# Instantaneous solutions to Maxwell's equations' potentials conversion?

• I
This page shows solutions for Maxwell's equations of the electric and magnetic potentials (Eqn.s (509) and (510)):
http://farside.ph.utexas.edu/teaching/em/lectures/node50.html
They are derived with the aid of a Green's function: http://farside.ph.utexas.edu/teaching/em/lectures/node49.html
They make use of retarded time.

I'm looking for a solution in terms of instantaneous time. If I swap the retarded time variables in 509 & 510 with instantaneous time, would those be mathematically valid solutions?

And if not, what should I modify?

Also, if my intuition is correct, although the retarded time involves the speed of light, any arbitrary speed will be a valid solution right?

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Orodruin
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I'm looking for a solution in terms of instantaneous time.
You cannot write them in terms of instantaneous time because, ultimately, what is physically relevant is not the charges and currents at the instantaneous time, it is the charges and currents on the past light-cone. Indeed, as a fully relativistic field theory, what is "instantaneous" depends on your inertial frame (or, more specifically, on your simultaneity convention).

If I swap the retarded time variables in 509 & 510 with instantaneous time, would those be mathematically valid solutions?
Definitely not.

Also, if my intuition is correct, although the retarded time involves the speed of light, any arbitrary speed will be a valid solution right?
No. In order to be a fully relativistic field theory, the speed has to be the invariant speed, i.e., the speed of light. (Note that the invariant speed is called the "speed of light" although it is more accurate to say that light moves at the invariant speed. This is clearly for historical reasons.)

You cannot write them in terms of instantaneous time because, ultimately, what is physically relevant is not the charges and currents at the instantaneous time, it is the charges and currents on the past light-cone. Indeed, as a fully relativistic field theory, what is "instantaneous" depends on your inertial frame (or, more specifically, on your simultaneity convention).

Definitely not.

No. In order to be a fully relativistic field theory, the speed has to be the invariant speed, i.e., the speed of light. (Note that the invariant speed is called the "speed of light" although it is more accurate to say that light moves at the invariant speed. This is clearly for historical reasons.)
I understand, but I'm not looking for a realistic solution, just a mathematically valid one, to use for comparison purposes.

Orodruin
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It is not a question of being "realistic" or not. It is a question of being a solution or not. The structure of Maxwell's equations are such that their solutions depend on the charges and currents on the past light cone, not on a simultaneous hypersurface.

• It is not a question of being "realistic" or not. It is a question of being a solution or not. The structure of Maxwell's equations are such that their solutions depend on the charges and currents on the past light cone, not on a simultaneous hypersurface.
So instantaneous potential solutions are mathematically impossible?

Orodruin
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So instantaneous potential solutions are mathematically impossible?
Yes. It is simply not how the equations work.

So instantaneous potential solutions are mathematically impossible?
Yes. It is simply not how the equations work.
i see, thanks. do you have a description or a link of the proof? much appreciated

i see, thanks. do you have a description or a link of the proof? much appreciated
cos I was thinking that there are relativistic and non-relativistic forms of solutions to Maxwell's equations

Orodruin
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cos I was thinking that there are relativistic and non-relativistic forms of solutions to Maxwell's equations
I am sorry, but it is unclear to me why you would think that. Maxwell's equations are inherently relativistic.

I am sorry, but it is unclear to me why you would think that. Maxwell's equations are inherently relativistic.
cos I was thinking, for example, some versions of the Biot-Savart law don't use retarded time but still satisfy Maxwell's equations. anyway, what's the proof against instantaneity?

Orodruin
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Instantaneity is simply not a physical concept in relativity. Besides, it is completely clear from the general solution (which only depends on the sources at the past light-cone) that this is the case, so I am not sure I understand your problem with this.

Instantaneity is simply not a physical concept in relativity. Besides, it is completely clear from the general solution (which only depends on the sources at the past light-cone) that this is the case
okay, but doesn't excluding instantaneous solutions require a mathematical proof regarding Maxwell's equations?

Orodruin
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okay, but doesn't excluding instantaneous solutions require a mathematical proof regarding Maxwell's equations?
No. Nothing is being excluded. It is a matter of the structure of the solution not allowing your instantaneous solutions. You already have the general solution and it is very clear from it that it depends on the past light-cone, not on a surface of simultaneity. You seem to be suggesting that there might exist additional solutions to these linear differential equations, but the solution is unique given the sources and the boundary conditions.

No. Nothing is being excluded. It is a matter of the structure of the solution not allowing your instantaneous solutions. You already have the general solution and it is very clear from it that it depends on the past light-cone, not on a surface of simultaneity. You seem to be suggesting that there might exist additional solutions to these linear differential equations, but the solution is unique given the sources and the boundary conditions.
ok, is the general solution you are referring to (509) and (510)? And the linear differential equations (506) and (507)?

haushofer
So instantaneous potential solutions are mathematically impossible?
You can choose a gauge(Coulomb gauge) in which the field equation for the scalar potential is the good old Poisson equation, however. See e.g. Travis Norsen's qm-book, page 20.

• You can choose a gauge(Coulomb gauge) in which the field equation for the scalar potential is the good old Poisson equation, however. See e.g. Travis Norsen's qm-book, page 20.
thank you, so, based on the Coulomb gauge, do you know what are the magnetic and electric potential solutions in terms of the charge and current densities?

haushofer