Worldline Curve vs Trajectory: What’s the Difference?

[SaintRodriguez]

Is a worldline a curve or a trajectory? Why?

 

[Dale]

Is a worldline a curve or a trajectory? Why?

What is the difference?

I would say “curve”, but if someone else said “trajectory” I wouldn’t correct them. I don’t know the difference in this context

 

[Ibix]

I would have said both. What's the distinction you're trying to make?

 

[SaintRodriguez]

What is the difference?

I would say “curve”, but if someone else said “trajectory” I wouldn’t correct them. I don’t know the difference in this context

A curve is the math object like a function and the trajectory is the set of images that the function (curve) mapped.

 

[Dale]

the set of images

What is this “set of images”? Are you just talking about the mathematical representation vs the physical thing that the math represents?

 

[PeterDonis]

@SaintRodriguez you might want to give some actual references that give the definitions of "curve" and "trajectory" that you are using.

 

[George Jones]

Is a worldline a curve or a trajectory? Why?

A curve is the math object like a function and the trajectory is the set of images that the function (curve) mapped.

I think that this can vary, i.e., depends on the reference being used. A reference that takes great care with the mathematics, "Semi-Riemannian Geometry With Applications to Relativity" by Barrett O'Neill, defines a particle to be a (particular type of) mapping (function), and the image of mapping to be the worldline of the particle. Other references might define the mapping itself to be the worldline, and references that take less care mathematically might blur the distinction between a mapping and the image of the mapping.

From O'Neill's book: "Definition. A material particle in $M$ is a timelike future pointing curve $\alpha : I \rightarrow M$ such that $\left| \alpha'\left(\tau\right) \right| = 1$ for all $\tau$ in $I$. The parameter $\tau$ is called the proper time of the particle. ... its image $\alpha\left(I\right)$ is a one-dimensional submanifold of $M$ called the worldline of $\alpha$."

Here, $M$ is the spacetime manifold, and $I$ is an interval of the real line.

 

[vanhees71]

Is a worldline a curve or a trajectory? Why?

That's a good question, because the terminology is a bit unclear in the physics literature. For me a curve is any smooth map between the real numbers (or an interval, if you have a finite curve) to a differentiable manifold, and spacetime is described in GR as such a differentiable manifold (with the extra properties making it a pseudo-Riemannian manifold, i.e., with a pseudometric and the uniquely defined torsion-free affine connection, compatible with this pseudometric). Another name for such a curve in relativity is "worldline".

The word "trajectory" I reserve for the solution of the equations of motion for a point particle (or the pseudo-photon interpretation of the eikonal approximation of electrodynamics in GR), which are timelike (lightlike) worldlines. If there are no forces, i.e., only gravity/aka spacetime curvature, then these are spacelike or timelike worldlines.

 

[Dale]

If there are no forces, i.e., only gravity/aka spacetime curvature, then these are spacelike or timelike worldlines.

Why would worldline refer only to force free trajectories?

 

[vanhees71]

Maybe that was formulated somewhat misleading. Of course also worldlines describing motion under the influence of interactions, e.g., the electromagnetic interaction, I would call trajectories.

 

[martinbn]

To me the word trajectory has the meaning of the path of a particle in space over some time. And is very non-relativistic. I would say stay with curves not to be misunderstood.