Four-Vector Problem: What Am I Missing?

[Wox]

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
$$\bar{v}(\tau)=\frac{d\bar{w}}{d\tau}=\frac{d\bar{w}}{dt}\frac{dt}{d\tau}=\frac{\bar{v}(t)}{\left\| \bar{v}(t)\right\|}=\frac{(c,\bar{u}(t))}{\frac{c}{\gamma}}=\gamma(1,\frac{\bar{u}(t)}{c})$$
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?

 

[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).

 

[PAllen]

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.

 

[PAllen]

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.

 

[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?

 

[PAllen]

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.

 

[Matterwave]

Are you talking about adding a factor of c into the metric? o.o

 

[PAllen]

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.

 

[Wox]

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?

 

[PAllen]

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.

 

[Wox]

Thanks, you've been a great help!

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)?

 

[PAllen]

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.

 

[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).

 

[Chestermiller]

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.

 

[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.]

 

[Matterwave]

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.

 

[Wox]

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.

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?

 

[PAllen]

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?

Sure there is a dot product. It is defined by the metric. For example, if the metric is diag(1,-1/c^2, -1/c^2,-1/c^2), then the dot product of X and Y is:

x0*y0 - x1*y1/c^2 - x2*y2/c^2 - x3*y3/c^2

 

[Wox]

Depends on your definition of the dot product, but I see what you mean. But I don't see why [Ux/Ut,Uy/Ut,Uz/Ut] would correspond to a spatial velocity.

 

[PAllen]

Depends on your definition of the dot product, but I see what you mean. But I don't see why [Ux/Ut,Uy/Ut,Uz/Ut] would correspond to a spatial velocity.

In a metric space there is one definition of the dot product. The Euclidean one looks the way it does solely because the Euclidean metric is diag(1,1,1).

If U is some tangent vector, and x,y,z,t are unit vectors for some frame, then the dot product of U with such unit vectors expresses U in that frame basis. Then Ux/Ut gives the x speed (well, actually, x-speed/c , but that is just as good). Look at the tangent vector itself expressed in your starting coordinates (c-normed, canonic metric; works the same in any other convention):

U = gamma(c,u)

Ux = gamma * ux is not the x speed; but note Ux/Ut = ux/c. This feature will be true in any other basis. In particular, in an orthonormal basis with U itself taken as the time unit vector, you get spatial speed of zero - the particle has no spatial speed in its own basis.