I What is a "rotating worldline"?

[etotheipi]

[Moderator's note: spin off from another thread due to topic change.]

includes rotating worldlines, which are accelerating

I wondered if somebody could explain what "rotating worldline" means?

- For the simple case of inertial observer, I imagine constant temporal basis vector $\mathbf{e}_0 = \mathbf{u}$ which is tangent to the worldline (which would just be, a geodesic/"straight line" through the spacetime), and spatial basis vectors in fixed orientations w.r.t. this $\mathbf{e}_0$. No rotation anywhere, here.

- You could alternatively have an observer moving at constant 4-velocity $\mathbf{e}_0$ but rotating his spatial basis vectors within a 3-dimensional hypersurface, i.e. $\mathbf{e}_i(\tau) = R_{ij}(\tau) \mathbf{e}_j(0)$, which constitutes a rotating frame.
- You could alternatively have an observer with non-zero 4-acceleration, so his temporal basis vector $\mathbf{e}_0 = \mathbf{e}_0(\tau)$ is now also a function of the arc-length parameter.

In the vein of the last point, I could imagine two separate worldlines "rotating around each other" in a sort of helix.

But I don't know what it would mean for a "worldline to rotate". Wouldn't it be better to instead say the basis vectors carried by the observer are rotating?

 

[vanhees71]

Besides the point, but wondered if somebody could explain what "rotating worldline" means?

- For the simple case of inertial observer, I imagine constant temporal basis vector $\mathbf{e}_0 = \mathbf{u}$ which is tangent to the worldline (which would just be, a geodesic/"straight line" through the spacetime), and spatial basis vectors in fixed orientations w.r.t. this $\mathbf{e}_0$. No rotation anywhere, here.

- You could alternatively have an observer moving at constant 4-velocity $\mathbf{e}_0$ but rotating his spatial basis vectors within a 3-dimensional hypersurface, i.e. $\mathbf{e}_i(\tau) = R_{ij}(\tau) \mathbf{e}_j(0)$, which constitutes a rotating frame.
- You could alternatively have an observer with non-zero 4-acceleration, so his temporal basis vector $\mathbf{e}_0 = \mathbf{e}_0(\tau)$ is now also a function of the arc-length parameter.

In the vein of the last point, I could imagine two separate worldlines "rotating around each other" in a sort of helix.

But I don't know what it would mean for a "worldline to rotate". Wouldn't it be better to instead say the basis vectors carried by the observer are rotating?

You can also construct rotation-free tetrads along arbitrary time-like worldlines of an observer using Fermi-Walker transport. A locally inertial observer must be freely falling, i.e., move along a time-like geodesics, and local inertial rest frames are constructed via the corresponding Fermi-Walker (i.e., non-rotating) transported tetrads (with the four-velocity as the temporal tetrad). In this case the Fermi-Walker transport is the same as parallel transport.

 

[martinbn]

I wondered if somebody could explain what "rotating worldline" means?

It seems like a nonstandard use of terminology. Unless he (the one who used it) explains what he means, there is no point worring about it.

 

[PeterDonis]

I wondered if somebody could explain what "rotating worldline" means?

I think it's just a confusion, at least in the particular case quoted.

 

[Grasshopper]

It seems like a nonstandard use of terminology. Unless he (the one who used it) explains what he means, there is no point worring about it.

The notion seems weird to me. Hopefully someone can elaborate on it, but if a worldline is the history of an object through spacetime, how can it rotate? Unless he means the path is a helix?

 

[MikeeMiracle]

Believe PBS Spacetime had a video on this within the past few months and the paths were both a helix.

 

[Halc]

I wondered if somebody could explain what "rotating worldline" means?

It very likely means the worldline of a rotating object, which might not itself be helical, but the worldlines of its components are, kind of like the strands of a twisted rope follow a helical shape despite it being a straight rope.

A worldline itself is a static set of events in spacetime, and as such does not move, sort of like the line left by a Bizzy Buzz Buzz toy pen (are you old enough to remember those?).

 

[Ibix]

It very likely means the worldline of a rotating object, which might not itself be helical, but the worldlines of its components are, kind of like the strands of a twisted rope follow a helical shape despite it being a straight rope.

Strictly that would be a worldtube. There's a (2+1)d Minkowski diagram that nearly illustrates what you are talking about in this post. That's actually a train on a circular track, so it's a 2d circle and hence a (2+1)d cylindrical shell on the diagram. A solid disc would be a (2+1)d solid cylinder on the diagram and would look like your rope analogy.

 

[pervect]

"Rotating" is always a bit vague. I don't think it applies to worldlines, but a region of space, which involves a worldline and some neighborhood of the worldline, not just the worldline itself.

One phrase I've seen in the literature that of the zero angular momentum observer, aka ZAMO, which is different than a worldline. In terms of tetrads, tetrads that are Fermi-Walker transported are often informally described as non-rotating.

Sometimes "rotating" is used in a different sense than the ZAMO, though. This sort of issue arises frequently in the Kerr space-time.