Lorentz transformation in non-inertial frame

[danjan1234]

I am doing some study about the lorentz transformation between non-inertial frames. I wonder if the tranformation is the same as in SR. I need to do the transformation of EM fields in a constantly rotational frame. Can anybody help me with this one. Is there anybook I can refer to? Many thanks.

 

[JustinLevy]

I am doing some study about the lorentz transformation between non-inertial frames. I wonder if the tranformation is the same as in SR.

Lorentz transformations relate inertial coordinate systems. If you start with an inertial coordinate system and apply a Lorentz transformation... you get another inertial coordinate system. In that sense (and I assume that is what you are getting at), no, the transformation to non-inertial frames are not the lorentz transformations.

However, just in case I assumed wrong (you did say "between" non-inertial frames), yes non-inertial frames can be related by lorentz transformations. Start with a non-inertial coordinate system, and apply a lorentz transformation and you will have another non-inertial coordinate system.

I need to do the transformation of EM fields in a constantly rotational frame. Can anybody help me with this one. Is there anybook I can refer to? Many thanks.

That gets very messy very quick. So this is important: Are you trying to solve problems in which things are rotating and involve EM interactions? Or are you truly interested in the math / the transformations themselves?

For the first, it is best just to analyze everything using inertial coordinates.

For the later, you need to first decide precisely what you even consider an EM field, for it gets nasty after this. Often one chooses either the components of the contravarient EM tensor, or the covarient EM tensor to define the fields. You choice is basically whether the maxwell source equations define the fields, or the source-free equations define the fields in some sense. It should also be noted that EM force equation on a charged particle will no longer have the same form either ... it is usually not possible to define the fields such that this form is retained (treating them as vector fields that is).

Furthermore, since you are dealing with non-linear transformations, the components of the metric will not be constant with respect to spacetime position. Therefore you also need to use covarient derivatives instead of regular derivatives.

In the end you will get the same prediction for experiments that you would obtain if you just had worked it out in an inertial coordinate system. So it really is worth asking what you are working towards here.

 

[atyy]

I agree with JustinLevy that this sounds like something a masochist would enjoy! Just in case that's really your idea of fun:

Relativistic contraction and related effects in noninertial frames
H. Nikolic
http://arxiv.org/abs/gr-qc/9904078

The discussion of the Sagnac effect may be relevant:
Relativity in the Global Positioning System
Neil Ashby
http://relativity.livingreviews.org/Articles/lrr-2003-1/

Gron and Hervik also discuss rotating frames in "Einstein's General Theory of Relativity: With Modern Applications in Cosmology" (Springer 2007).

 

[danjan1234]

Thank you guys for the replies. Actually, I am just trying to do the transformation for E&M fields. I think since they have no mass, there should be no difference between the transformations from inertial to non-inertial and from inertial to another inertial.

 

[Meir Achuz]

The LT for EM fields cannot be applied to non-inertial systems.

 

[danjan1234]

So. May I ask what is the transformation for EM fields from inertial to noninertial, for example, rotational frame.

 

[JustinLevy]

I think since [the E&M fields] have no mass, there should be no difference between the transformations from inertial to non-inertial and from inertial to another inertial.

You appear to not be understanding what a coordinate system even is. This probably stems from the shorthand that many of us use when we appear to define a coordinate system by just giving an object (clock A, an observer, the earth, a rocket, etc.). I can see how that would be misleading if you don't understand the implied conventions and definitions behind it.

So let's start there:
A coordinate system is a systematic way of labelling points in spacetime.

If an object is moving inertially, when we refer to the inertial system "of" that object, the implied convention is that we mean some inertial coordinate system in which the object is at rest. However, we can describe the motion of this object using any coordinate system, even a non-inertial one. A coordinate system is merely a choice.

So the coordinate system being inertial or non-inertial has NOTHING to do with whether you are describing the motion of inertial or non-inertial objects ... or whether the objects have mass or not (unless you are considering GR effects).

So. May I ask what is the transformation for EM fields from inertial to noninertial, for example, rotational frame.

Please reread my first post, for if you want to describe things with a non-inertial coordinate system you need to specify how you are defining the electric or magnetic field.

Also please let us know your current background:
Are you comfortable with metrics, four-vectors, and tensors?
Do you know what covarient vs contravarient is?
Do you know what a covarient derivative is?

This will help people know at what level to start their answers.

 

[danjan1234]

Also please let us know your current background:
Are you comfortable with metrics, four-vectors, and tensors?
Do you know what covarient vs contravarient is?
Do you know what a covarient derivative is?

I have seen these before when taking the classes. But I have never used then since that. Actually, I am only interested in the transformation itself, cause I need it in part of my reserch. Basically, I need to do the transformation of EM fields from the lab frame to the rotational frame(nonrelativistic).
First I thought I should do the relativisitc transforamtion and then take the low speed limit. But according to what you guys talked about, it's going to be painstaking. So I just would like to ask for the transformation form lab to rotational in nonrelativistic case.

 

[atyy]

You can do any transformation you want.

An inertial frame is simply one in which Maxwell's equations have their "standard form". Many inertial frames exist, all moving at constant velocity relative to each other. The Lorentz transformation gives the relationship between different inertial frames.

If you transform from an inertial frame to a non-inertial frame, the transformed Maxwell's equations will not have their "standard form". But you can do it, just make sure you transform the equations along with your coordinates.

 

[MikeLizzi]

“Because Einstein developed a whole new theory (his general theory of relativity, published in 1916) based upon the dynamical equivalence of an accelerated laboratory and a laboratory in a gravitational field, it is sometimes stated or implied that special relativity is not competent to deal with accelerated motions. This is a misconception. We can meaningfully discuss a displacement and all its time derivatives within the context of the Lorentz transformations.”**

**Page 153, "Special Relativity", by A.P. French, copyright 1968

So are the posters agreeing with A.P. French or not?

 

[atyy]

“Because Einstein developed a whole new theory (his general theory of relativity, published in 1916) based upon the dynamical equivalence of an accelerated laboratory and a laboratory in a gravitational field, it is sometimes stated or implied that special relativity is not competent to deal with accelerated motions. This is a misconception. We can meaningfully discuss a displacement and all its time derivatives within the context of the Lorentz transformations.”**

**Page 153, "Special Relativity", by A.P. French, copyright 1968

So are the posters agreeing with A.P. French or not?

Yes! In fact that is what Justin Levy first suggested - rather than use two frames, one Lorentz, and one rotating, to deal with the problem - it is probably easier to deal with rotations (which are accelerations) in a single Lorentz frame.

 

[JustinLevy]

If you are looking for things in the non-relativistic limit, you can just use the instantaneous inertial frame for a particular observer. That should work fine.

If not, can you give more details of what exactly you are trying to do?
It seems strange to insist on coordinate system dependent quantities in a non-inertial frame defined with non-linear transformations (really, why do you want such coordinate system dependent quantities?...they have little physical meaning here).

 

[MikeLizzi]

In the paragraphs following the above quote on pages 153 and 154 of French's book he derives a transformation for a' --> a.

I understand how to use those equations to determine the acceleration of a particle with respect to one ineritial ref frame when I know its acceleration in another ineritial ref frame.

But the original poster is trying to determine acceleration in a rotating ref frame (noninertial) when all he has are the params in an inertial ref frame. And in that ref frame there is no acceleration. I don't want to divert the course of this thread, but i folks are working up to describing how to do this, I would appreciate it.

...Looks like someone posted the answer while I wasn't looking...

 

[danjan1234]

Thank you guyes for all the posters.
I think the picture is getting clear to me.
I am thinking about a very simple case: A frame S has some acceleration a wrt it's co-moving inertial frame S'. The only difference b/t S and S' is the acceleration. Suppose there are some E&M fields in S, what are they going to be in S'.
I know there is definitely some difference. One example is that when I fix a charge in S, there is no B field in S. But in S', B exit due to radiation. LT doesn't apply since both frames have the same velocity, and the results given by LT will be the same.