I How can time only have one direction?

[student34]

Apart from anything else, if you accept your argument then there are zero spacelike dimensions.

What about the space 10 meters to my right. It may exist even though nothing can reach it from my position.

 

[student34]

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.

I was referring to the direction of the time axis. I did not know exactly how to explain it when you asked the question.

 

[vela]

I was referring to the direction of the time axis. I did not know exactly how to explain it when you asked the question.

Yeah, but it would have been useful to know why you think it can have only one direction in spacetime, because that's obviously not true. It seemed you were thinking of direction in a different sense which I wanted you to clarify.

 

[weirdoguy]

So I'll repeat my question:

So what textbooks are you working on right now?

 

[student34]

Yeah, but it would have been useful to know why you think it can have only one direction in spacetime, because that's obviously not true. It seemed you were thinking of direction in a different sense which I wanted you to clarify.

I thought that if time has more than one direction, then it should have more than one dimension.

 

[student34]

So I'll repeat my question:

Why do you need to know this so badly?

 

[phinds]

Why do you need to know this so badly?

Because we'd like to know where you are getting so much misinformation. When you are asked a direct question like that here on PF it is very bad form to ignore it **. How about you just answer the question?

** I've seen threads closed because the OP would not answer a question or questions.

 

[jbriggs444]

What about the space 10 meters to my right. It may exist even though nothing can reach it from my position.

Lots of things in your position can reach a position 10 meters to your right. Presumably you have reckoned the distance using an approximately inertial reference frame in which your chair is at rest. You can toss a ball over there. It may take one or two seconds to arrive as reckoned in your chair's rest frame.

Perhaps you were talking about an event located in your right hand just now and another event that is 10 meters to the right at the same time as reckoned in your chair's rest frame. Yes, you will not be able to toss a ball fast enough to travel between those two events.

You still cannot make the toss if you describe the same two events using a different inertial reference frame where they are neither simultaneous nor at a 10 meter separation from each other. They will be separated by the same space-time interval regardless. That interval will be space-like no matter what reference frame you choose.

You may have noticed correspondents in this thread using the terms "space-like" and "time-like". Those are useful notions.

 

[weirdoguy]

Why do you need to know this so badly?

Because what you wrote:

Believe me, I am trying.

[trying to learn from a textbook] seems unlikely. You ask questions that you wouldn't ask if you were working with a textbook. Or at least you would phrase them differently. So why being dishonest? Are you working with some textbook as you've been advised "thousands of times" or are you just don't want to take that advice, and instead wander around with loose thoughts based on tons of misunderstandings?

 

[vela]

I thought that if time has more than one direction, then it should have more than one dimension.

That's like saying the y-axis has more than one dimension because you can choose the orientation of coordinate axes arbitrarily. Once you make your choice of how to orient the axes, each axis points along one direction, i.e., on a diagram, you draw a line for each axis. Just because you can choose a different orientation doesn't mean each axis has more than one dimension. A different orientation corresponds to a completely separate set of axes.

 

[student34]

Because we'd like to know where you are getting so much misinformation. When you are asked a direct question like that here on PF it is very bad form to ignore it **. How about you just answer the question?

** I've seen threads closed because the OP would not answer a question or questions.

I have been reading a lot of publications related to this subject, and watching Khan videos, but that is not where the misinformation is coming from. My memory is bad, and I sometimes have slopping wording.

 

[vela]

You ask questions that you wouldn't ask if you were working with a textbook. Or at least you would phrase them differently. So why being dishonest?

I think you're being a bit judgmental here. Lots of students struggle with articulating what they're confused about. It's not being dishonest. It's just sometimes hard to put what's already confusing into words.

 

[weirdoguy]

I think you're being a bit judgmental here.

Well, maybe, I am sorry. But the amount of threads started by OP, and his reluctance to say on what textbook is he working on makes me really think this is going nowhere...

 

[student34]

That's like saying the y-axis has more than one dimension because you can choose the orientation of coordinate axes arbitrarily.

But it is not just one arbitrary choice, it seems to be infinite.

 

[vela]

But it is not just one arbitrary choice, it seems to be infinite.

How is that different from how you choose to orient the x- and y- axes when solving a mechanics problem? You also have an infinite number of ways you can choose to do that.

When you solve an inclined plane problem, you typically choose to orient the x-axis along the incline and the y-axis perpendicular to it. Does that mean the y-axis now points in an infinite number of directions because you could have chosen to orient the axes an infinite number of other ways? No. You make your choice, and it points in one direction.

In relativity, the same thing goes. In reference frame S, the t-axis points along one particular direction. In frame S', moving relative to S, the t'-axis points in a different direction but still in only one direction.

 

[student34]

How is that different from how you choose to orient the x- and y- axes when solving a mechanics problem? You also have an infinite number of ways you can choose to do that.

When you solve an inclined plane problem, you typically choose to orient the x-axis along the incline and the y-axis perpendicular to it. Does that mean the y-axis now points in an infinite number of directions because you could have chosen to orient the axes an infinite number of other ways? No. You make your choice, and it points in one direction.

In relativity, the same thing goes. In reference frame S, the t-axis points along one particular direction. In frame S', moving relative to S, the t'-axis points in a different direction but still in only one direction.

The difference is that the x axis is not different than the y axis. The time axis is intrinsically different than the spatial axis.

 

[vela]

You're making a pretty big leap here, and it would help if you could explain your reasoning. In what way is the time axis intrinsically different and why does it make a difference?

 

[phinds]

I have been reading a lot of publications related to this subject, and watching Khan videos, but that is not where the misinformation is coming from.

How do you know that? I suspect some of your misinformation IS coming from such sources. On line videos are usually just entertainment, not education and even decent ones are not a good source of learning.

To learn a subject you need to use a textbook AND work out the problems presented in the textbook. That's what I suggest you do, and, based on this thread, I have every confidence that you are not going to do that.

 

[student34]

You're making a pretty big leap here, and it would help if you could explain your reasoning. In what way is the time axis intrinsically different and why does it make a difference?

I am trying to say that space is different than time. I don't believe that is a controversial statement.

It makes a difference because now we csn have a space of time instead of just a line of time.

 

[PeterDonis]

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. A line is one dimensional. The space (or spacetime) in which the line exists might have more dimensions, depending on how many parameters it takes to specify the possible directions the line can go; but none of that affects how many dimensions the line itself has.

 

[PeterDonis]

I thought that if time has more than one direction, then it should have more than one dimension.

You thought wrong. In fact you aren't even thinking clearly about what a "dimension" is--despite repeated attempts to get you to do so. Some of the things you say appear to be groping in the right direction, but very slowly. For example:

I am trying to say that space is different than time. I don't believe that is a controversial statement.

A better way of stating this would be that spacelike vectors and curves are fundamentally different from timelike vectors and curves. And there is a third category in spacetime, null (or lightlike) vectors and curves, which are fundamentally different from both.

It makes a difference because now we csn have a space of time instead of just a line of time.

This is where you are sort of groping in the right direction; but you sidetrack yourself by not thinking clearly about what "time" is, what a "dimension" is, and what the different directions in spacetime of the worldlines of different observers in relative motion actually mean.

What you should do is step back from all that and, first of all, ask this question:

(Q1) How many distinct parameters does it take to describe all of the possible directions in spacetime that a timelike worldline (i.e., the worldline of an inertial observer) can have at a particular point?

The answer, of course, is "more than one". (I won't give the exact number right now because I want you to think about the question in those terms.) But why is it more than one?

Consider: suppose spacetime were 2-dimensional. That would mean the spacetime diagrams we draw on 2-dimensional sheets of paper (or the electronic equivalent) would be diagrams of actual spacetime, not just a 2-dimensional "slice" of 4-dimensional spacetime. And in a 2-dimensional spacetime, the answer to question Q1 above would be one. The "directions in spacetime" that a timelike worldline could have could be described by one parameter, which we could think of as the ordinary speed in the x-direction of that worldline in some fixed inertial frame. (Or, if we wanted our parameter to have the range $- \infty < p < \infty$ instead of $-1 < p < 1$, we could use the gamma factor or the rapidity.)

From the above, you should already be able to figure out the answer to question Q1 above for our actual 4-dimensional spacetime.

But now, take another step back, and ask a different question:

(Q2) How many timelike eigenvalues does the metric of spacetime have?

By "timelike eigenvalue" I just mean an eigenvalue of whichever sign we are using for timelike squared intervals in the metric. For example, if we write the metric as $d\tau^2 = dt^2 - dx^2 - dy^2 - dz^2$, timelike squared intervals are positive, so that is the timelike sign. (This "timelike convention" is more common in introductory books on SR and in particle physics. The opposite "spacelike convention" is more common in GR and more advanced relativity literature.)

The answer to Q2 above is one. That is, one always. It doesn't matter what frame you choose. It doesn't matter if spacetime is curved. It doesn't matter if you adopt some weird coordinate chart, even one in which you don't have one timelike and three spacelike coordinates as you do in a standard inertial frame. The metric always has precisely one timelike eigenvalue. And it always has precisely three spacelike eigenvalues.

What do the metric eigenvalues mean? They tell you what kinds of dimensions the space, or spacetime has. And that is why we say that spacetime has one timelike dimension and three spacelike dimensions, for four dimensions total: because of the eigenvalues of the metric. (Similarly, ordinary Euclidean 3-space has a metric with three spacelike eigenvalues--here we always adopt the spacelike convention so the metric is positive definite--and so we say it is a 3-dimensional space.) And note that, since the answers to Q1 and Q2 are different, the question of what kinds of dimensions the space or spacetime has is a different question from the question of how many different directions a particular kind of vector can point. Both are properties of the spacetime and its geometry; but they're different properties.

 

[PeterDonis]

those directions aren't orthogonal, so you aren't counting independent things.

Careful. The proper requirement is not orthogonality, but linear independence. And the kind of "independence" that leads to a count of the dimensions of spacetime is not "independence" of individual pairs of vectors.

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.

I would phrase this as: the fact that a standard inertial frame has four mutually orthogonal axes is an easy shorthand way to remember how many dimensions spacetime has: just count the axes. And the fact that one axis is timelike and three are spacelike gives a handy way to remember how many of what kinds of dimensions there are. But the ultimate count of dimensions doesn't come from that: it comes from counting the metric eigenvalues and their signs. The fact that it is possible to define a standard inertial frame with four mutually orthogonal axes, one timelike and three spacelike, is a consequence of the eigenvalues of the metric.

But counting dimensions that way is not the same as counting possible directions of curves in spacetime, as I made clear in post #81 just now. There are distinct, linearly independent timelike directions (vectors at a point) in spacetime even though there is only one timelike dimension (one timelike eigenvalue of the metric). But the criterion for "independence" of directions in spacetime is not and cannot be orthogonality: no pair of timelike vectors can possibly be orthogonal, but they can be linearly independent.

 

[vela]

I am trying to say that space is different than time. I don't believe that is a controversial statement.

Sure. A separation in time is measured using clocks, and a separation in space is measured using meter sticks. But I still don't see how you're getting from there to "therefore, a time axis points in more than one direction" and why any such argument wouldn't also apply to a spatial axis.

It makes a difference because now we csn have a space of time instead of just a line of time.

What do you mean by "a space of time"? I thought I knew what you meant, but I realized that perhaps you meant something different and this is where your confusion is stemming from.

 

[student34]

You thought wrong. In fact you aren't even thinking clearly about what a "dimension" is--despite repeated attempts to get you to do so. Some of the things you say appear to be groping in the right direction, but very slowly. For example:A better way of stating this would be that spacelike vectors and curves are fundamentally different from timelike vectors and curves. And there is a third category in spacetime, null (or lightlike) vectors and curves, which are fundamentally different from both.This is where you are sort of groping in the right direction; but you sidetrack yourself by not thinking clearly about what "time" is, what a "dimension" is, and what the different directions in spacetime of the worldlines of different observers in relative motion actually mean.

What you should do is step back from all that and, first of all, ask this question:

(Q1) How many distinct parameters does it take to describe all of the possible directions in spacetime that a timelike worldline (i.e., the worldline of an inertial observer) can have at a particular point?

The answer, of course, is "more than one". (I won't give the exact number right now because I want you to think about the question in those terms.) But why is it more than one?

Consider: suppose spacetime were 2-dimensional. That would mean the spacetime diagrams we draw on 2-dimensional sheets of paper (or the electronic equivalent) would be diagrams of actual spacetime, not just a 2-dimensional "slice" of 4-dimensional spacetime. And in a 2-dimensional spacetime, the answer to question Q1 above would be one. The "directions in spacetime" that a timelike worldline could have could be described by one parameter, which we could think of as the ordinary speed in the x-direction of that worldline in some fixed inertial frame. (Or, if we wanted our parameter to have the range $- \infty < p < \infty$ instead of $-1 < p < 1$, we could use the gamma factor or the rapidity.)

From the above, you should already be able to figure out the answer to question Q1 above for our actual 4-dimensional spacetime.

But now, take another step back, and ask a different question:

(Q2) How many timelike eigenvalues does the metric of spacetime have?

By "timelike eigenvalue" I just mean an eigenvalue of whichever sign we are using for timelike squared intervals in the metric. For example, if we write the metric as $d\tau^2 = dt^2 - dx^2 - dy^2 - dz^2$, timelike squared intervals are positive, so that is the timelike sign. (This "timelike convention" is more common in introductory books on SR and in particle physics. The opposite "spacelike convention" is more common in GR and more advanced relativity literature.)

The answer to Q2 above is one. That is, one always. It doesn't matter what frame you choose. It doesn't matter if spacetime is curved. It doesn't matter if you adopt some weird coordinate chart, even one in which you don't have one timelike and three spacelike coordinates as you do in a standard inertial frame. The metric always has precisely one timelike eigenvalue. And it always has precisely three spacelike eigenvalues.

What do the metric eigenvalues mean? They tell you what kinds of dimensions the space, or spacetime has. And that is why we say that spacetime has one timelike dimension and three spacelike dimensions, for four dimensions total: because of the eigenvalues of the metric. (Similarly, ordinary Euclidean 3-space has a metric with three spacelike eigenvalues--here we always adopt the spacelike convention so the metric is positive definite--and so we say it is a 3-dimensional space.) And note that, since the answers to Q1 and Q2 are different, the question of what kinds of dimensions the space or spacetime has is a different question from the question of how many different directions a particular kind of vector can point. Both are properties of the spacetime and its geometry; but they're different properties.

Thanks for all of this, and I will try to keep it in mind moving forward as best I can.

 

[student34]

Sure. A separation in time is measured using clocks, and a separation in space is measured using meter sticks. But I still don't see how you're getting from there to "therefore, a time axis points in more than one direction" and why any such argument wouldn't also apply to a spatial axis.

(I couldn't respond because this website wasn't working well with my cellphone)

Using the example in the OP, the object in the middle would calculate the time axis, of the objects moving away from it, pointing in different directions, no?

And to answer the second part, I would think that the time and space axis would have to be fixed if time is not going to point in a different direction for other observers in motion. Again, the example I gave would seem to have 3 different directions of time.

 

[student34]

What do you mean by "a space of time"? I thought I knew what you meant, but I realized that perhaps you meant something different and this is where your confusion is stemming from.

The worldlines of other objects seem to be able to fill an object's light cone, using multiple time axis. So any point in an object's light cone can be reached by lightlike worldlines. This seems to give time 2 dimensions, or space instead of just a line.

 

[jbriggs444]

The worldlines of other objects seem to be able to fill an object's light cone, using multiple time axis. So any point in an object's light cone can be reached by lightlike worldlines. This seems to give time 2 dimensions, or space instead of just a line.

Not exactly. I would not parameterize the space with [pairs of] lightlike worldlines. Instead, I would parameterize the space in terms of timelike directions.

The future light cone from a particular event includes a multitude of time-like directions. I count three dimensions. Any standard of rest corresponds to one of those directions. You have velocity in the x, y and z directions to define a standard of rest. So it takes three dimensions to identify a particular timelike direction.

If we are sticking to special relativity, "inertial reference frame" is nearly synonymous with both "standard of rest" and "timelike direction".

It is tempting to say "4 dimensions" since every event in the 4 dimensional future light cone is associated with a direction. But any given direction has infinitely many events all lined up on a straight world-line in that direction. So it is only three dimensions to identify a time-like direction/standard of rest/inertial reference frame.

It might be worth asking again -- do you even know what "dimension" means

 

[robphy]

In the plane, given a set of xy-axes, you have to specify two numbers to locate a point… a displacement along the given x-axis and a displacement along the given y-axis. (…Not two numbers along x-axes since you don’t have two x-axes.) This number of coordinates is the dimensionality of the plane…. but not the dimensionality of the x-axis.
One can certainly choose other orientations of the xy-axes. But the dimensionality of the plane is still two and the dimensionality of the given x-axis is still one.In a position vs time diagram, given a ty-plane associated with an inertial frame, you have to specify two numbers to locate an event… a displacement along the given t-axis (the reading of a clock) and a displacement along the given y-axis (the reading along a ruler). (…Not two numbers along t-axes since you don’t have two t-axes (you don’t have two clocks).) This number of coordinates is the dimensionality of the position-vs-time diagram…. but not the dimensionality of the t-axis.
One can certainly choose other spacetime-orientations of the ty-axes for different inertial frames. But the dimensionality of the position-vs-time diagram is still two and the dimensionality of the given t-axis is still one.

One can choose other pairs of axes… but that is a mathematical complication that will likely distract if the above is not first understood.

(crawl before walking and running…
note there is no need for special relativity to make the above points)

 

[student34]

Not exactly. I would not parameterize the space with [pairs of] lightlike worldlines. Instead, I would parameterize the space in terms of timelike directions.

The future light cone from a particular event includes a multitude of time-like directions. I count three dimensions. Any standard of rest corresponds to one of those directions. You have velocity in the x, y and z directions to define a standard of rest. So it takes three dimensions to identify a particular timelike direction.

If we are sticking to special relativity, "inertial reference frame" is nearly synonymous with both "standard of rest" and "timelike direction".

It is tempting to say "4 dimensions" since every event in the 4 dimensional future light cone is associated with a direction. But any given direction has infinitely many events all lined up on a straight world-line in that direction. So it is only three dimensions to identify a time-like direction/standard of rest/inertial reference frame.

It might be worth asking again -- do you even know what "dimension" means

Damn, I meant to put "timelike", not "lightlike". But anyways, I agree with what you are saying, but I still don't see it resolving my issue.

I do not know the rigorous definition of a dimension. I believe that I have a good idea though.

 

[phinds]

I do not know the rigorous definition of a dimension. I believe that I have a good idea though.

And have you made any attempt to find out the rigorous definition or do you prefer continuing to not know?