Electromagnetic field according to relativity

[PeterDonis]

I was actually thinking of the difference between coordinate system and reference frame

What difference? The only definition of "reference frame" that I can see being used in this thread is the one that is synonymous with "coordinate system". If you have a different definition in mind, then please state it explicitly (and, ideally, give a reference for it).

often I only care about the relative velocity of my reference frames and some (or all) of the details of the spatial coordinates fall under "don't care".

"Don't care" is very different from "don't exist".

 

[Ibix]

What difference? The only definition of "reference frame" that I can see being used in this thread is the one that is synonymous with "coordinate system".

MTW refer me to local Lorentz frames on p. 217, which defines a frame as a coordinate system with orthonormal coordinates with coordinate basis such that the metric is $\eta_{\alpha\beta}$ and $\Gamma^\alpha_{\beta\gamma}=0$, plus some other restrictions that I think are only germane to curved spacetime. That is more restrictive than "coordinate system", more restrictive than "frame" as used by fog37 in post #23 (and I think implicitly accepted by you in your response) which allows spherical polars for the spatial coordinates, and less restrictive than the definition of "frame" implied by pervect in #9 where he relates frames by translations and does not mention rotations.

"Don't care" is very different from "don't exist".

Fair enough - but even in this thread there's a degree of slop in how people are using "frame". So learning to accept a degree of variation in definitions (i.e., not getting too hung up on the precise definition of a term when it may not be generally used) and recognising when that does and does not have consequences is an important skill. You may recall resolving my confusion over Kruskal diagrams by pointing out that different sources label regions III and IV differently...

 

[PeterDonis]

MTW refer me to local Lorentz frames

Yes, this is a particular kind of coordinate system/frame. It is not a general definition of "coordinate system" or "reference frame".

even in this thread there's a degree of slop in how people are using "frame"

Which is why I asked you to state explicitly what definition you are using.

not getting too hung up on the precise definition of a term when it may not be generally used

I don't think the term "reference frame" can be said to be not generally used.

I agree that sometimes differences in definitions don't matter and aren't worth getting hung up on. In this thread, however, I think confusion over the meanings of terms is contributing to the OP's confusion over the physics involved. So I think in this case it's worth being more precise and explicit about the definitions of terms, and also about the physical implications. I tried to do some of that in post #29.

 

[vanhees71]

I was actually thinking of the difference between coordinate system and reference frame when I wrote that, specifically in the context of a spatially rotated Cartesian system. Perhaps context did not make that as clear as I thought. They are absolutely different coordinate systems. Are they different reference frames? I'd think the answer to that one is a little vaguer, because often I only care about the relative velocity of my reference frames and some (or all) of the details of the spatial coordinates fall under "don't care".

Perhaps I am being a bit casual. But I've certainly written of two observers at rest with respect to each other having the same rest frame without considering details of their spatial coordinates and basis vectors.

I'd also say that inertial reference frames which just differ by a static spatial rotation and/or a static space-time translation are to be taken as different reference frames, because the components of four-vectors and four-tensors change, but that's also kind of semantics.

Further an observer defines local (not necessarily inertial) reference frames, because you can always define a set of tetrades along his time-like worldline with the temporal basis vector given by $u^{\mu}$ (his four-velocity with $u_{\mu} u^{\mu}=1$) and three Minkowski-orthonormal space-like vectors. You can even make these "non-rotating" by Fermi-walker transporting an arbitrary initial tetrade.

 

[PeterDonis]

an observer defines local (not necessarily inertial) reference frames, because you can always define a set of tetrades along his time-like worldline with the temporal basis vector given by $u^{\mu}$ (his four-velocity with $u_{\mu} u^{\mu}=1$) and three Minkowski-orthonormal space-like vectors.

Note that this is a different definition of "reference frame" (the more technical term is "frame field") from the one that has been used up to now in this thread (and which you also used in the first paragraph of your post). This definition has nothing to do with coordinates (it can be defined without even making a coordinate choice at all).

 

[vanhees71]

Sure, I just wanted to clarify that there's the possibility to define reference frames in connection with an observer. I think in relativity it's important to keep in mind what an observer really observes and how different observers moving relative to each other observe the same physical situation.

 

[Ibix]

I don't think the term "reference frame" can be said to be not generally used.

I was meaning that a precise definition of "reference frame" is not generally used (i.e., the term has several formal meanings, as you say, and probably a slightly broader use in practice), not that the term "reference frame" is not widely used.

It looks a bit like there's room for an Insight here about coordinates and reference frames. And probably their relationship to observables.

 

[PeterDonis]

It looks a bit like there's room for an Insight here about coordinates and reference frames. And probably their relationship to observables.

Yes, good idea. Are you volunteering to write it?

 

[Ibix]

Yes, good idea. Are you volunteering to write it?

Do I remember correctly that you used to be a navy officer...?

I can try drafting something. I suspect it'll need a lot of input from others.

 

[PeterDonis]

Do I remember correctly that you used to be a navy officer...?

Yes.

I can try drafting something. I suspect it'll need a lot of input from others.

The FAQ and Insight development forum can help:

https://www.physicsforums.com/forums/faq-and-insight-blog-development.208/

 

[eltodesukane]

"the actual EM field is always time-varying"
No.

 

[Grossglockner]

I will give you a DIFFERENT REPLY. Maxwell's mathematical model of Electromagnetism has been derived from three basic experiments.

ONE FORCE discovered a few hundred years ago is that the force between two charged particles is proportional to the square of the inverse of the distance between the charges. The charges where macroscopic objects. The charged objects consisted of atoms which consisted of protons and neutrons and electrons. The Protons and neutrons consist of Quarks Gluons etc. All these particles move. The particles in one objects respond to forces radiated by the particles in the other object. True, these forces are very small, but they are not equal to zero. Thus these measurements though they could have been carried out to a large accuracy were still approximations. Gauss formulate this force mathematically. This mathematical formulation was named after Gauss.

ANOTHER FORCE is that a magnetic compass needle can be deflected by an electric current from a battery passing through a wire. The wire is composed of all the above described particles. The discovery of this phenomenon is accredited to Ampere. The mathematical model describing this phenomena was named after Ampere. It too is an approximation.

YET A THIRD FORCE was formulated by Michael Faraday. Faraday noticed that a spark jumped between the ends of a wire loop that has not been closed all the way when the wire loop is moved between the poles of a permanent magnet. That is, the electrical voltage between the wire loop ends was large enough to ionize the air between the wire ands and produce a spark. The wire consists of all the above described particles. In the permanent magnet the direction of the spin of the charged electrons are aligned to produce a magnetic field. The electrons also orbit about the atomic nuclei this too contributes to the magnetic field, but less than the "spin" of the electrons. All this electronic motion is actually described by the Quantum Mechanical model of nature which gives a different result than the classical electron motion model. This effect is also an approximation. Faraday formulated a mathematical model of the wire loop moving in the field of a permanent magnet. This Mathematical model was named after Faraday.

Maxwell used Gauss', Ampere's and Faraday's mathematical descriptions to formulate a consistent Mathematical model of Classical Electromagnetism, the Gauss Maxwell equation, the Ampere Maxwell equation, and the Faraday Maxwell equation. There is a fourth Maxwell equation that is similar to the Gauss Maxwell equation but for Magnetic fields.

Maxwell's Potentials, Electric and Magnetic Fields are Mathematical Constructs to describe the origin of the observable forces.

I have been working on Electro-Optics (electromagnetic) for about 40 years. In a lot of what I did the Classical Maxwell equations don't give an answer in agreement with experimental measurements. One has to use a Quantum Mechanical description of electromagnetism, Quantum Electrodynamics.

TO ANSWER YOUR QUESTION, in these experiments Faraday's wire loop had to move to produce a measurable effect. The charged particles (electrons) in Ampere's wire, too, had to move to deflect the magnetic needle.
Gauss' experiment that deals with charges and static electric fields does not require motion.

Thus, Maxwell's Electromagnetic Model, and all other our Models of Nature are just approximations. These models describe how Nature works to a Large accuracy for a range of Nature. But these Models are not absolute. The Mathematical models are some times subjects to contradictions. This will occur when the model is outside the Region of Nature where it is meant to apply. One should investigate the contradictions. But, again our Mathematical Models of Nature are not absolute God given laws, they are only approximations!

Philipp Kornreich
Professor Emeritus, Syracuse University.
E-Mail: pkornrei@syr.edu

 

[fog37]

Hello again,
Thank you for all your comments and teachings. I would like to summarize some of the key concepts discussed in this thread. I am still missing some clarity on certain definitions. Let's see...

1) Reference frame and coordinate system
A frame of reference is a construct represented by a set of physical reference points that at rest relative to each other. A reference frame is used to describe motion or physical phenomena from the point of view of an observer. I tend to think of observer and reference frame as synonym (habit hard to eradicate).

A reference frame is always equipped with both:

a) An arbitrary reference point called origin $O$ where we can imagine the observer to be located.

b) Coordinate system (this implicitly means a basis of independent vectors which can be orthogonal or not, unit length or not). The coordinate system serves to make space and time measurements by assigning unique coordinates (labels) to each space-time point. Examples: Cartesian, paraboloidal, conical, parabolic, prolate spheroidal, cylindrical, spherical, etc.). The points are the same, their labels change. Coordinate systems can be either global or local. Global means that the basis vectors don't change direction at different spatial points.

Examples:

• Two Cartesian coordinate systems at rest w.r.t. each other and with different origins $O# and$O'## represent different reference frames. Frames with different origins are always different reference frames.
• Two frames with the same origin but using different coordinate systems represent the same or a different reference frame?
• Two inertial reference frames (in relative motion at constant velocity) with the same coordinate system are different reference frames.

2) The electromagnetic field tensor F

• In classical electromagnetism, the electric and magnetic fields are considered as separate vectors
• In special relativity and four-space, the B and E fields are not vector fields: their components are the components of the antisymmetric Faraday field tensor Fμν (which truly has 6 distinct components).
• The components of E and B change from one inertial frame to another according to these equations (inertial frame moving along x direction):

The velocity v is the inertial observer's velocity. How are these transformation equations manipulated when we are considering the case of electrostatics, i.e. when there is only a static E field and zero B field?

Any corrections are welcome. Thank you.

 

[vanhees71]

That's a good question, but easy to answer by yourself just taking the above transformation formulae for the field components. Just do it for a Coulomb field of a point particle as and exercise (note that you have to finally express the $\vec{E}'$ and $\vec{B}'$ also in terms of the $(t',\vec{x}')$ for the spacetime components!).

 

[fog37]

Thank you vanhees71, will do.

Any major issue with how I understand the idea of reference frame/coordinate system? PeterDonis seemed, unless I misunderstood, to state that reference frame and coordinate system are somewhat synonyms so I have been reflecting on that...

 

[pervect]

Thank you vanhees71, will do.

Any major issue with how I understand the idea of reference frame/coordinate system? PeterDonis seemed, unless I misunderstood, to state that reference frame and coordinate system are somewhat synonyms so I have been reflecting on that...

I don't see any major issues if you restrict yourself to special relativity, i.e. flat space-time. You'll start to see issues if you try and extend this frame-work as-is to curved spaces or space-times.

There could be things I'm missing, but I don't see any "red flags" in the correct domain of applicability.

 

[PeterDonis]

A frame of reference is a construct represented by a set of physical reference points that at rest relative to each other.

This is still not correct as a matter of terminology, although you have captured some important concepts.

As has already been explained, there are two different meanings of the term "reference frame" that appear to be relevant for this discussion:

(1) One meaning of "reference frame" is simply as a synonym for "coordinate system". A coordinate system is an assignment of a 4-tuple of real numbers to each spacetime point, with certain technical requirements such as continuity that I don't think we need to go into detail about. From a coordinate system, it is always possible to derive a "coordinate basis", i.e., a set of 4 basis vectors at each spacetime point; however, such a basis will in general not be orthonormal, nor will it in general have any direct physical meaning. Also, the basis is not part of the definition of a coordinate system; it is simply something which can be derived if you have a coordinate system.

(2) The other meaning of "reference frame" is "frame field". A frame field is an assignment of an orthonormal tetrad (a "tetrad" is a set of four mutually orthogonal unit vectors, one timelike and three spacelike) to each spacetime point. Generally, such an assignment is made by finding a family of timelike worldlines that foliate the spacetime (or some region of the spacetime that is of interest); by "foliate" we mean that every point in the spacetime (or region) lies on exactly one worldline in the family. (The technical term for such a family of worldlines is "timelike congruence".) Then, at each point, the timelike unit vector of the tetrad is the tangent vector of the worldline that passes through the point, which can be interpreted as the 4-velocity of an observer following the worldline, and the spacelike unit vectors of the tetrad are assigned to describe three mutually orthogonal directions that the observer uses as axes for spatial measurements at that point, again with certain technical details about how the spacelike unit vectors of the tetrad are transported along the worldline that I don't think we need to go into detail about.

You seem to be combining elements of both of the above definitions into your concept of "reference frame". I would advise strongly against that. The two definitions are conceptually distinct and serve different purposes. Coordinate systems are very useful for actual mathematical computations, but they are often very poor or misleading guides for physical interpretation. Frame fields have an obvious direct physical interpretation in terms of a family of observers, but do not require any coordinates at all for their definition. Of course if you already have a coordinate system defined, you can always describe any frame field in terms of that coordinate system (by simply expressing the vectors of the tetrads in the chosen coordinates). However, the converse is not true: a frame field does not, in general, define a single global coordinate system on the spacetime (or region of spacetime) that it covers (although you can of course use a single tetrad at a given point to define local coordinates covering a small patch of spacetime centered on that point).

 

[fog37]

Thank you PeterDonis.

I guess I am still thinking of the basic concepts introduced in classical physics and special relativity (flat space-time). General relativity and its subtleties are still out of my league. That means that your point 1) is clear to me while 2) I am still digesting...

The concepts of reference frame and coordinate system are initially introduced in classical mechanics (dynamics and kinematics), which describes the slow world with speeds smaller than $c$. Reference frames, either inertial or noninertial, are introduced independently of the coordinate system that is chosen. For example, let's consider someone on a spacecraft and someone on planet earth. Both are observers that are, respectively, at rest with the spacecraft and at rest relative to planet earth. They can choose to describe the reality around them using whatever coordinate system they like (rectangular, cylindrical, spherical, etc.). The basis of unit vectors, which are perpendicular to coordinate surfaces, depend on which coordinate system we are using. At least this is what I am grasping from my mechanics book...

Thanks.

 

[PeterDonis]

Reference frames, either inertial or noninertial, are introduced independently of the coordinate system that is chosen.

At this point I think it would be helpful if you cited a specific reference that presents reference frames the way you describe.

At least this is what I am grasping from my mechanics book...

Which book?

 

[PeterDonis]

Reference frames, either inertial or noninertial, are introduced independently of the coordinate system that is chosen.

Here, again, you appear to be using "reference frame" to mean "observer". I strongly advise against conflating these two concepts. They are not the same. Note that neither meaning of "reference frame" that I described is tied to a single observer: coordinate systems aren't tied to any observer, and frame fields, while they can be defined using a family of observers, do not pick out any particular observer in that family.

Also, even given a single observer, the results of measurements that observer makes will depend on more than just which observer it is. The observer might have spatial axes oriented different ways, which would correspond to different tetrads if we are using frame fields, or different coordinate axes if we are using coordinates and want to align them with the observer's physical measuring devices. (Note that this orientation difference is independent of whether, for example, we are using Cartesian or polar coordinates; either way we still have to define axes as references, whether they are references for Cartesian distance coordinates or polar angular coordinates.)

 

[vanhees71]

I'm not sure, whether we discuss SR or GR.

In GR we deal with a differentiable pseudo-Riemannian manifold. There coordinates are usually defined on an open subset of the manifold (a socalled map). Then you have several maps covering the entire manifold. On the overlaps between any two maps there's a (local) diffeomorphism between the corresponding coordinates. All maps together build an atlas. Anything physical (or geometrical) is independent of the choice of maps and atlasses though and thus defined via invariant properties of tensors and tensor fields.

As @PeterDonis said above, the coordinates defined on a map always imply (local) reference frames, namely a set of basis vectors in the tangent space at any point covered by the map defined by the coordinate lines through this point. The most simple way to describe this reference frame is to use the coordinates of the map and the holonomous basis and co-basis vectors and the tensor components with respect to this holonomous basis. This is what's usually taught first in textbooks like Landau and Lifshitz vol. 2, and you get very far with this description in SR.

Sometimes you however need more sophisticated tools as the concept of frame fields also described by @PeterDonis above. There at any point you can define a time-like worldline through this point and construct a pseudo-orthonormal set of (tangent) basis vectors. This you need, e.g., to define spinors and spinor fields in GR. In SR that's not a big issue, because there you usually work with a global orthonormal basis defining at the same time a global Galilean reference frame.

 

[fog37]

Thank you!