I How can time only have one direction?

[PeterDonis]

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?

"Time axis" is a property of frames, not objects. So this question is unanswerable as you ask it.

I could try to guess what you actually mean, but I won't. Instead, you need to rephrase your question so it makes sense. It is very frustrating that, even after multiple threads with you on topics like this, you are still unable to use standard terminology in a standard way.

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.

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.

The worldlines of other objects seem to be able to fill an object's light cone

Objects don't have light cones. Events do. So this is nonsense as you state it.

See my remarks above about rephrasing things so they make sense and using standard terminology in a standard way.

using multiple time axis.

Nonsense. You've already been told why.

This seems to give time 2 dimensions, or space instead of just a line.

Nonsense. You've already been told why.

I agree with what you are saying, but I still don't see it resolving my issue.

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?

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

Then why can't you answer the question you've already been asked, multiple times, by giving whatever "good idea" you have?

 

[PeterDonis]

it takes three dimensions to identify a particular timelike direction.

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).

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.

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.

 

[vela]

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

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]

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)

Ok, I believe that I understand what you are saying. I will strip my issue down to make it as clear as possible.

If we can, assume only a universe of one space dimension and one time dimension. Two rocks suddenly separate left and right from event e. After 1 second, we want to draw a Minkowski diagram, and it of course is a V shape.

Now how can we make a single graph to satisfy the V? If this universe actually has only one time and one space dimension, how can we now "superimpose" (sorry but I do not know the correct terminology. I hope it is obvious what I am trying to say) a graph onto the V?

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.

 

[robphy]

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.

Each rock has its own time axis in a spacetime diagram, just like two different surveyors on a plane have their own x-axes (their own forward directions).

On a plane, the projection of one rock’s unit x vector is shorter than one on the other rock’s x-axis.

 

[PeterDonis]

how can we make a single graph to satisfy the V?

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.

As is the procedure, we give both rocks' worldlines the time axis.

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.

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.

How a length looks in a spacetime diagram does not always correspond to the actual length.

The lengths along worldlines are invariant. No spacetime diagram can make all lengths "look" like their actual lengths, because the piece of paper you are drawing the diagram on has Euclidean geometry, but spacetime itself has Minkowski geometry. So any diagram will distort some lengths.

And with all that said, your actual description of what the V will look like (leaving aside that how it looks is distorted compared to actual lengths) is wrong. If you just go by what each arm of the V looks like, and we assume that the actual lengths of the arms are equal (i.e., we mark the ends of each arm by the same elapsed proper time of the rock whose worldline is along that arm), then the arm corresponding to the time axis will look shorter than the other, not longer.

 

[PeterDonis]

Each rock has its own time axis in a its own spacetime diagram drawn in its own rest frame.

Please see the corrections in the above quote. We don't want to confuse the OP any more than he already is.

 

[robphy]

In response to @PeterDonis,
let me also suggest an edit of the analogy:
two different surveyors on a plane have their own x-axes (their own forward directions) in their own xy-planes.

 

[PeterDonis]

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.

If the "xy-planes" are the diagrams the surveyors are drawing, yes. But they're both surveying the same actual space.

 

[robphy]

If the "xy-planes" are the diagrams the surveyors are drawing, yes. But they're both surveying the same actual space.

Yup.

 

[Nugatory]

As is the procedure, we give both rocks' worldlines the time axis. Which Worldline gets the time axis?

It doesn't matter as far as counting dimensions is concerned. Up/down on the sheet of graph paper is forwards/backwards in the time dimension, left/right is the space dimension.

 

[student34]

"Time axis" is a property of frames, not objects. So this question is unanswerable as you ask it.

This was from your other post to the ultimate question of this thread, "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?"

I think I have narrowed down closer to my confusion. Here is my conclusion from my premise.

P1: Either rock can claim to be travelling through only time and no space.
Conclusion: Then it would seem to be true to say that both are travelling through only time and no space.

Of course my conclusion must be wrong, but I don't know why. I don't know how it can be either, but not both.

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.

I meant that each gets its own time axis in its own diagram.

How a length looks in a spacetime diagram does not always correspond to the actual length.

The lengths along worldlines are invariant. No spacetime diagram can make all lengths "look" like their actual lengths, because the piece of paper you are drawing the diagram on has Euclidean geometry, but spacetime itself has Minkowski geometry. So any diagram will distort some lengths.

And with all that said, your actual description of what the V will look like (leaving aside that how it looks is distorted compared to actual lengths) is wrong. If you just go by what each arm of the V looks like, and we assume that the actual lengths of the arms are equal (i.e., we mark the ends of each arm by the same elapsed proper time of the rock whose worldline is along that arm), then the arm corresponding to the time axis will look shorter than the other, not longer.

Oh right, I made a mistake. The point that I am trying to make is that the structure changes depending on which rock gets the time axis.

 

[PeterDonis]

I don't know how it can be either, but not both.

Because "traveling through time and not space" is frame-dependent. Each one, to make his claim that he is only traveling through time and not space, has to choose his own rest frame. But those frames are different. There is no frame in which they are both traveling through time and not space.

The point that I am trying to make is that the structure changes depending on which rock gets the time axis.

How the V looks changes. But the actual, physical lengths of its arms do not.

 

[jbriggs444]

The point that I am trying to make is that the structure changes depending on which rock gets the time axis.

The structure does not change. The description changes. One can describe all three rocks from the point of view of the inertial rest frame of any of the three rocks. Or from any other inertial frame not associated with any of the rocks.

There is a three dimensional space from which one can choose inertial frames that all correctly describe the three rock system.

All of the descriptions work. None of these descriptions are any more correct than any other.

 

[PeterDonis]

Thread closed for moderation.

 

[Dale]

After a brief discussion this thread will remain closed