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The Galilean principle of relativity:

The laws of classical mechanics apply in all inertial reference systems

OR

No experiment carried out in an inertial frame of reference can determine the absolute velocity of the frame of reference

These two statements written above are equivalent.

Maxwell's equations were discovered later. My question is (1) how did Maxwell's equations contradict the Galilean principle of relativity?

Furthermore if one studies the two postulates of Einstein's special theory of relativity, they can be simply translated as follows:

Postulate 1: Galileo was right.

Postulate 2: Maxwell was right.

(2) How did the Maxwell equations retain the same form in all inertial frames by obeying the Lorentz transformation?

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    $\begingroup$ (1) Maxwell equations predict that electromagnetic waves (light) travel with constant speed $c$, which is independent of the reference frame. This contradicts Galileo's transformations (not the principle of relativity though), according to which the velocity of light has to achieve additional contributions when passing to other frames of reference. $\endgroup$ – Prof. Legolasov Sep 21 '16 at 10:45
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    $\begingroup$ I don't think this is a very clear question. To 1, the answer is simply that they do. If you perform a galilean transformation ($t'=t$, $x'=x-vt$), Maxwell's equations don't retain the same form. To 2, the answer is simply that if you perform a lorentz transformation on Maxwell's equations (detailed on wikipedia here: en.wikipedia.org/wiki/… ), Maxwell's equations do retain the same form. I don't know if anyone here wants to write out all four equations on $\vec{E}$ and $\vec{B}$ and go through the motions, step-by-step, explicitly. $\endgroup$ – user12029 Sep 21 '16 at 10:47
  • $\begingroup$ @NeuroFuzzy: could you please elaborate the algebra (If you perform a galilean transformation (t'=t, x′=x−vt), Maxwell's equations don't retain the same form) or could you please specify a link where it is elaborate? $\endgroup$ – user103515 Sep 21 '16 at 10:53
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    $\begingroup$ @NeuroFuzzy no........to your last point, they can easily be found on wikipedia, and a thousand other places. $\endgroup$ – user108787 Sep 21 '16 at 11:10
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    $\begingroup$ You should check out this video, The speed of light is not about light. It discusses the invariance issues with the different transformations, and how it was resolved, all without getting too deep into the math and keeping it on a conceptual level. $\endgroup$ – Cody Sep 21 '16 at 16:23
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A Galilean set of frames are an obvious/common sense way of viewing motion if we assume the validity of 3 also apparently obvious postulates.

  1. All clocks measure time at the same rate, independent of their velocity.

  2. Objects have no limit on their potential velocity.

  3. Rulers have the same length (difference in position between the lengths at a common time), independent of their velocity.

When Maxwell formulated/compiled his equatons, implying that light speed was invariant in every frame, Einstein was forced to consider the implications of this for Galilean transformations and their "obvious" underlying assumptions.

If light speed is invariant in all frames, then something has to give to preserve that invariance, and the 3 assumptions above needed to be abandoned to preserve Maxwell's laws.

How the Maxwell equations retained the same form in all inertial frames by obeying Lorentz transformation?

By the development of the Faraday tensor $F_{\mu v}$ based on a vector potential $\vec A $ and a scalar potential $\Phi $ .

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  • $\begingroup$ "When Maxwell formulated/compiled his equatons, implying that light speed was invariant in every frame..." No. Actually Maxwell's theory implied the opposite (the speed of light relative to the observer varies with the speed of the observer): pitt.edu/~jdnorton/papers/Chasing.pdf "That [Maxwell's] theory allows light to slow and be frozen in the frame of reference of a sufficiently rapidly moving observer." $\endgroup$ – Pentcho Valev Sep 22 '16 at 5:19
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My question is (1) how Maxwell's equations contradicted Galilean principle of relativity.

Maxwell's equations have wave solutions that propagate with speed $c = \frac{1}{\sqrt{\mu_0\epsilon_0}}$.

Since velocity is relative (speed c with respect to what?), it was initially thought that the what is an luminiferous aether in which electromagnetic waves propagated and which singled out a family of coordinate systems at rest with respect to the aether.

If so, then light should obey the Galilean velocity addition law. That is, a lab with a non-zero speed relative to the luminiferous aether should find a directionally dependent speed of light.

However, the Michelson–Morley experiment (original and follow-ups) failed to detect such a directional dependence. Some implications are

(1) there is no aether and electromagnetic waves propagate at an invariant speed. This conflicts with Galilean relativity for which two observers in relative uniform motion will measure different speeds for the same electromagnetic wave. This path leads to special relativity theory.

(2) there is an aether but it is undetectable. This path leads to Lorentz aether theory.

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  • $\begingroup$ In 1887 (prior to FitzGerald and Lorentz advancing the ad hoc length contraction hypothesis) the Michelson-Morley experiment UNEQUIVOCALLY confirmed the variable speed of light predicted by Newton's emission theory of light and refuted the constant (independent of the speed of the light source) speed of light predicted by the ether theory and later adopted by Einstein as his special relativity's second postulate. $\endgroup$ – Pentcho Valev Sep 22 '16 at 5:35
  • $\begingroup$ If the Galilean velocity addition law is broken, why does it mean that the laws of Newtonian Mechanics need to be adjusted? Maybe this question is trivial but I don't see the direct implication. $\endgroup$ – philmcole Feb 6 '18 at 19:32
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The difference between Galilean and special relativity is the details of how spacetime coordinates change between reference frames. The Galilean transformation $t'=t,\,\mathbf{x}' =\mathbf{x}-\mathbf{v}t$ relates reference frames of relative velocity $\mathbf{v}$. This implies that, if $A$ has velocity $\mathbf{u}$ relative to $B$ and $B$ has velocity $\mathbf{v}$ relative to $C$, $A$ has velocity $\mathbf{u}+\mathbf{v}$ relative to $C$. This implies no speed can be invariant across reference systems. For example, if I shine a torch while on a train that's going past you, you and I should disagree on the speed of the torch's light.

However, Maxwell's theory contains waves of speed $c:=1/\sqrt{\mu_0\varepsilon_0}$, so cannot apply in all reference frames if they are related as per Galileo's formulae. In a region with no electric charges or currents, Maxwell's equations imply the wave equations $$\nabla^2\mathbf{B}=c^{-2}\partial_t^2\mathbf{B},\,\nabla^2\mathbf{E}=c^{-2}\partial_t^2\mathbf{E}.$$

Special relativity still claims physical laws are the same in all reference frames, but they relate their coordinates differently, viz. $$t'=\frac{t-\mathbf{v}\cdot\mathbf{x}/c^2}{\sqrt{1-v^2/c^2}},\,\mathbf{x}' =\frac{\mathbf{x}-\mathbf{v}t}{\sqrt{1-v^2/c^2}}.$$One can show that a change in reference frames preserves the above speed-$c$ wave equations.

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From a mathematical point it is rather simple: Considering by $A^\mu$ the usual 4-vector potential and assuming the Lorenz gauge $ \partial_{\mu}A^\mu = 0$ the Maxwell equations of vacuum write as $\square A^{\mu} =\partial_{\nu}\partial^{\nu}A^{\mu} = 0$. However the D'Alembert operator $\partial_{\nu}\partial^{\nu} = \frac{1}{c^2}\partial_t^2 - \partial_x^2 -\partial_y^2 - \partial_z^2$ is invarariant under a linear transformation given by a matrix ${\Lambda^{\sigma}}_{\tau}$ if and only if ${\Lambda^{\sigma}}_{\mu}{\Lambda^{\tau}}_{\nu}g^{\mu\nu} = g^{\sigma\tau} = \mathrm{diag}(1,-1,-1,-1)$. These are precisely the Lorentz transformations. However, the Galilei transformations do not form a subgroup of these.

The basic idea of this approach is the idea that the physical laws (and therefore the corresponding differential operators) have to keep their form under valid frame transformations. But it is then postulated that (somehow vice versa) all transformations keeping the form (Lorentz-transformations, that is) are actually valid changes of reference frames.

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Imagine a stationary electron sitting next to a long length of wire with current flowing through it. Since the wire is neutrally charged, there is no electric force on the electron, and since the electron is stationary, there is no magnetic force.

Now imagine the whole system is moving lengthwise at a constant velocity. All of a sudden the electron is moving through a magnetic field and experiences a force. This seems to be a contradiction.

In relativity, this will be answered by the differing length contractions of the positive (protons)/negative (electrons) parts of the wire, creating an electric force on the electron that balances the magnetic force. This also serves to show the difficulty of distinguishing electric from magnetic forces, as one may become the other in a different reference frame.

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protected by Qmechanic Sep 21 '16 at 17:57

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