Generalized Lorentz Transformation for an Accelerated Frame of Reference

[Project]

This paper by Robert A. Nelson derives an exact, explicit coordinate transformation between an inertial frame of reference and a frame of reference having an arbitrary time-dependent, nongravitational acceleration and an arbitrary time-dependent angular velocity.
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JMAPAQ000028000010002379000001

The sophisticated formalism of this article is beyond my mathematical training. I would like to study a simplified, non-rotating, 1+1-dimensional version of the transformation. Can it be written out for someone at the undergraduate level? What is the exact, explicit coordinate transformation between an inertial frame of reference and an arbitrarily accelerated frame of reference for one spatial dimension?

 

[pervect]

Would you be happy with a uniform acceleration? If you don't mind having a constant acceleration

this thread

presents the coordinate system of an accelerated observer as part of the notes for a problem in relativity that I thought was interesting. (YMMV: the thread never got a reply from anyone :-)).

One thing that I don't mention specifically in this thread (because it wasn't specifically relevant to the problem I was presenting) is that an acclelerated observer cannot assign coordinates to all of space-time, due to the existence of a Rindler horizon. For an observer accelerating at 1 light year/year^2 (which is approximately 1g), anything that is more than 1 light year "below" the observer will not be visible to him, as no light from this region can reach him as long as he continues to accelerate. The boundary between this inaccessible region of space-time and the region the observer can actually see is known as the "Rindler Horizon". It is very similar in a lot of respects to the event horizon of a black hole.

The uniform acceleration case is simpler than the arbitrary acceleration case you asked for because one doesn't have to worry about Thomas precession (the rotation issue you refer to).

Unfortunately I don't derive the results I present in this thread, but you can find the derivation in Misner, Thorne, Wheeler, "Gravitation". It's still a fairly tough read, you'll need some familiarity with four vectors to follow MTW's derivation.

There's another thread on this board where I derive the relativistic rocket equations from the velocity addition formula. I'm not sure if you are interested in this at this point, if you are let me know. You can check out the sci.physics.faq on the relativistic rocket which presents the equations for the motion of relativistic rocket without deriving them (all I added was a simple derivation for some of the equations presented in the following URL). This is related to, but simpler than, finding the coordinate system of an accelerated observer, because it only describes the motion of an accelerated observer, not his coordinate system.

http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html

 

[Project]

Thanks pervect, but I believe I understand constant proper acceleration. That's why I'm now interested in variable acceleration.

Do you know or have any interest in knowing the transformation equations for an arbitrarily accelerated frame of reference for one spatial dimension?

 

[pervect]

I'll apologize in advance if this isn't very clear. All I can say is that it is a complex enough derivation that you might want to actually get the textbook and study it. "Gravitation" , by Misner, Thorne, Wheeler (henceforece MTW) has the fullest discussion that I know of of the coordinate system of an accelrating observer and that's what I'll base my reply on.

MTW discusses arbitrary accelerations in "Gravitation", using a highly geometric approach. They start off with the simple case of a constant acceleration, and point out that in that case the 4-acceleration is always perpendicular to the 4-velocity, i.e. $$\vec{a} \cdot \vec{u} = 0$$.

Hopefully these terms are familiar - if not, the 4-velocity for an observer in a coordinate system based on (t,x,y,z) is

dt/dtau, dx/dtau, dy/dtau, dz/dtau

and the 4-acceleration is

d^2t/dtau^2, d^2x/dtau^2, d^2y/dtau^2, d^2z/dtau^2

here tau is the proper time of the accelerating observer and t,x,y,z are coorinate times and positions.

When constructing a coordinate system for a constantly accelerating observer, one needs to define an orthonormal basis. One picks a co-moving observer (one who has the same velocity as the accelerating observer and the same position, but who is not himself accelerating) to construct the orthonomal basis of vectors (ONB) that define the coordinate system. The time vector of the ONB is just the 4-velocity of the co-moving observer, and one picks 3 orthogonal space vectors.

This approach yields the results I already presented.

For an arbitrary accelerating observer, the situation is very similar, the 4-accleration is perpendicular to the 4-velocity, and the time vector of the ONB is the 4-velocity of the co-moving observer. However, the spatial vectors must be Fermi-Walker transported. This means that they must not be allowed to rotate in the a ^ u plane. Here ^ is the "wedge product", $$a^x u^y - a^yu^x$$. Rotations in 4-d are most conveniently described by rotations around planes, not around axes.

I suspect this didn't help a lot, but again all I can say is that the topic is complex enough to warrant a textbook, not just a short post. The textbook discussion is not nearly as terse as my post,though it does use the same geometric language (4-velocities and 4-accelerations). The material I presented in this short post requires an entire chapter to explain properly, and MTW takes the time and space to do it, spending several pages on the topic of rotations in 4-d spacetime alone.

If you don't happen to have "Gravitation", try the index of whatever text you do have or can find for the term "Fermi-Walker transport".

 

[Project]

pervect,

I was hoping that someone would just post the simplified answer for 1+1 dimensions. I don't need to derive the transformation. I just want to know what it looks like and understand all its terms in the simplified case of one spatial dimension. MTW isn't easy for me like it is for you.

 

[pervect]

MTW definitely isn't an easy read - I've probably had my copy for 10 years, and still haven't read all of it. But it does have a good treatment of the accelerated observer.

Also, it's worth reading for the "fine print". Something that I probably should have mentioned earlier, but I didn't, was the fact that the the coordinate system of an accelerated observer is well behaved only in a small local region. Specifically, a single event in space-time might have more than one set of coordinates if one tries to extend to coordinate system to cover too large a region. This is in addition to the point that I made earlier that the Rindler horizion prevents one from assigning coordinates or even seeing certain regions of space-time.

The rule of thumb is to say that the coordinate system is valid for distances less than c^2/a, where a is the acceleration of the observer. This rule is a bit too restrictive in some cases, but guarantees that the coordinate system will be well behaved.

The problem really is complex enough that it's worth devoting an entire chapter of a textbook to it.

 

[Project]

MTW definitely isn't an easy read - I've probably had my copy for 10 years, and still haven't read all of it. But it does have a good treatment of the accelerated observer.

OK. So what's the answer for the elementary simplification to one spatial dimension?

the coordinate system of an accelerated observer is well behaved only in a small local region. Specifically, a single event in space-time might have more than one set of coordinates if one tries to extend to coordinate system to cover too large a region. This is in addition to the point that I made earlier that the Rindler horizion prevents one from assigning coordinates or even seeing certain regions of space-time.

Correct, but I already realize that from a straightforward study of constant proper acceleration.

The rule of thumb is to say that the coordinate system is valid for distances less than c^2/a, where a is the acceleration of the observer. This rule is a bit too restrictive in some cases, but guarantees that the coordinate system will be well behaved.

If you don't know the answer, just say so. There's no shame in not knowing basic facts about one spatial dimension, only in evading the truth.

The problem really is complex enough that it's worth devoting an entire chapter of a textbook to it.

I'm not looking for a lengthy discussion of this topic or a derivation, just a pair of simple equations.

 

[pervect]

OK. So what's the answer for the elementary simplification to one spatial dimension?

Correct, but I already realize that from a straightforward study of constant proper acceleration.

If you don't know the answer, just say so. There's no shame in not knowing basic facts about one spatial dimension, only in evading the truth.

I'm not looking for a lengthy discussion of this topic or a derivation, just a pair of simple equations.

I really don't think that referring you to a source that discusses your problem, and giving you some warnings about the unusal properties and limitations of the coordinate system of an accelerated observer is "evading" anything. In fact, presenting a solution without the proper discussion of the limits of applicability would be in my mind much more misleading.

I didn't realize that you were totally unintrested in doing your own work, but I'm beginning to get that message. So, just ignore everything I said about where you can find the tools needed to do the job yourself, and keep on looking for someone to give you your answer all neatly wrapped up in a blanket. I do hope that the person who hands you said answer doesn't make any mistakes, because it doesn't appear that you are going to be doing much double-checking.

 

[Project]

I really don't think that referring you to a source that discusses your problem, and giving you some warnings about the unusal properties and limitations of the coordinate system of an accelerated observer is "evading" anything.

I know all about accelerated frames of reference for constant proper acceleration, the Rindler horizon and how to compute it. It's not the answer to the question I asked. My opening post begins with a source that I don't understand and I've already said that MTW is too difficult for me.

In fact, presenting a solution without the proper discussion of the limits of applicability would be in my mind much more misleading.

I also know what it means for a solution to be valid in a sufficiently small neighborhood.

I didn't realize that you were totally unintrested in doing your own work, but I'm beginning to get that message. So, just ignore everything I said about where you can find the tools needed to do the job yourself, and keep on looking for someone to give you your answer all neatly wrapped up in a blanket. I do hope that the person who hands you said answer doesn't make any mistakes, because it doesn't appear that you are going to be doing much double-checking.

If I had posted a request that someone write out for me what the Lorentz transformation is in the easy case of one spatial dimension and to explain what all the terms meant, I bet that would generate a greatly detailed response. You have no idea about my abilities to check or not check the validity of any given answer.

 

[selfAdjoint]

So from you original link, all I can see is his abstract, since I am not a subscriber to the Journal of Mathematical Physics. But that abstract seems clear enough: he has to handle the Thomas Precession that comes up with accelerated frames and he has to rotate the axes of the accelerated frame relative to the observation frame. As I say, I can't critique/explicate the math he uses to do this, since I can't see it. So let's break it down. Is your problem with Thomas precession? Or where is it?

 

[pervect]

I can't see the original paper either, also not having a subscription. But if Project is truly interested in only the 1space+1time case, then knowing the 4-velocity and the position of the accelerated observer as a function of time in an inertial coordinate system should be sufficient to define the local coordinate system of the accelerated observer. The 4-velocity will give the time-vector of the accelerated observer, and there is only one vector orthogonal to this 4-velocity, which must then give the space vector of the accelerated observers ONB. So he won't really need to worry about Thomas precession.

This ONB then describes how to "map" the accelerated observers coordinates into the inertial space-time.

So what remains is the process of writing down the differential equations to describe the 4-velocity and position of the arbitrarily accelerated observer, and actually solving them (I rather doubt that there are any closed form solutions, though, which is what Project seems to be demanding.) I'd assume that Project could do the "writing down" part from any of several sources that have been discussed - the original paper, MTW's Gravitation, or even the old phyiscs forum thread

https://www.physicsforums.com/showthread.php?t=58291&highlight=pervect

And on that note, I think I've said everything useful that I have to say at the moment.