacetimes time is NOT a dimension and NOT an independent variable.

No I am not.

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.

So then, by the same logic, you don't consider the three spatial "dimensions" to be dimensions either?

 

[Passionflower]

So then, by the same logic, you don't consider the three spatial "dimensions" to be dimensions either?

Correct.

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.

 

[pervect]

One suggested that time is a physical dimension of space/time, just like the normal 3 physical dimensions. It said that we are traveling through the time dimension at the speed of light, and that whenever we accelerate we are just "redirecting" velocity from the time vector into a spatial vector.

There's good news and bad news about this. The good news is that it's a popularization by a well known author with good credentials, and can be interpreted in a manner that makes sense.

The bad news about this interpretation is that it's ambiguous at best, and I think it's fair to say that it subtly redefines the word "velocity" to mean something other than what the word generally means. Furthermore, by the time you get around to addressing or discussing this particular ambiguity, you've mostly likely lost the original audience that the popularization was intended for. And probably created some long, meandering thread if it was posted to PF.

Specifically, the sort of velocity the author is talking about would be the rate of progression through coordinate time with respect to proper time. This sort of "velocity" isn't standard, but it's been called "celerity" or "proper velocity". There's a paper that discusses this this in elementary that I forget the URL for, meanwhile Wiki has some discussion that is not nearly as clear. http://en.wikipedia.org/w/index.php?title=Proper_velocity&oldid=490863337

If someone wants to disagree and create the usual long meandering thread about this, Id suggest to avoid hijacking the original poster's question it might be best to open a new thread and link it.

Anyway - that's my $.02, and I hope it helps more that it confuses.

 

[Dale]

So then, by the same logic, you don't consider the three spatial "dimensions" to be dimensions either?

Correct.

Then it seems like this is a purely semantic discussion. If you don't consider the three spatial dimensions to be dimensions, then whatever you mean by your use of the word "dimension" it isn't the same thing that most other people have in mind when they use the word.

I think such a re-definition of the word "dimension" is a bad idea, but it isn't strictly wrong since words get redefined all the time. So I won't argue about it.

 

[elfmotat]

Correct.

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.

"Dimension" has a specific meaning - a meaning which you are ignoring. The dimension of a space is the minimum number of basis vectors necessary to span the entire space. I'm not sure what your particular definition is, but it is not a common one.

Also, by your definition, time was considered to be the only dimension before 1905 because you need to use the Pythagorean theorem to measure distances.

 

[Passionflower]

Also, by your definition, time was considered to be the only dimension before 1905 because you need to use the Pythagorean theorem to measure distances.

Not really, in Galilean spacetime one can use the Pythagorean theorem to measure 4d distance, however that distance is not time.

In case of Galilean spacetime the path length between two events does not measure time, time in Galilean spacetime is the difference between the time coordinates of the two events. And those time coordinates are the same for all observers so clearly there is an independent time dimension in Galilean spacetime.

However in case of Minkowskian or Lorentzian spacetime the path length determines the time between two events.

Clearly time is NOT an independent variable in Minkowskian or Lorentzian spacetime, however it is in Galilean spacetime.

I suppose one can think of time as some axis in a coordinate system but a coordinate system is not the same spacetime it is a chart of spacetime.

And using this chart becomes problematic when observers accelerate, more problematic when spacetime is curved and globally useless when spacetime is non-stationary.

I think the first thing to learn when introduced to relativity is that space and time are dynamic variables not static dimensions.

 

[TrickyDicky]

Not really, in Galilean spacetime one can use the Pythagorean theorem to measure 4d distance, however that distance is not time.

In case of Galilean spacetime the path length between two events does not measure time, time in Galilean spacetime is the difference between the time coordinates of the two events. And those time coordinates are the same for all observers so clearly there is an independent time dimension in Galilean spacetime.

However in case of Minkowskian or Lorentzian spacetime the path length determines the time between two events.

Clearly time is NOT an independent variable in Minkowskian or Lorentzian spacetime, however it is in Galilean spacetime.

I suppose one can think of time as some axis in a coordinate system but a coordinate system is not the same spacetime it is a chart of spacetime.

And usig this chart becomes more problematic when observers accelerate and globally useless when spacetime is non-stationary.

Given the casual tone of these comments and your previous posts in this thread, I'm not sure if they are coming from some misunderstanding or if they try to convey some radical departure from the conventional views about dimensions and spacetime.
Would you please give your definition of dimension, and do you care to clarify if you see the distinction between proper and coordinate time and which of them you do not consider a dimension? Finally, how many dimensions exist in your opinion, then?

 

[yuiop]

Let me see if I have this right. Given the spacetime interval between two events:

[tex] \tau =\sqrt{\Delta x_0^2 - \Delta x_1^2 - \Delta x_2^2 - \Delta x_3^2} = \sqrt{\Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2}[/tex]

(using units of c=1) the quantity on the left (the proper time) is not a dimension, is an invariant and is an independent variable, while the coordinate time Δt is a dimension, is observer dependent, is not an independent variable and is interchangeable with the spatial dimensions?

 

[Passionflower]

Yuiop,

Let's consider an arbitrary curved the path length between two events in an arbitrary curved non-stationary spacetime.

Care to express this in terms of coordinate time?
And even if it were possible what kind of "time" is this coordinate time?

 

[yuiop]

Yuiop,

Let's consider an arbitrary curved the path length between two events in an arbitrary curved non-stationary spacetime.

Care to express this in terms of coordinate time?
And even if it were possible what kind of "time" is this coordinate time?

I dunno, I was the one asking questions and not preaching (for a change :tongue:)

Time is a mystery to me. My guess is that reasonably intelligent beings would try to map the world the see around them and would come up with some form of coordinate system to try and make sense of how distant events relate to each other, even a complex non stationary spacetime. On the other hand in a non stationary spacetime, they will probably meet some horrible singularity pretty soon and have better things to do with their "time" than map their surroundings.

Anyway, your loaded question "And even if it were possible what kind of "time" is this coordinate time?" seems to suggest that you do not think that coordinate time not useful for anything, and presumably to you coordinate distance is equally useless in arbitrary curved non stationary spacetime.

In my last post I was trying to suggest that my understanding is that proper time is the more useful quantity, being an invariant and your post seems to suggest that you tacitly agree. I was only trying to make clear the differences between proper time and coordinate time and how they relate to dimensions, but I do not claim to be 100% clear on these issues, so that is why I asked the question. Care to elucidate on your position and educate me?

 

[Mentz114]

Let's consider an arbitrary curved the path length between two events in an arbitrary curved non-stationary spacetime.

Care to express this in terms of coordinate time?
And even if it were possible what kind of "time" is this coordinate time?

Is this what you mean ?

An arbitrary worldline can be parametrized by the t-coordinate ( t-parameter ?)
[tex]
\dot{x}^\mu \equiv \frac{\partial x^\mu}{\partial t}
[/tex]
and the proper length is
[tex]
\tau = \int_{t_0}^{t_1} \left(g_{00}- g_{mn}\ \dot{x}^m \dot{x}^n \right)^{1/2}\ dt
[/tex]
where m,n are spatial indexes.

( I don't like the Tex rendering but I can't fix it )

[edit]Sorry, I made a mistake which I hope I've corrected now.

 

[Passionflower]

The t-coordinate?
As in "this one big leg out of four"?

 

[Mentz114]

The t-coordinate?
As in "this one big leg out of four"?

I've based this on the metric (with c=1) which defines the proper interval,
[tex]
\left(\frac{d\tau}{dt}\right)^2 = g_{00}-g_{mn}\frac{dx^m }{dt}\frac{dx^n}{ dt}\ \Rightarrow \ d\tau=\left( g_{00}-g_{mn}\frac{dx^m }{dt}\frac{dx^n}{ dt} \right)^{1/2}\ dt
[/tex]
but it is late and I may have made an elementary mistake ( like assuming a diagonal metric ).

 

[elfmotat]

I've based this on the metric (with c=1) which defines the proper interval,
[tex]
\left(\frac{d\tau}{dt}\right)^2 = g_{00}-g_{mn}\frac{dx^m }{dt}\frac{dx^n}{ dt}\ \Rightarrow \ d\tau=\left( g_{00}-g_{mn}\frac{dx^m }{dt}\frac{dx^n}{ dt} \right)^{1/2}\ dt
[/tex]
but it is late and I may have made an elementary mistake ( like assuming a diagonal metric ).

I believe it should be [itex]d\tau =\sqrt{-(g_{00}+g_{ij}\dot{x}^i\dot{x}^j)}dt[/itex]

g₀₀ includes the "-" sign.

 

[A.T.]

And those time coordinates are the same for all observers so clearly there is an independent time dimension in Galilean spacetime.

Would you please give your definition of dimension,

Sounds like he is suggesting that only frame independent quantities can be called a "dimension".

For me a "dimension" is just an element of a certain geometrical interpretation of the observed quantities, including frame dependent ones. In different geometrical interpretations you have different sets of dimensions:

- coordinate_time + 3 x space
- proper_time + 3 x space

Which of them is more useful depends on the application.

 

[harrylin]

[..] I've seen several different definitions of time.
One suggested that time is a physical dimension of space/time, just like the normal 3 physical dimensions. It said that we are traveling through the time dimension at the speed of light, and that whenever we accelerate we are just "redirecting" velocity from the time vector into a spatial vector. It also suggested that matter has a "shape" in the time dimension. It also went on to suggest, based on Einsteins work, that the past, present and future all exist simultaneously and that for some unknown reason we just "experience" the forward arrow of time. [..]

That is called the "block time" interpretation:
https://www.physicsforums.com/showthread.php?t=595021

Others have said that while the passage of time may be relativistically tied to space, that when people refer to it as a "dimension" they simply mean in the mathematical sense (as in an independent variable).

It's tied to speed and to distance to gravitational mass. Time is defined as a dimension in the mathematical sense - at least originally.
- http://www.bartleby.com/173/17.html

Anyway, I'm sure I've butchered these theories and made myself sound completely ignorant. What I'm curious about is the latest thinking into the physicality of time and if we have any idea why relativistic velocities have an impact on how an object experiences time.

As you can see from the discussion, "time" is still very much open for interpretation. As on this topic hardly anything new has been discovered for almost a century, I don't believe that the very latest thinking is necessarily better than earlier thinking.

 

[Dale]

Sounds like he is suggesting that only frame independent quantities can be called a "dimension".

Agreed, which is not related to most people's idea of what a dimension is. I simply don't see the point in engaging in a semantic argument, particularly when the definitions are not clearly spelled out.

 

[PhilDSP]

Time is a dimension and independent variable in Galilean spacetime however Einstein showed that in Minkowski and Lorentzian spacetimes time is NOT a dimension and NOT an independent variable.

Would you settle for the terms quasi-dimension and quasi-independent or maybe pseudo-dimension and pseudo-independent?

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)

 

[TGlad]

Surely the 4 elements in Minkowski space-time are independent variables / degrees of freedom / dimensions.
The specific formula for the distance measure is irrelevant to that. You can always vary one variable without the others changing, and get a different event in space-time.

Personally I think of time as an imaginary dimension, since its square is negative.
But equations can be interpreted in many ways and with many different definitions of terms. Not much point in arguing over different interpretations, as long as they fit the equations.

 

[Passionflower]

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]

Surely the 4 elements in Minkowski space-time are independent variables / degrees of freedom / dimensions.
The specific formula for the distance measure is irrelevant to that. You can always vary one variable without the others changing, and get a different event in space-time.

Personally I think of time as an imaginary dimension, since its square is negative.
But equations can be interpreted in many ways and with many different definitions of terms. Not much point in arguing over different interpretations, as long as they fit the equations.

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.T.]

Others, keep singing the mantra that time is the fourth dimension in relativity,

Not in in relativity, but in certain geometrical interpretations of relativity.

 

[A.T.]

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".

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

 

[Passionflower]

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

Only when you want a distance over a mixture of spatial dimensions, but the distance in a particular dimension stays the same for all observers.

In Galilean spacetime time is absolute, that is time is the same for all observers, that makes it a dimension of spacetime, just like any particular spatial direction of spacetime is a dimension. However in relativity time and space is not absolute, it can differ completely from one observer to another, clearly that implies it is not a dimension of spacertime.

Some keep insisting that coordinate time (and oddly enough nobody speaks about coordinate space), which is a dimension of a chart of spacetime, not a dimension of spacetime, is the real time even when there are situations where this coordinate time cannot possibly represent any physical clock.

But old habits die hard and old dogs cannot be taught new tricks.

 

[A.T.]

Some keep insisting that coordinate time ... is the real time.

Who? And what does "real" mean anyway?

 

[Dale]

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.

The words are only wrong because you changed the definition. That is just a strawman.

 

[yuiop]

However in relativity time and space is not absolute, it can differ completely from one observer to another, clearly that implies it is not a dimension of spacertime..

I assume by "it" you mean time. But as you you said space is not absolute, it follows by your reasoning that space is not a dimension of spacetime either. Do you disagree with the notion that spacetime is 4 dimensional? If you do agree that spacetime is 4 dimensional, what would you suggest those 4 dimensions comprise of?