Retarded time and advanced time (and binning)

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Discussion Overview

The discussion revolves around the propagation of electromagnetic emissions through the atmosphere, specifically focusing on the concepts of retarded time and advanced time in relation to observer time. Participants explore the implications of multi-branch functions resulting from these time definitions and how to properly account for contributions to observer time when data is binned.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant describes the challenge of calculating observer time from retarded time when dealing with a charge distribution moving at relativistic speeds, noting the complexity introduced by varying refractive indices.
  • The participant expresses uncertainty about how to handle multiple retarded times that correspond to the same observer time, particularly in the context of data binning and sampling rates.
  • Another participant provides a link to a resource on the propagation of electromagnetic waves, suggesting it may help with basic understanding, but does not delve into the specifics of the original question.
  • A third participant indicates they will seek a better explanation and review relevant materials, implying a collaborative approach to understanding the topic.
  • A fourth participant suggests considering parametric equations and emphasizes the importance of identifying key points along the curve, though the connection to the original question remains vague.

Areas of Agreement / Disagreement

Participants do not reach a consensus on how to handle the contributions from multiple retarded times to the same observer time bin. There are varying levels of understanding and approaches suggested, but no definitive solutions are presented.

Contextual Notes

The discussion highlights potential limitations in the participants' understanding of the mathematical treatment of multi-branch functions and the implications of data binning on the results. Specific assumptions about the behavior of electromagnetic emissions in non-vacuum media are not fully explored.

complexconjugate
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TL;DR
How to calculate electromagnetic radiation contributions for a given observer
Hi!

I am dealing with the propagation of electromagnetic emissions through the atmosphere. To quickly outline:
Imagine some charge distribution flying at relativistic speeds and inducing a four-potential A. We define some retarded time at which the charges are at some positions, and calculate the observer time (the advanved time for the emission time) by integrating over the refractive index of the atmosphere. So far so good.

If you plot the retarded time t* over observer time t, you get these multi-branch functions, like in this plot:
zotero_3OPqowuHet.png

t is the observer time, t* is the emission time, t_B is just some plot relevant parameter. The different line styles correspond to different refractive indices (solid = realistic, dashed = 1, dotted = 1.0003).

If I want to calculate the contribution for an observer at time t, t* gives multiple possible solutions. If I want to go the other way and want to calculate where the emission for a given t* ends up, I feel like I can in essence just iterate over all my possible t* and tally up when in time it arrives.

As you can see in the plot, many retarded times are compressed into very short observer times and I am not sure how to deal with that properly. If I had infinite precision it would be fine, but my data will be binned in time, and (for the solid line) emissions from t* = 10³ to t* = 10² end up in the same t bin. Do I add all of these? That was my first instinct but then the bin values depend on my sampling rate of t*. Do I scale the contributions to each bin with dt*/dt or something similar? This feels both trivial and somehow just beyond my grasp...

Further, does someone have experience with taming these multibranch functions? Again, would I just add the different t* contributions into their shared t bin?

I hope my question is clear (and belongs here), let me know if you need more info.

PS, I didn't want to open several threads in a short time, but if anyone has Jefimenko's equations for non-vacuum media or literature in that direction, I'd be very grateful.

Cheers!
 
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complexconjugate said:
TL;DR Summary: How to calculate electromagnetic radiation contributions for a given observer
https://byjus.com/physics/propagation-of-electromagnetic-waves/#:~:text=The propagation of electromagnetic waves through the vacuum happens at,the speed in the vacuum.

This should definitely help with the basic understanding of it.

Also if you have some more time


This is a lecture that was given and it is just incredibly explained. Explaining stuff is not where I shine at, in order to explain something good you have to know it cold. As I have a basic understanding of it I don't know it cold so someone else will chime in shortly will a way better explanation also.
 
I'll also keep my eye out for a better explanation as I will be reviewing some materials.
 
My hint
Parametric equations
x=f(t), y=g(t)

Think about it,
I= is the interval of the perimeter in this case T
T= is the time

You've got to find the important points which are the ones at both ends of the line

X=__ •t•10³, Y=__•t•10²


Hope that helped some and again soon some smarter people should start to arrive and chime in.
 

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