I How can time only have one direction?

[vela]

Can someone help me understand how time has only one direction?

It would help if you could explain to everyone what precisely you mean by "time has only one direction."

 

[jbriggs444]

You said, "But I can imagine the same space time as a collection of all events on a different family of non-intersecting worldlines all sharing a different common direction.".

Yours are aligned in one direction. Mine are aligned in a different direction. Yours do not intersect with yours. Mine do not intersect with mine. What problem do you imagine?

 

[student34]

Worldlines don't "occupy time". They are curves in spacetime.

I think you need to take a big step back and think carefully about what you actually mean by the word "time". I suspect, as @Ibix said in post #12, that you actually mean several different things but are treating them as if they were the same. That's only going to confuse you, as indeed it seems to have done.

Ok, maybe this is where I am wrong. In special relativity, I thought that an object at rest travels though only time and no space. When I said "occupy time" I meant that the worldline would seem to "occupy" the time axis kind of the same way an electron occupies a position in space.

 

[student34]

What kind of "space" do you think the worldlines form?

Maybe subspace would have been a be a better term.

 

[phinds]

I thought that an object at rest

At rest relative to what? Your statement seems to imply absolute rest, which of course does not exist.

 

[student34]

At rest relative to what? Your statement seems to imply absolute rest, which of course does not exist.

Relative to itself

 

[phinds]

Relative to itself

EVERYTHING is at rest relative to itself, so your "at rest" includes everything/everyone.

 

[PeterDonis]

In special relativity, I thought that an object at rest travels though only time and no space.

There is no such thing. Worldlines are curves in spacetime. They are not curves in "time" or "space". Also, as @phinds has pointed out, there is no such thing as "at rest". You can pick a particular frame in which the object is at rest, its rest frame. But that doesn't mean the object is "at rest" in any absolute sense.

When I said "occupy time" I meant that the worldline would seem to "occupy" the time axis kind of the same way an electron occupies a position in space.

In the object's rest frame, you could say that the object's worldline is the time axis; that is a valid description. But it doesn't mean any of the things you appear to think it means.

Relative to itself

As @phinds has pointed out, everything is at rest relative to itself. So this is vacuous.

 

[student34]

EVERYTHING is at rest relative to itself, so your "at rest" includes everything/everyone.

I meant inertial frame of reference.

 

[PeterDonis]

I meant inertial frame of reference.

Which inertial frame?

 

[student34]

In the object's rest frame, you could say that the object's worldline is the time axis; that is a valid description. But it doesn't mean any of the things you appear to think it means.

Can't we break down the vector components of the object, namely of space and time?

 

[phinds]

I meant inertial frame of reference.

I have no idea what you intend with that statement. Seriously, I don't think you do either. I GUESS that you mean you meant to say that you were describing something that was at rest in an inertial FOR. That doesn't help your case at all.

Student34, you are clearly floundering around here. Trying to learn physics by semi-random questions on an internet forum is not a good approach. Get some textbooks and get a grounding in the fundamental concepts of cosmology and you'll see that his entire thread has been pretty much a waste of time.

 

[student34]

Get some textbooks and get a grounding in the fundamental concepts of cosmology and you'll see that his entire thread has been pretty much a waste of time.

Believe me, I am trying.

 

[PeterDonis]

Can't we break down the vector components of the object, namely of space and time?

You can take the 4-vector that is tangent to the object's worldline, and give its four components in any inertial frame. One component will be the "time" component and the other three will be the "space" components.

All that is true, but what's the point? What do you think you are accomplishing by doing this?

 

[Ibix]

Can't we break down the vector components of the object, namely of space and time?

Sure, but the vector components are different in different frames. What one frame breaks down as "no movement in space, just a timelike component" another will break down as "moving through space" with a different timelike component. The principle of relativity says both are equally valid. Once again, they're using different definitions of what "time" is, because there is a lot of room for choice in how you divide spacetime into space and time, and the two frames are different choices.

 

[student34]

You can take the 4-vector that is tangent to the object's worldline, and give its four components in any inertial frame. One component will be the "time" component and the other three will be the "space" components.

All that is true, but what's the point? What do you think you are accomplishing by doing this?

I am trying to understand how time can only be one dimension when it appears to have its vector components pointing in many different directions.

 

[PeterDonis]

I am trying to understand how time can only be one dimension when it appears to have its vector components pointing in many different directions.

This entire sentence is a confusion.

Vector components don't "point" anywhere. Vectors do.

Different timelike vectors point in different directions in spacetime. That does not mean "time" points in different directions.

The sense in which time is "one dimension" has nothing to do with how many directions timelike vectors can point.

Where are you getting your understanding of all this from?

 

[weirdoguy]

Believe me, I am trying.

So what textbooks are you working on right now?

 

[vanhees71]

In my post, I tried to explain how the worldlines of time appear to fill a space.

What you mean is the definition of a (local) reference frame in terms of a congruence of time-like worldlines. This is a way to define a "time slicing", i.e., you use the time-like world lines, which cover some open connected piece of spacetime, to define a set of space-like hypersurfaces, of which the tangent vectors at the time-like worldlines are the "normal vectors".

This congruence of time-like worldlines are parametrized by an arbitrary scalar parameter, which you can use as a scalar time coordinate. One possibility is to use the proper time of each of these worlelines. Time itself is, of course, always parametrized by one parameter. Only in this way you get a (local) "causality structure" of spacetime, and that's why in GR the metric has the signature $(1,3)$ or equivalently $(3,1)$ (depending on your choice of the west- or east-coast convention). It cannot be something like $(2,2)$, because then you'd have something like "two-dimensional time", but which sense should this make?

 

[Ibix]

I am trying to understand how time can only be one dimension when it appears to have its vector components pointing in many different directions.

Draw two dots on a piece of paper. You can get from one to the other along a straight line. (Edit: ugh, new phone, new drawing package, ignore the small dot below the right hand large dot.)

Draw a third point off to one side.

No matter how much you move in the direction of your line, you can't reach that third point starting from either of the first two. But if you draw a line perpendicular to the first line, you can reach the third point in two moves, one in the direction of the first line and one in the direction of the second.

This is what it means to be a 2d surface: you can pick two directions in the surface and you can get from any place on the surface to any other by moving first in one of those directions and then in the other. (Aside: that lacks rigour, but it'll do for now.) You can check this for yourself - add a fourth point and you will be able to get from any of the points to any other by moving some distance parallel to the first line and some distance parallel to the second.

The lines are not dimensions. How many of them you need is the dimensionality of the space. For example (assuming your screen is vertical), if you now draw a spot on the wall behind it, you can't get to that spot with moves in the plane of the screen. You need to add a third direction perpendicular to the plane of the screen. Now you can get from any point anywhere in space to any other (although you may need a drill to actually do it) in three moves, one along each of your directions. Three directions needed, so we say space is 3d.

Finally, you can change from the notion of points to the notion of events - a place in space at a given time. You can get from any place to any other using three moves, but you may need to wait - which might be seen as a fourth move. This is, of course, true even if space is 3d and time is something completely different, as is assumed in Newtonian physics. To be a dimension, there has to be some flexibility - if I'd drawn those first two dots in different places on the diagram the lines would be different but the reasoning the same. But in Newtonian physics you have no such freedom - everyone shares a unique notion of time.

Minkowski's insight was that Einstein's maths meant that there is flexibility in the notion of time, that you can just pick four arbitrary orthogonal directions and connect any two events in four moves along those four directions. We aren't constrained to share the same fourth direction. And if you do pick four arbitrary orthogonal directions, exactly one of them will be timelike.

And that's what we mean when we say time is one dimension in spacetime. That you can get from any event to any other in four moves (some may be zero-length moves if you happen to be lucky), and exactly one of those moves will be timelike.

 

[student34]

The sense in which time is "one dimension" has nothing to do with how many directions timelike vectors can point.

I am surprised by what you say here. If I have a line, and it can go in an infinite number of directions, doesn't this have something to do with how many dimensions it has?

 

[weirdoguy]

doesn't this have something to do with how many dimensions it has?

No. Line is one-dimensional, period.

 

[Ibix]

I am surprised by what you say here. If I have a line, and it can go in an infinite number of directions, doesn't this have something to do with how many dimensions it has?

No. It tells you that the line is embedded in a two-or-more dimensional space (or spacetime). The directions a line can point are a function of the directions available, not of the line.

 

[weirdoguy]

I am surprised by what you say here.

So maybe you should tell us what do you think dimension is?

 

[student34]

Draw two dots on a piece of paper. You can get from one to the other along a straight line. (Edit: ugh, new phone, new drawing package, ignore the small dot below the right hand large dot.)
View attachment 319578
Draw a third point off to one side.
View attachment 319579No matter how much you move in the direction of your line, you can't reach that third point starting from either of the first two. But if you draw a line perpendicular to the first line, you can reach the third point in two moves, one in the direction of the first line and one in the direction of the second.
View attachment 319580
This is what it means to be a 2d surface: you can pick two directions in the surface and you can get from any place on the surface to any other by moving first in one of those directions and then in the other. (Aside: that lacks rigour, but it'll do for now.) You can check this for yourself - add a fourth point and you will be able to get from any of the points to any other by moving some distance parallel to the first line and some distance parallel to the second.

The lines are not dimensions. How many of them you need is the dimensionality of the space. For example (assuming your screen is vertical), if you now draw a spot on the wall behind it, you can't get to that spot with moves in the plane of the screen. You need to add a third direction perpendicular to the plane of the screen. Now you can get from any point anywhere in space to any other (although you may need a drill to actually do it) in three moves, one along each of your directions. Three directions needed, so we say space is 3d.

Finally, you can change from the notion of points to the notion of events - a place in space at a given time. You can get from any place to any other using three moves, but you may need to wait - which might be seen as a fourth move. This is, of course, true even if space is 3d and time is something completely different, as is assumed in Newtonian physics. To be a dimension, there has to be some flexibility - if I'd drawn those first two dots in different places on the diagram the lines would be different but the reasoning the same. But in Newtonian physics you have no such freedom - everyone shares a unique notion of time.

Minkowski's insight was that Einstein's maths meant that there is flexibility in the notion of time, that you can just pick four arbitrary orthogonal directions and connect any two events in four moves along those four directions. We aren't constrained to share the same fourth direction. And if you do pick four arbitrary orthogonal directions, exactly one of them will be timelike.

And that's what we mean when we say time is one dimension in spacetime. That you can get from any event to any other in four moves (some may be zero-length moves if you happen to be lucky), and exactly one of those moves will be timelike.

Thanks for this. It helps clarify things for me.

With what you say above in mind, imagine a typical object in an inertial frame of reference. We have defined a time axis for it. But there is a different object in motion, and it has a different time axis. Won't there be 2 directions, something like what you did above with the drawing of the intersection to make 2 dimensions?

This is where I am getting the notion of time have 2 dimensions.

 

[jbriggs444]

With what you say above in mind, imagine a typical object in an inertial frame of reference.

An inertial frame of reference is not what you think it is. Objects are not "in" frames of reference. Objects are "at rest" in particular frames of reference.

We can use a single frame of reference to describe the motion of multiple objects not all moving in the same direction at the same speed.

 

[Ibix]

With what you say above in mind, imagine a typical object in an inertial frame of reference. We have defined a time axis for it. But there is a different object in motion, and it has a different time axis. Won't there be 2 directions, something like what you did above with the drawing of the intersection to make 2 dimensions?

Yes, but those directions aren't orthogonal, so you aren't counting independent things. That doesn't matter for counting the dimensionality of the space overall, but for more detail you need orthogonal directions.

Apart from anything else, if you accept your argument then there are zero spacelike dimensions. Or I could pick two spacelike directions and say there were zero timelike dimensions. Or two null vectors and say there are no timelike or spacelike dimensions at all! The way out of the confusion is to require that the basis directions be orthogonal if you are trying to classify your space/spacetime by looking at them.

 

[student34]

An inertial frame of reference is not what you think it is. Objects are not "in" frames of reference. Objects are "at rest" in particular frames of reference.

We can use a single frame of reference to describe the motion of multiple objects not all moving in the same direction at the same speed.

Thanks. I will try to remember the correct wording.

 

[student34]

Yes, but those directions aren't orthogonal, so you aren't counting independent things. That doesn't matter for counting the dimensionality of the space overall, but for more detail you need orthogonal directions.

Ok, I feel like this thread is progressing.

Before I comment on the later part of this post, I have to say that I am not sure exactly what you mean by I am not counting independent things.

 

[vela]

With what you say above in mind, imagine a typical object in an inertial frame of reference. We have defined a time axis for it. But there is a different object in motion, and it has a different time axis. Won't there be 2 directions, something like what you did above with the drawing of the intersection to make 2 dimensions?

I wish you had taken the time to answer my earlier question. So apparently what you mean by "time has only one direction" is that the time axis points in only one direction in spacetime.

You're confusing two uses of the word direction. One refers to the arrow of time. We observe that cause always precedes effect. Time doesn't go backwards. We never observe an effect occurring before its cause. If one observer sees event A as the cause of event B, all inertial observers will agree that A happens before B.

The other notion is the direction that the time axis is oriented in spacetime, which depends on the reference frame. That's a different notion of direction. Don't take them to be the same thing.