One more talk about the independence of Einstein's SR axioms

[Sagittarius A-Star]

More precisely, P1 + P2 + Coulomb + 4-momentum conservation + the Lorentz force law -> Maxwell.

To my understanding, they derive the Lorentz force law from Coulomb's law + transformation law for force.

 

[aclaret]

maybe also of interest to realize that possible derive the photon speed $c$ just by noticing that photons travel along null geodesics of spacetime $\mathscr{E}$. If observer (4-velocity is $u$, 4-acceleration is $a$) at $O \in \mathscr{E}$ and photon at $M \in \mathscr{E}$, velocity $V \in E$ of photon measured with his local frame is nothing but $V = c(1 + a \cdot \overrightarrow{OM})n - \omega \times_u \overrightarrow{OM}$, where $n$ is unit vector in local rest space of observer, defining direction of photon propagation with respect to observer. Inertial observer characterised by $a = \omega = 0$, or simply, $|V|_g = c$.

exercise for the brave...! prove that equation for $V$ is correct ;)

anyway, much nicer to derive this result by first making some assumes about the geometry of spacetime, and not other way around... don't u think :)

 

[PeterDonis]

To my understanding, they derive the Lorentz force law from Coulomb's law + transformation law for force.

Ah, yes, you're right.

 

[vanhees71]

But regarding Maxwell not much, according to the following paper.

Source:
http://richardhaskell.com/files/Special Relativity and Maxwells Equations.pdf

From that you get P1 + P2 + Coulomb's law + conservation of 4-momentum -> Maxwell

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.

Now, how do you get within SRT to Maxwell's theory for the electromagnetic interaction. What you have is the fundamental symmetry of Minkowski space (proper orthochronous Poincare transformations) and the paradigm that everything should be described by local field theories (an argument dating back to Faraday's qualitative insight based on his observations on electricity and magnetism). Then you can systematically study which representations of the symmetry group you can build with fields, fulfilling the constraints of causality. As it turns out, within classical physics, these are the massive and massless representations in terms of tensor fields (including of course scalar and vector fields too).

That's already pretty nicely constraining the possible types of fields and the corresponding action functionals but you need some input from experiment to know, which field might describe the phenomena in question. In the case of electrodynamics everything hints at a massless vector field with 2 polarization degrees of freedom propagating with the speed of light. What then fits of all the representations in form of local tensor-field realizations of the Poincare group is a massless vector field, which necessarily must be a gauge field in order to avoid unphysical continuous intrinsic degrees of freedom and ending up with the said to polarization states. A massless vector field has two helicity states rather than 3 spin states, and these build the basis of the observed polarization states (left- and right-circular polarizations, from which you can build any general elliptic polarization state you like). In addition we know that the em. interactions also obeys the symmetry under spatial reflections, which leaves you with practically one choice for the free-field Langrangian if you restrict yourself with the lowest dimensions.

Coupling this to matter you need a conserved charge and the corresponding Noether current, being determined from the additional constraint of gauge symmetry. In this way you end up with Maxwell's electrodynamics, but as you see, you need far more empirical input than just Einstein's two postulates on the space-time structure.

For me that greatest ingeniuty of Einstein's approach to solve the problem of the violation of Gaileo symmetry of Maxwell's equations was to extract the minimal needed assumption from Maxwell's theory to modify the space-time model such that Maxwell's equations can be symmetric under changes between inertial frames of reference, namely this additional postulate of the independence of the speed of light from the motion of the light source, leading to an additional fundamental constant of nature, the "limiting speed" of Minkowski space (the choice of the value of this speed is just convention defining the system of units, as is the case within the SI units since 1983 and of course also in the newest version of 2019, where almost all base units are defined by choosing particular values of all the fundamental natural constants, except $G$, which simply is too difficult to measure with sufficient condition today to be included in the list of fixed values to determine the base units of the SI).

That the em. field is indeed a massless vector field and that the speed of light is indeed the limiting speed of Minkowski space is now to be interpreted as a question of experiment. Today there's no experimental hint that this assumption is wrong. Usually the empirical status is given by an upper bound for a possible photon mass which is $m_{\gamma} < 10^{-18} \text{eV}/c^2$.

 

[TeethWhitener]

Here's a more recent article saying much the same thing in English.

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)?

There’s nothing in the calculus of variations that restricts the Lagrangian to only being a function of $x^{\mu}$ and $\dot{x}^{\mu}$; it’s only an empirical observation/physical law. So it almost feels like the first postulate (at least in the linked paper) is saying “the laws of physics are the same in all frames where the laws of physics are the same.” I dunno, maybe I’m missing something, but it feels off to me.

 

[strangerep]

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)?

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.

With further analysis, one can extract the de Sitter, Poincare and Galileo algebras as physically plausible possibilities. (The first one is the reason why some people advocate so-called "de Sitter Special Relativity", which has an extra universal constant with dimensions of inverse time, as well as the usual universal constant $c$.)

 

[Dale]

This turns out to be the group of fractional linear transformations.

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]

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.

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.)

More careful analysis of the FL version of velocity boosts (cf. Kerner, Manida, -- refs I've given previously) shows that these transformations are ok if restricted to lightcone interiors. Similarly for the de Sitter (or anti de Sitter) transformations in which either spatial or temporal translations are not well-defined everywhere on Minkowski spacetime.

The crucial step, then, is to understand that the usual abstract concept of Minkowski spacetime is really just a homogeneous space for a particular group of symmetry transformations, i.e., the Poincare group. If one starts from a larger group (or possibly semigroup), and carefully constructs a homogeneous space from scratch, one can reach a physically plausible spacetime geometry -- in the sense that physically measurable intervals remain sensible, and only unphysical intervals (i.e., spacelike intervals) involve pathogical (ill-defined) transformations. But now I'm already straying a little into unpublished research so I'll leave it at that.

 

[Dale]

only unphysical intervals (i.e., spacelike intervals) involve pathogical (ill-defined) transformations

I will be interested to see your paper when it comes out, but spacelike intervals are not unphysical.

 

[strangerep]

I will be interested to see your paper when it comes out, but spacelike intervals are not unphysical.

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]

That depends how you define "unphysical". I mean it as "not directly measurable"

Understood. That is clear, but I wouldn’t accept that definition.

 

[brotherbobby]

I must confess I haven't read all the responses in this thread, but there is a basic thing in here.

If Maxwell's equations are covariant in all inertial frames, it implies that there is no preferred frame in which it only is (a.k.a the ether). It was thought that light only has a speed $c$ in ether. For all frames which move with a velocity $\vec v$ with respect to ether, the speed of light gets altered by the vector relation $\vec c' = \vec c - \vec v$.

Now if Maxwell's equations have to remain covariant in all frames (postulate 1), the speed of light has to remain the same in all frames (postulate 2), as Maxwell's equations don't allow for the a change in the speed of light. Far as I can see, the second postulate follows as a consequence from the first.

 

[A.T.]

Far as I can see, the second postulate follows as a consequence from the first.

You assumed Maxwell & the first postulate, not just the first postulate. See:
https://www.physicsforums.com/threa...e-of-einsteins-sr-axioms.1000017/post-6460698

 

[brotherbobby]

You assumed Maxwell & the first postulate, not just the first postulate.

The first postulate says "all" laws of physics are covariant in inertial frames, "all" implying the laws of mechanics and electrodynamics.

 

[A.T.]

The first postulate says "all" laws of physics are covariant in inertial frames, "all" implying the laws of mechanics and electrodynamics.

The first postulate doesn't say which laws are correct laws.

 

[A.T.]

I must confess I haven't read all the responses in this thread,...

You should at least read Einstein's own explanation:
https://www.physicsforums.com/threa...e-of-einsteins-sr-axioms.1000017/post-6460938

 

[brotherbobby]

You should at least read Einstein's own explanation:
https://www.physicsforums.com/threa...e-of-einsteins-sr-axioms.1000017/post-6460938

I copy and paste for you from Einstein's 1905 paper. The red and green underlinings are mine.

 

[brotherbobby]

But P1 doesn't list the specific laws that are true. It just says that whatever laws we use, they should take the same form in all inertial frames. That by itself, without assuming specific laws, doesn't imply a certain transformation. It's only P2 that rules out the Galilei transformation.

Not correct. The first postulate accepts Maxwell's equations along with the laws of mechanics as valid in all inertial frames of reference. Einstein was in effect "raising" Galileo's relativity principle to include electrodynamics too, in opposition to what most physicists at that time. They were happy to have the relativity principle valid for mechanics, and not valid for electrodynamics.

 

[A.T.]

The first postulate accepts Maxwell's equations along with the laws of mechanics as valid in all inertial frames of reference.

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.

 

[Sagittarius A-Star]

I copy and paste for you from Einstein's 1905 paper. The red and green underlinings are mine.

View attachment 278995

If P2 is needed depends on, if you formulate P1 to also inherit Maxwells theory. See also:

The following considerations are based on the Principle of Relativity and on the Principle of Constancy of the velocity of light, both of which we define in the following way.

1. The laws according to which the states of physical systems alter are independent of the choice, to which of two co-ordinate systems (having a uniform translatory motion relative to each other) these state changes are related.

2. Every ray of light moves in the "stationary" co-ordinate system with the definite velocity V, the velocity being independent of the condition, whether this ray of light is emitted by a body at rest or in motion.

Source

 

[brotherbobby]

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.

Yes you are right there. The first postulate does not tell us what laws to modify. However, from Einstein's 1905 paper, it is clear that he was raising the principle of relativity from mechanics to the whole of physics, namely, electrodynamics. That was his greatness - which would not have been if he was merely re-stating what Galileo had said 300 years earlier.
Also, the dramatic nature of a statement does not mean it has to be independent. The constancy of the speed of light is dramatic and amongst the first of many things in special relativity that takes us by surprise. It is understandable therefore that we would like to give it a special status. But it is follows from the first postulate, which, at least from a logical standpoint, is all we need.

 

[A.T.]

Yes you are right there. The first postulate does not tell us what laws to modify. However, from Einstein's 1905 paper, it is clear that he was raising the principle of relativity from mechanics to the whole of physics, namely, electrodynamics.

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.

 

[Dale]

Not correct. The first postulate accepts Maxwell's equations along with the laws of mechanics as valid in all inertial frames of reference. Einstein was in effect "raising" Galileo's relativity principle to include electrodynamics too, in opposition to what most physicists at that time. They were happy to have the relativity principle valid for mechanics, and not valid for electrodynamics.

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/0302045

The second postulate is necessary because it rejects the Galilean transform. Another way to reject the Galilean transform would be to assert Maxwell's equations. Regardless, something beyond the first postulate is required. A third way to reject the Galilean transform would be through experiment (my preference).

 

[brotherbobby]

Both the Galilean transform and the Lorentz transform are compatible with the first postulate.

The Galilean transformations do not leave Maxwell's equations covariant. The first postulate asserts that the laws of Mechanics and Electrodynamics must both be left covariant.

 

[A.T.]

The Galilean transformations do not leave Maxwell's equations covariant. The first postulate asserts that the laws of Mechanics and Electrodynamics must both be left covariant.

But the first postulate doesn't assert that the laws of Electrodynamics are actually Maxwell's equations.

 

[brotherbobby]

But the first postulate doesn't assert that the laws of Electrodynamics are actually Maxwell's equations.

True there - I understand this difficulty. I suppose I am going with Resnick here - Robert Resnick - Introduction to Special Relativity.

I will present his argument briefly. There are three issues at hand that rose. Galilean transformations, the relativity principle and Maxwell's equations. We can at best have any two of them satisfied. Most physicists dropped the relativity principle (for electrodynamics). They asserted a special frame (ether) in which Maxwell's equations are valid. Some, notably Hertz, dropped Maxwell's equations! More correctly, he tried to modify Maxwell's equations so that it would fit both Galilean transformations and the relativity principle for electrodynamics. And some, like Poincare and Einstein, adopted Maxwell's equations and felt that it is in fact the Galilean transformations that need to be altered. Lorentz role here is tricky. However, his belief in the "length contraction" violates the relativity principle. A bar moving relative to ether would be contracted in ether's frame, but a bar at rest in ether's frame would not be contracted for a moving frame, or so he felt. Hence it might be correct to say Lorentz too was part of the first group.

In order to solve our riddle here, we need to find out to what extent is Resnick justified. Was Einstein thinking about Maxwell's equations when he mentioned "electrodynamic and optical laws"? I argue that he did. For one, electrodynamics was only recently "tied" to optics back then - so he felt the need to take the trouble to put the two of them together. Today, we know that light waves and x-rays as electromagnetic waves so well that we wouldn't bother to do it. Hence, in mentioning both in the same vein, Einstein was making recourse to Hertz's experiment (e.m. waves). I do not think he wondered if some other equations of electrodynamics, other than those of Maxwell's, would turn out to be the true ones in the (pseudo) flat space of Minkowski.

 

[A.T.]

I suppose I am going with Resnick here...

Why not with Einstein himself?

The special relativity theory rests on the following postulate which also holds valid for the Galileo-Newtonian mechanics.

If a co-ordinate system K be so chosen that when referred to it the physical laws hold in their simplest forms, these laws would be also valid when referred to another system of co-ordinates K′ which is subjected to an uniform translational motion relative to K. We call this postulate "The Special Relativity Principle".
...
The Special Relativity Theory does not differ from the classical mechanics through the assumption of this postulate, but only through the postulate of the constancy of light-velocity in vacuum which, when combined with the special relativity postulate, gives in a well-known way, the relativity of synchronism as well as the Lorentz transformation, with all the relations between moving rigid bodies and clocks.

Source

Was Einstein thinking about Maxwell's equations when he mentioned "electrodynamic and optical laws"? I argue that he did.

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.

 

[vanhees71]

The Galilean transformations do not leave Maxwell's equations covariant. The first postulate asserts that the laws of Mechanics and Electrodynamics must both be left covariant.

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.

The modern idea is to have the symmetry principles and then deduce the form of the physical laws from them, and if you only take the special principle of relativity together with the Euclidicity of space and homogeneity of time for all inertial observers you have two options as a space-time model, namely Galilei-Newton space-time or Einstein-Minkowski space-time. You can now build all kinds of mathematical models to describe the phenomena obeying the one or the other symmetry defining these space-time models. The only way to decide, which one is correct is empirical evidence, and it's well known that already "Newtonian electrodynamics" can be ruled out in favor of Maxwell's equations, which respects the Poincare symmetry of Minkowski space-time.

It's not so important which postulates you use to find the space-time model but which one describes nature best!

 

[Nugatory]

Somewhere further up in this thread, you said

I must confess I haven't read all the responses in this thread, but there is a basic thing in here.

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.

The first postulate says "all" laws of physics are covariant in inertial frames, "all" implying the laws of mechanics and electrodynamics.

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.

It's probably better to consider both postulates in historical context. Imagine yourself explaining special relativity to a group of physicists at the turn of the last century:

Folks, we all know that we've had a problem since 1865: Maxwell's electrodynamics only works covariantly with Galilean relativity if there is some luminous ether; but this approach is increasingly unsatisfactory and Michelson/Morley aren't helping any. I have a solution, if you'll just accept two things: Principle of relativity, which we already all agree about; and that we can go all in on that principle and apply it to E&M as well. Now let's see where this line of thought takes us

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.

 

[Ibix]

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.

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.

 

[Dale]

The Galilean transformations do not leave Maxwell's equations covariant. The first postulate asserts that the laws of Mechanics and Electrodynamics must both be left covariant.

You are still assuming that Maxwell’s equations are the laws of electrodynamics. That is an additional assumption beyond the first postulate.

 

[vanhees71]

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.

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.

 

[Dale]

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).

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]

If I'd always knew, where I've read things... By a bit of googling, I found this paper, which contains a summary of Helmholtz's theory on electrodynamics:

https://doi.org/10.1086/350399

Another good book on the history of electrodynamics, containing a long appendix on Helmholtz's electrodynamics and its relation to the various theories discussed from around 1870 on is

O. Darrigol, Electrodynamics from Ampere to Einstein, Oxford University Press (2000)

 

[Sagittarius A-Star]

That seems like it was probably one of the first "test theories" ever.

Tests seem to decide about $k$ in:
https://puhep1.princeton.edu/~mcdonald/examples/velocity.pdf