Ok, now I think I grasped what you said earlier in this thread: a Langevin congruence picks out a "specific" 2D local spacelike plane at each point P along congruence's worldlines (it is the 2D plane as defined in my previous post).PeterDonis said:This is correct. However, that still leaves an infinite number of such vectors.
I mean, then it's obvious what vector at ##Q## is "the same as" one at ##P## in the relevant sense. In fact, if you work in polars it's completely trivial.Ibix said:Why not just express everything in the disc inertial rest frame?
If you mean the inertial frame in which the disc's center is at rest, that is the frame in which the analysis in the Insights article is conducted (in cylindrical coordinates).Ibix said:Why not just express everything in the disc inertial rest frame?
Yes, but it's still a 2D plane. That's an infinite number of possible spacelike directions.cianfa72 said:a Langevin congruence picks out a "specific" 2D local spacelike plane at each point P along congruence's worldlines
But there are an infinite number of possible ##b## members for which this will work. Which one do you pick?cianfa72 said:Then, given a pair of congruence's worldlines say ##a## and ##b##, there is only one geodesic of flat spacetime pointing in a (spacelike) direction that lies in those local planes at point P on ##a## that intersect the ##b## member.
My point was different: to me congruence's member ##a## and ##b## are given/fixed in "advance".PeterDonis said:But there are an infinite number of possible ##b## members for which this will work. Which one do you pick?
I was aiming the comment more at @cianfa72. Surely what "the same vector at a different event on a Langevin worldline" means (in the relevant sense) is obvious in those coordinates.PeterDonis said:If you mean the inertial frame in which the disc's center is at rest, that is the frame in which the analysis in the Insights article is conducted (in cylindrical coordinates).
What does this mean? The congruence has an infinite number of worldlines in it. How are you fixing any pair of them in advance?cianfa72 said:to me congruence's member ##a## and ##b## are given/fixed in "advance".
There is no unique concept of "the same vector at a different event" even in flat spacetime. The advantage flat spacetime gives you is that parallel transport is not path dependent. But parallel transport is still only one of a number of possible transport laws that can be used to define "the same vector at a different event", and not all of them give the same answers even in flat spacetime.Ibix said:Surely what "the same vector at a different event on a Langevin worldline" means (in the relevant sense) is obvious in those coordinates.
Maybe I wasn't clear. In Langevin congruence take two members say ##a## and ##b## (e.g. the worldlines of two clocks riding on a ring). From a point P on worldline ##a## take a spacelike direction such that the spacelike geodesic (in flat spacetime) in that direction intersects the member ##b##. Call ##L## the spacetime lenght of such geodesic segment.PeterDonis said:What does this mean? The congruence has an infinite number of worldlines in it. How are you fixing any pair of them in advance?
This is not correct. Work it out and see. In fact the parallel transported spacelike direction at Q will in general not intersect worldline ##b## at all.cianfa72 said:I believe the answer is simply the parallel transport in flat spacetime of the spacelike direction picked at point P from P to Q.
Sure. So I was proposing a methodology that defines "the same vector at a different event" in a way that matches what I suspect @cianfa72 intends. I think the intention is that if a Langevin observer, ##a##, points at another Langevin observer, ##b##, and keeps pointing at them then a vector orthogonal to ##a##'s four velocity and in the worldsheet of their arm should be "the same" by this definition. I think that if you work in the obvious polar coordinate system in the disc COM's inertial rest frame and its coordinate basis, this condition is trivial to express.PeterDonis said:There is no unique concept of "the same vector at a different event" even in flat spacetime.
That is a requirement for what the OP wants to do, yes. The question is, if we have a vector at event P on ##a##'s worldline that does this, how do we transport that vector along ##a##'s worldline to another event Q so it keeps doing this? So far we have shown that neither parallel transport nor Fermi-Walker transport meet this requirement.Ibix said:the intention is that if a Langevin observer, ##a##, points at another Langevin observer, ##b##, and keeps pointing at them
This is a requirement, yes.Ibix said:a vector orthogonal to ##a##'s four velocity
I'm not sure what this means.Ibix said:and in the worldsheet of their arm should be "the same" by this definition.
Then can you express it?Ibix said:I think that if you work in the obvious polar coordinate system in the disc COM's inertial rest frame and its coordinate basis, this condition is trivial to express.
An observer sits in a chair that is bolted to a disc of radius ##R## that rotates at ##\omega##. He is thus constrained to face the rotation axis. His four velocity expressed in non-rotating cylindrical polars centered on the rotation axis is $$\begin{eqnarray*}v^a&=&(v^t,v^r,v^\phi,v^z)\\&=&(\gamma_\omega,0,\gamma_\omega\omega ,0)\end{eqnarray*}$$where ##\gamma_\omega## is the Lorentz gamma factor associated with speed ##\omega R## and the metric is ##\mathrm{diag}(1,-1,-r^2,-1)##. At any event he can define three unit spacelike vectors, #m$$\begin{eqnarray*}r'^a&=&(0,1,0,0)\\PeterDonis said:Then can you express it?
From the Insight ##\gamma_\omega## should be ##1 / \sqrt{1 – \omega^2 R^2}## even though the signature chosen there is ##\mathrm{diag}(-1,1,r^2,1)##.Ibix said:An observer sits in a chair that is bolted to a disc of radius ##R## that rotates at ##\omega##. He is thus constrained to face the rotation axis. His four velocity expressed in non-rotating cylindrical polars centered on the rotation axis is $$\begin{eqnarray*}v^a&=&(v^t,v^r,v^\phi,v^z)\\&=&(\gamma_\omega,0,\gamma_\omega\omega ,0)\end{eqnarray*}$$where ##\gamma_\omega## is the Lorentz gamma factor associated with speed ##\omega R## and the metric is ##\mathrm{diag}(1,-1,-r^2,-1)##. At any event he can define three unit spacelike vectors, $$\begin{eqnarray*}r'^a&=&(0,1,0,0)\\
\phi'^a&=&(\gamma_\omega \omega R,0,\gamma_\omega/R,0)\\
z'^a&=&(0,0,0,1)\end{eqnarray*}$$using the same coordinate system. These are mutually orthogonal and also orthogonal to his four velocity, so these four four vectors form a tetrad at the event.
By "point" on the rim of the disc above (bold is mine) you mean the associated Langevin timelike congruence's worldline/member, right ?Ibix said:If there is a mark painted somewhere on the disc the observer can point to it (if very, very strong), or even point in the direction where he sees it to be (that's a different direction, but I don't care here). If he keeps pointing at it, I think this action matches the intuitive notion of "pointing in the same direction" in this case. The observer could always express the direction he's pointing as ##Ar'^a+B\phi'^a##, and we can define a one-parameter family of geodesics with these tangent vectors extending from the observer at all events he passes through. These geodesics will all pass through the same point (i.e. Langevin worldline) on the rim of the disc. The segments of geodesic from observer to point will all have the same length, and any pair of geodesics separated at the observer by Langevin proper time ##\Delta \tau## will also be separated by Langevin proper time ##\Delta\tau## at their other ends.
Ibix said:Note that the direction is not only expressible as a sum of those two tetrad vectors, the coefficients ##A## and ##B## are constants. Thus the tetrad field ##v^a, r'^a,\phi'^a,z'^a## is what we want to transport along the Langevin worldlines. This is not parallel transport (which would keep the tetrad fixed in an inertial frame) nor Fermi-Walker transport (which would precess with respect to these).
We said that the tetrad field that includes the Langevin worldline timelike 4-velocity and the spacelike vectors ##r'^a## and ##\phi'^a## at a point P is not Fermi-Walker transported since ##D_Fr'^a \neq 0## and ##D_F\phi'^a \neq 0##. Why do you think that a Langevin congruence's worldline must F-W its tetrad ?Ibix said:Finally, I've been careful to not call my observer a Langevin observer. He is following a Langevin worldline but he is not Fermi-Walker transporting his tetrad, and I am not sure if Langevin observers are formally points or tetrads.
Yes, exactly as I said in the exact bit you highlighted.cianfa72 said:By "point" on the rim of the disc above (bold is mine) you mean the associated Langevin timelike congruence's worldline/member, right ?
I don't think it must. I don't know if observer, in this context, is typically defined to include a "natural" tetrad. It might be because there is interesting behaviour of the tetrad. But it might not. I don't know. That's why the sentence you quoted says "I am not sure".cianfa72 said:Why do you think that a Langevin congruence's worldline must F-W its tetrad ?
Ah ok. To me an observer was just a synonym for "timelike worldline" in the congruence (Langevin congruence in this specific case) without reference to any tetrad/frame it carries with itself.Ibix said:I don't think it must. I don't know if observer, in this context, is typically defined to include a "natural" tetrad. It might be because there is interesting behaviour of the tetrad. But it might not. I don't know. That's why the sentence you quoted says "I am not sure".
In other words, he is facing in the direction that, in the notation of the Insights article, would be ##- \hat{p}_2##. In your notation, it would be ##- r'^a##. The tetrad you define is the same one that is defined in the Insights article. And as the Insights article shows, this tetrad is indeed not Fermi-Walker transported along the worldlines of the congruence.Ibix said:He is thus constrained to face the rotation axis.
Ibix said:I don't know if observer, in this context, is typically defined to include a "natural" tetrad.
Both meanings can be found in the literature. But regardless of how you define "observer", we have a tetrad (or more precisely a tetrad field--the Insights article calls it a frame field, which is the same thing). Does this help answer the question the OP is trying to address?cianfa72 said:To me an observer was just a synonym for "timelike worldline" in the congruence (Langevin congruence in this specific case) without reference to any tetrad/frame it carries with itself.
Ok, by definition of F-W derivative, ##D_F\hat{p}_0 = 0## for the 4-velocity ##\hat{p}_0## of any worldline in any timelike congruence. Then the Insights article shows that ##D_F\hat{p}_2 \neq 0, D_F\hat{p}_3 \neq 0## hence that "overall" tetrad field defined there is not F-W transported along the Langevin congruence's worldlines.PeterDonis said:The tetrad you define is the same one that is defined in the Insights article. And as the Insights article shows, this tetrad is indeed not Fermi-Walker transported along the worldlines of the congruence.
As pointed out earlier in this thread, fixed two Langevin congruence's members ##a## and ##b## (e.g. the timelike worldlines of ring riding clocks), there are spacelike directions belonging to the 2D local planes spanned from frame basis spacelike vectors ##\{ \hat{p}_2, \hat{p}_3 \}## at each point P along the worldline ##a## such that the spacelike geodesics drawn in each of those directions intersect for sure the member ##b##.PeterDonis said:Does this help answer the question the OP is trying to address?
Yes, there is such a spacelike direction for a given worldline ##b## at each point P along worldline ##a##. How is the vector that defines this spacelike direction transported along worldline ##a##?cianfa72 said:there are spacelike directions belonging to the 2D local planes spanned from frame basis spacelike vectors ##\{ \hat{p}_2, \hat{p}_3 \}## at each point P along the worldline ##a## such that the spacelike geodesics drawn in those directions intersect for sure the member ##b##.
First question: at each point P along worldline ##a##, is the spacelike geodesic pointing along a direction in ##\{ \hat{p}_2, \hat{p}_3 \}## plane with the property that intersects the worldline ##b## unique ? I believe yes, it is.PeterDonis said:Yes, there is such a spacelike direction for a given worldline ##b## at each point P along worldline ##a##. How is the vector that defines this spacelike direction transported along worldline ##a##?
Of course it is. Consider: at how many points will the timelike curve ##b## intersect the spacelike ##\{ \hat{p}_2, \hat{p}_3 \}## plane?cianfa72 said:at each point P along worldline ##a##, is the spacelike geodesic pointing along a direction in ##\{ \hat{p}_2, \hat{p}_3 \}## plane with the property that intersects the worldline ##b## unique ? I believe yes, it is.
Yes, we've established how it's not transported. I am asking you how it is transported.cianfa72 said:such unique vector at each point P along worldline ##a## is not F-W transported or Parallel transported (according flat spacetime) along the worldline itself.
I would say it does: we have a rigorous definition of "the same vector" at different events on the worldline, and given that definition the answer to the question in the OP is yes. From the symmetry it's also obvious that using other definitions of "the same vector" the answer will be "no" in general.PeterDonis said:we have a tetrad (or more precisely a tetrad field--the Insights article calls it a frame field, which is the same thing). Does this help answer the question the OP is trying to address?
The question you are answering in your first paragraph here--do we have a well defined meaning for "same vector"--is not the question the OP is trying to answer, although it is necessary to establish the overall framework of the scenario the OP is using.Ibix said:I would say it does
Only if you can define how the Langevin congruence transports its basis vectors. What transport law does it use?Ibix said:"does 'the combination of rotation and time translation that generates the Langevin congruence' answer your 'how is the vector transported' question"
As @Ibix pointed out, the vector field we're interested in along worldline ##a## is given as linear combination ##\hat{P} =A \hat{p}_2 + B \hat{p}_3## where ##A,B## are constant. Therefore we need to find a sort of derivative ##D_Z## along Langevin congruence's worldlines such that ##D_Z\hat{P}=0##.PeterDonis said:Only if you can define how the Langevin congruence transports its basis vectors. What transport law does it use?
Yes, any such vector will work. One of them, though--the one with ##A = -1## and ##B = 0##, has a particular useful property in terms of the Langevin worldlines. Can you see what it is? (Hint: the relevant spacelike vector for the Rindler congruence, the one that lies in the 2-plane you described earlier, has the same property.)cianfa72 said:As @Ibix pointed out, the vector field we're interested in along worldline ##a##, is given as linear combination ##\hat{P} =A \hat{p}_2 + B \hat{p}_3## where ##A,B## are constant.
Yes.cianfa72 said:Therefore we need to find a sort of derivative ##D_Z## along Langevin congruence's worldlines such that ##D_Z\hat{P}=0##.
Parallel transport plus a rotation around the z axis?PeterDonis said:Only if you can define how the Langevin congruence transports its basis vectors. What transport law does it use?
Fermi-Walker transport plus a z axis rotation described by the vorticity vector is one way to describe it, yes. (Note that the z axis rotation only applies to the vectors you call ##r'^a## and ##\phi'^a##.)Ibix said:Parallel transport plus a rotation around the z axis?
Its useful property is that it always points inward in the radial direction ##- \partial_R##, so the transport law we are looking for is such that transports it along the Langevin worldlines.PeterDonis said:Yes, any such vector will work. One of them, though--the one with ##A = -1## and ##B = 0##, has a particular useful property in terms of the Langevin worldlines. Can you see what it is?
Sorry, the 2-plane I described earlier (namely the 2-plane spanned from ##\{ \hat{p}_2, \hat{p}_3 \}##) was for Langevin congruence not for Rindler one.PeterDonis said:(Hint: the relevant spacelike vector for the Rindler congruence, the one that lies in the 2-plane you described earlier, has the same property.)
The point here is that, for example, in Minkowski standard global inertial chart the Langevin congruence's worldline ##b## is represented by an helix and of course spacelike planes in flat spacetime by planes in that chart. Therefore in that chart (hence in spacetime) there is only one intersection event/point between them.PeterDonis said:Of course it is. Consider: at how many points will the timelike curve ##b## intersect the spacelike ##\{ \hat{p}_2, \hat{p}_3 \}## plane?
Ok, but that's still a coordinate-dependent specification. Is there some invariant that points in that direction? (Hint: the Insights article tells you the answer.)cianfa72 said:Its useful property is that it always points inward in the radial direction ##- \partial_R##
I know that. I was pointing out a similarity between the Langevin congruence and the Rindler one, in terms of a particular invariant that picks out a spacelike direction that satisfies the property you want it to satisfy.cianfa72 said:the 2-plane I described earlier (namely the 2-plane spanned from ##\{ \hat{p}_2, \hat{p}_3 \}##) was for Langevin congruence not for Rindler one.
Yes, it is the covariant derivative of the 4-velocity of Langevin congruence's worldlines (i.e. their proper acceleration), namely $$PeterDonis said:Ok, but that's still a coordinate-dependent specification. Is there some invariant that points in that direction? (Hint: the Insights article tells you the answer.)
Ok, so the same as above for Rindler congruence, namely the particular invariant is the congruence's worldlines proper acceleration $$PeterDonis said:I was pointing out a similarity between the Langevin congruence and the Rindler one, in terms of a particular invariant that picks out a spacelike direction that satisfies the property you want it to satisfy.