#PhysicsFactlet (283)
Lorentz transformations pre-date Special Relativity. How is that even possible?
A thread.
Trigger warning for typos (hopefully just in the text and not in the equations) and carefree manipulations of equations 😉
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Lorentz transformations pre-date Special Relativity. How is that even possible?
A thread.
Trigger warning for typos (hopefully just in the text and not in the equations) and carefree manipulations of equations 😉
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The historical route is interesting but complicated, so I will leave that story for someone more qualified to write it. What I want to look at is: how do we get the Lorentz transformations without knowing anything about special relativity?
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A requirement we want all physical theories to satisfy is the "principle of relativity", i.e. the fact that the laws of Physics are the same in every frame of reference. Were this not the case, each of us would experience a different universe, making life quite complicated.
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This principle takes slightly different forms for different theories. For Newtonian mechanics, the requirement is that, if you move from one "inertial" frame of reference to another, you still see the same forces/accelerations.
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Galilean relativity tells us how your experience of the world around you changes if you move to a different inertial frame in Newtonian mechanics. Time is absolute (i.e. the same in every frame of reference), but perceived position changes with velocity.
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This description matches with our everyday experience, and satisfies the relativity principle for Newtonian mechanics, so it went unchallenged for centuries.
But in the XIX century we started understanding electromagnetic phenomena, and things got complicated.
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But in the XIX century we started understanding electromagnetic phenomena, and things got complicated.
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Maxwell equations work great to describe electromagnetic phenomena. But what happens when we apply Galilean relativity to them?
With the benefit of hindsight we can look at this problem in a smart way, but this requires some extra machinery.
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With the benefit of hindsight we can look at this problem in a smart way, but this requires some extra machinery.
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There are 4 Maxwell equations for the fields, but if we rewrite them in terms of the scalar and vector potentials φ and A⃗, we can do with just 2 equations.
(Notice that we are going to assume we are in vacuum, with no charges nor currents.)
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(Notice that we are going to assume we are in vacuum, with no charges nor currents.)
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You might be used to denote the scalar potential as V, but in my mind V is the symbol for the units of a electric potential, and if you are not careful you end up writing stuff like V=3V to say that your potential is equal to 3 Volts. So I will use φ to avoid confusion.
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Maxwell equations in terms of the potentials are quite complicated, but keep in mind that we still have the freedom to choose our gauge.
If we choose the Lorenz (beware, Lorenz is a different person than Lorentz 🙃) things get MUCH simpler.
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If we choose the Lorenz (beware, Lorenz is a different person than Lorentz 🙃) things get MUCH simpler.
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Ok, now we are (finally) ready to see what happens when we apply Galilean relativity to Maxwell equations, i.e. to see how the operator in brackets above (which is the same for both equation) changes when we change frame of reference.
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The answer? It doesn't work.
Maxwell equations look different in different (inertial) frames of references. We now have 3 options: we can decide Maxwell equations are wrong, we can decide Galilean relativity is wrong, or we can decide the relativity principle is false.
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Maxwell equations look different in different (inertial) frames of references. We now have 3 options: we can decide Maxwell equations are wrong, we can decide Galilean relativity is wrong, or we can decide the relativity principle is false.
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The last option is not really palatable. So the choice is between the first two. Let's see what happens if we assume that Maxwell equations are right and it is Galilean relativity that needs to be modified.
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What we want is to find a (possibly simple) new rule on how to move from one inertial frame of reference to the next that leaves Maxwell equations unchanged. We would also be happy if it doesn't play havoc with Newtonian mechanics.
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We have 4 parameters (a₁, a₂, a₃, and a₄) to find. First of all we apply again the chain rule to find out how the second derivative with respect of time and the second derivative with respect with position change.
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Then we use this result to impose that Maxwell equations should not change.
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Which results in 3 equations and 4 variables.
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Before we look for a 4th equation we can introduce a new dummy variable and make the change of variable a₁= cosh (a₅), which allows us to rewrite the change of variables in a compact way.
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Our 4th equation is that the point at x'=0 has coordinates x= v t, and thus
0=cosh(a₅) v t −c sinh(a₅) t →
→v/c =tanh(a₅)→
→a₅ =artanh(v/c)
which give us our desired transformations!
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0=cosh(a₅) v t −c sinh(a₅) t →
→v/c =tanh(a₅)→
→a₅ =artanh(v/c)
which give us our desired transformations!
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Are we done? Not really.
The problem is that, in order to rewrite Maxwell equations in an easy-to-manage form, we chose the Lorenz gauge, so we must impose that those transformations won't change the gauge.
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The problem is that, in order to rewrite Maxwell equations in an easy-to-manage form, we chose the Lorenz gauge, so we must impose that those transformations won't change the gauge.
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This results in a new set of formulas telling us how the potentials must change to keep both Maxwell equations and the Lorenz gauge untouched.
The process is exactly the same as above, so we can skip a few steps.
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The process is exactly the same as above, so we can skip a few steps.
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And we are finally done!
We got Lorentz transformations just by asking how coordinates (and potentials) should change when we move from one inertial frame of reference to another, assuming that Maxwell equations are correct.
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We got Lorentz transformations just by asking how coordinates (and potentials) should change when we move from one inertial frame of reference to another, assuming that Maxwell equations are correct.
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Obviously this doesn't tell us whether this is the correct path, or if Maxwell equations are the one that needs to be modified to accommodate Galilean relativity.
For that we need to wait for Special Relativity to be developed and tested 🙂
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For that we need to wait for Special Relativity to be developed and tested 🙂
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Ok, a very long thread with a ton of equations and no animations.
To apologize here is an old(ish) visualization of an applications of the equations above 🙂
To apologize here is an old(ish) visualization of an applications of the equations above 🙂
https://twitter.com/j_bertolotti/status/1275465886581903360
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