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?PeterDonis said:The sense in which time is "one dimension" has nothing to do with how many directions timelike vectors can point.
student34 said: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.student34 said: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?
student34 said:I am surprised by what you say here.
Thanks for this. It helps clarify things for me.Ibix said: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.
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.student34 said:With what you say above in mind, imagine a typical object in an inertial frame of reference.
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.student34 said: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?
Thanks. I will try to remember the correct wording.jbriggs444 said: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.
Ok, I feel like this thread is progressing.Ibix said: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.
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.student34 said: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?
What about the space 10 meters to my right. It may exist even though nothing can reach it from my position.Ibix said:Apart from anything else, if you accept your argument then there are zero spacelike dimensions.
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 said: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.
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.student34 said:I was referring to the direction of the time axis. I did not know exactly how to explain it when you asked the question.
weirdoguy said:So what textbooks are you working on right now?
I thought that if time has more than one direction, then it should have more than one dimension.vela said: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.
Why do you need to know this so badly?weirdoguy said:So I'll repeat my question:
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?student34 said:Why do you need to know this so badly?
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.student34 said:What about the space 10 meters to my right. It may exist even though nothing can reach it from my position.
student34 said:Why do you need to know this so badly?
[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?student34 said:Believe me, I am trying.
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 said:I thought that if time has more than one direction, then it should have more than one dimension.
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.phinds said: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 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 said: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?
vela said:I think you're being a bit judgmental here.
But it is not just one arbitrary choice, it seems to be infinite.vela said:That's like saying the y-axis has more than one dimension because you can choose the orientation of coordinate axes arbitrarily.
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.student34 said:But it is not just one arbitrary choice, it seems to be infinite.
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 said: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.
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.student34 said: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.
I am trying to say that space is different than time. I don't believe that is a controversial statement.vela said: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?
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.student34 said: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?
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:student34 said:I thought that if time has more than one direction, then it should have more than one dimension.
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.student34 said:I am trying to say that space is different than time. I don't believe that is a controversial statement.
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.student34 said:It makes a difference because now we csn have a space of time instead of just a line of time.
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.Ibix said:those directions aren't orthogonal, so you aren't counting independent things.
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.Ibix said: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.
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.student34 said:I am trying to say that space is different than time. I don't believe that is a controversial statement.
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 said:It makes a difference because now we csn have a space of time instead of just a line of time.
Thanks for all of this, and I will try to keep it in mind moving forward as best I can.PeterDonis said: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.
(I couldn't respond because this website wasn't working well with my cellphone)vela said: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.
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.vela said: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.
Not exactly. I would not parameterize the space with [pairs of] lightlike worldlines. Instead, I would parameterize the space in terms of timelike directions.student34 said: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.
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.jbriggs444 said: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
And have you made any attempt to find out the rigorous definition or do you prefer continuing to not know?student34 said:I do not know the rigorous definition of a dimension. I believe that I have a good idea though.
"Time axis" is a property of frames, not objects. So this question is unanswerable as you ask it.student34 said: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?
This is all nonsense, and the reason why has already been explained to you. There is no point in continuing to go around in circles.student34 said: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.
Objects don't have light cones. Events do. So this is nonsense as you state it.student34 said:The worldlines of other objects seem to be able to fill an object's light cone
Nonsense. You've already been told why.student34 said:using multiple time axis.
Nonsense. You've already been told why.student34 said:This seems to give time 2 dimensions, or space instead of just a line.
Then what's the point of further discussion? How many times do we have to see you making the same errors and giving you the same corrections?student34 said:I agree with what you are saying, but I still don't see it resolving my issue.
Then why can't you answer the question you've already been asked, multiple times, by giving whatever "good idea" you have?student34 said:I do not know the rigorous definition of a dimension. I believe that I have a good idea though.
To put it another way: the future-pointing unit timelike vectors at a point in spacetime form a three-dimensional vector space (more precisely, they form a three-dimensional subspace of the space of all possible vectors at a point in spacetime).jbriggs444 said:it takes three dimensions to identify a particular timelike direction.
That is why we need to put the qualifier "unit vectors" in the statement I made above. That is what "divides out", so to speak, the variation along each individual direction, and leaves only the variation in directions.jbriggs444 said: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.
I can't say I agree that you have a good idea since you keep suggesting time has two dimensions. Reread @robphy's post above. What he wrote directly conflicts with your claim. If you truly understand his post, you should be able to understand the errors in your reasoning.student34 said:I do not know the rigorous definition of a dimension. I believe that I have a good idea though.
Ok, I believe that I understand what you are saying. I will strip my issue down to make it as clear as possible.robphy said: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 said:As is the procedure, we give both rocks' worldlines the time axis. Which Worldline gets the time axis? And because of time dilation, it seems that which ever rock gets the time axis is the side of the V that has a longer worldline than the other.
You can't. There are an infinite number of possible "graphs" (spacetime diagrams) that will "satisfy" the V, corresponding to the infinite number of possible inertial frames in which to draw the diagrams.student34 said:how can we make a single graph to satisfy the V?
No, that is not "the procedure". A given inertial frame can only have one time axis. So you have to pick one. You can't draw a single diagram with both rocks' worldlines as the time axis.student34 said:As is the procedure, we give both rocks' worldlines the time axis.
How a length looks in a spacetime diagram does not always correspond to the actual length.student34 said:because of time dilation, it seems that which ever rock gets the time axis is the side of the V that has a longer worldline than the other.
Please see the corrections in the above quote. We don't want to confuse the OP any more than he already is.robphy said:Each rock has its own time axis inaits own spacetime diagram drawn in its own rest frame.
If the "xy-planes" are the diagrams the surveyors are drawing, yes. But they're both surveying the same actual space.robphy said:In response to @PeterDonis,
let me also suggest an edit the analogy:
two different surveyors on a plane have their own x-axes (their own forward directions) in their own xy-planes.
Yup.PeterDonis said:If the "xy-planes" are the diagrams the surveyors are drawing, yes. But they're both surveying the same actual space.
