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

[PeterDonis]

the natural way is to start with the postulates:
P1 + P2 -> Maxwell

But P1 + P2 does not give you Maxwell. The Lorentz transformations by themselves, which is basically what P1 + P2 amounts to, do not give you Maxwell's Equations; those are by no means the only possible equations which are Lorentz invariant.

 

[A.T.]

But P1 + P2 does not give you Maxwell.

P1 + P2 tells you that Maxwell is correct and doesn't need modifications while Newton does.

 

[PeterDonis]

P1 + P2 tells you that Maxwell is correct

Yes, but if you already know Maxwell is correct, you can use that to deduce P2, as @vanhees71 said. But you cannot use P2 to deduce Maxwell.

More generally, either P1 by itself or P1 + P2 does not tell you what the laws of physics are. It only tells you that, whatever they are, they must be invariant under transformations between inertial frames (P1) and those transformations must be Lorentz transformations (P2). If all you're interested in is those general constraints on the laws of physics, then P1 + P2 is sufficient. But to know what the laws of physics actually are, you need to know more than just P1 + P2.

 

[Sagittarius A-Star]

But to know what the laws of physics actually are, you need to know more than just P1 + P2.

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

Special Relativity and Maxwell’s Equations
Richard E. Haskell
…
19. Discussion
In the preceding sections we have derived Maxwell’s equations directly from special relativity and Coulomb’s law.

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

 

[PeterDonis]

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

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

Whether the things that need to be added to P1 + P2 in this case are "not much" is a matter of opinion.

 

[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.