I Affine parameter and non-geodesic null curves

JimWhoKnew
Messages
181
Reaction score
90
TL;DR Summary
Is there a sensible way to define an affine parameter for non-geodesic null curves?
Consider the curve (thanks to SE) in flat spacetime, given in Cartesian coordinates by$$x^μ(λ)=\left(λ , R\cos\frac{\lambda⁡}{R} , R\sin\frac{\lambda}{⁡R} ,0\right)$$where ##~R~## is a positive constant. At each point$$\dot x^\mu \dot x_\mu=0$$so it is a null curve but not a geodesic (not a straight line). It also satisfies$$\ddot x^\mu \dot x_\mu=0 \quad .$$If I got the calculation right, it turns out that for any reparametrization ##~\lambda'~## , where ##~\lambda'(\lambda)~## is an arbitrary monotonic function, ##~\dot x^\mu \dot x_\mu=\ddot x^\mu \dot x_\mu=0~## holds in this particular case (dots here w.r.t. ##~\lambda'~##).

Is there a sensible way in which we can define an affine parameter for non-geodesic null curves like this, such that certain parametrizations are affine while others are not?

Edit: (We have criteria for parameters to "be affine" in the cases of timelike/spacelike curves and null geodesics. Is the non-geodesic null curve an exception?)
 
Last edited:
Physics news on Phys.org
For the particular example in OP, I think that the time coordinate ##~t~## of the specific reference frame can be regarded as an affine parameter. Because of the symmetry (the Euclidean length traced in each uniform interval ##~\Delta t~## is the same).
 
Thread 'Can this experiment break Lorentz symmetry?'
1. The Big Idea: According to Einstein’s relativity, all motion is relative. You can’t tell if you’re moving at a constant velocity without looking outside. But what if there is a universal “rest frame” (like the old idea of the “ether”)? This experiment tries to find out by looking for tiny, directional differences in how objects move inside a sealed box. 2. How It Works: The Two-Stage Process Imagine a perfectly isolated spacecraft (our lab) moving through space at some unknown speed V...
Does the speed of light change in a gravitational field depending on whether the direction of travel is parallel to the field, or perpendicular to the field? And is it the same in both directions at each orientation? This question could be answered experimentally to some degree of accuracy. Experiment design: Place two identical clocks A and B on the circumference of a wheel at opposite ends of the diameter of length L. The wheel is positioned upright, i.e., perpendicular to the ground...
According to the General Theory of Relativity, time does not pass on a black hole, which means that processes they don't work either. As the object becomes heavier, the speed of matter falling on it for an observer on Earth will first increase, and then slow down, due to the effect of time dilation. And then it will stop altogether. As a result, we will not get a black hole, since the critical mass will not be reached. Although the object will continue to attract matter, it will not be a...

Similar threads

Replies
10
Views
2K
Replies
28
Views
3K
Replies
8
Views
1K
Replies
10
Views
5K
Replies
16
Views
9K
Back
Top