Retarded and advanced potentials

[menniandscience]

"of the two types of solutions which the Maxwell equations yield for the wave
equation, the retarded and advanced potentials, only the retarded field seems
to have a physical meaning,"

let's start please with basic (and detailed as possible for the knowledgeable layman! p.s-which equation is it?) explanation of the quote above, and I will investigate further accordingly.
thank you! :)

 

[Vanadium 50]

Retarded potentials have effects after their causes. It's the other way around for advanced.

 

[PeterDonis]

Summary: electricity, time

let's start please with basic (and detailed as possible for the knowledgeable layman! p.s-which equation is it?)

You marked this topic as "A" level; that means you are expected to already have graduate level knowledge of the subject matter, and any responses you get would be suited to that level of knowledge. But at that level of knowledge you should already know the answer to your question.

What is your actual background in this topic? That will allow us to determine a more appropriate thread level.

the quote above

Where is this quote from? Please give a specific reference.

 

[Delta2]

You have to read this
https://en.wikipedia.org/wiki/Mathe...lectromagnetic_field#Potential_field_approachdown to the section "Extension to quantum electrodynamics" (not including)
You need a good knowledge of vector calculus in order to understand the equations presented there and what is being said below.

As is being said in that Wikipedia article, Maxwell equations in the potential formulation and with the Lorentz gauge are:
one for the scalar potential $\phi$
$$\nabla^2\phi-\frac{1}{c^2}\frac{\partial^2\phi}{\partial t^2}=-\frac{\rho}{\epsilon_0}$$
and one for the vector potential $\vec{A}$
$$\nabla^2\vec{A}-\frac{1}{c^2}\frac{\partial^2\vec{A}}{\partial t^2}=\mu_0\vec{J}$$.

These two equations are particular cases of the inhomogeneous three dimensional wave equation, and so they tell us that the two potentials are waves, and also reveal a causal relationship between the potentials and the charge and current density.
They tell us that the scalar potential $\phi$ is caused by the charge density $\rho$ and vector potential $\vec{A}$ has as a cause (as a source if we might say) the current density $\vec{J}$.

These two equations have the following solutions, the first for $\phi$
$$\phi(\vec{r},t)=\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r'},t')}{|\vec{r}-\vec{r'}|}d^3\vec{r'}$$
and the second for $\vec{A}$
$$\vec{A}(\vec{r},t)=\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r'},t')}{|\vec{r}-\vec{r'}|}d^3\vec{r'}$$

We can still see from the above equations that the potentials $\phi$ and $\vec{A}$ have as sources (causes) the charge density $\rho(\vec{r'},t')$ and the current density $\vec{J}(\vec{r'},t')$ respectively.

It can be proven mathematically that the "source time" $t'$ has two possible solutions and it is
$t'=t+\frac{|r-r'|}{c}$ (the advanced time)
OR
$t'=t-\frac{|r-r'|}{c}$ (the retarded time)

From these two solutions only the retarded time has a clear physical meaning because it means that the potentials at a point of distance $r$ at time $t$ is caused by the respective source (charge or current density) at an earlier time $t-\frac{|r-r'|}{c}$ (basic causal relationship is that past causes the present and present causes the future). If we take the advanced time as solution it would mean that the future causes the present and the past.

 

[phyzguy]

I find the discussion of the retarded and advanced potentials by Feynman at this link to be fascinating.

 

[vanhees71]

Retarded potentials have effects after their causes. It's the other way around for advanced.

This is often misunderstood. Sommerfeld more carefully called the usually called retardation condition the "irradiation condition" (I don't know if this is the correct translation of the German "Ausstrahlungsbedingung").

After all the microscopic Maxwell equations are time-reversal invariant. Take the usual Hertzian-dipole solution, which is the "retarded solution" of an oscillating dipole with outgoing waves. This is of course the physically correct solution to describe the emission of electromagnetic waves from a "point like" source. That's so, because it's very easy to "prepare": Just put some antenna somewhere and run an AC through it. The radiation energy is of course moving radially outward from the dipole source, as is shown by the Poynting vector $\vec{E} \times \vec{B}$.

Nevertheless, the "time-reversed field" is again a perfectly valid "causal solution" of the Maxwell equations: It describes a situation, where some dipole-field is moving toward a dipole with its energy absorbed by the dipole. Now it's clear, why this is usually not the wanted solution: It's FAPP impossible to prepare such a situation! You'd need to somehow put sources (charge and current distributions) far away from the dipole and somehow excite these sources precisely such as to give this "time-reversed dipole field".

Once more the apparent "arrow of time" is determined rather "thermodynamically", i.e., by the complexity of the preparation procedure necessary to realize the "time-reversed field" as compared to the usual situation of out-going waves from a compact charge-current distribution.

 

[menniandscience]

This is often misunderstood. Sommerfeld more carefully called the usually called retardation condition the "irradiation condition" (I don't know if this is the correct translation of the German "Ausstrahlungsbedingung").

After all the microscopic Maxwell equations are time-reversal invariant. Take the usual Hertzian-dipole solution, which is the "retarded solution" of an oscillating dipole with outgoing waves. This is of course the physically correct solution to describe the emission of electromagnetic waves from a "point like" source. That's so, because it's very easy to "prepare": Just put some antenna somewhere and run an AC through it. The radiation energy is of course moving radially outward from the dipole source, as is shown by the Poynting vector $\vec{E} \times \vec{B}$.

Nevertheless, the "time-reversed field" is again a perfectly valid "causal solution" of the Maxwell equations: It describes a situation, where some dipole-field is moving toward a dipole with its energy absorbed by the dipole. Now it's clear, why this is usually not the wanted solution: It's FAPP impossible to prepare such a situation! You'd need to somehow put sources (charge and current distributions) far away from the dipole and somehow excite these sources precisely such as to give this "time-reversed dipole field".

Once more the apparent "arrow of time" is determined rather "thermodynamically", i.e., by the complexity of the preparation procedure necessary to realize the "time-reversed field" as compared to the usual situation of out-going waves from a compact charge-current distribution.

thank you. do you agree with the following paragraph: "The resulting wave appears to have a preferred time direction, because it respects causality. However, this is only an illusion. Indeed, it is always possible to reverse the time direction by simply exchanging the labels emitter and absorber. Thus, the apparently preferred time direction results from the arbitrary labelling. "
https://en.wikipedia.org/wiki/Wheeler–Feynman_absorber_theory
unfortunately i couldn't under all of your response. can you say in other words? i would like to understand without equations representation, what is the process that takes place. i will write my understanding and maybe you (and others) can take it from there.

there is an electric charged particle that emits electromagnetic wave, get to location Xat the speed of light. now the equations say, that there is another possibility that the wave left the particle before its emission. it left the moving particle and returned to the particle, right?

 

[menniandscience]

You have to read this
https://en.wikipedia.org/wiki/Mathe...lectromagnetic_field#Potential_field_approachdown to the section "Extension to quantum electrodynamics" (not including)
You need a good knowledge of vector calculus in order to understand the equations presented there and what is being said below.

As is being said in that Wikipedia article, Maxwell equations in the potential formulation and with the Lorentz gauge are:
one for the scalar potential $\phi$
$$\nabla^2\phi-\frac{1}{c^2}\frac{\partial^2\phi}{\partial t^2}=-\frac{\rho}{\epsilon_0}$$
and one for the vector potential $\vec{A}$
$$\nabla^2\vec{A}-\frac{1}{c^2}\frac{\partial^2\vec{A}}{\partial t^2}=\mu_0\vec{J}$$.

These two equations are particular cases of the inhomogeneous three dimensional wave equation, and so they tell us that the two potentials are waves, and also reveal a causal relationship between the potentials and the charge and current density.
They tell us that the scalar potential $\phi$ is caused by the charge density $\rho$ and vector potential $\vec{A}$ has as a cause (as a source if we might say) the current density $\vec{J}$.

These two equations have the following solutions, the first for $\phi$
$$\phi(\vec{r},t)=\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r'},t')}{|\vec{r}-\vec{r'}|}d^3\vec{r'}$$
and the second for $\vec{A}$
$$\vec{A}(\vec{r},t)=\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r'},t')}{|\vec{r}-\vec{r'}|}d^3\vec{r'}$$

We can still see from the above equations that the potentials $\phi$ and $\vec{A}$ have as sources (causes) the charge density $\rho(\vec{r'},t')$ and the current density $\vec{J}(\vec{r'},t')$ respectively.

It can be proven mathematically that the "source time" $t'$ has two possible solutions and it is
$t'=t+\frac{|r-r'|}{c}$ (the advanced time)
OR
$t'=t-\frac{|r-r'|}{c}$ (the retarded time)

From these two solutions only the retarded time has a clear physical meaning because it means that the potentials at a point of distance $r$ at time $t$ is caused by the respective source (charge or current density) at an earlier time $t-\frac{|r-r'|}{c}$ (basic causal relationship is that past causes the present and present causes the future). If we take the advanced time as solution it would mean that the future causes the present and the past.

THANKS; you added the equations, which are hard for me to understand but the last few lines are great help! can you elaborate more? make an example? analogy? are we talking about electron moving, magnet, electromagnet wave from the antenna?
are you also dealing with philosophy of physics/time?

 

[Delta2]

THANKS; you added the equations, which are hard for me to understand but the last few lines are great help! can you elaborate more? make an example? analogy? are we talking about electron moving, magnet, electromagnet wave from the antenna?
are you also dealing with philosophy of physics/time?

The equations are valid for all cases of current and charge densities, including the case of current and charge density for an electron moving, as well as the current and charge densities at an antenna that emits waves. They don't apply for the case of moving magnet as in that case the charge and current densities are not well defined.
I am not doing much of philosophy of physics/time except to emphasize the fact we know from our common sense and everyday experience that the "cause" comes before in time and the "effect" comes later in time, or that the past causes the present and the present causes the future as I said.

 

[menniandscience]

The equations are valid for all cases of current and charge densities, including the case of current and charge density for an electron moving, as well as the current and charge densities at an antenna that emits waves. They don't apply for the case of moving magnet as in that case the charge and current densities are not well defined.
I am not doing much of philosophy of physics/time except to emphasize the fact we know from our common sense and everyday experience that the "cause" comes before in time and the "effect" comes later in time, or that the past causes the present and the present causes the future as I said.

thanks again :)
can you elaborate on what is the meaning of the potential that is advancing. what is that advances, is it a virtual photons, a disturbance in an electromagnetic field, a charged particle. and it released before its emission and get to the source or to another point in space? if possible for you to say it like a story would be very helpful

 

[Delta2]

thanks again :)
can you elaborate on what is the meaning of the potential that is advancing. what is that advances, is it a virtual photons, a disturbance in an electromagnetic field, a charged particle. and it released before its emission and get to the source or to another point in space? if possible for you to say it like a story would be very helpful

The potential whether it is advanced or retarded satisfies the wave equation, therefore it is a wave that travels through space and time. It is basically a disturbance in the electromagnetic field that travels throughs space and time, like a water wave is a disturbance that travels through a water pool. The thing with advanced potential is that the source that produced the wave is from the future.

For example: Imagine we see a tsunami wave , We know that this tsunami wave was produced by an earthquake that happened somewhere underwater in the past. If we see a tsunami wave and I tell you "You see this tsunami wave? It is from an earthquake that will happen tomorrow" what will you tell me? Cause that's what an advanced potential wave is all about.

 

[menniandscience]

The potential whether it is advanced or retarded satisfies the wave equation, therefore it is a wave that travels through space and time. It is basically a disturbance in the electromagnetic field that travels throughs space and time, like a water wave is a disturbance that travels through a water pool. The thing with advanced potential is that the source that produced the wave is from the future.

For example: Imagine we see a tsunami wave , We know that this tsunami wave was produced by an earthquake that happened somewhere underwater in the past. If we see a tsunami wave and I tell you "You see this tsunami wave? It is from an earthquake that will happen tomorrow" what will you tell me? Cause that's what an advanced potential wave is all about.

Great! much thanks!
now about the equation. the distance between A and B let's say 100 km. the speed, 50 km per hour. the time to make the distance - 2 hours.
(B-A)/50=2 (distance/speed=time).
now how time, in the maxwill equation, gets a negative value when there is no such thing as negative time or distance? if you can explain please in the simple more intuitive and less technical i did 2 lines above and not in the maxwill one, please.
thanks again!

 

[menniandscience]

It can be proven mathematically that the "source time" $t'$ has two possible solutions and it is
$t'=t+\frac{|r-r'|}{c}$ (the advanced time)

you can show me on that equation, how the mathematics works that the time of the origin fits time of the arrival PLUS time it took the wave to get there.

 

[HomogenousCow]

The thing with advanced potential is that the source that produced the wave is from the future.

Like Vanhees said this isn't really true, it's simply a spherical wave that's converging inwards instead of outwards.

 

[Delta2]

Like Vanhees said this isn't really true, it's simply a spherical wave that's converging inwards instead of outwards.

You can view it that way but then the source isn't the source (emitter) anymore it is something like a sink absorber).
Read the link posted by @phyzguy at post #5 (it is only a paragraph located towards the end of that page), Feynman agrees with my view.

 

[HomogenousCow]

You can view it that way but then the source isn't the source (emitter) anymore it is something like a sink absorber).
Read the link posted by @phyzguy at post #5 (it is only a paragraph located towards the end of that page), Feynman agrees with my view.

What's wrong with the localized charge/currents being a sink? Seems much more intuitive to me to think of the retarded solution as an ingoing wave than some acausal solution, that doesn't seem very useful.

 

[Delta2]

What's wrong with the localized charge/currents being a sink? Seems much more intuitive to me to think of the retarded solution as an ingoing wave than some acausal solution, that doesn't seem very useful.

I don't know if it is possible to make some sort of spherical waves that converge inwards to a specific point, towards a sink that is, vanhees71 said its hard to make it happen.

 

[Delta2]

Great! much thanks!
now about the equation. the distance between A and B let's say 100 km. the speed, 50 km per hour. the time to make the distance - 2 hours.
(B-A)/50=2 (distance/speed=time).
now how time, in the maxwill equation, gets a negative value when there is no such thing as negative time or distance? if you can explain please in the simple more intuitive and less technical i did 2 lines above and not in the maxwill one, please.
thanks again!

Sorry I don't understand how we get a negative time, if we set as time t=0 the moment that the source starts emitting waves then all times are positive. If we set as time t=0 the moment that the wave arrives at a point of distance r from the source, then yes the time that the source emited this wave is negative $t=-\frac{r}{c}$ but this just means it was emitted in the past

you can show me on that equation, how the mathematics works that the time of the origin fits time of the arrival PLUS time it took the wave to get there.

the math behind the advanced and retarded time are not so easy, read this Wikipedia article on the screened Poisson equation
https://en.wikipedia.org/wiki/Screened_Poisson_equation
The screened Poisson equation is similar to the wave equation after we have make a Fourier transform with respect to the time variable in the wave equation. The only difference is that the screening constant will be imaginary and will have two possible values $\lambda_1=+i\frac{\omega}{c}$ and $\lambda_2=-i\frac{\omega}{c}$ where c is the speed of light and $\omega$ the Fourier transform variable and $i$ the imaginary unit. The advanced time corresponds to the $\lambda_1$ while the retarded time to the $\lambda_2$ .

 

[menniandscience]

Sorry I don't understand how we get a negative time, if we set as time t=0 the moment that the source starts emitting waves then all times are positive. If we set as time t=0 the moment that the wave arrives at a point of distance r from the source, then yes the time that the source emited this wave is negative $t=-\frac{r}{c}$ but this just means it was emitted in the past

the math behind the advanced and retarded time are not so easy, read this Wikipedia article on the screened Poisson equation
https://en.wikipedia.org/wiki/Screened_Poisson_equation
The screened Poisson equation is similar to the wave equation after we have make a Fourier transform with respect to the time variable in the wave equation. The only difference is that the screening constant will be imaginary and will have two possible values $\lambda_1=+i\frac{\omega}{c}$ and $\lambda_2=-i\frac{\omega}{c}$ where c is the speed of light and $\omega$ the Fourier transform variable and $i$ the imaginary unit. The advanced time corresponds to the $\lambda_1$ while the retarded time to the $\lambda_2$ .

thanks again. I think the mathematics became too difficult for me, unfortunately. although I'm strong believer that understanding of things has almost nothing to do with math...

but anyways, thank you, and the rest