Why can we normalize the tangent vector product to ±1 by affine reparameterization?

[binbagsss]

If $\sigma$ is an affine paramter, then the only freedom of choice we have to specify another affine parameter is $a\sigma+b$, a,b constants. [1]

For the tangent vector, $\xi^{a}=dx^{a}/du$, along some curve parameterized by $u$

My book says that ' if $\xi^{a}\xi_{a}\neq 0$, then by suitable affine parameterization we can arrange such that $\xi^{a}\xi_{a}=\pm1$,

Question:

What does it mean by some suitable affine parameterization? so say if $\sigma$ is a affine parameter and we do not have $\xi^{a}\xi_{a}=\pm1$, is it saying that we can use [1] and carefully choose $a$ and $b$ such that this is the case?

I've often seen proper time used such that $\xi^{a}\xi_{a}=\pm1$ is the case.
Why is this?

Or Is this part of the definition of proper time, are there any other 'known' parameters for which $\xi^{a}\xi_{a}=\pm1$ or is the affine parameter for which this holds unique?

Thanks in advance.

 

[Orodruin]

Yes, the point is that if $\xi^a \xi_b \neq 0$, then you can pick $a$ such that $\xi^a \xi_a = \pm 1$ ($b$ does not really enter into it, it is just a translation along the curve). The point is that this parametrises the curve using the curve length as parameter. In Minkowski space, for time-like curves, this means parametrising the curve with the proper time.

 

[binbagsss]

Yes, the point is that if $\xi^a \xi_b \neq 0$, then you can pick $a$ such that $\xi^a \xi_a = \pm 1$ ($b$ does not really enter into it, it is just a translation along the curve). The point is that this parametrises the curve using the curve length as parameter. In Minkowski space, for time-like curves, this means parametrising the curve with the proper time.

Ok. So if i use an affine parameter $\tau+b$ I Still achieve $\xi^a \xi_a =\pm 1$?

For a null geodesic we can't use $\tau$ as it is always zero.
However am I correct in thinking that the paramter $\tau+b$ would be plausible?

Thanks.

 

[Orodruin]

Ok. So if i use an affine parameter $\tau+b$ I Still achieve $\xi^a \xi_a =\pm 1$?

For a null geodesic we can't use $\tau$ as it is always zero.

Yes, if you have $\tau' = \tau + b$, you would get $d\tau'/d\tau = 1$ and thus $dx^\mu/d\tau = dx^\mu/d\tau'$. It is just a change of what you call "proper time equals zero".

However am I correct in thinking that the paramter $\tau+b$ would be plausible?

No, it would not be. Just by the same argumentation as above. You would still be trying to parameterise with the proper time, just with a different definition of proper time equal to zero. It does not work for light-like curves. You can still find an affine parameter, but it cannot be based on curve length (i.e., proper time).

 

[binbagsss]

Yes, if you have $\tau' = \tau + b$, you would get $d\tau'/d\tau = 1$ and thus $dx^\mu/d\tau = dx^\mu/d\tau'$. It is just a change of what you call "proper time equals zero".

No, it would not be. Just by the same argumentation as above. You would still be trying to parameterise with the proper time, just with a different definition of proper time equal to zero. It does not work for light-like curves. You can still find an affine parameter, but it cannot be based on curve length (i.e., proper time).

Is a re-scale of prper time ok, so $\lambda=a\tau$?