To my understanding, they derive the Lorentz force law from Coulomb's law + transformation law for force.PeterDonis said:
Sagittarius A-Star said:
Fortunately SRT does not only necessarily lead just to electromagnetism but is just a spacetime model which is flexible enough to apply to all of physics. You can argue that GR, needed to include the gravitational interaction, really goes beyond it, but you can as well argue that gravity fits to the general scheme how to describe the interactions within field theories using the gauge principle with the specialty that what's gauged in the case of gravitation is the spacetime (Poincare) symmetry itself and not some intrinsic symmetry of the fields, which is the case for the other fundamental interactions.Sagittarius A-Star said:
The argument in this paper feels weird, almost circular. The author is taking the first postulate as “the laws of physics are the same in all inertial reference frames,” but the issue I see is how to define “inertial.” It’s typically the frame of an observer with no net forces acting on them. Ok fine, but in this paper, that basically means that coordinate transformations are functions of ##x^{\mu} ## and ##v^{\mu}## only. Doesn’t this implicitly include a law of physics (namely that ##F\propto a## and therefore coordinate transformations are not functions of second or higher order time derivatives of position)?Ibix said:
A more general approach involves finding the maximal group of coordinate transformations that preserve zero acceleration, i.e., such that under ##(t,x)\to(t',x')## we have $$a := \frac{d^2x}{dt^2} ~=~ 0 ~~~ \Leftrightarrow ~~~ a' := \frac{d^2x'}{dt'^2} ~=~ 0 ~.$$ This turns out to be the group of fractional linear transformations. There's no point involving an acceleration parameter because it's zero by assumption.TeethWhitener said:
Although that is the most general group, I think that taking the affine transformations is better. The non-affine portions of the fractional linear transformations don't make a lot of sense physically.strangerep said:
That's what (almost) everyone thinks, and is probably the reason why there's so little relevant literature. (This is strange in itself because there's heaps of literature on the conformal group, yet it also has transformations containing denominators which can become zero.)Dale said:
I will be interested to see your paper when it comes out, but spacelike intervals are not unphysical.strangerep said:
That depends how you define "unphysical". I mean it as "not directly measurable", and only (potentially) useful in physics by reference to an assumed mathematical model. I.e., one measures timelike and/or lightlike intervals experimentally and then infers spacelike intervals only by reference to the abstract model known as Minkowski spacetime.Dale said:
Understood. That is clear, but I wouldn’t accept that definition.strangerep said:
You assumed Maxwell & the first postulate, not just the first postulate. See:brotherbobby said:
The first postulate says "all" laws of physics are covariant in inertial frames, "all" implying the laws of mechanics and electrodynamics.A.T. said:
The first postulate doesn't say which laws are correct laws.brotherbobby said:
brotherbobby said:
A.T. said:
A.T. said:
One of those set of laws needed modifications to achieve this, and the first postulate is agnostic about which. It's only the second postulate that implies Galileo-Newtonian mechanics needs modifications, not Maxwell's equations.brotherbobby said:
brotherbobby said:
SourceEinstein (also 1905 - §2) said:
A.T. said:
Yes, but again, the first postulate by itself doesn't say what to change about the (then) known laws to achieve that generalization. Only the second postulate does, so it cannot follow from the first postulate only.brotherbobby said:
No, @A.T. is correct. The first postulate alone is insufficient. Both the Galilean transform and the Lorentz transform are compatible with the first postulate. The first postulate alone does not allow you to select the Lorentz transform over the Galilean transform. See: https://arxiv.org/abs/physics/0302045brotherbobby said:
Dale said:
But the first postulate doesn't assert that the laws of Electrodynamics are actually Maxwell's equations.brotherbobby said:
True there - I understand this difficulty. I suppose I am going with Resnick here - Robert Resnick - Introduction to Special Relativity.A.T. said:
Why not with Einstein himself?brotherbobby said:
SourceEinstein (1916) said:
Instead of making up what Einstein was thinking about, why not read his own statement quoted above. It explicitly says that his first postulate by itself is consistent with Galileo-Newtonian mechanics.brotherbobby said:
It's clear that the special principle of relativity together with the assumption that Maxwellian electromagnetism obeys it (together with some other symmetries about the space-time model) leads uniquely to Minkowski space as space-time model. Einstein only took the most simple assumption out of Maxwellian electromagnetism, namely that there is a fundamental constant with the dimension of speed, as a postulate.brotherbobby said:
If you still haven't, you might want to - there's an element of talking past one another in the last few posts in thread.brotherbobby said:
Indeed it does. However, neither "postulate" really deserves that name. They aren't phrased with sufficient rigor, and @vanhees71 points out above that the original German could as reasonably have been translated to some other English word.brotherbobby said:
That's pretty close to what Einstein was doing in 1905. We wouldn't approach a modern audience with that argument; like you I tend to interpret the second postulate as just "And I really mean the first postulate, no exceptions for E&M" and prefer to incorporate the "no exceptions" bit into the first postulate. But that's largely a matter of personal taste in how to present the argument; an unnecessary postulate in a formal mathematical system is an abomination, but in a physical theory how we package the initial assumptions is less important.
I like that way of putting it. In 1905, cutting edge physics assumed that Maxwell's equations were an approximation to, or special case of, an as-yet-undiscovered Galilean-covariant theory of electromagnetism. So, at that time, Einstein absolutely needs a separate postulate that brings something of Maxwell's electromagnetism into the argument.vanhees71 said:
You are still assuming that Maxwell’s equations are the laws of electrodynamics. That is an additional assumption beyond the first postulate.brotherbobby said:
At the end of the 19th century (beginning 20th century) there were also much more pragmatic approaches, and those lead finally to progress. I think among the most important contributions to the development of (S)RT was the research program by Helmholtz, who simply wrote down a set of equations with some free parameters which included all the theories on electromagnetism of the time (including action-at-a-distance theories a la Weber as well as Maxwell's). One of the most important outcomes was the discovery of em. waves by H. Hertz and finally the decision in favor of Maxwell's theory. I think there were not many physicists left around 1900 who thought there should be a Galilei invariant electrodynamics. Rather the majority believed in some kind of aether in the sense that there's a preferred inertial system defined by that inertial system, where the Maxwell equations are valid. The only trouble with that of course was that there was still no consistent formulation of the theory for moving media, and that's where Einstein ingeniously found the right "Copernicanian turn": One must give up the idea of a preferred frame of reference, because in fact not a single observation was known to really show the "asymmetries" in electromagnetic phenomena "usually assumed" and thus the necessity to change the space-time model and the fundamental equations of mechanics correspondingly rather than to assume a preferred frame of reference or even to give up Maxwell's empirically well established theory in favor of some Newtonian electrodynamics, and even more ingeniously he just picked the most simple and most general feature of the Maxwell equations, namely the above mentioned additional natural constant (additional as compared to Newton's mechanics, where the only natural constant known was the universal gravitational constant) of the dimensions of a speed and the fact that within Maxwell's electrodynamics it's the phase velocity of the electromagnetic waves in vacuo. Together with the empirical fact that light is also nothing else than such an em. wave, it's thus simply the speed of light in vacuum which must be invariant under transformations from one inertial frame to the other. As @Dale says the validity of Maxwell's equations (or more generally the existence of a fundamental speed as a natural constant) is in addition to the special principle of relativity, which indeed (together with the other symmetry assumptions, i.e., the Euclidicity of space for any inertial observer and the homogeneity of time) allows for both Galilei-Newton spacetime and Minkowski spacetime. The additional assumption Einstein has chosen was in a sense the minimal one, and he showed in his famous paper of 1905 that indeed this is sufficient for the Maxwell equations to be form-invariant under the now to be used transformations from one inertial frame to another, namely the Lorentz instead of the Galilei transformations.Ibix said:
Oh, I didn't know about that. Do you have a reference handy? That seems like it was probably one of the first "test theories" ever. I think that test theories are in some ways as important as the actual theories they test, but certainly they get less attention.vanhees71 said:
Tests seem to decide about ##k## in:Dale said: