I How can time only have one direction?

[student34]

A basic example to explain what brings about my question is when considering two objects moving away from each other with an object at rest in the middle. In all 3 objects' frames of reference they are going through their respective time axises at the speed of light.

It would appear that time has at least 3 directions in this example. Can someone help me understand how time has only one direction?

 

[phinds]

Time is a scalar but you're trying to treat it like a vector. Localized time is called "proper time" and always moves forward at one second per second (regardless of how it might look to some non-local observer).

 

[PeterDonis]

I would appear that time has at least 3 directions in this example. Can someone help me understand how time has only one direction?

None of these are "directions of time"; the concept "direction of time" doesn't have a well-defined meaning.

What you have here are three different timelike curves, with three different timelike tangent vectors (vectors parallel to the curves) at the point where they meet (the common spacetime origin of the three rest frames of the three objects). The three different timelike tangent vectors define three different timelike directions in spacetime at the given point.

 

[jbriggs444]

A basic example to explain what brings about my question is when considering two objects moving away from each other with an object at rest in the middle. In all 3 objects' frames of reference they are going through their respective time axises at the speed of light.

It would appear that time has at least 3 directions in this example. Can someone help me understand how time has only one direction?

The idea that objects "move at the speed of light" in their own rest frame is rather meaningless. It sounds good in a pop science video. But that is about it. Objects have 3-velocities of zero in their own rest frames. Four-velocities have a fixed norm by definition. The fact that the norm is what it is says nothing about the four velocity.

But that is not what you ask about.

Let us take your three objects. Each of them has a timelike trajectory (like all massive objects). We assume that the three trajectories share a common intersection event. You express surprise that the three trajectories each share a common general direction in time.

But that is silly. There were six trajectories to choose from. You could have chosen the half-trajectory for object a going toward the intersection from the past or the half-trajectory for object a going away from the intersection into the future. The same for objects b and c. But you imply that you have chosen the three half-trajectories going into the future. Those are all future-directed because you chose them that way.

Presumably you have chosen to work in a spacetime that is time-orientable so that a clear distinction between future and past light cones anywhere can be extended unambiguously through the rest of spacetime everywhere else. [I think this is a consequence of time orientability]

See https://arxiv.org/pdf/gr-qc/0202031.pdf which goes a bit above my comfort level.

 

[PeterDonis]

a clear distinction between future and past light cones anywhere can be extended unambiguously through the rest of spacetime everywhere else. [I think this is a consequence of time orientability]

It's the definition of time orientability. Pretty much all other properties used by physicists assume a time orientable spacetime.

 

[strangerep]

See https://arxiv.org/pdf/gr-qc/0202031.pdf which goes a bit above my comfort level.

From section 4 of that paper:

It is not possible to define a spinor field on a non-time orientable spacetime [8 (Geroch)].
Since fermions exist, this well-known result has been cited as evidence that time must be
orientable.

So far, so good. But then... (with my emboldening]:

However the argument relies on a realist interpretation of the wavefunction and the false assumption that a wavefunction is defined at each spacetime point. In fact a wavefunction is a function defined on a 3N-dimensional configuration space where, N, is the number of particles.

Hadley does not justify the statement shown in bold above. Indeed, spinors are known to be just as physically real as, say, vectors. Imho, his invocation of wavefunctions here is spurious.

 

[pervect]

A basic example to explain what brings about my question is when considering two objects moving away from each other with an object at rest in the middle. In all 3 objects' frames of reference they are going through their respective time axises at the speed of light.

It would appear that time has at least 3 directions in this example. Can someone help me understand how time has only one direction?

We usually call the things that the three objects have that are different "4-velocities", but it is correct to say that each of the 4-velocity represents the time axis of the corresponding frame of reference, and it is also correct to say that they are all different.

It is also correct to say that the 4-velocites are all different and are not the same. I'm not quite sure why you think they should be or need to be the same.

"Time" can have a lot of meanings, it can be confusing to pick the right word to prevent ambiguities.

Possibly you expect there to be a "future" and a "past", and that's why you think that time should have only one direction? I'd say that it's better to model that concept with light cones, in which case you do have only two light cones, a future light cone and a past light cone. In special relativity and in most reasonable GR geometries, there is a common and global notion of "past" and "future" light cones, but things can get more complicated in some screwy GR geometries.

 

[vanhees71]

From section 4 of that paper:So far, so good. But then... (with my emboldening]:

Hadley does not justify the statement shown in bold above. Indeed, spinors are known to be just as physically real as, say, vectors. Imho, his invocation of wavefunctions here is spurious.

I never understood this discussion about many-body states in the position representation. Also in classical point-particle mechanics the configuration space of $N$ particles is $3N$ dimensional, i.e., you have functions of configuration-space variables and their time derivatives as observables and "states" in statistical mechanics are phase-space-distribution functions of $6N$ phase-space variables.

Why then is it so strange that in quantum mechanics we have wave functions with $3N$ configuration-space arguments?

Of course in relativistic QFT we don't have wave functions with a fixed number of particles but quantum fields and Fock states of (asymptotic) free particles.

The orientationability of time is just taken as an axiom in all of physics, i.e., there's a past and a future light cone containing the events "in the past" which can be causes to an event or for which the event in question can be the cause for an event "in the future". In SR that's globally in GR in general only locally defined.

 

[LittleSchwinger]

Hadley does not justify the statement shown in bold above. Indeed, spinors are known to be just as physically real as, say, vectors. Imho, his invocation of wavefunctions here is spurious.

Agreed. Geroch's argument is about Spinor fields, which are either operators in the quantum theory or just fields in the classical case. Nothing to do with wavefunctions.

 

[student34]

It is also correct to say that the 4-velocites are all different and are not the same. I'm not quite sure why you think they should be or need to be the same.

Then it would seem to me that time has more than one dimension. With the example in mind, time appears to be able to intersect with itself. I don't know the exact definition of a 2d space, but if it can intersect with itself everywhere and be parallel with itself everywhere, then doesn't that mean that it has at least 2 dimensions?

And I suppose I am assuming that spacetime is continuous. I believe that Einstein made this claim, but I do not know if it is assumed in the standard view of spacetime today.

 

[phinds]

Then it would seem to me that time has more than one dimension.

Time is one dimension. How can a dimension have more than one dimension? Could length have more than one length?

 

[Ibix]

It would seem to me that time has more than one dimension.

No. You are mixing up several uses of the word "time" and getting confused. When you are talking about the proper time along a worldline, as you appear to be doing here, time is just a scalar and doesn't really have a "direction" per se. The worldline has a direction, its tangent vectors have directions, but the proper time doesn't.

This is a very different use of "time" from the dimension, which is just one of the dimensions in spacetime. That doesn't have a direction either.

A third usage of the word "time" is coordinate time, which is part of a coordinate grid we superpose on spacetime. That does (sort of) have a direction, at least in the sense that the associated basis vectors are everywhere defined and have direction. But different frames define different directions that they call time. This still doesn't mean that time has more than one dimension, any more than the fact that the direction you call forward and the direction I call forward may be different implies that forwards is not a direction but a plane.

 

[student34]

Time is one dimension. How can a dimension have more than one dimension? Could length have more than one length?

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

 

[student34]

No. You are mixing up several uses of the word "time" and getting confused. When you are talking about the proper time along a worldline, as you appear to be doing here, time is just a scalar and doesn't really have a "direction" per se. The worldline has a direction, its tangent vectors have directions, but the proper time doesn't.

This is a very different use of "time" from the dimension, which is just one of the dimensions in spacetime. That doesn't have a direction either.

A third usage of the word "time" is coordinate time, which is part of a coordinate grid we superpose on spacetime. That does (sort of) have a direction, at least in the sense that the associated basis vectors are everywhere defined and have direction. But different frames define different directions that they call time. This still doesn't mean that time has more than one dimension, any more than the fact that the direction you call forward and the direction I call forward may be different implies that forwards is not a direction but a plane.

I understand what you are saying, but I still come to the same conclusion as in my post. The worldlines of time seem to fill a 2d space, don't they?

 

[Ibix]

The worldlines of time

This is nonsense. Time does not have worldlines - objects (and light) have worldlines.

 

[student34]

This is nonsense. Time does not have worldlines - objects (and light) have worldlines.

I meant the worldlines that occupy time.

 

[robphy]

It is also correct to say that the 4-velocites are all different and are not the same. I'm not quite sure why you think they should be or need to be the same.

Then it would seem to me that time has more than one dimension.

Note that your conclusion would also apply to Galilean/Newtonian physics.
(Note: A spacetime diagram is a position-vs-time diagram.)

 

[Ibix]

I meant the worldlines that occupy time.

A set of worldlines occupy space as well, and it's the combination of space and time that is multi-dimensional. A "congruence" is the term used when you are filling spacetime with a set of worldlines, and it's a standard GR tool. It doesn't mean "time is more than one dimension".

 

[jbriggs444]

I meant the worldlines that occupy time.

Interpreted literally, that is nonsense. World lines do not "occupy time". But I think that I understand. It amounts to a choice of coordinate system.

You imagine space time as a the collection of all events on a family of parallel world lines? Like a set of parallel hairs filling a volume?

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. Neither imagining is more valid than the other.

Different coordinate systems covering the same space-time can make different choices for the direction corresponding to world lines with fixed spatial coordinates.

 

[student34]

(Note: A spacetime diagram is a position-vs-time diagram.)

Yes, but isn't spacetime considered to be 4 dimensions? We still need space don't we?

 

[student34]

A set of worldlines occupy space as well, and it's the combination of space and time that is multi-dimensional. A "congruence" is the term used when you are filling spacetime with a set of worldlines, and it's a standard GR tool. It doesn't mean "time is more than one dimension".

Like I said, I do not know the exact definition of a dimension, but the light cone seems to fill a 2d space. Since this is already known, I would like to know why it does not meet the criteria of a 2d space.

 

[Ibix]

I would like to know why it does not meet the criteria of a 2d space.

It does. In fact the interor of a lightcone is a 4d space (only 2d are shown on a Minkowski diagraml. But this has nothing to do with time.

 

[robphy]

Yes, but isn't spacetime considered to be 4 dimensions? We still need space don't we?

Yes... so our diagram is (d+1)-dimensional for d spatial dimensions,
which can [and is] also done for Galilean/Newtonian kinematics.

My point is your comments aren't restricted to special or general relativity,
they also appear in Galilean relativity.
In other words, forget special relativity... do you have a problem with Galilean relativity?

 

[student34]

Interpreted literally, that is nonsense. World lines do not "occupy time". But I think that I understand. It amounts to a choice of coordinate system.

You imagine space time as a the collection of all events on a family of parallel world lines? Like a set of parallel hairs filling a volume?

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. Neither imagining is more valid than the other.

Different coordinate systems covering the same space-time can make different choices for the direction corresponding to world lines with fixed spatial coordinates.

How could non of them intersect?

 

[student34]

It does. In fact the interor of a lightcone is a 4d space (only 2d are shown on a Minkowski diagraml. But this has nothing to do with time.

I meant the worldlines in a light cone.

 

[jbriggs444]

How could non of them intersect?

How could none of what intersect? You have populated spacetime with a bunch of world lines. You say that these world lines fill it. If they intersect, you've done the tesselation inefficiently.

 

[student34]

Yes... so our diagram is (d+1)-dimensional for d spatial dimensions,
which can [and is] also done for Galilean/Newtonian kinematics.

My point is your comments aren't restricted to special or general relativity,
they also appear in Galilean relativity.
In other words, forget special relativity... do you have a problem with Galilean relativity?

I only want to focus on special relativity for now.

 

[student34]

How could none of what intersect?

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.".

 

[PeterDonis]

I meant the worldlines that occupy time.

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.

I do not know the exact definition of a dimension

Then I think you need to take another big step back and think carefully about what you actually mean by "dimension".

You can't expect to reason well if you don't even understand the words you are using.

 

[PeterDonis]

I meant the worldlines in a light cone.

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

 

[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.