Parametrizing a path through spacetime

[guitarphysics]

It is customary, when discussing a particle's motion through spacetime, to talk about its path $x^{\mu}(\lambda)$, where $x^{\mu}$ are the the spacetime coordinates of the particle in some frame, and $\lambda$ is some parameter. I have a doubt regarding this parameter. Everywhere I've looked, people seem to say "this parameter can, for example, be the particle's proper time $\tau$". And then they proceed to, for some reason, only use this very specific example (proper time) as the parameter for the particle's path (or any parameter of the form $\tau'=a\tau+b$). So my question is: are there any other physically distinct parameters that can be used? (By physically distinct I mean something that isn't of the form $a\tau+b$; that doesn't rely on the proper time.) If so, why is it that the proper time is almost always used?
Thanks in advance.

 

[vanhees71]

Proper time is nice, because it's (with a grain of salt) the parametrization of the world line in terms of its "length" measured from the initial point. This makes some equations simpler. There's however nothing to object against the use of a completely independent parameter which doesn't even need to have a physical meaning at all.

 

[m4r35n357]

Everywhere I've looked, people seem to say "this parameter can, for example, be the particle's proper time $\tau$". And then they proceed to, for some reason, only use this very specific example (proper time) as the parameter for the particle's path (or any parameter of the form $\tau'=a\tau+b$). So my question is: are there any other physically distinct parameters that can be used? (By physically distinct I mean something that isn't of the form $a\tau+b$; that doesn't rely on the proper time.) If so, why is it that the proper time is almost always used?
Thanks in advance.

This is a tiny bit tangential but it might help if you have access to MTW. $25.6 discusses in some depth the choice of affine parameter for light trajectories, where proper time is emphatically not valid.

[ASIDE: In my light pulse simulations, I have used a scaled affine parameter determined by setting $E = 1$, so that the impact parameter $b = L / E = L$. This scaling just affects the "speed" of the simulation.]

 

[Orodruin]

This is a tiny bit tangential

But the tiny bit has the same length all along the geodesic as long as the parametrisation is affine!

 

[Dale]

I like Carroll's notes on this. See P 70 in chapter 3.

https://www.preposterousuniverse.com/grnotes/

 

[stevendaryl]

It is customary, when discussing a particle's motion through spacetime, to talk about its path $x^{\mu}(\lambda)$, where $x^{\mu}$ are the the spacetime coordinates of the particle in some frame, and $\lambda$ is some parameter. I have a doubt regarding this parameter. Everywhere I've looked, people seem to say "this parameter can, for example, be the particle's proper time $\tau$". And then they proceed to, for some reason, only use this very specific example (proper time) as the parameter for the particle's path (or any parameter of the form $\tau'=a\tau+b$). So my question is: are there any other physically distinct parameters that can be used? (By physically distinct I mean something that isn't of the form $a\tau+b$; that doesn't rely on the proper time.) If so, why is it that the proper time is almost always used?
Thanks in advance.

There are two things that are special about proper time (and other parameters linearly related to it):

1. It's guaranteed to increase monotonically along the path (for slower-than-light objects, anyway).
2. It's an "affine" parameter.

The technical significance of the latter is that associated with a parametrized path $x^\mu(\lambda)$ is a tangent vector, or "velocity" $U^\mu = \frac{dx^\mu}{d\lambda}$. Moving along a geodesic, if it is parametrized using an affine parameter, then $U$ is locally unchanging in both magnitude and direction. In contrast, with a non-affine parameter, $U$ may change in magnitude as you move along the path.

 

[guitarphysics]

Thanks everyone for the responses!
Two followup (related) questions- 1) In Carroll's notes, he says that you could pick some other parameter $\alpha$. What could this parameter be, more specifically? (This is sort of asking my first question again: you all told me reasons why the proper time is more appropriate; but could someone give an example of this parameter $\alpha$? This is the part that's confusing to me- maybe I don't understand what a parameter is clearly enough.) Basically my question is: is there any other quantity that monotonically increases along the path of a particle, but isn't some function $f(\tau)$? And related to that (to see if I'm understanding): if consider a parameter $\lambda=a\tau^2+b\tau+c$, then this works as a parameter for a path, but it's not affine so it won't be very friendly. Is that right?

And 2) stevendaryl, why is it that proper time is an affine parameter, and thus defines an unchanging four-velocity? Is it because in the MCRF, $dx^{0}/d\lambda=1$ (or some constant number, depending on the parameter), and $dx^i/d\lambda=0$?

 

[pervect]

It is customary, when discussing a particle's motion through spacetime, to talk about its path $x^{\mu}(\lambda)$, where $x^{\mu}$ are the the spacetime coordinates of the particle in some frame, and $\lambda$ is some parameter. I have a doubt regarding this parameter. Everywhere I've looked, people seem to say "this parameter can, for example, be the particle's proper time $\tau$". And then they proceed to, for some reason, only use this very specific example (proper time) as the parameter for the particle's path (or any parameter of the form $\tau'=a\tau+b$). So my question is: are there any other physically distinct parameters that can be used? (By physically distinct I mean something that isn't of the form $a\tau+b$; that doesn't rely on the proper time.) If so, why is it that the proper time is almost always used?
Thanks in advance.

Here's a super short answer. You certainly could parameterize a curve in terms of an arbitrary parameter s, rather than one based on proper time $\tau$.. It's very easy to construct such a non-affine parameterization given an affine one. Given that we have a parameterziation of a curve in terms of proper time that maps proper time $\tau$ into a point on the curve $\vec{x}$ i we can re-label our points with a different parameter s. This re-labelling process defines some function $s = f(\tau)$ and $\tau = f^{-1}(s)$, and it defines a map between s and $\vec{x}$ that maps the same points to the same vectors, only the labels on the points have changed.

But if we do this, a rather important equation, the Geodesic equation, namely:
$$\frac{d^2 x^i}{ds^2} + \Gamma^i{}_{jk} \frac{dx^j}{ds}\frac{dx^k}{ds}$$
will no longer be correct. So if we wish to use the Geodesic equation, we need to make sure we've used an affine parameterization of the curve and not a general one, the Geodesic equation has assumed that we've used an affine parameterization of the curve.

 

[Orodruin]

Here's a super short answer.

With a super short addition: Not using an affine parametrisation will result in the equation

$$\frac{d^2 x^i}{ds^2} + \Gamma^i{}_{jk} \frac{dx^j}{ds}\frac{dx^k}{ds}$$

having a right-hand side proportional to the tangent itself. (There should be a RHS equal to zero in the equation, otherwise it is just the expression for the covariant derivative of the tangent in the tangent direction.) This can be derived directly from finding the EL equations for the path length.

 

[samalkhaiat]

It is customary, when discussing a particle's motion through spacetime, to talk about its path $x^{\mu}(\lambda)$, where $x^{\mu}$ are the the spacetime coordinates of the particle in some frame, and $\lambda$ is some parameter. I have a doubt regarding this parameter. Everywhere I've looked, people seem to say "this parameter can, for example, be the particle's proper time $\tau$". And then they proceed to, for some reason, only use this very specific example (proper time) as the parameter for the particle's path (or any parameter of the form $\tau'=a\tau+b$). So my question is: are there any other physically distinct parameters that can be used? (By physically distinct I mean something that isn't of the form $a\tau+b$; that doesn't rely on the proper time.) If so, why is it that the proper time is almost always used?
Thanks in advance.

The proper time action integral
$$S = - m \int d\tau = - m \int \sqrt{g_{ab} dx^{a} dx^{b}} ,$$
is invariant under arbitrary change of parametrisation
$$\tau \to \lambda = \lambda (\tau) .$$
This is clear because $d\tau = (d\tau / d\lambda) d\lambda$ is independent of $\lambda$. So, you can rewrite the action as
$$S = - m \int \ d\lambda \ \sqrt{g_{ab} \frac{dx^{a}}{d\lambda} \frac{dx^{b}}{d\lambda}} .$$
Parametrisation-invariance means that the action is independent of what you choose to parameterise the path $x^{a}$. This is, however, not the case for the geodesic equation. Indeed, if you change the proper-time according to $\tau \to \lambda (\tau)$, the geodesic equation transforms into

$$\frac{d^{2}x^{a}}{d\lambda^{2}} + \Gamma^{a}_{bc} \frac{dx^{b}}{d\lambda} \frac{dx^{c}}{d\lambda} = - \frac{d^{2}\lambda / d\tau^{2}}{(d\lambda / d\tau)^{2}} \frac{dx^{a}}{d\lambda} .$$
Thus, the geodesic equation remains invariant only under a class of parametrisation defined by
$$\frac{d^{2}\lambda}{d\tau^{2}} = 0 \ \ \Rightarrow \ \ \lambda = a \tau + b .$$
This is the class of affine parameters: parameters related to the proper-time $\tau$ by an affine transformation $\tau \to \sigma = a \tau + b$ are called affine parameters.