Relativistic velocity terminology

[md65536]

Suppose we're considering the velocity of a point P relative to an observer O.

 

P's velocity can be expressed as a change in the distance to P as measured by O, divided by the change in time as measured by O.

It can also be expressed as a change in the distance measured by P, divided by the change in time of P.

These two velocities are the same value; the speed that O measures P approaching is the same speed that P measures O approaching.

 

If we're talking about everything from O's perspective, do these 2 velocities have different names? That is "change in locally defined distance over local time" vs "change in remotely defined distance over remotely defined time"?

 

 

 

There is also the idea of dividing O's distance by the change in P's clock. Does this value have a name?

It usually comes up when one mixes frames, or tries to calculate P's velocity in terms of rest distance instead of relativistic distance.

"rest velocity" comes to mind but that term is obviously nonsensical. Does this "invalid velocity" value have a practical application other than in mistakes? I'd like to refer to it as a useful value.

 

Thanks.

[Iggy]

Suppose we're considering the velocity of a point P relative to an observer O.

 

P's velocity can be expressed as a change in the distance to P as measured by O, divided by the change in time as measured by O.

It can also be expressed as a change in the distance measured by P, divided by the change in time of P.

These two velocities are the same value; the speed that O measures P approaching is the same speed that P measures O approaching.

 

I agree, in particular with the part I bolded. I have heard it called the principle of reciprocity. If two things are moving relative to one another with constant velocity then v' = -v. Each calculates the velocity of the other equally.

 

EDIT... Yeah, here is the: reciprocity principle ...EDIT

 

 

If we're talking about everything from O's perspective, do these 2 velocities have different names? That is "change in locally defined distance over local time" vs "change in remotely defined distance over remotely defined time"?

 

How do you mean "the two velocities"? If everything is from O's perspective then the velocity of O is zero and the velocity of P is some value [latex]v[/latex].

 

Then, by the principle of reciprocity, from P's perspective O would have the velocity [latex]-v[/latex].

 

There is also the idea of dividing O's distance by the change in P's clock. Does this value have a name?

It usually comes up when one mixes frames, or tries to calculate P's velocity in terms of rest distance instead of relativistic distance.

"rest velocity" comes to mind but that term is obviously nonsensical. Does this "invalid velocity" value have a practical application other than in mistakes? I'd like to refer to it as a useful value.

 

I don't know of any name. I agree it would be mixing frames, and it could have values greater than c so probably is avoided.

Edited September 20, 2011 by Iggy

[md65536]

Thanks for the link. I'll have to read more of it.

How do you mean "the two velocities"? If everything is from O's perspective then the velocity of O is zero and the velocity of P is some value [latex]v[/latex].

The two velocities would be P's change in O's measurement of distance divided by O's delta time, and P's change in P's measurement of distance divided by P's delta time.

I guess that's the same as "velocity measured in O's frame" and "velocity measured in P's frame".

I guess that any observer would agree on what P measures...

and trying to express this from O's perspective is an unnecessary complication?

 

I don't know of any name. I agree it would be mixing frames, and it could have values greater than c so probably is avoided.

Good point. It's not quite a velocity at all. If it has no name I'll just refer to its description; it represents the change in rest distance per unit of relativistic time. But this isn't a robust definition so I'll have to be careful!

 

 

[Iggy]

Thanks for the link. I'll have to read more of it.

The two velocities would be P's change in O's measurement of distance divided by O's delta time, and P's change in P's measurement of distance divided by P's delta time.

I guess that's the same as "velocity measured in O's frame" and "velocity measured in P's frame".

I guess that any observer would agree on what P measures...

and trying to express this from O's perspective is an unnecessary complication?

Yeah, I think that makes sense.

[Schrödinger's hat]

Thanks for the link. I'll have to read more of it.

The two velocities would be P's change in O's measurement of distance divided by O's delta time, and P's change in P's measurement of distance divided by P's delta time.

I guess that's the same as "velocity measured in O's frame" and "velocity measured in P's frame".

I guess that any observer would agree on what P measures...

and trying to express this from O's perspective is an unnecessary complication?

 

Good point. It's not quite a velocity at all. If it has no name I'll just refer to its description; it represents the change in rest distance per unit of relativistic time. But this isn't a robust definition so I'll have to be careful!

 

I'm not sure if this comes up often enough to have a succinct name.

If I wanted to refer to it succinctly I'd just use the mathematical terminology:

Things like [math] \frac{dx_P}{dt_P'}[/math] or [math]\frac{dx_P'}{dt_P}[/math]

Define 'the primed frame is the frame in which O is at rest, the non-primed frame is the frame in which P is at rest' etc, then read

Read "the change in the three-position of P in the non-primed frame with respect to the change in the time coordinate of P in the primed frame" etc.

 

You could also use proper times:

[math]\frac{dx_P}{d\tau_O}[/math]

These are all clear, concise and unambiguous, but one would have to be careful about transforming these between frames. (proper times are often better in this regard, as [math]x_a(\tau_{b1}) + \frac{dx_a}{d\tau_b}\Delta \tau_b[/math] will alwaysrepresent a set event.

 

 

Generally the easiest quantity to work with is a four vector.

Ie. A four-displacement from an origin: [math]X_P[/math], with coordinates ct,x,y,z

Then you can define four velocity as either [math]\frac{dX_P}{dt_P}[/math]

This needs some care, because dt is frame dependant. So another definition of four-velocity is often used:

[math]\frac{d X_P}{d\tau_P}[/math], which is the change in the four-displacement for each tick of a clock on P.

 

You can transform this vector exactly as you do X, so it's easier to think about, too (no memorizing a separate velocity addition formula).

Edited September 21, 2011 by Schrödinger's hat

[DrRocket]

 

Generally the easiest quantity to work with is a four vector.

Ie. A four-displacement from an origin: [math]X_P[/math], with coordinates ct,x,y,z

Then you can define four velocity as either [math]\frac{dX_P}{dt_P}[/math]

This needs some care, because dt is frame dependant. So another definition of four-velocity is often used:

[math]\frac{d X_P}{d\tau_P}[/math], which is the change in the four-displacement for each tick of a clock on P.

 

4-velocity is particularly simple, since the speed of anything, not just light, is c.

[Schrödinger's hat]

4-velocity is particularly simple, since the speed of anything, not just light, is c.

Just to clarify, DrRocket is using [math]\frac{dX}{dt}[/math] (where X is the four-position). I have seen both referred to as four-velocity, but I believe this is the more common. Edit: Wrong, see DrRocket's comment and my response

The other thing that's useful with [math]\frac{dX}{dt}[/math] is it's defined for light (where [math]\frac{dX}{d\tau}[/math] is not),

but my personal preference is for [math]\frac{dX}{d\tau}[/math]. It seems neater not to have factors of gamma floating about, and you don't have to re-normalize after Lorentz transforms.

I also like [math]P=m\frac{dX}{d\tau}[/math] rather than [math]P=m\gamma\frac{dX}{dt}[/math].

The disadvantage is the magnitude is [math]\gamma c[/math]

 

I find it also helps with my intuitive picture of Minkowski space.

I think of [math]\tau[/math] as the parameter which takes an object through spacetime along its worldline.

[math]\frac{dX}{d\tau}[/math] can then be thought of as analogous to a velocity* through spacetime according to this parameter.

 

 

 

*in this interpretation, velocity isn't a perfectly accurate concept because spacetime does not evolve, nor do the worldlines of objects in it.

Edited September 22, 2011 by Schrödinger's hat

[DrRocket]

 

*in this interpretation, velocity isn't a perfectly accurate concept because spacetime does not evolve, nor do the worldlines of objects in it.

 

Sure it is.

 

A worldline is just a curve in spacetime. The velocity vector at a point on a parameterized curve is a perfectly well-defined object.

 

If you parameterize a world line by arc length divided by c then the tangent vector is just velocity with respect to proper time. The magnitude is always c.

[Schrödinger's hat]

Sure it is.

 

A worldline is just a curve in spacetime. The velocity vector at a point on a parameterized curve is a perfectly well-defined object.

 

If you parameterize a world line by arc length divided by c then the tangent vector is just velocity with respect to proper time. The magnitude is always c.

Oops, had a bit of a brain failure there. Apologies.

 

 

Ignore this:

Just to clarify, DrRocket is using [math]\frac{dX}{dt}[/math] (where X is the four-position). I have seen both referred to as four-velocity, but I believe this is the more common.

 

And this one:

The disadvantage is the magnitude is [math]\gamma c[/math]

(the magnitude is c as DrRocket said. I had my factor of gamma wrong, the magnitude of [math]\frac{dX}{dt}[/math] is [math]\frac{c}{\gamma}[/math]).

 

 

That said, I think my comment about the concept of velocity not fitting exactly still stands. Mathematically it is the logical extension of the concept, and fits exactly, but when most people think about things from a three dimensional perspective, velocity usually implies evolution -- ie. a three dimensional object moving to a different location. Thinking about things in these terms can lead to confusion when dealing with objects as world-lines.

Edited September 22, 2011 by Schrödinger's hat

[csmyth3025]

Sure it is.

 

A worldline is just a curve in spacetime. The velocity vector at a point on a parameterized curve is a perfectly well-defined object.

 

If you parameterize a world line by arc length divided by c then the tangent vector is just velocity with respect to proper time. The magnitude is always c.

(bold added by me)

 

-and-

 

4-velocity is particularly simple, since the speed of anything, not just light, is c.

These concepts are really hard for me. Do I understand correctly that the velocity of any object (or massless particle) along its spacetime world line always equals 299,792,458 km/s? (as measured by an observer on the object using his own "ruler" and his own clock - i.e., his own proper time)

 

I think I must be missing something fundamental here.

 

Chris

 

Edited to add parenthetical clause to question

Edited September 22, 2011 by csmyth3025

[IM Egdall]

(bold added by me)

 

-and-

 

 

These concepts are really hard for me. Do I understand correctly that the velocity of any object (or massless particle) along its spacetime world line always equals 299,792,458 km/s? (as measured by an observer on the object using his own "ruler" and his own clock - i.e., his own proper time)

 

I think I must be missing something fundamental here.

 

Chris

 

Edited to add parenthetical clause to question

 

This is talking not about velocity through space but velocity through spacetime. This is called the 4-velocity because it is the velocity over three space and one time dimension. When you calculate this velocity, you find it always equals the speed of light c or 299,792,458 km/s.

 

Motion through time only: Say you are sitting in a chair. You are at rest with respect to that chair. But time is going by. So in the chair reference frame, you are moving only through time. And in this frame of reference, time is going by as fast at it can. Why? There is no time dilation when you are at rest.

 

Motion through time and space: When you move with respect to that chair, you are now moving though time and space. Time is going by and you are moving though space. But there is time dilation now. In fact, the faster you move though space, the slower your time runs (with respect to time in the chair reference frame). So time and space kind of balance out. The more through space the less though time. The net result is the same velocity though spacetime.

 

Motion though space only: Now light travels at the speed of light c with respect to that chair reference frame. And at c, time dilation is maxed out. That is motion though time is zero. So a photon of light travels through space only. At what speed does a photon travel through space? At speed c.

 

So actually, the whole thing balances for motion through spacetime. No motion through space means all motion through time (no time dilation). Some motion through space means less motion through time ( time dilation). All motion through space means no motion through time (max. time dilation). Since the space and time velocities balance out, they are all the same velocity through spacetime.

 

And the velocity for motion though space only is the speed of light c. Therefore it is the same speed of light c for all motions through spacetime.

 

So when you are sitting in that chair, with respect to the chair, you are going at the speed of light (though spacetime)!

 

I think this is not easy to absorb at first, but I found that thinking about it over time, it helped me understand why all motion through spacetime is at the speed of light.

 

(Edited for spelling)

Edited September 22, 2011 by IM Egdall

[csmyth3025]

Thanks for that explanation. I think I actually understand it. This should make it a bit easier for me to understand some of the other concepts in relativity theory that confuse me.

 

Chris