Instantaneous solutions to Maxwell's equations' potentials conversion?

In summary: No. Nothing is being excluded. It is a matter of the structure of the solution not allowing your instantaneous solutions.
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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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.)

Orodruin said:
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.

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.

Orodruin said:
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?

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

Orodruin said:
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

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.

Orodruin said:
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?

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.

Orodruin said:
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?

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.

Orodruin said:
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)?

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.

haushofer said:
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?

Wel, I guess that you can find those solutions in any book on electromagnetism, but E and B will be retarded. My point was that the gauge field can show behaviour which smells like "instantanious behaviour", but because they are merely redundant degrees of freedom, nobody cares.

haushofer said:
Wel, I guess that you can find those solutions in any book on electromagnetism, but E and B will be retarded. My point was that the gauge field can show behaviour which smells like "instantanious behaviour", but because they are merely redundant degrees of freedom, nobody cares.
thanks, so far instantaneous potential solutions have eluded me after searching. I have found some stuff, but they aren't solely in terms of J and ρ.

1. What are Maxwell's equations and their significance in physics?

Maxwell's equations are a set of four equations that describe the fundamental principles of electricity and magnetism. They are significant because they provide a unified understanding of these two phenomena and have been crucial in the development of modern technology.

2. What is the conversion process for potentials in Maxwell's equations?

The conversion process involves using the electric and magnetic fields to calculate the scalar and vector potentials, which are then used to determine the electric and magnetic fields. This allows for a more efficient and elegant solution to Maxwell's equations.

3. How do instantaneous solutions to Maxwell's equations differ from other solutions?

Instantaneous solutions refer to solutions that are valid at a specific moment in time, as opposed to time-dependent solutions which describe the behavior of fields over a period of time. Instantaneous solutions are useful for understanding the behavior of fields in a specific situation, whereas time-dependent solutions are necessary for predicting the behavior of fields over time.

4. What are the practical applications of instantaneous solutions to Maxwell's equations?

Instantaneous solutions to Maxwell's equations have numerous practical applications, including in the design and optimization of electronic circuits, electromagnetic wave propagation, and the development of new technologies such as wireless communication and radar systems.

5. Are there any limitations to using instantaneous solutions to Maxwell's equations?

While instantaneous solutions are useful for understanding specific situations, they do not provide a complete picture of the behavior of fields over time. Additionally, they may not accurately capture the effects of non-linearities or boundary conditions. Therefore, it is important to use a combination of instantaneous and time-dependent solutions to fully understand the behavior of electromagnetic fields.

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