[Dimensional (OP)]

Note: Title changed from "Does light have more than one direction?".
Assuming light time has a direction, an object in reference to itself travels only through time in a certain direction. If an object travels away from the object at rest, then according to the object at rest, it will have a spatial component to its direction.

I can't figure out how these objects can both be travelling through time in only one direction.

[Origin]

I can't figure out how these objects can both be travelling through time in only one direction.

Because the time dimension is orthogonal to all 3 spatial dimensions.

[evan_au]

Does light have more than one direction?

In Quantum Theory, light can be represented by a wave function.
- This wave function varies over time and space
- With the curious effect that everyone who measures its speed locally (in their lab) measures the speed as c
- However, people in different frames of reference will measure the direction differently, since there is no universal set of coordinates.
- The amplitude of the wavefunction determines the probability of finding a photon in a particular place.

If we simplify this by measuring everything relative to the "same" frame of reference...
- If we have an isotropic light source, the wavefunction is spherically symmetrical - ie the light has "every" direction. There is equal probability of detecting the photon in any particular 1mm² area
- If we have a laser beam, the wavefunction might be strongly directed along the "+x" axis, with very little divergence. In this case, the wavefunction is very asymmetric, and you could say that the light has a direction along the +x axis. You are much more likely to detect a photon in a 1mm² area near the +x axis.
- Both wavefunctions spread out as time goes on, so the chance that a photon will land in a particular 1mm² area decreases the farther you are from the source (ie the longer you wait after the light is emitted).
- But you can't determine which direction a particular photon traveled without detecting it, and detecting it means that it is no longer traveling in that direction.

[Dimensional (OP)]

I can't figure out how these objects can both be travelling through time in only one direction.

Because the time dimension is orthogonal to all 3 spatial dimensions.

I don't understand how orthogonality explains how there appears to be 2 different forward directions of time in my example.

[Dimensional (OP)]

Does light have more than one direction?

In Quantum Theory, light can be represented by a wave function.
- This wave function varies over time and space
- With the curious effect that everyone who measures its speed locally (in their lab) measures the speed as c
- However, people in different frames of reference will measure the direction differently, since there is no universal set of coordinates.
- The amplitude of the wavefunction determines the probability of finding a photon in a particular place.

If we simplify this by measuring everything relative to the "same" frame of reference...
- If we have an isotropic light source, the wavefunction is spherically symmetrical - ie the light has "every" direction. There is equal probability of detecting the photon in any particular 1mm² area
- If we have a laser beam, the wavefunction might be strongly directed along the "+x" axis, with very little divergence. In this case, the wavefunction is very asymmetric, and you could say that the light has a direction along the +x axis. You are much more likely to detect a photon in a 1mm² area near the +x axis.
- Both wavefunctions spread out as time goes on, so the chance that a photon will land in a particular 1mm² area decreases the farther you are from the source (ie the longer you wait after the light is emitted).
- But you can't determine which direction a particular photon traveled without detecting it, and detecting it means that it is no longer traveling in that direction.

Thanks for your help, but can we leave out QM, if possible. 

I have seen this explanation that you gave here "people in different frames of reference will measure the direction differently, since there is no universal set of coordinates.".

My problem with this is that we are already assuming a direction of time when we define an object to be at rest.  And then when another object is moving relative to it, that object can be defined to be at rest too. 

Is there an infinite array of directions of light?  If so, how can the time dimension be considered only one dimension?

[evan_au]

we are already assuming a direction of time when we define an object to be at rest

An object at rest displays no direction of time. It is stationary and unchanging.
- A moving object (in our frame of reference) with no external forces  displays a progress of time: At time t=a it is at position y=1, at a later time t=2a, it is at position y=2.
- A moving object (in our frame of reference) with no external forces  does not display a direction of time: If we assume that time is reversed (including for the observer):  At time t=2a, it is at position y=2, and at a subsequent time t=a it is at position y=1 (remember, time is going backwards!). But from the (time-reversed) view of the observer, he still see: At time t=a it is at position y=1, at a later time t=2a, it is at position y=2

Is there an infinite array of directions of light?

No, 2 dimensions are enough to measure the directions of light (assuming it is a tiny particle emitted from a point source, rather than a distributed wave as per QM).
- These 2 dimensions can be constructed in different ways, but one common example is spherical coordinates, which represents direction by two angles, φ and θ, measured relative to some coordinate system. If we treat them like latitude and longitude, φ can vary from -180° to +180°, and θ can vary from -90° to +90°.
- In spherical coordinates, there is a third coordinate r (radial distance), which can be used to represent the speed of light (which is the same for all local observers).
https://en.wikipedia.org/wiki/Spherical_coordinate_system

In QM/Many Worlds interpretation, the wave function of the universe may have infinite dimensions, (certainly a very large number of dimensions, like 10⁵⁰⁰).
See: https://en.wikipedia.org/wiki/Hilbert_space

[Eternal Student]

Hi.

   Now I'm certain that I'm not following exactly what you see as a problem.   @Origin gave one very short and reasonable answer to something that you might have been asking about.  @evan_au  has given another answer that has possibly shed some light down an alleyway in a very convoluted part of town that I'm really not sure you were asking about.

I'd like to just comment on one thing you said in the first post because there's one thing there which might, just might, be what is confusing you:

..an object in reference to itself travels only through time.....

   Which I have understood to mean that any ordinary object (one with some mass) has a rest frame.  In the co-ordinates of that rest frame the object is only travelling through time.
   That's fine, it holds for everything  (everything with mass as stated).

However you added something at the front of the sentence about light....

Assuming light has a direction, an object in reference to itself travels only through time in a certain direction.

   That's the main thing where the principle does not apply.   Light has no mass and must always travel at the speed c in any inertial reference frame.  So light has no valid rest frame.  Hence, there is no valid frame where you can say it is only travelling through time.

     I'm not sure if you were thinking that light itself can somehow point the way to the direction of time.  It just sounded a Little bit like it.   Whatever frame of reference you use, light is never travelling just through time.    So a ray of light can  go  --> that way   while  an object  goes  <-- this way and it means nothing more than that the spatial direction that the light was travelling in was opposite to that of the object.   You are simply unable to find a valid reference frame where the light was only travelling through time.

   There you go, a different alleyway illuminated.  If none of the replies so far seem to be going to the part of town you were really asking about, then it might be worth starting again and explaining your concerns in more detail.

Best Wishes.

[Halc]

I will attempt to clarify some stuff, but I'm not really sure what is being asked.

Assuming light has a direction

Well it does, else I'd not be able to shine my laser pointer at a specific spot on the wall. A normal light bulb shines pretty much in all directions, so there is not one direction of light. I suspect this is not what you're asking. The whole business of what light does has little to do with say the orientation of a time dimension.

an object in reference to itself ...

First of all, light does not define a frame, so one cannot define any of the usual values relative to a pulse of light. A rock works better.

travels only through time in a certain direction.

In the co-ordinates of that rest frame the object is only travelling through time.

Travel is defined as a change in spatial location over time, resulting in it no longer being at the first location, so 'travel through time' is not defined at all.  Within spacetime, a rock traces a worldline and is present at all locations along that line, so it doesn't 'travel' the line. Ditto with a light pulse, which traces a light-like woldline, but does not meaningfully travel along it. Travel is something you do in space over time, but not in spacetime.

If an object travels away from the object at rest, then according to the object at rest, it will have a spatial component to its direction.

Yes.

I can't figure out how these objects can both be travelling through time in only one direction.

Travel is a change in location in space, so travel through space. You seem to be mixing presentist terms (travel) with block terms (spacetime, time as a dimension). This is going to result in meaningless statements.

can we leave out QM, if possible.

Agree. You made no mention of photons, so we can keep it classical.

I have seen this explanation that you gave here "people in different frames of reference will measure the direction differently, since there is no universal set of coordinates.".

My problem with this is that we are already assuming a direction of time when we define an object to be at rest.  And then when another object is moving relative to it, that object can be defined to be at rest too.

A specification of an inertial frame (defining a specific object to be at rest say) defines the orientation of the time dimension,. The direction of it (the arrow of time) specifies which way along this arbitrary line is the positive direction, and entropy determines that. Space has no preferred direction, so if you define an x axis, either direction will suffice as the positive direction.

Is there an infinite array of directions of light?  If so, how can the time dimension be considered only one dimension?

There is no limit to the number of orientations one assign to the time dimension. This is the same as saying there is no limit to the number of valid inertial frames. Each frame defines one orientation for that dimension (and the spatial ones too maybe). But no matter which frame you select, only one dimension in it is the time dimension. Only one spatial orientation is the x axis, and so forth, despite the unlimited allowed choices for the orientation of it. If you choose a different orientation for any of these, you've defined a different coordinate system (frame).

An object at rest displays no direction of time.

Not sure what you mean by this. By defining a certain object to be at rest, you've defined the orientation of the time axis. It can be oriented no other way.

It is stationary and unchanging.

It very much can be changing. It is aging. Maybe the paint is fading, or the coffee is cooling, all of which are "displays of progress of time". But its spatial location is unchanging, which is what I think you mean by this.

- A moving object (in our frame of reference) with no external forces  does not display a direction of time: If we assume that time is reversed (including for the observer):  At time t=2a, it is at position y=2, and at a subsequent time t=a it is at position y=1 (remember, time is going backwards!). But from the (time-reversed) view of the observer, he still see: At time t=a it is at position y=1, at a later time t=2a, it is at position y=2

But the coffee gets hotter in the reverse direction, violating thermodynamics. The coffee was specifically chosen for its entropy and defined arrow of time.

In QM/Many Worlds interpretation, the wave function of the universe may have infinite dimensions, (certainly a very large number of dimensions, like 10⁵⁰⁰).
See: https://en.wikipedia.org/wiki/Hilbert_space

You have a reference for that? The wiki link didn't mention a large quantity of dimensions, but then I don't know my Hilbert space mathematics all that well. I usually think of a dimension that can be measured with a scalar such as 5 meters or 12 seconds or something. Hilbert space uses the word in a different way, not giving meaning say to 'the distance from here to the world where I cut my finger this morning'.

PS, MWI reference is still dragging QM into this, despite the request to keep it classical.

[Dimensional (OP)]

we are already assuming a direction of time when we define an object to be at rest

An object at rest displays no direction of time. It is stationary and unchanging.
- A moving object (in our frame of reference) with no external forces  displays a progress of time: At time t=a it is at position y=1, at a later time t=2a, it is at position y=2.
- A moving object (in our frame of reference) with no external forces  does not display a direction of time: If we assume that time is reversed (including for the observer):  At time t=2a, it is at position y=2, and at a subsequent time t=a it is at position y=1 (remember, time is going backwards!). But from the (time-reversed) view of the observer, he still see: At time t=a it is at position y=1, at a later time t=2a, it is at position y=2

I hope this isn't frustrating, but now I am more confused.  If an object at rest has no direction in time and an object in motion has no direction in time, then what would have a direction in time?

[Dimensional (OP)]

Hi.

   Now I'm certain that I'm not following exactly what you see as a problem.   @Origin gave one very short and reasonable answer to something that you might have been asking about.  @evan_au  has given another answer that has possibly shed some light down an alleyway in a very convoluted part of town that I'm really not sure you were asking about.

I'd like to just comment on one thing you said in the first post because there's one thing there which might, just might, be what is confusing you:

..an object in reference to itself travels only through time.....

   Which I have understood to mean that any ordinary object (one with some mass) has a rest frame.  In the co-ordinates of that rest frame the object is only travelling through time.
   That's fine, it holds for everything  (everything with mass as stated).

However you added something at the front of the sentence about light....

Assuming light has a direction, an object in reference to itself travels only through time in a certain direction.

   That's the main thing where the principle does not apply.   Light has no mass and must always travel at the speed c in any inertial reference frame.  So light has no valid rest frame.  Hence, there is no valid frame where you can say it is only travelling through time.

     I'm not sure if you were thinking that light itself can somehow point the way to the direction of time.  It just sounded a Little bit like it.   Whatever frame of reference you use, light is never travelling just through time.    So a ray of light can  go  --> that way   while  an object  goes  <-- this way and it means nothing more than that the spatial direction that the light was travelling in was opposite to that of the object.   You are simply unable to find a valid reference frame where the light was only travelling through time.

   There you go, a different alleyway illuminated.  If none of the replies so far seem to be going to the part of town you were really asking about, then it might be worth starting again and explaining your concerns in more detail.

Best Wishes.

Oh no, I meant to put time, not light.  I can't believe I didn't catch that in my review before posting the OP.

[Eternal Student]

Hi.

Travel is defined as a change in spatial location over time, resulting in it no longer being at the first location, so 'travel through time' is not defined at all.

   I'll concede that the language I used was sloppy.
   There is a 4-velocity that an object has at any given co-ordinate time t.   It's a bit sloppy  - but none-the-less it is often done - that the 4-velocity is just considered or described as a velocity through spacetime.
    I would have thought that @Dimensional has probably seen that sort of language elsewhere.    Here's some proof that it's done and prolific among PopSci articles:

Headline title of this article:   We All Travel Through Spacetime at the Speed of Light
https://medium.com/predict/we-all-travel-through-spacetime-at-the-speed-of-light-d60cb389dfc2

A sentence picked out from an article from the University of Orgeon:
The ultimate result from a spacetime view for the Universe is the realization that all objects move at one 'speed', the speed of light through space and time. Consider an object with mass at rest versus an object with zero mass (a photon). The object with mass moves through time, but not space (it is at rest). This 'motion' through spacetime is all time, no space.
http://abyss.uoregon.edu/~js/cosmo/lectures/lec06.html
   
(In fairness, the one from University of Orgeon had some 'quotation' marks around some words  but the next sentence or two was so bad that I've just got to include it here.....
Likewise, a photon has no time. It uses all of its spacetime velocity in the spatial direction and has none leftover for time.  )
   I can pull up more examples... but life is short and it may not be helping the OP.

Best Wishes.

[Dimensional (OP)]

Is there an infinite array of directions of light?  If so, how can the time dimension be considered only one dimension?

There is no limit to the number of orientations one assign to the time dimension. This is the same as saying there is no limit to the number of valid inertial frames. Each frame defines one orientation for that dimension (and the spatial ones too maybe). But no matter which frame you select, only one dimension in it is the time dimension. Only one spatial orientation is the x axis, and so forth, despite the unlimited allowed choices for the orientation of it. If you choose a different orientation for any of these, you've defined a different coordinate system (frame).

I don't know how, but I kept accidently putting "light" instead of "time".  But here you are answering the question that I meant to ask anyways, so I just want to comment on it.

You said, "There is no limit to the number of orientations one assign to the time dimension.".  I don't understand how this can be.  How can there be more that 2 orientations in a single dimension? 

For example, we define a rock to be at rest therefore only travelling along the time dimension.  Then there is a particle moving away from it.  In a different reference, the particle is at rest.  To make it easy to imagine, let's just have an x and t coordinate.  How can they both not have an x component?  Doesn't at least one of them have to have an x component.

[Halc]

I'll concede that the language I used was sloppy.

And I'll similarly concede that the 'sloppy' language is actually in use, at least in pop articles. My reluctance to qualify it as motion is the point I mentioned that motion implies no longer being at the origin if one is at the destination. It implies a 'current' position, which has is just going to lead to confusion in a spacetime model.

I don't know how, but I kept accidently putting "light" instead of "time".

OK, I saw that after I did my post. It makes your post more clear.

You said, "There is no limit to the number of orientations one assign to the time dimension.".  I don't understand how this can be.  How can there be more that 2 orientations in a single dimension?

You're thinking of which way to assign the arrow (the positive direction) once the orientation has been selected. I'm speaking of the orientation of the dimension itself, not the end to which you assign + or -. So imagine you have a meter-stick that you want to represent the x axis. You can align it east-west, north-south, or anywhere in between, perhaps not level with the ground. There is no limit to the different orientations you can give it. But yes, once you've chosen an arbitrary orientation, then you still have two choices as to which end of the stick you're going to designate as positive.
The time dimension is like that. You pick two events (points in spacetime). You say 'this' event and clack two rocks together to define the event.  Then you wander off some arbitrary place and clack the same two rocks a 2nd time to define a second event. The one unique line through spacetime connecting those two events is now defined. You've chosen a totally arbitrary orientation for your time dimension since the two events you chose are completely arbitrary.

For example, we define a rock to be at rest therefore only travelling along the time dimension.

Per E-S's post, I accept this language with reservations.

Then there is a particle moving away from it.  In a different reference, the particle is at rest.

Yes, in that different reference, the time axis is oriented differently in spacetime, not parallel to the one where the rock is stationary.
 

To make it easy to imagine, let's just have an x and t coordinate.  How can they both not have an x component?

In any one given frame, they can't both be stationary. Without a frame specification, the x component of any object's velocity is undefined. So your statement above is undefined because the frame isn't specified. A given object might be moving north in one frame and east in another.

Doesn't at least one of them have to have an x component.

In a given frame, yes, but since none was specified, no x component is defined.


All this (entire topic) is in the realm of Galilean relativity, physics over 400 years old. None of this touches on what Einstein added last century.

[Dimensional (OP)]

 

Then there is a particle moving away from it.  In a different reference, the particle is at rest.

Yes, in that different reference, the time axis is oriented differently in spacetime, not parallel to the one where the rock is stationary.

But I thought that there can only be one time axis. 

All this (entire topic) is in the realm of Galilean relativity, physics over 400 years old. None of this touches on what Einstein added last century.

If we were talking about an x/y spatial coordinate of the paths of the worldlines that the particle and the pebble takes, I wouldn't have this confusion.  I would see that it does not matter which worldline got the y coordinate because it would not change anything. 

However, I thought that we would not be able to do that because the time dimension t is indeed different than the spatial dimension x.  In that sense saying that something is only moving along the time axis or not would seem to actually make an objective difference.

[Eternal Student]

Hi.

All this (entire topic) is in the realm of Galilean relativity, physics over 400 years old. None of this touches on what Einstein added last century.

   Not entirely true.   ( @Dimensional , you shouldn't worry about being too far behind the current times ).
Newton is generally credited as the first to use the term "spacetime".   That's under 400 years ago, say 350 years.   His spacetime wasn't at all like the later spacetime of Einstein of course.
   Also I like the comment @Halc made about choosing the orientation of a time axis...

...You pick two events (points in spacetime).... The one unique line through spacetime connecting those two events is now defined... (and that can be your time axis)....

   That's (more or less) true but not at all obvious.   How long did you take to come up with that?
   What's more important is just that no-one considered a time axis to be anything like that until Einstein came along with his relativity and spacetime.  It was previously always assumed that time was some uniquely defined thing,  there was a universal time axis for everything, everywhere.

But I thought that there can only be one time axis.

    Well done @Dimensional ,  you are right up to where Einstein and his spacetime took over.

   Let's have two objects and call them "Billy Blue" or "Blue" for short,    and "Betty Black" ("Black" for short), this will be better than calling them A and B because they will end up using something that is blue or black.   If  Blue has some (non-zero) velocity relative to Black in some reference frame, then there is no way you can find a frame where both of them are at rest at the same time.   There is a rest frame for Blue,  in this frame Blue is at rest but Black would be moving.    There is a rest frame for Black, in which Black is at rest but then Blue would be moving.

    As outlined by @Halc ,  the time axis isn't always in precisely the same orientation in every reference frame.   In the rest frame of Black the time axis is orientated a certain way through spacetime.  We going to call this "straight up".   There's no point starting with a load of words, you'll want a diagram to visualise this, so here is one:
 

spacetime.jpg (6.76 kB . 257x260 - viewed 1991 times)

  Look at the black axis,  we have a conventional set of axis with the x-axis on the horizontal and the time axis running "straight up".   
    Meanwhile the time axis for the rest frame of Blue is the blue sloping line,  it runs at an angle, partly up the old time axis and partly along the old x-axis.  This is what we mean by the time axis of another reference frame running in a different direction or having a different orientation.   It's not like the whole axis runs along some spatial direction,  if you were a person at rest in the Black frame then there is no way you can point in some physical direction and say that is "along the time axis".   You can only point things out that are in space.  The time axis is "in time" and not in physical space.  If someone asks you to point in the direction of time then you might as well point to your watch and make some gestures with your hands and face as if you are playing a game of charades.   Now the diagram has just represented space and time on a flat 2-D surface and in that diagram, time has something we can call a direction and think of like a direction in space and it's what I am going to call "straight up".  Well to be precise the time axis is "straight up" in the rest frame of Black.
    Hopefully you can start to see where I'm going with this....   The rest frame of Blue has a very different orientation of the time axis,  it's the blue line which runs at a slope instead of being "straight up".   Like I said, it's not a real direction in physical space, it just looks like a direction when we represent it on the diagram.   If you were a person at rest in frame Blue then you still can't point your hand in the direction of that time axis.   The whole concept of "a direction" or orientation of a time axis is just telling you that the time axis in one frame of reference is actually a hybrid or combination of what was the time and space axis in another frame of reference.   (The diagram isn't something I made up off the top of my head, it's based on a conventional space-time diagram that's been used for years although it's simplified because I haven't needed to show you what happens to the x-axis for Blue).
   Now, without worrying too much about the details but just by looking at the diagram... can you see that if Billy Blue has all of his 4-velocity along the blue time axis, then that does not mean that he has all of his 4-velocity along the black axis?  Indeed, if something is "moving" (in the sense of having a 4-velocity) only along the blue axis then it is moving only in time when regarded with the blue axis.  However, it is moving partly through time but also partly through space when regarded with the black axis.   (You can make similar arguments for something moving only along the black time axis, that's only moving through time when regarded with the back axis but is moving partly through space when regarded with the blue time axis).
   Anyway, this is the sense in which two objects can be "moving" only through time in their own rest frame, without contradicting the idea that they do have some ordinary spatial velocity in the frame of the other.  The key really is that the two time axis do not have the same orientation.

   I hope that helps a bit.  Best Wishes.

[Colin2B]

@Eternal Student
It would be worth expanding for @Dimensional that in addition to “The key really is that the two time axis do not have the same orientation” that we can draw the diagram from the perspective of either object and each considers their time ‘orientation’ to be the vertical and the other’s sloping. Relativity rules OK.
Apologies if you or @Halc have already done that and I’ve missed it 

[evan_au]

wave function of the universe may have infinite dimensions, (certainly a very large number of dimensions, like 10⁵⁰⁰⁾.

You have a reference for that?

I occasionally listen to Sean Carrol's Mindscape podcast, and he uses a line something like this in his monthly "Ask Me Anything" (AMA) episodes when people ask about the wave function of the universe, or whether the universe is infinite or finite. Such questions have popped up several times.
- As I understand it, in QM, the state space of wave functions exist in Hilbert Space https://en.wikipedia.org/wiki/Hilbert_space#Quantum_mechanics
- Hilbert space may have an infinite or finite number of dimensions.
- Nobody knows whether the universe is infinite or finite, but Hilbert space can represent both
- The extremely large number that I vaguely recall for a finite dimensional Hilbert Space might represent the number of states in our observable universe (making no claims about the number of states in the non-observable universe).
https://www.preposterousuniverse.com/podcast/

@evan_au  ...has possibly shed some light down an alleyway in a very convoluted part of town that I'm really not sure you were asking about.

Guilty as charged - but the original question did ask about the direction of light - and QM accurately describe the way that light evolves in time and space.
- It's just a pity that the original question was not intended to be about light at all

an object is aging. Maybe the paint is fading, or the coffee is cooling,... thermodynamics & entropy

The OP did not ask about entropy, either.
But if the original question is now understood to be about the direction of time (rather than the direction of light), then entropy is definitely the primary consideration.

[Dimensional (OP)]

Hi.

All this (entire topic) is in the realm of Galilean relativity, physics over 400 years old. None of this touches on what Einstein added last century.

   Not entirely true.   ( @Dimensional , you shouldn't worry about being too far behind the current times ).
Newton is generally credited as the first to use the term "spacetime".   That's under 400 years ago, say 350 years.   His spacetime wasn't at all like the later spacetime of Einstein of course.
   Also I like the comment @Halc made about choosing the orientation of a time axis...

...You pick two events (points in spacetime).... The one unique line through spacetime connecting those two events is now defined... (and that can be your time axis)....

   That's (more or less) true but not at all obvious.   How long did you take to come up with that?
   What's more important is just that no-one considered a time axis to be anything like that until Einstein came along with his relativity and spacetime.  It was previously always assumed that time was some uniquely defined thing,  there was a universal time axis for everything, everywhere.

But I thought that there can only be one time axis.

    Well done @Dimensional ,  you are right up to where Einstein and his spacetime took over.

   Let's have two objects and call them "Billy Blue" or "Blue" for short,    and "Betty Black" ("Black" for short), this will be better than calling them A and B because they will end up using something that is blue or black.   If  Blue has some (non-zero) velocity relative to Black in some reference frame, then there is no way you can find a frame where both of them are at rest at the same time.   There is a rest frame for Blue,  in this frame Blue is at rest but Black would be moving.    There is a rest frame for Black, in which Black is at rest but then Blue would be moving.

    As outlined by @Halc ,  the time axis isn't always in precisely the same orientation in every reference frame.   In the rest frame of Black the time axis is orientated a certain way through spacetime.  We going to call this "straight up".   There's no point starting with a load of words, you'll want a diagram to visualise this, so here is one:
 

spacetime.jpg (6.76 kB . 257x260 - viewed 1991 times)

  Look at the black axis,  we have a conventional set of axis with the x-axis on the horizontal and the time axis running "straight up".   
    Meanwhile the time axis for the rest frame of Blue is the blue sloping line,  it runs at an angle, partly up the old time axis and partly along the old x-axis.  This is what we mean by the time axis of another reference frame running in a different direction or having a different orientation.   It's not like the whole axis runs along some spatial direction,  if you were a person at rest in the Black frame then there is no way you can point in some physical direction and say that is "along the time axis".   You can only point things out that are in space.  The time axis is "in time" and not in physical space.  If someone asks you to point in the direction of time then you might as well point to your watch and make some gestures with your hands and face as if you are playing a game of charades.   Now the diagram has just represented space and time on a flat 2-D surface and in that diagram, time has something we can call a direction and think of like a direction in space and it's what I am going to call "straight up".  Well to be precise the time axis is "straight up" in the rest frame of Black.
    Hopefully you can start to see where I'm going with this....   The rest frame of Blue has a very different orientation of the time axis,  it's the blue line which runs at a slope instead of being "straight up".   Like I said, it's not a real direction in physical space, it just looks like a direction when we represent it on the diagram.   If you were a person at rest in frame Blue then you still can't point your hand in the direction of that time axis.   The whole concept of "a direction" or orientation of a time axis is just telling you that the time axis in one frame of reference is actually a hybrid or combination of what was the time and space axis in another frame of reference.   (The diagram isn't something I made up off the top of my head, it's based on a conventional space-time diagram that's been used for years although it's simplified because I haven't needed to show you what happens to the x-axis for Blue).
   Now, without worrying too much about the details but just by looking at the diagram... can you see that if Billy Blue has all of his 4-velocity along the blue time axis, then that does not mean that he has all of his 4-velocity along the black axis?  Indeed, if something is "moving" (in the sense of having a 4-velocity) only along the blue axis then it is moving only in time when regarded with the blue axis.  However, it is moving partly through time but also partly through space when regarded with the black axis.   (You can make similar arguments for something moving only along the black time axis, that's only moving through time when regarded with the back axis but is moving partly through space when regarded with the blue time axis).
   Anyway, this is the sense in which two objects can be "moving" only through time in their own rest frame, without contradicting the idea that they do have some ordinary spatial velocity in the frame of the other.  The key really is that the two time axis do not have the same orientation.

   I hope that helps a bit.  Best Wishes.

Yes, that helps a lot.  But this all seems to be a logical nightmare. 

Going back to the question that I meant to ask in the OP, it seems as though time has more than one direction.  And I suppose that this is all explained by people who understand the mathematics of the Minkowski space better than me.  And I understand that that the Minkowski metric works wayyy differently than the Euclidean metric inside of a light cone, but I still wonder though, how can time have multiple/infinite axis and still be one dimension.  But I guess that is a different topic.

I guess for now I have to figure out how it is logically possible for one dimension to have multiple axis.

[evan_au]

Minkowski metric ... how can time have multiple/infinite axis and still be one dimension.

Sometimes, when light cones are illustrated near an event horizon, they show that inside the twisted spacetime of a black hole, the space dimension becomes more time-like, leading any object into a one-way trip to the singularity.

However, as a corollary of these simplified diagrams (illustrating 4D spacetime as a 3D graph on a 2D screen), it looks like there are now two dimensions of time.

Could this be the source of the confusion?

[Dimensional (OP)]

Minkowski metric ... how can time have multiple/infinite axis and still be one dimension.

Sometimes, when light cones are illustrated near an event horizon, they show that inside the twisted spacetime of a black hole, the space dimension becomes more time-like, leading any object into a one-way trip to the singularity.

However, as a corollary of these simplified diagrams (illustrating 4D spacetime as a 3D graph on a 2D screen), it looks like there are now two dimensions of time.

Could this be the source of the confusion?

Well, maybe.  But I do know that those diagrams are not a proper geometric model of what is actually happening in the Minkowski space. 

Having said that, there does seem to be multiple time dimensions, at least two.  I know this is wrong, but I do not know exactly why.