# Four-vector problem

by Wox
Tags: four-velocity, special relativity
 P: 71 The four-velocity as defined for example here, is given by $$U=\gamma(c,\bar{u})$$ but I get $$U=\gamma(1,\frac{\bar{u}}{c})$$ Consider the timelike curve $\bar{w}(t)=(ct,\bar{x}(t))$ with velocity $\bar{v}(t)=(c,\bar{x}'(t))\equiv (c,\bar{u}(t))$ and the arc-length (proper time) $$\tau\colon I\subset \mathbb{R}\to \mathbb{R}\colon t\mapsto \int_{t_{0}}^{t}\left\| \bar{v}(k)\right\|dk$$ for which (by First Fundamental Theorem of Calculus (1), the Minkowskian inner product (2) and the definition of the Lorentz factor (3) ) $$\Leftrightarrow \frac{d\tau}{dt}=\left\| \bar{v}(t)\right\|=\sqrt{c^{2}-\bar{u}^{2}(t)}\equiv \frac{c}{\gamma}$$ then the velocity of the curve after arc-length (proper time) parameterization, is given by $$\bar{v}(\tau)=\frac{d\bar{w}}{d\tau}=\frac{d\bar{w}}{dt}\frac{dt}{d\tau }=\frac{\bar{v}(t)}{\left\| \bar{v}(t)\right\|}=\frac{(c,\bar{u}(t))}{\frac{c}{\gamma}}=\gamma(1,\f rac{\bar{u}(t)}{c})$$ I would think that my $\bar{v}(\tau)$ is the four-velocity but in fact $\bar{v}(\tau)=\frac{U}{c}$ where U the four-velocity as defined in textbooks. What am I missing?
 P: 1,937 If you are using units where c is not 1, then certainly you want the 4-velocity to be normalized to c (or -c depending on signature of the metric), and not 1 which is not in units of velocity (if, again, c is not set to 1).
 PF Patron Sci Advisor P: 4,473 I think it is purely a convention. I learned that U is meant to be a unit vector, always, even when c is not taken to be 1. Some people like norm of U = c, some like 1. Norm of one amounts to units of time rather than distance for positions. However, since you start with 4-position in units of distance, you should get a U whose norm is c. The flaw is your d tau/ dt computation. It is 1/gamma not c/gamma. Then you get U with norm of c. Specifically, your integral formula is not right. Given your units for v, the integral is c * tau, not tau. This, then, is the initial (and only) error, from which all else follows.
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P: 4,473

## Four-vector problem

 Quote by Matterwave If you are using units where c is not 1, then certainly you want the 4-velocity to be normalized to c (or -c depending on signature of the metric), and not 1 which is not in units of velocity (if, again, c is not set to 1).
I'll add one more thing. The idea of norming U to c has led to Brian Greene's "speed through space-time equals c", which has led to numerous confusions and debates on these forums, as well as facilitating cranks. The convention of Einstein and Bergmann that U has norm 1, irrespective of the value of c sidesteps all of this lunacy.
 P: 1,937 If you parameterize your curve with the proper time in seconds, and your proper distances are measured in meters, don't you necessarily get the norm condition in U to be c? What I mean is, if you use units of distance the same as your units of time, aren't you necessarily setting c=1?
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 Quote by Matterwave If you parameterize your curve with the proper time in seconds, and your proper distances are measured in meters, don't you necessarily get the norm condition in U to be c? What I mean is, if you use units of distance the same as your units of time, aren't you necessarily setting c=1?
No. The conventions:

position: (t, x/c, y./c, z/c)
line element: d tau^2 = d t^2 - (dx^2 + dy^2 + dz^2)/c^2
and the consequence that U = d(position) / d tau has norm 1

in no way have c=1.
 P: 1,937 Are you talking about adding a factor of c into the metric? o.o
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 Quote by Matterwave Are you talking about adding a factor of c into the metric? o.o
Don't know what you are asking. The two common conventions for the metric are:

ds^2 = dx^2 + dy^2 + dz^2 - c^2 t^2

and

d tau^2 = dt^2 - dx^2/c^2 - dy^2/c^2 - dz^2/c^2

I have always used the latter.
P: 71
 Quote by PAllen Given your units for v, the integral is c * tau, not tau.
Ok, so because $\bar{w}(t)=(ct,\bar{x}(t))$ is in space units, $\left\| \bar{v}(t)\right\|$ has units space/time and the integral is in space units. Then the arc-length (proper time) parameterization in time units is given by
$$\tau\colon I\subset \mathbb{R}\to \mathbb{R}\colon t\mapsto \frac{1}{c}\int_{t_{0}}^{t}\left\| \bar{v}(k)\right\|dk$$
and the four-velocity in space/time units
$$\bar{v}(\tau)=\gamma(c,\bar{u}(t))$$
Is this the correct explanation? As I understand, the four-velocity in the other convention (norm=1) is unitless, isn't it? So how is it used then?
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P: 4,473
 Quote by Wox Ok, so because $\bar{w}(t)=(ct,\bar{x}(t))$ is in space units, $\left\| \bar{v}(t)\right\|$ has units space/time and the integral is in space units. Then the arc-length (proper time) parameterization in time units is given by $$\tau\colon I\subset \mathbb{R}\to \mathbb{R}\colon t\mapsto \frac{1}{c}\int_{t_{0}}^{t}\left\| \bar{v}(k)\right\|dk$$ and the four-velocity in space/time units $$\bar{v}(\tau)=\gamma(c,\bar{u}(t))$$ Is this the correct explanation? As I understand, the four-velocity in the other convention (norm=1) is unitless, isn't it? So how is it used then?
Yes, this is correct.

The magnitude of U really has no real meaning in either convention. All information about measured velocity in any basis (frame) is contained in the direction of U as a tangent vector. The norm 1 convention makes this explicit: it is literally a unit tangent vector to a world line.

There are pros and cons to either convention.
P: 71
Thanks, you've been a great help!

 Quote by PAllen The magnitude of U really has no real meaning in either convention.
And what about the magnitude of space-component $\gamma\bar{u}(t)$ of the four-velocity? I'm not quite sure how all this relates to some physical reality. Can I interpret the space-component of the four-velocity as the classical velocity (at least when not choosing the norm=1 convention)?
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 Quote by Wox Thanks, you've been a great help! And what about the magnitude of space-component $\gamma\bar{u}(t)$ of the four-velocity? I'm not quite sure how all this relates to some physical reality. Can I interpret the space-component of the four-velocity as the classical velocity (at least when not choosing the norm=1 convention)?
Let's say you have U as a tangent vector, considered a coordinate independent quantity. You want to know the spatial velocity measured in some frame defined by a 4 orthonormal unit vectors, one timelike the others spacelike. You take U dot <x unit vector>/ U dot <t unit vector>, same for y and z. Clearly, the norm of U drops out.

In the coordinates you initially used to express U, the spatial velocity is just the u you started with (which you would get by executing the procedure above). The quantity gamma*u would be rather meaningless: the rate of change of distance in a given frame by a particle's proper time. This could exceed c by a large factor.
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 Quote by PAllen No. The conventions: position: (t, x/c, y./c, z/c) line element: d tau^2 = d t^2 - (dx^2 + dy^2 + dz^2)/c^2 and the consequence that U = d(position) / d tau has norm 1 in no way have c=1.
Error here. In this convention you simply have position: (t,x,y,z)

Then U, with norm 1, becomes: gamma * (1,u)

This is not necessarily taking c=1, because the covariant metric diagonal is still (1, -1/c^2, -1/c^2,-1/c^2). Contravariant diagonal obviously (1,-c^2,-c^2,-c^2).
 PF Patron Thanks P: 2,974 [QUOTE=Wox;3804757]The four-velocity as defined for example here, is given by $$U=\gamma(c,\bar{u})$$ but I get [tex] U=\gamma(1,\frac{\bar{u}}{c}) I have a simpler answer for you. Some people use the first version (and your derivations of this version are correct), and some people use the second version, which is what I call the dimensionless four velocity. Both versions are OK to use, provided you tell which one you are using in advance. I think that most physicists prefer using the dimensionless version (see e.g., MTW), although I personally prefer the dimensional version. When you use the dimensionless version, the 4 velocity is equal to a unit vector in the time direction of the object's rest frame. When you use the dimensional version, the 4 velocity is c times a unit vector in the time direction of the object's rest frame. I hope this is helpful.
 PF Patron Sci Advisor P: 4,473 Probably no one interested anymore, but I have clarified a few things in my own mind. Given the desire to express a 4 - tangent vector to a world line in terms of u = (dx/dt, dy/dt, dz/dt) with conventional meanings, two separate conventions affect the form it takes: - how you norm it - is your metric canonic (all +1,-1) or not (you have c^2 or 1/c^2 in your metric). The signature of the metric is irrelevant for this situation. With canonic metric, you have a factor of c in your tangent vector components, so you have two natural choices: gamma * (c, u) // norm c; dimensions distance/time; from d (ct,x,y,z) / d tau gamma * (1, u/c) // norm 1; dimensionless; from d (t, x/c, y/c, z/c) / d tau With non-canonic metric, you have only one natural form, with norm 1: gamma * (1, u) // mixed dimensions, as is characteristic of non-canonic metric // from d (t,x,y,z) / d tau With non-canonic metric, a form with norm c is simply unnatural. It happens that I learned SR with non-canonic metric, 4-velocity being a unit vector, and the concept of 'speed through spacetime' not remotely meaningful. It appears that almost all modern books use canonic metric. [EDIT: For emphasis, note something I derived in an earlier post: the norm of c or 1 plays no role at all in computing any observable. You could even normalize to 42 and it would make no difference. Only the direction in 4-space of the tangent vector plays any role in computing observables.]
P: 1,937
 Quote by PAllen Probably no one interested anymore, but I have clarified a few things in my own mind. Given the desire to express a 4 - tangent vector to a world line in terms of u = (dx/dt, dy/dt, dz/dt) with conventional meanings, two separate conventions affect the form it takes: - how you norm it - is your metric canonic (all +1,-1) or not (you have c^2 or 1/c^2 in your metric). The signature of the metric is irrelevant for this situation. With canonic metric, you have a factor of c in your tangent vector components, so you have two natural choices: gamma * (c, u) // norm c; dimensions distance/time; from d (ct,x,y,z) / d tau gamma * (1, u/c) // norm 1; dimensionless; from d (t, x/c, y/c, z/c) / d tau With non-canonic metric, you have only one natural form, with norm 1: gamma * (1, u) // mixed dimensions, as is characteristic of non-canonic metric // from d (t,x,y,z) / d tau With non-canonic metric, a form with norm c is simply unnatural. It happens that I learned SR with non-canonic metric, 4-velocity being a unit vector, and the concept of 'speed through spacetime' not remotely meaningful. It appears that almost all modern books use canonic metric. [EDIT: For emphasis, note something I derived in an earlier post: the norm of c or 1 plays no role at all in computing any observable. You could even normalize to 42 and it would make no difference. Only the direction in 4-space of the tangent vector plays any role in computing observables.]
Yes, this what I meant when I said "are you talking about adding a factor of c to the metric?" I always learned to us diag(-1,1,1,1) for my metric so any factors of c's I need are in the 4-vectors themselves.
P: 71
 Quote by PAllen You want to know the spatial velocity measured in some frame defined by a 4 orthonormal unit vectors, one timelike the others spacelike. You take U dot / U dot , same for y and z. Clearly, the norm of U drops out.
Not sure what you mean: there is no dot product in Minkowskian space-time... Do you mean that [Ux/Ut,Uy/Ut,Uz/Ut] is the spatial 3-velocity? Why?
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