Velocity has units of frequency?

[decraig]

In the common representation, the mathematical structure of general relativity builds upon vectors. Specifically, Riemann geometry incorporates the notion of tangent vectors. (See "vector space".)

 

The four-velocity generalizes three dimensional velocity, v to include time:

 

[math]V = (\gamma c, \gamma v)[/math]

 

To be useful in curved spacetime, we need basis vectors. Velocity is represented as a contravariant vector with the basis normalized to distance.

 

It has dimensions of frequency[1/T]

 

D: distance

T: time

 

The covariant velocity has dimensions of [D^2/T].

 

(I'd put it all in LaTex, but the parser on this forum seems to be broken.)

 

Is anyone else uncomfortable with the units? Or perhaps thinks it could be done better?

Edited December 20, 2013 by decraig

[ajb]

In some chosen local coordinates, 4-velocity is the rate of change of "4-position" with respect to the proper time along the curve. You scale the time by a factor of c so our "4-position" has units of length. Proper time has units of time and so the 4-velocity has units of length/time.

 

The norm of the 4-velocity is c.

[decraig]

The elements of the 4-velocity have dimension [D/T].

 

The vector itself with bases included has dimension [1/T] in the usual representation.

 

But if we use a metric tau^2, then the vector has units [D/T^2].

[ajb]

 

Okay, we have something like [math]C'(\tau) = V(C(\tau))[/math], that is we define a vector field in terms of the integral curves. This gives us the units length per time (with the c's in the right places).

 

But I now see what you are saying...

 

I think you are right the vector has units of 1/time. As a differential operator that makes sense.

 

I don't think this is a problem. In a given coordinate system the velocity vector action on the coordinate functions by replacing them with a velocity. The units make sense.

[decraig]

 

I'm not a mathematician. Could you define your terms and operators,please?

Edited December 22, 2013 by decraig

[ajb]

C here is a curve and \tau the affine parameter describing that curve. This is a standard definition of a vector using a curve.

 

EDIT: Let me be a little more careful here. A curve does not carry any dimensions and so the expression I gave you earlier has units of 1/time (taking time as the affine parameter).

 

Locally we can always write this as (using standard abuses of notation)

 

[math]\frac{d C\circ x^{\mu}}{d\tau}(\tau) = \frac{d x^{\mu}(\tau)}{d \tau}[/math]

 

which now gives us the units of length/time for the components.

Edited December 22, 2013 by ajb

[decraig]

ok, ok. My oversight. I couldn't figure out your use of the prime. Not a change of coordinates but a schematic derivative.

 

In any case, It makes mathematical sense, of course. But can you describe the physical frequency?

Edited December 29, 2013 by decraig

[Colic]

ok, ok. My oversight. I couldn't figure out your use of the prime. Not a change of coordinates but a schematic derivative.

 

In any case, It makes mathematical sense, of course. But can you describe the physical frequency?

It means time is measured in meters, same as any other dimension.

Edited December 29, 2013 by Colic

[ajb]

In any case, It makes mathematical sense, of course. But can you describe the physical frequency?

I don't think it has any obvious physical interpretation. You pick coordinates and from there you have an interpretation. This is a generic feature of GR, it is often not easy to have physical interpretations of things we are used to thinking in flat space-time with lots of symmetries.

[decraig]

I've been looking for a generally covariant representation of the 4-velocity as a k-form of the velocity, that when self-integrated over k dimensions results in a velocity valued pseudo scalar that is, at the least, Lorentz invariant.

 

This would be a velocity analog of the 3-form charge-current density.

Edited January 4, 2014 by decraig