I read on some forum that the electric field of an accelerated charge in its rest frame is the same as for a static charge in an inertial frame. This doesn't make any sense to me, because that would mean that the charge does not radiate in its rest frame. However, radiation can be seen as photons, particles of light. And it seems to me that if a particle exists in one frame, it must exist in all frames, accelerated or not.
So I would answer an accelerated charge radiates in all frames, whether the frame of observation is accelerated or not.
Does an accelerated charged....?
The answer of this question may be given when we know the relations between the space-time coordinates of the accelerating frame and the rest frame.
Regards
Ramesh Chandra
I meant that the charge is accelerated with respect to some inertial frame F, so we know that it radiates in F. Now, the frame F' attached to the particle is accelerated. In F' the particle is at rest, hence not accelerated. Does it radiate in F'?
Dear V. G. Rousseau,
The frame F is not an inertial frame whereas, the frame F' is an inertial frame for the charged particle.
Regards
Ramesh Chandra
In this problem the frame F is inertial by definition. In other words, it is a frame where Newton's 1st law holds. Now, by definition of the problem, the charge, and its rest frame F', are accelerated with respect to F. I agree that F is accelerated from the viewpoint of the charge, but F still remains an inertial frame, whereas F' is not. The charge must feel an acceleration. Does it radiates in F'?
Dear V. G. Rousseau,
What are the relations between the coordinates of the frames F and F' ?
Regards
Ramesh Chandra
Well, by definition the coordinates of F' as expressed in F are the coordinates of the charge (ct,x, y, z), since F' is the rest frame of the particle. Now x, y, z are non-linear functions of t since the particle is accelerated. In F', the coordinates of the particle are (ct',0,0,0), so t' is the proper time of the particle. Therefore x'=y'=z'=0. Now the relationship between t and t' is not a simple dilation formula, since there is an acceleration. I guess that instead the dilation formula needs to be integrated.
Lienard-Wiechert fields can be broken to two parts (See Jackson). Velocity dependent and acceleration dependent. If you choose a reference frame where acceleration vanishes, you still will get a non vanishing velocity dependent part.
The location of the observer is completely ignored in your discussion. You cannot have F’ as the rest frame of the charged particle with the observer sitting in another frame F. And to get the observer to sit in F’ is impossible because the charged particle travels at fraction of the speed of light. So in reality there is no such thing as a rest frame of the charged particle, because it is never at rest with respect to the observer. This is the reason we have zero point energy and the mass gap.
@ Vikram Zaveri
I'm talking here about a though experiment. It doesn't matter whether the speed of the charge is close to the speed of light or not, we can always imagine an observer attached to it. Actually, it doesn't even have to be a thought experiment. We can imagine an astronaut in a rocket that carries a static charge and that is accelerating towards Mars. In his rocket, the charge is not accelerated. For us on Earth, the charge is accelerated, and according to Maxwell equations, it must radiate. My question is whether the astronaut sees radiations or not.
@ V. G. Rousseau,
The question of this thread is no doubt interesting.I think the answer may not be found with imaginary speculative physics but the question may lead one to reality. Maxwell’s equations does not account for quantum aspect of the radiation. When a charge is accelerated, it may not radiate till certain quantum of energy is built up, the acceleration experienced by the rocket cannot be significant to make the charged particle eject a quanta. So neither people on earth nor the astronaut would see any radiation. Mathematical analysis of such arguments can eliminate many speculations.
Dear V. G. Rousseau,
If it is necessary to use the kinematics of classical physics, then we cannot consider that an electric charge (even in non accelerated motion) always generates an electric field in the reference frame where it is continuously at rest because the formulas of transformations do not make it possible to obtain a magnetic field in a reference frame where the source is moving.
And if we do not have to use classical kinematics then we must first specify how we define an electric field in a reference frame where a particle is continually accelerated because it does not make sense neither in special relativity nor in general relativity.
If the accelerated charge has been under constant acceleration since eternity and continues with this until eternity, in a Minkowsky universe, no radiation will be observed. Just like a charge at rest near the surface of earth, experiencing our constant acceleration of gravity, will not be seen to radiate. This becomes different when the acceleration is time dependent, in direction or magnitude (or both). Some analysis of the forever constantly accelerating system can be done, since this (with some qualifications) is equivalent to a charge at rest. Due to the invariance of Maxwell's equations under conformal transformations.
Dear Kare,
yes, provided that the application of the Equivalence Principle holds...
QUESTION: "So I would answer an accelerated charge radiates in all frames, whether the frame of observation is accelerated or not."
the acceleration has to be non stationary (DIPOLE)... or rather radiation depends on its derivative, then it is obvious that photons are emitted and their existence is frame independent.
An accelerated charge doesn't have a globally defined rest frame; that's what ``accelerated'' means-that's why it radiates. A uniformly moving charge does have a globally defined rest frame. The calculations can be found here: https://www.rand.org/content/dam/rand/pubs/research_memoranda/2006/RM2820.pdf
Stefano> ...provided that the application of the Equivalence Principle holds...
For this case I don't give a sh*t about the equivalence principle! All that needs to be done is to solve the Maxwell equations in the relevant coordinate system(s). For forever constant acceleration this is Rindler coordinates (which only covers a wedge of Minkowski space, a Rindler universe), and their relation to Minkowski coordinates.
And it should^* not even be necessary to solve the Maxwell equations: It should be sufficient to find a solution ansatz with the help of a conformal transformation of the Coulomb electric field around a charge at rest, and check by insertion that this ansatz really solves the Maxwell equations. I phrase it this way because a conformal transformation of the point charge leads to the electromagnetic fields from two charges, of opposite signs, under constant acceleration. But these charges live in two no-communicating Rindler universes^**, so it should be possible to separate their respective contributions.
^*) I use a lot of should here, since I have not carried out the calculations, only imagined how they can be done.
In general, I would say that the Equivalence Principle, as a guiding principle for model building, now mostly belongs to the history of science. The current standard method is to write down an action functional for the system under scrutiny, imposing symmetry principles like position and orientation invariance of the action, and (in the classical case) derive a consistent set of equations of motions for the system by use of the principle of stationary action. Through this formalism one may derive local equivalence principles, to their extent they are valid, by analysing the equations in an evolving set of local Lorentz frames, with corrections which depend on distance from a selected world-line.
Note added: I did a fast search to find a huge literature on this, starting with a 1909 paper by Max Born, Annalen der Physik 30, 1 (1909). Some other relevant papers are Bondi and Gold, Proceedings of the Royal Society of London. series A, Mathematical and Physical Sciences 229, 416-424 (1955), Fulton and Rohrlich, Annals of Physics 9, 499-517 (1960), and David G. Boulware, Annals of Physics 124, 169-188 (1980). A more recent paper is Camila de AlmeidaAlberto Saa, American Journal of Physics 74, 154 (2006), with many more references, as also can be found in the other papers.
I seems to me that the difference in conclusions can be traced to the non-uniqueness of solutions to the Maxwell equations (unless one specifies initial conditions). The conformal transformation does lead to a solution, but not necessarily a physically acceptable one.
^**) This is not correct, for all relevant wedges of Minkowski space.
The accelerated observer will not see any radiation, and an inertial observer must see something which is consistent with this (by just applying a coordinate transformation). But there is a region (wedge) of Minkowski spacetime which are not accessible to the accelerated observer, but to inertial ones, and for which it makes a difference whether we have one (the real problem) or two (the conformally transformed problem) accelerated charges. So there is a region for radiation to go, unnoticed by the accelerated observer.
There doesn't exist a globally defined Lorentz transformation, that maps the worldline of a non-uniformly moving point electric charge to the worldline of a uniformly moving charge and, also, makes the electric and magnetic fields, for which the charge is the source, vanish everywhere (and everywhere includes infinity, since the boundary conditions are important-that's what ``globally defined'' means).
The above paragraph is the rewriting of the statement ``the electric charge doesn't radiate''.
And one very easy way to see this is that a globally defined Lorentz transformation that maps the worldline of a non-uniformly moving electric charge to the worldline of a uniformly moving charge doesn't exist. This can be shown by writing the most general global Lorentz transformation and imposing the corresponding conditions. And the reasons ought to be well known.
This is a homework problem in classical electrodynamics, whose solution, by another method, is in Coleman's paper.
The correct formulation is that there doesn't exist a globally defined frame, where the charge doesn't radiate.
Locally, at any given point, it's possible to define the corresponding inertial frame, where the charge is at rest and, therefore, doesn't radiate.
This point is made very nicely in Coleman's paper, https://www.rand.org/content/dam/rand/pubs/research_memoranda/2006/RM2820.pdf (that discusses constant acceleration in detail, too).
Dear Kare,
"The current standard method is to write down an action functional for the system under scrutiny "
I agree, I wrote ironically about the application of the EP...or worse the EEP..
"Through this formalism one may derive local equivalence principles, to their extent they are valid, . "
virtually never valid..
"I did a fast search to find a huge literature on this, starting with a 1909 paper by Max Born, Annalen der Physik 30, 1 (1909). "
yes, there was already a similar question in RG reporting a lot of papers on the subject..
In principle it is an ill posed problem, the boundaries are not defined: infact any physical problem has a beginning and one end, a charge cannot be accelerated "ad libitum" to infinite speed, sooner or later it has to slow down and will emit radiation (if stopped it will emit a lot of radiation, the energy expressed by the larmor formula which was not previously emitted by the charge)
While constantly accelerated, the charge should not radiate, since any radiation has a frequency and
while the charge is accelerated: how would it possibl eto associate a frequency, on which base? Should it radiate at all frequencies??
At what frequency, if not at all frequencies? What quanta of radiation should it emit at a generic acceleration g?
Stefano> virtually never valid..
If we consider small systems, like the effects of gravity on a Cesium 133 atom, I think we can consider the acceleration of gravity equivalent to the acceleration in a rocket in flat space; in both cases of negligible magnitude.
Stefano> ... sooner or later it has to slow down and will emit radiation
Yes, and also when the acceleration is turned on. If the constant acceleration phase last a long time, it is possible to believe that an observer following the acceleration will measure an essentially constant field around the charge. From an inertial frame this should look like an electromagnetic field bound to the charge, but with a time-dependent shape. But one would have to make the calculations, or investigate the many formulas available in the literature, to be sure.
Dear Kare,
So basically non-inertiality, for charges, does not necessarily means emission of EMwaves,
" From an inertial frame this should look like an electromagnetic field bound to the charge, but with a time-dependent shape "
though there is always a change in the EM fields detected from an IRF, which thing in a static gravitational configuration does not occur...
An observer, in free fall along with a charge, doesn't detect radiation, precisely because the observer can't detect that the charge is moving non-uniformly.
The statement that a charge is moving non-uniformly means that it's moving non-uniformly with respect to the observer that's doing the observing.
Therefore the statement that non-inertial motion of charges doesn't imply production of electromagnetic waves is wrong.
Just do the calculation. You'll find that the electromagnetic field of charges, whose motion isn't inertial, is given by expressions that are superpositions of electromagnetic waves. So yes, non-inertial motion of the charges means that the electromagnetic field isn't static and that means that it is described by electromagnetic waves.
Then do a general coordinate transformation and find the fields in any other frame. There's nothing conceptually new about this.
The reason is trivial: Maxwell's equations are wave equations with source terms-and for prescribed sources, they're linear. So it's completely straightforward to obtain the solution for uniform motion of a point source as well as for a uniformly accelerating point source-Coleman's paper does the calculations in detail, if one wants to look them up and not do them for oneself.
(Incidentally, the reason static electric and magnetic fields aren't considered as such is because it's well known that they aren't stable configurations, by themselves.)
Dear Stam,
so let me understand, you affirm that a uniformly accelerated charge will shoot EM radiation towards an IRF due to its accelerated motion in any case.
"So yes, non-inertial motion of the charges means that the electromagnetic field isn't static and that means that it is described by electromagnetic waves."
sorry but I do not agree with the latest. An electromagnetic field which is not static is not necessarily described by EM waves at all, it is the EM wave that is described by the means of a particular arrangement of E and M fields.
The EM waves are a consequence of the transitories (of a non stationary) of Electric or Magnetic field.
In other words if the phenomenon is totally described through a Fourier transform there is no emission of waves, if it is described by a Laplace transform it has transitories which necessarily involve emission of waves, the energy is released in form of waves.
Dear V. G. Rousseau, you wrote:
>
Because I am interested in this subject (https://www.researchgate.net/post/Has_this_interpretation_of_the_Machs_principle_already_been_explored), can you specify the source of this statement?
No, I don't remember, that was 3 months ago on a random forum that popped when I did a Google search. Sorry!
About Maxwell's equations
During the year 2002, after studying electrostatic, magnetostatic, electromagnetism and special relativity, I am very disturbed by the physical interpretation of Maxwell's equations in the presence of sources (and the continuity equation) which is used to construct the Poynting theorem concerning the conservation of energy in an electromagnetic field. I decide to perform calculations where the Maxwell equations in vacuum will no longer be considered as a postulate, but will be sought in the case of a point source:
- by using the known laws for an Alpha experimenter who finds that the source is constantly at rest
- by using formulas of transformations of the physical quantities between the measurements carried out by Alpha and those carried out by a Beta experimenter who notes a motion of the point source.
\Step 1. In fact, when the motion of Alpha with respect to Beta is rectilinear and uniform, it is a question of using Coulomb's law, the fundamental relation of the dynamic (appropriate modification of Newton's second law) establishes in special relativity, and the Lorentz transformation of space and time coordinates. And without any surprise, we obtain the expression of the Lorentz electromagnetic force and Maxwell equations in a vacuum.
This is not surprising since, in the article "The Principle of Relativity and the Fundamental Equations of Mechanics (1906)", Max Planck establishes the
appropriate modification of Newton's second law in order to ensure compatibility between Maxwell's theory in a vacuum and the principle of relativity. He uses in his calculations the transformation of electric and magnetic fields that had been established by Albert Einstein in his 1905 article to ensure the covariance of the Maxwell-Ampère and Maxwell-Faraday equations in a vacuum. However, we can notice that all these developments do not require to make very general considerations on the nature of the motion of the sources which must generate electromagnetic fields in Galilean reference systems. Everything would be the same if no source was accelerated.
\Step 2. If it was possible to generalize the Lorentz transformation so that each experimenter could be associated with a single three-dimensional space where he can state that a body is constantly at rest or not, then one could replicate algorithm of step 1 to deduce the nature of the fields generated by a source whose motion is not rectilinear and uniform with respect to the observer.
We can quickly notice that the mathematical model chosen by the theory of general relativity to study families of observers with arbitrary motions (end-to-end connection of locally Euclidean spaces to cover the region to be studied) does not allow to conceive the sought generalization of Lorentz transformation.
\Step 3. We can already do some calculations while waiting for the completion of step 2. In a Galilean frame of reference, we can consider a point source whose motion is rectilinear and uniform with velocity v_{a} in the time interval ]t_{a}, t_{b}[ , and with velocity v_{c} in the time interval ]t_{c}, t_{d}[ , and with velocity v_{e} in the time interval ]t_{e}, t_{f}[ ...
To determine the action of this source on a test particle, one must be able to reproduce the algorithm of step 1 on each of the intervals where the source has a uniform motion, provided you take into account the time required for the propagation of the influence between the source and the test particle. Thus, for the sake of approximation, when the motion of the source is arbitrary, it is supposed that its influence on test particles is always of electromagnetic nature (Lorentz force on the test particles), and that the force vector exerted at an event on a test particle depends on the instantaneous velocity that the source had at the event where was emitted the influence that has propagated to the receiver. We then obtain that the electromagnetic field in the vacuum verifies the Maxwell equations said in the presence of sources but the quantities which represented the densities of charge and current become functions of the acceleration of the source.
We can notice that in the classical theory, we assume not only that the fields are always of electromagnetic nature (Lorentz force on the test particles) independently of the nature of the motion of the source, but in addition we suppose that the Maxwell vacuum equations are still valid and that is how the Liénard-Wiechert potentials are obtained:
October 2003: https://forums.futura-sciences.com/physique/3632-electromagnetisme-relativiste.html
October 2003: https://forums.futura-sciences.com/physique/3630-relativite-restreinte-generale.html
December 2003: https://forums.futura-sciences.com/physique/4559-changement-de-referentiel.html
April 2019: https://www.researchgate.net/post/Can_Maxwells_equations_be_excluded_from_electromagnetism2
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