Title: Does action at distance in electromagnetism violate energy conservation?
Tags: electromagnetism;energy-conservation;classical-electrodynamics
Question: Consider two charges A and B separated at distance D. charge B is attached on spring and can move towards and away from charge A. Now charge A is brought closer to charge B and then it is taken back to its original position. Work done in this process is zero because of conservative forces. If this action is done in time interval less than D/c, then charge B does not feel any force or reaction during this time. Now as this reaction force reaches to charge B, it will oscillate. In this process work done on charge A is zero but charge B is having energy due to oscillation. Is energy conservation violated in this retarded action ?
Comment: By action at distance, I mean retarded interactions in electromagnetism.

Comment: Electromagnetism is not action at a distance.

Comment: conservation laws hold for inertial frames,. the action " charge A is brought closer to charge B and then it is taken back to its original position. " "brought closer and taken back" are non inertial frames.


Here is another answer: There is no "action at a distance" in electromagnetism, when you have the full theory. Charges affect the electromagnetic field, locally. The parts of the electromagnetic field affect other parts, locally. And the electromagnetic field affects charges, locally.
This is one of the consequences of Maxwell's equations in differential form:
\begin{align}
\nabla \cdot \mathbf{E} &amp; = \frac{\rho}{\epsilon_0}, &amp; \nabla \cdot \mathbf{B} &amp; = 0, \\
\nabla \times \mathbf{E} &amp; = -\frac{\partial \mathbf{B}}{\partial t}, &amp; \text{and } \nabla \times \mathbf{B} &amp; = \mu_0 \mathbf{J} + \mu_0\epsilon_0 \frac{\partial \mathbf{E}}{\partial t},
\end{align}
when combined with the formulae for the Lorentz force (density)
\begin{align}
\mathbf{F} &amp; = q\mathbf{E} + q\mathbf{v}\times\mathbf{B} \text{ or} \\
\mathbf{f} &amp; = \rho\mathbf{E} + \mathbf{J} \times \mathbf{B} \text{ (force density)}.
\end{align}
Energy only appears to be lost, to you, because you're not accounting for the energy and momentum carried by the electromagnetic field. You can look at the components of the electromagnetic stress-energy tensor to figure out how much energy, momentum, and stresses are in the electromagnetic field at any time. When you add up all of the energy and momentum, you'll find it's quite conserved.
Now, if you think that this is just a hack, it is the most straightforward model to understand how the pressure light sails experience is produced (recently demonstrated).

Here is another answer: Wheeler and Feynman developed the action-at-a-distance theory for
electromagnetic field theory. This theory is usually called as the
absorber theory (1945-1949) [1,2]. The absorber theory is build
on the the top of action-at-a-distance theory [3,4,5]. The absorber
theory is further developed by John Cramer as transactional interpretation
of quantum mechanics (1980) [6,7]. The key of the action-at-a-distance
theory,absorber theory, transactional interpretation is that the action
is build at least between two objects: the source and sink. The source
can be a transmitting antenna, a light source. The sink can be a receiving
antenna or light sink. The source can have radiation of the retarded
wave. The sink can have the radiation of the advanced wave.
A the above theory from action-at-a-distance to transactional theory
is qualitative theory but not a quantitative theory. Hence all these
theory have not associated to energy conservation.
However the theory of action-at-a-distance, absorber theory and transactional
interpretation theory is further developed by the mutual energy theorem [8,9,10] (1987)
and [11,12,13,14] (2017), which is a quantitative theory. The the mutual
energy theorem, the energy conservation law for N current elements
\begin{equation}
\sum_{i=1}^{N}\sum_{j=1,j\neq i}^{N}\intop_{t=-\infty}^{\infty}dt\iiint_{V}(\boldsymbol{J}_{i}\cdot\boldsymbol{E}_{j})dV\equiv\sum_{i=1}^{N}\sum_{j=1,j<i}\intop_{t=-\infty}^{\infty}dt\iiint_{V}(\boldsymbol{J}_{i}\cdot\boldsymbol{E}_{j}+\boldsymbol{J}_{j}\cdot\boldsymbol{E}_{i})dV=0\label{eq:10}
\end{equation}
The above conservation law is self-explanatory. Since $\iiint_{V}(\boldsymbol{J}_{i}\cdot\boldsymbol{E}_{j})dV$
is the power of current element $\boldsymbol{J}_{j}$ give to $\boldsymbol{J}_{i}$.
$\iiint_{V}(\boldsymbol{J}_{j}\cdot\boldsymbol{E}_{i})dV$ is the
power of current element $\boldsymbol{J}_{i}$ give to $\boldsymbol{J}_{j}$.
If $\boldsymbol{J}_{i}$ get some power $\boldsymbol{J}_{j}$ will
loss the same amount. Hence, in general there is,
\begin{equation}
\intop_{t=-\infty}^{\infty}dt\iiint_{V}(\boldsymbol{J}_{i}\cdot\boldsymbol{E}_{j})dV=-\intop_{t=-\infty}^{\infty}dt\iiint_{V}(\boldsymbol{J}_{j}\cdot\boldsymbol{E}_{i})dV\label{eq:20}
\end{equation}
Hence we have the above energy conservation law. In mutual energy
theory. The above energy conservation law is supported by the mutual
energy principle,
\begin{equation}
-\sum_{i=1}^{N}\sum_{j=1,j\neq i}^{N}\iint_{\Gamma}(\boldsymbol{E}_{i}\times\boldsymbol{H}_{j})\cdot\hat{n}d\Gamma=\sum_{i=1}^{N}\sum_{j=1,j\neq i}^{N}\iiint_{V}(\boldsymbol{J}_{i}\cdot\boldsymbol{E}_{j}+\boldsymbol{E}_{i}\cdot\frac{\partial}{\partial t}\boldsymbol{D}_{j}+\boldsymbol{H}_{i}\cdot\frac{\partial}{\partial t}\boldsymbol{B}_{j})dV\label{eq:30}
\end{equation}
Mutual energy principle is equivalent to N group of Maxwell equations,
\begin{equation}
\left(\begin{array}{c}
\nabla\times\boldsymbol{E}_{i}=-\frac{\partial}{\partial t}\boldsymbol{B}_{i}\\
\nabla\times\boldsymbol{H}_{i}=\boldsymbol{J}_{i}+\frac{\partial}{\partial t}\boldsymbol{D}_{i}
\end{array}\ \ \ \ \ \ \ \ i=1,\cdots N\right)\label{eq:40}
\end{equation}
That means the the mutual energy principle is the sufficient and necessary
condition of the above N group of Maxwell equations. Hence, the mutual
energy conservation law of action-at-a-distance theory should be the
above mutual energy principle.
It should be noticed that the above mutual energy principle as a energy
conservation law have major difference with the energy conservation
law of the traditional electromagnetic field theory. In traditional
electromagnetic field theory the energy conservation law is the Poynting
theorem, which is
\begin{equation}
-\iint_{\Gamma}(\boldsymbol{E}\times\boldsymbol{H})\cdot\hat{n}d\Gamma=\iiint_{V}(\boldsymbol{J}\cdot\boldsymbol{E}+\boldsymbol{E}\cdot\frac{\partial}{\partial t}\boldsymbol{D}+\boldsymbol{H}\cdot\frac{\partial}{\partial t}\boldsymbol{B})dV\label{eq:50}
\end{equation}
If there are N current elements $\boldsymbol{J}_{i}$ $i=1,...N$.
There should be superposition law
\begin{equation}
\boldsymbol{E}=\sum_{i=1}^{N}\boldsymbol{E}_{i}\label{eq:60}
\end{equation}
\begin{equation}
\boldsymbol{E}=\sum_{i=1}^{N}\boldsymbol{E}_{i}\label{eq:70}
\end{equation}
\begin{equation}
\boldsymbol{J}=\sum_{i=1}^{N}\boldsymbol{J}_{i}\label{eq:80}
\end{equation}
Substitute the above to the Poynting theorem we obtained,
\begin{equation}
-\sum_{i=1}^{N}\sum_{j=1}^{N}\iint_{\Gamma}(\boldsymbol{E}_{i}\times\boldsymbol{H}_{j})\cdot\hat{n}d\Gamma=\sum_{i=1}^{N}\sum_{j=1}^{N}\iiint_{V}(\boldsymbol{J}_{i}\cdot\boldsymbol{E}_{j}+\boldsymbol{E}_{i}\cdot\frac{\partial}{\partial t}\boldsymbol{D}_{j}+\boldsymbol{H}_{i}\cdot\frac{\partial}{\partial t}\boldsymbol{B}_{j})dV\label{eq:90}
\end{equation}
These can be called Poynting theorem of N current elements. This formula
is very close to the mutual energy principle, but there are clear
difference, the difference is at the summation. There is
\begin{equation}
\sum_{i=1}^{N}\sum_{j=1}^{N}\ \ \ \ \ \ vs\ \ \ \ \ \ \sum_{i=1}^{N}\sum_{j=1,j\neq i}^{N}\label{eq:100}
\end{equation}
The theory of Mutual energy believe the summation $\sum_{i=1}^{N}\sum_{j=1,j\neq i}^{N}$
is correct, and $\sum_{i=1}^{N}\sum_{j=1}^{N}$ is wrong. $\sum_{i=1}^{N}\sum_{j=1}^{N}$
over estimated the energy of the system. If we use $\sum_{i=1}^{N}\sum_{j=1}^{N}$
this summation the energy have additional terms which are,
\begin{equation}
-\sum_{i=1}^{N}\iint_{\Gamma}(\boldsymbol{E}_{i}\times\boldsymbol{H}_{i})\cdot\hat{n}d\Gamma=\sum_{i=1}^{N}\iiint_{V}(\boldsymbol{J}_{i}\cdot\boldsymbol{E}_{i}+\boldsymbol{E}_{i}\cdot\frac{\partial}{\partial t}\boldsymbol{D}_{i}+\boldsymbol{H}_{i}\cdot\frac{\partial}{\partial t}\boldsymbol{B}_{i})dV\label{eq:110}
\end{equation}
The above is all self-energy terms. The mutual energy theory believe
self-energy terms cannot transfer energy. Hence N current-element
Poynting theorem over-estimate the energy of the system.
From the follower of action-at-a-distance-theory:the mutual energy
theory offers a corrected the energy conservation law. The traditional
electromagnetic field theory offers a wrong Poynting theorem of N
current elements which over estimates the energy of the electromagnetic
system.
This over estimation is the cause of wave and particle duality problem
of electromagnetic field theory and quantum mechanics.
Any way the action-at-a-distance can offer correct energy conservation
law, but the traditional electromagnetic field theory especially the
Poynting theorem of N current elements offers an over-estimated energy
conservation law!
Reference:
[1] Wheeler. J. A. and Feynman. R. P. Rev. Mod. Phys., 17:157,
1945. https://authors.library.caltech.edu/11095/1/WHErmp45.pdf
[2] Wheeler. J. A. and Feynman. R. P. Rev. Mod. Phys., 21:425,
[3] K. Schwarzschild. Nachr. ges. Wiss. Gottingen, pages 128,132,
[4] H. Tetrode. Zeitschrift fuer Physik, 10:137, 1922.
[5] A. D. Fokker. Zeitschrift fuer Physik, 58:386, 1929.
[6] John Cramer. The transactional interpretation of quantum mechanics.
Reviews of Modern Physics, 58:647_688, 1986.
[7] John Cramer. An overview of the transactional interpretation.
International Journal of Theoretical Physics, 27:227, 1988.
[8] Shuang ren Zhao. The application of mutual energy theorem
in expansion of radiation fields in spherical waves. ACTA Electronica
Sinica, P.R. of China, 15(3):88_93, 1987.
[9] Shuangren Zhao. The application of mutual energy formula in
expansion of plane waves. Journal of Electronics, P. R. China, 11(2):204_208,
March 1989.
[10] Shuangren Zhao. The simplification of formulas of electromagnetic
fields by using mutual energy formula. Journal of Electronics, P.R.
of China, 11(1):73_77, January 1989.
[11] Shuang ren Zhao. A new interpretation of quantum physics:
Mutual energy flow interpretation. American Journal of Modern Physics
and Application, 4(3):12_23, 2017.
[12] Shuang-ren Zhao, Photon Can Be Described as the Normalized
Mutual Energy Flow. Journal of Modern Physics Vol.11 No.5, May 202.
DOI: 10.4236/jmp.2020.115043
[13] Shuang-ren Zhao, A solution for wave-particle duality using
the mutual energy principle corresponding to Schrödinger equation,
DOI - 10.1490/ptl.dxdoi.com/08-02tpl-sci
[14] Shuang-ren Zhao, Derive the Huygens Principle through the
mutual energy flow theorem, https://hal.archives-ouvertes.fr/hal-02270471

Here is another answer: No. There is energy in the electromagnetic field that you aren’t considering. When you include both the energy of moving charges and the field energy, energy is conserved in all electromagnetic interactions. So is linear momentum and angular momentum.
Electromagnetic fields not only make these quantities conserved globally: they make them conserved locally in any infinitesimal region of space you care to consider.
Comment for this answer: Oscillation energy produced at charge B is also depends on magnitude of charge B. Because more charge, more force and hence more oscillating amplitude of charge B. We can take as much large magnitude of charge B as we want. This does not affect radiation energy at charge A, but it will increase energy of oscillation of charge B. So, radiation energy can not account for whatever energy is produced at the charge B.
Comment for this answer: The total energy of the charged particles and the fields is conserved. This follows from Maxwells equation and the Lorentz force law. See [Wikipedia](https://en.wikipedia.org/wiki/Electromagnetic_stress–energy_tensor#Conservation_laws).

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