Understanding Time and Space as One entity.

[belliott4488]

Hi, Bill - I've been distracted for the past couple of days, so I haven't had a chance to respond to your last response to me. I was going to ... but then I saw this:

If Rachel were traveling along the x-axis of Bob's coordinate system, wouldn't she be traveling perpendicular to Bob's time?

I realize this is a nit-pick, but by what differential quantity does x change if time is perpendicular to it?

No, no, no! And again, I say, NO! This is NOT a nit-pick - it's the very crux of what we're discussing, so I'm deferring a response to your last response to me until we've straightened this out. (Note that the quotation above was a response to JesseM.)

We arbitrarily define the x-axis in Bob's frame to be in the direction of Rachel's motion, since we are free to do that. Therefore she travels along Bob's x-axis by definition - but this does NOT mean she is traveling perpendicular to Bob's time axis in space-time. The points on the x-axis all have the same time coordinate (t=0), so "motion" - i.e. a world-line - perpendicular to the t-axis is not possible - all space-time points on the x-axis are simultaneous. As Rachel moves along the direction of the x-axis (in 3-d space), she is at a succession of (x,t) points, where the ratio of the changes in each coordinate , dx/dt, is simply her velocity. That ratio also gives the slope of her path on the space-time diagram (inverse slope, actually, since slope would be dt/dx).

Bob is stationary in his frame, so his x-coordinates don't change, so his world line consists of the points along his time axis (this is true for all observers in their own rest frames). Rachel moves along a slanted path somewhere between vertical (v=0) and 45 degrees (v=c=1, since we've been implicitly assuming that we're using units where c=1).

It's critical that you see this. If you already do, then please excuse my pounding the table, but otherwise let's try to reach agreement on this before moving on.

 

[Antenna Guy]

Hi, Bill - I've been distracted for the past couple of days, so I haven't had a chance to respond to your last response to me. I was going to ... but then I saw this:

If Rachel were traveling along the x-axis of Bob's coordinate system, wouldn't she be traveling perpendicular to Bob's time?

I realize this is a nit-pick, but by what differential quantity does x change if time is perpendicular to it?

No, no, no! And again, I say, NO! This is NOT a nit-pick - it's the very crux of what we're discussing, so I'm deferring a response to your last response to me until we've straightened this out.

Well.. I didn't think "nit-pick" was quite the right term, but I see you understand what I was getting at.

We arbitrarily define the x-axis in Bob's frame to be in the direction of Rachel's motion, since we are free to do that.

Well, since there are three elements in Bob's frame (Rachel + two photons), am I not to assume that everything is traveling along Bob's "x" (where "x" is arbitrary)? If one of the two photon's were traveling along Bob's "y", would that change how it maps into Bob's "space" if all three start at O?

Therefore she travels along Bob's x-axis by definition - but this does NOT mean she is traveling perpendicular to Bob's time axis in space-time. The points on the x-axis all have the same time coordinate (t=0), so "motion" - i.e. a world-line - perpendicular to the t-axis is not possible - all space-time points on the x-axis are simultaneous.

This is where we diverge. Rachel's time is not equivalent to Bob's time, and the difference between the two is the mechanism whereby Rachel becomes displaced (in the direction of "space") in Bob's frame. If you have some other rationalization of this, I'm all ears.

As Rachel moves along the direction of the x-axis (in 3-d space), she is at a succession of (x,t) points, where the ratio of the changes in each coordinate , dx/dt, is simply her velocity.

If I were to generalize velocity to dr/dt (and refer to r as "space"), which direction must time go for me to get the correct dt to go with my dr?

That ratio also gives the slope of her path on the space-time diagram (inverse slope, actually, since slope would be dt/dx).Bob is stationary in his frame, so his x-coordinates don't change, so his world line consists of the points along his time axis (this is true for all observers in their own rest frames).

Since we still seem to be operating on different wavelengths, let me draw the diagram a different way. We have an x-axis that Bob, Rachel and the two photons might travel along, and all four start at O at t=0 - proceding at constant velocity. Each velocity vector has a (positive-definite) magnitude in either the +x or -x direction. At any time t the velocity vectors have the same magnitude they had at t=0, so I can simply lay them on the x-axis in the appropriate direction to show how they relate at any given time. Let's say that I'd like to see how those velocity vectors relate to x at t=1. For Rachel and one of the photons, t=1 is in the direction of +x, and the other photon observes time to procede in the direction of -x. Thus, from a (positive-definite) velocity times time equals distance standpoint, the diagram contains two different senses of time relative to O (clarification: recall that I define v=dx/dt).

Like you, my available time has been somewhat limited lately. Compounding this is the fact that I'll be traveling over the next week - so please be patient waiting for further replies from me. I'm curious read what you had going before I got you side-tracked.

Regards,

Bill

 

[belliott4488]

Well, since there are three elements in Bob's frame (Rachel + two photons), am I not to assume that everything is traveling along Bob's "x" (where "x" is arbitrary)? If one of the two photon's were traveling along Bob's "y", would that change how it maps into Bob's "space" if all three start at O?

I'm not sure why you're introducing photons (the two lines on the diagram are usually there as references to define Bob's future space cone), but it's no problem to include them.

To answer your question, if one photon travels in the x direction and the other in the y direction, then we need a 3-d space-time diagram, i.e. spanned by t, x, and y. Rachel and one photon have world lines in the x-t plane, and the other photon has its world line in the y-t plane. The two photon lines are at 45 deg. to the t axis, and Rachel's line is steeper.

Therefore she travels along Bob's x-axis by definition - but this does NOT mean she is traveling perpendicular to Bob's time axis in space-time. The points on the x-axis all have the same time coordinate (t=0), so "motion" - i.e. a world-line - perpendicular to the t-axis is not possible - all space-time points on the x-axis are simultaneous.

This is where we diverge. Rachel's time is not equivalent to Bob's time, and the difference between the two is the mechanism whereby Rachel becomes displaced (in the direction of "space") in Bob's frame. If you have some other rationalization of this, I'm all ears.

We've said nothing about Rachel's time or any other observations she makes, so let's leave them out for now; we're just concerned with Bob's observations and how we can deduce them from the space-time diagram.

As for the "mechanism whereby Rachel becomes displaced," I'd say that mechanism is simply her motion. She is moving, i.e. her x-coordinate in Bob's frame is changing as a function of time, so therefore she is "displaced" over time. I don't know why you bring in the difference between Rachel's and Bob's frames of reference; they're certainly different, but that is a result of her motion, not the cause of it.

If I were to generalize velocity to dr/dt (and refer to r as "space"), which direction must time go for me to get the correct dt to go with my dr?

I don't think I understand this r variable. What do you mean r is "space"? x is the variable the defines Rachel's position in Bob's frame, and it does so unambiguously - why introduce anything else? The only meaning I can see for r is that it is the magnitude of x (or the absolute value of x if we're using just one space dimension), but why introduce it? You throw away the sign of x, and thus introduce an unnecessary ambiguity in your knowledge of Rachel's position. In any case, however, since we do know Rachel's position, we can still discuss the r variable. Since we picked the +x direction to be in the direction of her motion, however, then her x positions after t=0 will all be positive (assuming she starts at O at t=0), so r is simply equal to x.

As for the direction of t, +t is in the direction of the future, by almost universal convention. Why change that? Once you've picked your directions for +x and +t, you have labeled all space-time points unambiguously, and you can trace Rachel's history and future (assuming she continues her inertial motion) in an equally unambiguous way.

Since we still seem to be operating on different wavelengths, let me draw the diagram a different way. We have an x-axis that Bob, Rachel and the two photons might travel along, and all four start at O at t=0 - proceding at constant velocity. Each velocity vector has a (positive-definite) magnitude in either the +x or -x direction. At any time t the velocity vectors have the same magnitude they had at t=0, so I can simply lay them on the x-axis in the appropriate direction to show how they relate at any given time. Let's say that I'd like to see how those velocity vectors relate to x at t=1.

When you talk about velocity vectors, are you talking about three-vectors? It sounds like it. Keep in mind, though, that plotting a velocity vector against axes that have units of length is kind of arbitrary, since you have to specify how you're drawing the velocity units relative to the length units. Generally you'd draw velocity vectors in velocity space, where the axis units are units of, well ... velocity. We can do so, and align the velocity axes with the space axes, which is natural, but understand that we're overlaying two different spaces: one where vectors between points correspond to displacements, and one where vectors between points correspond to velocities.

For Rachel and one of the photons, t=1 is in the direction of +x, and the other photon observes time to procede in the direction of -x.

What do you mean "t=1 is in the direction of +x"? Are you going to a space-time diagram now? If so, then there is no point "t=1"; there are an infinite number of points with t-components equal to 1: they form a line parallel to the x-axis and correspond to all the simultaneous events in Bob's frame that occur at t=1.

Now, if what you meant is the direction of the three-velocity vectors at time t=1, well - why would they be any different than they were at any other time? There's no acceleration here, right? so no one's velocity is changing. You defined two of them to point in the +x direction and one to point in the -x direction, so they still do.

Thus, from a (positive-definite) velocity times time equals distance standpoint, the diagram contains two different senses of time relative to O (clarification: recall that I define v=dx/dt).

Okay - (positive-definite) velocity magnitude times time does give you distance, i.e. the magnitude of the position vector, but again, why are you limiting yourself to this when we have all the information we need to determine position unambiguously?

Rachel moves in the +x direction at velocity +v, so her position at any time t is simply x = vt. Similarly, photon 1 (the one that moves in the same direction as Rachel) moves at velocity +c, so its position is given by x=ct. Photon 2 has velocity -c, so its position is x = -ct.

This is easy to draw in the space-time diagram as two 45 degree lines from O for the two photons - one to the left and one to the right, and a steeper slanted line going up to the right (positive slope) for Rachel. (Again, the photon lines are at 45 degrees because we are using units where c=1.)

So far, there's really no relativity in this at all - it's all just basic Physics. The relativity will come in when we talk about how Rachel sees all this - but in Bob's frame there's nothing unusual at all.