The Nature of Time: Should It Be Considered a 4th Dimension?

[JesseM]

In pre-relativistic kinematics and dynamics time is indeed a dimension but not in relativity.

Once again it seems you are inventing your own language, rather than using the standard language of physicists. What are the precise criteria for something to be treated as "a dimension" in physics, according to you? Would you disagree that any variable one chooses--temperature, say--can be considered a dimension?

Time in space-time is proper time which is not expressed as a dimension

What does the phrase "not expressed as a dimension" mean you you, exactly?

but is expressed by the metric of space-time, and this metric is composed of four separate dimensions.

You could similarly say that space in ordinary 2D euclidean geometry is expressed by a metric with two dimensions--but this wouldn't justify the statement that an x-axis and a y-axis placed on this space cannot themselves be described as "spatial dimensions", it's standard terminology in mathematics to refer to them that way.

As I wrote before, the hypersurfaces of constant proper time of space-time are hyperbolic.

If you take a bunch of clocks radiating out from a single event at different velocities, with each reading t=0 where their worldlines intersect this event, and then draw a hypersurface based on the event of each clock reading the same proper time t=T, then sure, you get a hyperbola. But what does this have to do with whether time is "a dimension"?

These hypersurfaces could only overlap the hypersurfaces of constant t (for the commonly called "time" dimension) in the case the speed of light would be infinite.

I still don't get what point you think you're making here. If you like you are free to use a coordinate system where the t-coordinate is based on the proper time along worldlines radiating out from a single event (although the coordinate system can only cover the future and past light cone of that event), but you'll still need four coordinates to pinpoint any event in the region covered by the coordinate system, and there'll still be an unambiguous notion of whether the separation between two events is timelike, spacelike or lightlike (though I think in this coordinate system it'd be possible for two events to have the same t-coordinate but a timelike separation). And the conventional coordinate systems used in SR can also be understood in terms of the proper time on physical clocks, except that instead of using a collection of physical clocks radiating out from a single point in spacetime at different velocities, you have a collection of clocks at rest with respect to each other and synchronized according to the Einstein synchronization convention. In this case if you look at the hypersurface of constant proper time (the event on each clock's worldline where it has ticked some time T since t=0), then you have the standard surface of simultaneity of an inertial coordinate system in SR. Leaving aside the question of why you think your choice of coordinate system shows "time is not a dimension", do you think that your choice of coordinate system, based on the proper time of clocks radiating out from a single event and all set to read the same time where their worldlines intersect that event, is somehow more "physical" than this one, based on the proper time of clocks at rest with respect to each other and synchronized according to the Einstein clock synchronization convention?

Actually, if you want to, in relativity, you can do away with time. The theory is diffeomorphism invariant and that means that each instance in time is simply the same thing just in another format.

I don't understand what diffeomorphism invariance has to do with "doing away with time", or what you mean by "each instance in time"--each instance of what, exactly? Would you agree that the question of whether two events are timelike separated, spacelike separated, or lightlike separated is a physical issue which is not affected by your choice of coordinate system?

 

[MeJennifer]

Jesse, I am trying to explain why none of the four dimensions of space-time represent time, but that instead the metric of space-time represents time.

I am not arguing the philosophy of what a dimension is.
Let's keep these two things separate.

 

[pmb_phy]

In mathematical terms:

\tau = \int \sqrt {dt^2 - dx^2/c^2 - dy^2/c^2 - dz^2/c^2}

If we let c \rightarrow \infty we can see that \tau = t as is the case in pre-relativistic kinematic and dynamic models.

I don't see what that matters in this thread? Please clarify.

"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."

Ya got to love that Minkowski. Did you read that article? Minkowski went on to say in that same article

But I still respect the dogma that both space and time have independent significance.

The union is the metric.

That makes no sense to me.

Thanks

Pete

 

[MeJennifer]

Pete, perhaps it helps if you can explain your views on this, then we can perhaps understand why and how we differ.

Suppose we have a space-time of say 7 observers. Now do you think that the t-dimension of this space-time expresses time in relativity?

I can readily see it does so in pre-relativistic kinematics and dynamics, afteral those theories postulate a notion of absolute time, so the t-dimension is indeed time.

But in relativity, clearly there is no such thing as absolute time, each of the 7 observers of can measure time quite differently.

So how do you conclude that the t-dimension represents time?

 

[masudr]

I can only repeat what I have said previously, as I feel that this is the source of the problem:

It must be stressed here that time being the 4th dimension is coordinate time. This is very different from the time that clocks will measure (the so-called proper time): that is proportional to lengths of paths in spacetime and can involve as much space as they do time.

MeJennifer is simply saying that time is proper time; and I'm sure we all here recognise that the metric is needed to define proper time. I think that's what is meant by the metric unifying the two.

 

[robphy]

Maybe the unqualified term "time" should be replaced by:
"proper time" when associated with [the spacetime arclength of] an observer's worldline [and events on that worldline],
"coordinate time" (or "time-coordinate") when associated with an observer's coordinate system [which can be used to label events not on the observer's worldline],
"clock reading" when referring to particular event.

Note that the metric by itself doesn't give us the notion of proper time. It is the metric and the choice of particular timelike path that does.


What this discussion needs [as well as many discussions in this forum] are more precise-definitions (when needed) and less loose talk.

 

[MeJennifer]

Note that the metric by itself doesn't give us the notion of proper time. It is the metric and the choice of particular timelike path that does.

Since,

\tau = \int \sqrt {dt^2 - dx^2/c^2 - dy^2/c^2 - dz^2/c^2}

I do not see any reason why it would be wrong to say that the metric in relativity gives us the the notion of proper time in relativity. The value obviously depends on the path but the way it is summed is by the metric.

 

[robphy]

Since,

\tau = \int \sqrt {dt^2 - dx^2/c^2 - dy^2/c^2 - dz^2/c^2}

I do not see any reason why it would be wrong to say that the metric in relativity gives us the the notion of proper time in relativity.

By itself, the metric does not.
There is an integral over a path-to-be-specified to be done.

So, the metric is just one of the needed structures that "gives us the the notion of proper time in relativity".

 

[robphy]

The value obviously depends on the path but the way it is "summed over" is by the metric.

The sum is over infinitesimal neighboring segments [the integrand].
The metric came in when determining what each segment contributes.

 

[MeJennifer]

The sum is over infinitesimal neighboring segments.

That is exactly right!

The metric came in when determining what each segment contributes

The metric determines what each segment contributes!

Anyway we are arguing miniscule details.

 

[robphy]

That is exactly right!


The metric determines what each segment contributes!

Anyway we are arguing miniscule details.

Yes, infinitesimal ones.
But these add up to precise statements.

 

[masudr]

Note that the metric by itself doesn't give us the notion of proper time. It is the metric and the choice of particular timelike path that does.

Yes, fair enough.

 

[MeJennifer]

Yes, infinitesimal ones.
But these add up to precise statements.

Indeed, a metric is an interval between two infinitesimally nearby events.

 

[masudr]

Indeed, a metric is an interval between two infinitesimally nearby events.

No, g(dx,dx) is!

 

[robphy]

Indeed, a metric is an interval between two infinitesimally nearby events.

No, g(dx,dx) is!

To add to masudr's comment,
the metric is a tensor g_ab that maps two vectors in the tangent space to a real number.

An interval (i.e., the square-interval or line-element g(dx,dx) ) is a scalar.

 

[MeJennifer]

To add to masudr's comment,
the metric is a tensor g_ab that maps two vectors in the tangent space to a real number.

An interval (i.e., the square-interval or line-element g(dx,dx) ) is a scalar.

You are really not saying anything different than what I am saying Robphy.

And if you want to be exact, the metric tensor, which is not the same as the metric, is quite useless unless you describe the metric coefficients with it.

 

[pervect]

A metric defines the interval between two nearby points, but the metric itself isn't an interval, because as robphy points out the metric is a rank 2 tensor, not a scalar.

 

[pmb_phy]

Pete, perhaps it helps if you can explain your views on this, then we can perhaps understand why and how we differ.

Let's define spacetime then. spacetime is a 4-dimensional manifold. Each point in spacetime represents an event that occurs in nature. This event is the 4-tuple (ct, x, y, z) = (ct, r) where each component describes one part of the event. The three spatial coordinates,r, describe the spatial portion of the event (i.e. where it happened) and the other represents the temporal component (i.e. when the event happened). A frame of reference is a set of coordinates in which one sets up a system of clocks and rods. All the rods are in sync in that frame. A components of two events have a physical significance. The difference between temporal readings on a single clock represents the proper time of that clock. The difference between the temporal readings of two different clocks read at the same time in a frame is the coordinate difference of time. The difference can be non-zero in a frame moving relative to the frame in which our clock at rest.

According to relativity the same event n another frame

Suppose we have a space-time of say 7 observers. Now do you think that the t-dimension of this space-time expresses time in relativity?

Yes. And you don't?

Pete

 

[MeJennifer]

Let's define spacetime then. spacetime is a 4-dimensional manifold. Each point in spacetime represents an event that occurs in nature. This event is the 4-tuple (ct, x, y, z) = (ct, r) where each component describes one part of the event. The three spatial coordinates,r, describe the spatial portion of the event (i.e. where it happened) and the other represents the temporal component (i.e. when the event happened).

What makes you think that is the case?
Different observers can make different slices of space-time into space and time, it completely depends on their relative orientations.

The absolute orientation or the coordinate values have no significance in relativity only their relative values.

You can think of each of these 7 observers having a different orientation in space-time, like nuts and bolts in a box, the absolute orientation does not matter in the least, since there is no absolute orientation, but their relative orientation will determine how they slice space-time into space and time.

Yes. And you don't?

No, I don't.
Each of those 7 observers can have their own unique measure of time (e.g. proper time), they could all be the same but it does not have to be the same. And also here their measure of time depends on their relative orientations in space-time.
There is no absolute space and no absolute time in relativity.

 

[pmb_phy]

What makes you think that is the case?
Different observers can make different slices of space-time into space and time, it completely depends on their relative orientations.

That was a frame dependant definition which is valied even though it is not an invariant definition.

No, I don't.
Each of those 7 observers can have their own unique measure of time (e.g. proper time), they could all be the same but it does not have to be the same. And their measure of time depends on their relative orientations in sapce-time.
There is no absolute space and no absolute time in relativity.

Nobody ever claimed otherwise, especially me. But it has nothing to do with the definition that I gave.

MJ - I think we've come to an impass where we'd just keep saying the same thing over and over. If you wish to PM me and convince me in PM then I'll return to this thread. I myself am not 100% satisfied with the definition that I gave above but have been unable to readily find one in the texts I have that I like.

Take care MJ

Best wishes

Pete

 

[MeJennifer]

The topic is already quite long, so perhaps we should call it quits.

For myself, the best way to understand relativity is in a coordinate free and Lorentz invariant way.

To me, to understand relativity in terms of three spatial dimensions, e.g. a plane of simultaneity, is like looking at shadows on the wall and be amazed at the "strange" kinematics of those "objects".

But, of course, everybody has their own preferred way of understanding it.

 

[quasar987]

Are we talking about Hausdorff spaces yet?

 

[MeJennifer]

Are we talking about Hausdorff spaces yet?

 

[rbj]

what does the sign indicate? ...

happily, Garth responded to this because for me to would begin to step beyond my competence.

I am interested in hearing more about the arrow of time in a black hole.

actually there is this arrow of time pretty much everywhere. although i can put my car in 1^st gear and go in the +x direction and put it in reverse and go in the -x direction, my clock only ticks in the +t direction. it never goes in the -t direction. that's the arrow of time, i think. there is much more to this concept like causality, i s'pose.

as my car indicates, there is no "arrow of space" in general, but it was pointed out to me that moving from outside a black hole to inside might be an arrow of space. can't put the car in reverse and back out of a black hole.edit: Holy Crap! i didn't realize that this thread got so long. i guess i was responding to a pretty stale post. sorry.

 

[masudr]

Are we talking about Hausdorff spaces yet?

I thought we settled that... the metric does not define nearness; we use a more useful distance function to define topological nearness.

 

[tehno]

I'm afraid I'm not sure what exactly is the issue of the thread?
Maybe ,the questions are "in what way time can be considered
4^th dimension in relativity?" or "what's intuitive meaning of
the term time in relativity or ,generaly,in physics ?".
Some posters already answered first question,but some of
the posters are overcomplicating in doing so (like reffering to tensors,
completely unnecessary in flat spacetime of special relativity).
Time isn't independent variable in relativity,nor it is like
"mysterious extra dimension itself".It shouldn't be confused with
additional spatial dimension of 4D hiperspace either (Jennifer is correct,
that's different).
Main reason behind speaking of 4-dimensionality in relativity is mathematical
description.
Origin can be found in difference between Galilean transformation
and Lorentz transformation.
Both transformations provide functional relation between
coordinates (x,y,z,t)<-->(x',y',z',t') of two inertial frames ,in
uniform motion.So,how would you explain it to a layman?
Here's my way ( motion is along x-axis):
Galilean tr.:
t&#039;= t;x&#039;=x-vt,y&#039;= y,z&#039;=z

Lorentz tr.:
t&#039;=\frac{1}{\sqrt{1-\beta^2}}(t-\frac{\beta}{c}x);<br /> x&#039;=\frac{1}{\sqrt{1-\beta^2}}(x-vt),y&#039;=y,z&#039;=z

Now,if we consider:
t&#039;=f_{1}(x,t),x&#039;=f_{2}(x,t)

we see that in Lorentz tr. functions f_{1},f_{2} are
both functions in 2 variables.In Galilean tr. this not the case (only
f_{2} is function in 2 variables)!
Therefore,if Galilean relativity charaterisation , by this standard,corresponds
somehow to "1+2=3",special relativity charaterization must be "2+2=4".
Of course ,this is just a funny analogy,very far from rigorous mathematical
treatment but layman may get a core idea.

 

[saderlius]

happily, Garth responded to this because for me to would begin to step beyond my competence.
actually there is this arrow of time pretty much everywhere. although i can put my car in 1^st gear and go in the +x direction and put it in reverse and go in the -x direction, my clock only ticks in the +t direction. it never goes in the -t direction. that's the arrow of time, i think. there is much more to this concept like causality, i s'pose.
as my car indicates, there is no "arrow of space" in general, but it was pointed out to me that moving from outside a black hole to inside might be an arrow of space. can't put the car in reverse and back out of a black hole.
edit: Holy Crap! i didn't realize that this thread got so long. i guess i was responding to a pretty stale post. sorry.

Don't sweat it, I'm still reading and studying posts, but at a slow pace.
This is actually exactly my purpose for posting- to explore the nature of time in comparison to the nature of space. Part of the reason i received an infraction in my previous thread was for my claim that the arrow of space actually is time, making time a primitive component of space, the latter of which has 2 arrows.(left/right etc) I'm still trying to test this idea, but as others have warned, i should take it to a philosophy forum.
thanks,
sad

 

[saderlius]

I'm afraid I'm not sure what exactly is the issue of the thread? Maybe ,the questions are "in what way time can be considered
4^th dimension in relativity?" or "what's intuitive meaning of
the term time in relativity or ,generaly,in physics ?". Some posters already answered first question,but some of the posters are overcomplicating in doing so (like reffering to tensors,completely unnecessary in flat spacetime of special relativity). Time isn't independent variable in relativity,nor it is like
"mysterious extra dimension itself".It shouldn't be confused with
additional spatial dimension of 4D hiperspace either (Jennifer is correct,
that's different).Main reason behind speaking of 4-dimensionality in relativity is mathematical description. Origin can be found in difference between Galilean transformation and Lorentz transformation. Both transformations provide functional relation between coordinates (x,y,z,t)<-->(x',y',z',t') of two inertial frames ,in uniform motion.So,how would you explain it to a layman?
Here's my way ( motion is along x-axis): Galilean tr.:
t&#039;= t;x&#039;=x-vt,y&#039;= y,z&#039;=z
Lorentz tr.:
t&#039;=\frac{1}{\sqrt{1-\beta^2}}(t-\frac{\beta}{c}x);<br /> x&#039;=\frac{1}{\sqrt{1-\beta^2}}(x-vt),y&#039;=y,z&#039;=z
Now,if we consider:
t&#039;=f_{1}(x,t),x&#039;=f_{2}(x,t)
we see that in Lorentz tr. functions f_{1},f_{2} are
both functions in 2 variables.In Galilean tr. this not the case (only
f_{2} is function in 2 variables)!
Therefore,if Galilean relativity charaterisation , by this standard,corresponds
somehow to "1+2=3",special relativity charaterization must be "2+2=4".
Of course ,this is just a funny analogy,very far from rigorous mathematical
treatment but layman may get a core idea.

Ah yes that's much simpler than some previous posts. I see in Galilean trans. time is treated as universal between 2 reference frames, but in Lorentz trans., respective velocity determines the time dynamic.
interesting... it is easy to see from the equation that time is intimately articulated with space. Wouldn't another word for "spacetime" be "motion"?
cheers,
sad

 

[Andrew Mason]

At the risk of adding to the confusion: I don't think it is that difficult to understand why a fourth dimension of time is required. If one is assigning co-ordinates to events, one has to add a fourth co-ordinate specifying the time of the event. That is all that is meant by "time" being the fourth dimension.

What Einstein discovered was that two events with the same time co-ordinates but different spatial co-ordinates in one inertial frame of reference did not have the same time co-ordinates in another inertial frame of reference. He noted that the quantity \Delta x^2 + \Delta y^2 + \Delta z^2 - c^2\Delta t^2 (the space-time interval) was the same in all inertial frames.

But the fact that this space-time interval is invariant is not what makes time a dimension. It just blurs the distinction between the time and space dimensions (since what may appear to one observer as spatial separation may be seen by another as a time separation).

AM

 

[robphy]

At the risk of adding to the confusion: I don't think it is that difficult to understand why a fourth dimension of time is required. If one is assigning co-ordinates to events, one has to add a fourth co-ordinate specifying the time of the event. That is all that is meant by "time" being the fourth dimension.

What Einstein discovered was that two events with the same time co-ordinates but different spatial co-ordinates in one inertial frame of reference did not have the same time co-ordinates in another inertial frame of reference. He noted that the quantity \Delta x^2 + \Delta y^2 + \Delta z^2 - c^2\Delta t^2 (the space-time interval) was the same in all inertial frames.

But the fact that this space-time interval is invariant is not what makes time a dimension. It just blurs the distinction between the time and space dimensions

AM

Good points.

(since what may appear to one observer as spatial separation may be seen by another as a time separation).

You probably mean to say that [for example]
what may appear to one observer as purely-spatial separation may be seen by another to have, in addition to a [different] spatial separation, a time separation.

This is probably why Minkowski introduced the ideas of "space-like" and "time-like" when he formulated the notion of "space-time".