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

[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? :tongue:

 

[MeJennifer]

Are we talking about Hausdorff spaces yet? :tongue:

 

[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? :tongue:

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.:
$$t'= t;x'=x-vt,y'= y,z'=z$$

Lorentz tr.:
$$t'=\frac{1}{\sqrt{1-\beta^2}}(t-\frac{\beta}{c}x); x'=\frac{1}{\sqrt{1-\beta^2}}(x-vt),y'=y,z'=z$$

Now,if we consider:
$$t'=f_{1}(x,t),x'=f_{2}(x,t)$$

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.:
$$t'= t;x'=x-vt,y'= y,z'=z$$
Lorentz tr.:
$$t'=\frac{1}{\sqrt{1-\beta^2}}(t-\frac{\beta}{c}x); x'=\frac{1}{\sqrt{1-\beta^2}}(x-vt),y'=y,z'=z$$
Now,if we consider:
$$t'=f_{1}(x,t),x'=f_{2}(x,t)$$
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".

 

[Andrew Mason]

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.

Yes. For a purely spatial separation (simultaneous events separated by a distance) in one frame the space-time interval is positive. For events separated only by time, the space-time interval is negative. Since the space-time interval is invariant (same in all frames) a pure spatial separation of events in one frame will appear to be separated in both space and time in all other frames. The spatial separation between such events will always be greater than the distance traveled by light in the time separation between the events.

AM

 

[lalbatros]

Why should time be considered as a 4th dimension?

Just counting: 1 for x, 2 for y, 3 for y, 4 for time.
you don't need more to define the position of a (classical) particle.
But physics needs more information sometimes, like the spin, the charge, the color.
These attributes however can be separated from the 4 spatial coordinates, it seems.

 

[masudr]

But physics needs more information sometimes, like the spin, the charge, the color.
These attributes however can be separated from the 4 spatial coordinates, it seems.

Well for one thing, those other attributes don't take on a continuous range of values and also are specific to certain interactions.

 

[windscar]

Say for instance, I was going to invite you out to the bar, and say all the drinks were on me. Then I decided that as a trick, I would give you the address on a cordinate plane made from the city. To find the location, you would have to figure out were the bar was on this coordinate plane. Well, the city is relativaly flat so that rules out one dimension. And you figure, per say, that it is at X=5 and y=10. You get excited and go there to get your free drinks, but you find out that I am not even there... Likely way for me to get out of it right? No, I just thought you would know it would be tomorrow, but you arrived that night. So, we both where at the same location, but we were there at different times. Therefore, it was the time that separated us, not the space. If two events were not separated in some way, then we would run into each other at the bar at no matter what time we arrived there. It is simply saying that time separates events through some "distance" in order for them not to overlap. So on my cordinate plane, I assumed all events took place the next day, and you assumed all the points on that plane were the points takeing place today. And the difference between those two planes would be a higher dimension of time, that allows both to exist seperatly.

 

[saderlius]

Say for instance, I was going to invite you out to the bar, and say all the drinks were on me. Then I decided that as a trick, I would give you the address on a cordinate plane made from the city. To find the location, you would have to figure out were the bar was on this coordinate plane. Well, the city is relativaly flat so that rules out one dimension. And you figure, per say, that it is at X=5 and y=10. You get excited and go there to get your free drinks, but you find out that I am not even there... Likely way for me to get out of it right? No, I just thought you would know it would be tomorrow, but you arrived that night. So, we both where at the same location, but we were there at different times. Therefore, it was the time that separated us, not the space. If two events were not separated in some way, then we would run into each other at the bar at no matter what time we arrived there. It is simply saying that time separates events through some "distance" in order for them not to overlap. So on my cordinate plane, I assumed all events took place the next day, and you assumed all the points on that plane were the points takeing place today. And the difference between those two planes would be a higher dimension of time, that allows both to exist seperatly.

hrm, a very useful analogy, thanks. I think i understand the practicality involved in the use of time as a "dimension" of a system which allows for separation, just as space is also a dimension which allows for separation of events. This says more to me about a practical perspective of time than it does about the actual nature of time, the latter being more what I'm interested in. But that, as others have said, might be more properly discussed in a philosophy forum.
said,
sad

 

[windscar]

hrm, a very useful analogy, thanks. I think i understand the practicality involved in the use of time as a "dimension" of a system which allows for separation, just as space is also a dimension which allows for separation of events. This says more to me about a practical perspective of time than it does about the actual nature of time, the latter being more what I'm interested in. But that, as others have said, might be more properly discussed in a philosophy forum.
said,
sad

Your welcome and your right. The true nature of time would be better discussed in a philosophy forum, because there really isn't anything in physics that tells about about the true nature of time. It is like the bull in the china shop analogy. You know that if a raging bull goes in it is going to destroy everything in the shop until there is nothing left in one peice, but according to the laws of physics time should be able to run equally in both directions. So why don't you ever see bull's comeing out of destroyed china shops backwards with everything in tack? The problem is that, there is nothing to show times arrow, that events pass by only one way forward in time. The Arrow of Time is an all right book, and I suggest reading it if you want to gain more insight about time itself and how it is used in physics and some of the problems faced with it and times arrow.