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zwoodrow
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Has anyone ever seen a proof that lorentz transforms are the only transforms for maxwells equations to remain invariant between two reference frames moving at a uniform velocity with respect to each other?
zwoodrow said:Has anyone ever seen a proof that lorentz transforms are the only transforms for maxwells equations to remain invariant between two reference frames moving at a uniform velocity with respect to each other?
Meir Achuz said:The proof is simple.
Max's equations contain the constant c.
This leads to the speed of light being c in any coordinate system.
The LT is the only transformation between two frames moving at a uniform velocity with respect to each other that preserves the speed c.
Meir Achuz said:The proof is simple.
Meir Achuz said:Max's equations contain the constant c.
This leads to the speed of light being c in any coordinate system.
Meir Achuz said:The LT is the only transformation between two frames moving at a uniform velocity with respect to each other that preserves the speed c.
The addition of any charged object changes the permittivity (from a constant that produces the speed of light to a tensor producing dispersion and a different group velocity)
The full set of transformations that leave Maxwell's equations form-invariant is the Poincare group, which is larger than the group of Lorentz boosts.
... that the form of Maxwell's equations are invariant under not only the ten dimensional Poincare group (generated by boosts, rotatations, and translations) but also under the E^{1,3} conformal group, which happens to be SO(2,4), a fifteen dimensional Lie group. The extra generators of the Lie algebra can be given as a scaling transformation and certain "toral rolls" (higher dimensional generalizations of certain Moebius transformations); the corresponding transformations are conformal but do not preserve the Minkowski interval.
The addition of any charged object changes the permittivity (from a constant that produces the speed of light to a tensor producing dispersion and a different group velocity)
Yes, look at the equation for Gauss' law. It is almost always given in 2 variations, one for use in a vacuum using the symbol E (electric displacement) and the second with D (electric displacement field) for general use.Naty1 said:Is this generally accepted?? Never knew about it if accurate...
The presence of a medium provides a preferred system, the rest system of the medium, so the Lorentz transformation does not have the same meaning as it does in vacuum.PhilDSP said:That's an interesting proof, but it only applies to a pure vacuum. The addition of any charged object changes the permittivity (from a constant that produces the speed of light to a tensor producing dispersion and a different group velocity)
clem said:The presence of a medium provides a preferred system, the rest system of the medium, so the Lorentz transformation does not have the same meaning as it does in vacuum.
The Maxwell equations are a set of four partial differential equations that describe the behavior of electric and magnetic fields. They are invariant under other transforms, meaning that they have the same form regardless of the coordinate system used to describe them. This is because the equations are based on fundamental physical principles that are independent of the specific coordinates chosen.
The Maxwell equations are invariant under a variety of transforms, including Galilean transformations, Lorentz transformations, and general coordinate transformations. These transforms are used to describe different reference frames, such as moving or accelerating frames, and help to ensure that the equations hold true in all situations.
The invariance of the Maxwell equations under other transforms is a crucial aspect of their usefulness in describing the behavior of electromagnetic fields. It allows the equations to be applied in a wide range of situations and reference frames, making them a powerful tool for understanding and predicting the behavior of electromagnetic phenomena.
Yes, the invariance of Maxwell equations under other transforms can be mathematically demonstrated using tensor analysis. This involves showing that the equations have the same form and properties in different coordinate systems, which is a key characteristic of invariance.
The principle of relativity states that the laws of physics should be the same in all inertial reference frames. The invariance of Maxwell equations under other transforms is consistent with this principle, as it ensures that the equations hold true in all reference frames and do not depend on a specific frame of reference.