Follow-up Einstein Definition of Simultaneity for Langevin Observers

[cianfa72]

TL;DR

Extending a 1D spacelike line to a 2D spacelike surface orthogonal to Langevin worldlines at a single radius.

Reading again this old thread Einstein Definition of Simultaneity for Langevin Observers, I'd like to ask about the following.

Consider a disk rotating at fixed angular velocity $\Omega$. In the inertial coordinate system the center of the disk is at rest, the worldlines of the Langevin observers at any fixed $r$ are timelike helical lines around the cylinder of radius $r$ -- we use a (2+1)d spacetime model charted on 3D euclidean space as usual.

That thread explains how to "unwrap" the cylinder into a flat sheet using the (1+1)d Minkowski metric on it inherited from the "ambient" (2+1)d Minkowski metric. Langevin worldlines are timelike worldlines on it (i.e. parallel straight lines pointing inside the light cone at any point along them). We can draw then the 1D spacelike lines orthogonal to them.

Now, since the Langevin worldlines are "slanted" in the picture on the flat sheet, their (Minkowski) orthogonal lines won't be drawn as intersecting them at 90 degree in the picture. By wrapping again around the cylinder, such spacelike orthogonal lines on the flat sheet picture will become non-closing helical lines.

Repeating again the same for increasing values of $r$, we'll get a 2D non-achronal spacelike surface which however, as explained there, won't be orthogonal in all directions to the full set of Langevin worldlines (i.e. Langevin congruence) at any point along them. This follows from the fact that the Langevin congruence has non-zero vorticity hence is not hypersurface orthogonal. So far so good.

Consider now a single 1D spacelike non-closing helical line (1D spacelike surface). As @DrGreg explained, one can extend such line to a 2D spacelike surface that's orthogonal to (less than a complete revolution of) the Langevin worldlines at the corresponding/relevant radius (but not orthogonal to Langevin worldlines at a different radius). However it can't be extended too far, otherwise it would meet the same Langevin worldline twice.

My doubt: does the above mean that extending it to far would meet twice the same Langevin worldline apart from/excluding those of Langevin observers at the specific radius used in the construction? Thanks.

 

[PeterDonis]

does the above mean that extending it to far would meet twice the same Langevin worldline apart from/excluding those of Langevin observers at the specific radius used in the construction?

No. "The same Langevin worldline" means a worldline that's one of those at the radius you're working with.

 

[cianfa72]

No. "The same Langevin worldline" means a worldline that's one of those at the radius you're working with.

Ah ok. So starting from one single 1D spacelike line Minkowski orthogonal to the Langevin worldlines on the cylinder of given radius $r$, one can define/build from it a 2D spacelike surface orthogonal in all directions to the Langevin worldlines at that fixed radius. However when extended it will intersect twice those same Langevin worldlines. That means such a spacelike surface isn't achronal.

 

[cianfa72]

@DrGreg in that thread argued that the spacelike surface defined by the set of 1D spacelike helices, each orthogonal in the tangential direction to Langevin worldlines at a given radius, isn't orthogonal to the Langevin congruence (as implied by non-zero vorticity of the congruence). He gave the reason that the "slope" of 1D spacelike helices at a given radius on the cylinder increases proportionally with the radius.

The conclusion was that the spacelike surface obtained by "joining" such spacelike helices can't be orthogonal in the radial direction to Langevin worldlines.

Why the argument given above imply the conclusion ?

 

[cianfa72]

Why the argument given above imply the conclusion ?

Thinking about it, based on what explained here, I believe the key point is as follows.

The Langevin frame field has as timelike unit vector field the tangent 4-velocity $p_0$ at any point along the Langevin congruence's worldlines. In cylindrical coordinates for Minkowski spacetime $(R,Z,\Phi,T)$ it is
$$p_0 = \frac {1} {\sqrt{1 - \omega^2 R^2}} \partial_T + \frac {\omega R} {\sqrt{1 - \omega^2 R^2}} \frac {1} {R} \partial_{\Phi}$$
Vector fields $\{ p_2,p_3 \}$ (excluding the inessential $p_1$ vector field along $Z$ coordinate) form a basis for the orthogonal complement to $p_0$ at the tangent space at each point. The integral curves of $p_3$ in those coordinates are spacelike helical curves orthogonal to the intersecting Langevin worldlines.

However, $\{ p_2,p_3 \}$ as vector fields are not integrable since they do not meet the Frobenius condition (i.e. they fail to form a Lie algebra). Indeed the spacelike surface the integral curves of $p_3$ lie on isn't orthogonal everywhere to the Langevin congruence.