C represents the speed limit of the universe rather than the speed of light

[CosmicVoyager]

Greetings,

I read in a relativity paper or book (I think one of Einsteins. I am trying to find it again.) that c is actually represents the speed limit of the universe, and because light has no mass it travels at that speed. That c was arrived at independently from the speed of light.

I think looking at it this way is extremely helpful in understanding things. Everyone seems to ask questions such as "Why can't you go faster than the speed of light?" As though light were somehow the cause, but light traveling at speed c is actually a consequence of c being the limit.

Does anyone happen to know the paper or book?

Thanks

 

[bcrowell]

FAQ: Is the c in relativity the speed of light?

Not really. The modern way of looking at this is that c is the maximum speed of cause and effect. Einstein originally worked out special relativity from a set of postulates that assumed a constant speed of light, but from a modern point of view that isn't the most logical foundation, because light is just one particular classical field -- it just happened to be the only classical field theory that was known at the time. For derivations of the Lorentz transformation that don't take a constant c as a postulate, see, e.g., Morin or Rindler.

One way of seeing that it's not fundamentally right to think of relativity's c as the speed of light is that we don't even know for sure that light travels at c. We used to think that neutrinos traveled at c, but then we found out that they had nonvanishing rest masses, so they must travel at less than c. The same could happen with the photon; see Lakes (1998).

Morin, Introduction to Classical Mechanics, Cambridge, 1st ed., 2008

Rindler, Essential Relativity: Special, General, and Cosmological, 1979, p. 51

R.S. Lakes, "Experimental limits on the photon mass and cosmic magnetic vector potential", Physical Review Letters 80 (1998) 1826, http://silver.neep.wisc.edu/~lakes/mu.html

 

[CosmicVoyager]

FAQ: Is the c in relativity the speed of light?

Not really. The modern way of looking at this is that c is the maximum speed of cause and effect. Einstein originally worked out special relativity from a set of postulates that assumed a constant speed of light, but from a modern point of view that isn't the most logical foundation, because light is just one particular classical field -- it just happened to be the only classical field theory that was known at the time. For derivations of the Lorentz transformation that don't take a constant c as a postulate, see, e.g., Morin or Rindler.

One way of seeing that it's not fundamentally right to think of relativity's c as the speed of light is that we don't even know for sure that light travels at c. We used to think that neutrinos traveled at c, but then we found out that they had nonvanishing rest masses, so they must travel at less than c. The same could happen with the photon; see Lakes (1998).

Morin, Introduction to Classical Mechanics, Cambridge, 1st ed., 2008

Rindler, Essential Relativity: Special, General, and Cosmological, 1979, p. 51

R.S. Lakes, "Experimental limits on the photon mass and cosmic magnetic vector potential", Physical Review Letters 80 (1998) 1826, http://silver.neep.wisc.edu/~lakes/mu.html

If the speed of light does not equal c, is it correct to read e=mc^2 as "energy equals mass times the speed of light squared"?

Is the point of the c in that equation to say that the amount of energy a certain amount of mass equals depends on the speed of light, or is the point to say it depends on the maximum speed of cause and effect?

 

[Pengwuino]

If the speed of light does not equal c, is it correct to read e=mc^2 as "energy equals mass times the speed of light squared"?

Is the point of the c in that equation to say that the amount of energy a certain amount of mass equals depends on the speed of light, or is the point to say it depends on the maximum speed of cause and effect?

It is the speed of light. All of relativistic physics is built up using the idea that the speed of light and the maximum speed limit in the universe are the same quantities. In fact, it might be better to say that the maximum speed an object can travel is 3x10^8 m/s and it can only travel at this speed if it is massless. The photon is massless, ergo, light travels at that speed.

We actually can never physically demonstrate that the photon is massless, however. All we have is what current experiment tells us. For example, Jackson's 3rd edition of Classical Electrodynamics notes that, at the time of his writing, the mass of the photon was experimentally shown to be less than 4x10^-51kg. So at the time, we knew it was less than that. I'm sure we have a better upper limit now. Now if we were to find out that the photon is NOT massless, then what we consider the speed of light now would then be considered simply the speed limit of the universe and light would travel just a tad tad tad bit below that.

Come to think of it, though, everything I know is built off of assuming they're one in the same (that the photon is massless, ergo the speed limit of the universe and the speed of light are one in the same). If we actually did find a tiny but non-zero mass of the photon, I believe the 'c' in E=mc^2 would not be the speed of light, but it would be this maximum speed limit.

 

[CosmicVoyager]

It is the speed of light. All of relativistic physics is built up using the idea that the speed of light and the maximum speed limit in the universe are the same quantities. In fact, it might be better to say that the maximum speed an object can travel is 3x10^8 m/s and it can only travel at this speed if it is massless. The photon is massless, ergo, light travels at that speed.

We actually can never physically demonstrate that the photon is massless, however. All we have is what current experiment tells us. For example, Jackson's 3rd edition of Classical Electrodynamics notes that, at the time of his writing, the mass of the photon was experimentally shown to be less than 4x10^-51kg. So at the time, we knew it was less than that. I'm sure we have a better upper limit now. Now if we were to find out that the photon is NOT massless, then what we consider the speed of light now would then be considered simply the speed limit of the universe and light would travel just a tad tad tad bit below that.

Come to think of it, though, everything I know is built off of assuming they're one in the same (that the photon is massless, ergo the speed limit of the universe and the speed of light are one in the same). If we actually did find a tiny but non-zero mass of the photon, I believe the 'c' in E=mc^2 would not be the speed of light, but it would be this maximum speed limit.

"It is the speed of light."

I was surprised to read that."If we actually did find a tiny but non-zero mass of the photon, I believe the 'c' in E=mc^2 would not be the speed of light, but it would be this maximum speed limit."

But then you said that. If you understand my what I'm asking then "It is the speed of light." is definitely not true. That would be like saying c is the speed gravity propagates. It is the other way around. Gravity propagates at c. The things light and gravity and anything else that travel at the same speed have in common is what the c in e=mc^2 is. This may seem subtle, but I think it is extremely important. Realizing this is an epiphany.

I would like to know if other experts can confirm that if the speed of light is not exactly c, then energy does not exactly equal mass times the speed of light squared, but energy still equals mass times c squared.

 

[Pengwuino]

I think I probably am being a bit confusing. So let's make it concrete. If we find the photon has a non-zero mass, then the speed of light would not be the same as the maximum speed limit. The number 'c' in E = mc^2, would be the maximum speed limit, not the speed of light because as far as I know, you can derive special relativity independently of electrodynamics.

 

[CosmicVoyager]

I think I probably am being a bit confusing. So let's make it concrete. If we find the photon has a non-zero mass, then the speed of light would not be the same as the maximum speed limit. The number 'c' in E = mc^2, would be the maximum speed limit, not the speed of light because as far as I know, you can derive special relativity independently of electrodynamics.

Excellent! Thanks :-)

 

[CosmicVoyager]

So I wonder what a more accurate, hopefully a perfectly accurate, way of reading e=mc^2 is.

Energy equals mass times what squared? And the speed limit might not apply to the quantum world, right? How c is defined should incorporate that fact, and also the fact that there might be particles already traveling faster than c.

 

[Pengwuino]

So I wonder what a more accurate, hopefully a perfectly accurate, way of reading e=mc^2 is.

Energy equals mass times what squared? And the speed limit might not apply to the quantum world, right? How c is defined should incorporate that fact, and also the fact that there might be particles already traveling faster than c.

Times... the speed massless particles travel?

The speed of light does apply to relativistic quantum mechanics. What makes you think it does not apply?

Working in the realm of particles that live beyond the speed of light is not something I know anything about, however. It's completely out of mainstream research and I wouldn't even know if our theories still make sense in that realm.

 

[CosmicVoyager]

The speed of light does apply to relativistic quantum mechanics. What makes you think it does not apply?

I was thinking of c as being the speed limit of one things affecting another thing, and of things in quantum mechanics such as instantaneous action at a distance and teleportation.

 

[CosmicVoyager]

Times... the speed massless particles travel?

But isn't it possible for there to be particles already traveling faster than light?

 

[harrylin]

Greetings,

I read in a relativity paper or book (I think one of Einsteins. I am trying to find it again.) that c is actually represents the speed limit of the universe, and because light has no mass it travels at that speed. That c was arrived at independently from the speed of light.

I think looking at it this way is extremely helpful in understanding things. Everyone seems to ask questions such as "Why can't you go faster than the speed of light?" As though light were somehow the cause, but light traveling at speed c is actually a consequence of c being the limit.

Does anyone happen to know the paper or book?

Thanks

Indeed light is not the cause of the speed with which it propagates. However, the speed of propagation of electromagnetic waves and light in vacuum was found to be a constant, c. And the assumption that this is exactly the case is part of the light postulate, which served as a boundary condition for the derivations of the Lorentz transformations. So, the c in the Lorentz transformations was not arrived at independently of the speed of light. This is explained here:

http://www.bartleby.com/173/7.html
http://www.bartleby.com/173/11.html

However, the outcome of that derivation is that "c plays the part of an unattainable limiting velocity", as explained in the same book:

http://www.bartleby.com/173/12.html

Cheers,
Harald

 

[CosmicVoyager]

Indeed light is not the cause of the speed with which it propagates. However, the speed of propagation of electromagnetic waves and light in vacuum was found to be a constant, c. And the assumption that this is exactly the case is part of the light postulate, which served as a boundary condition for the derivations of the Lorentz transformations. So, the c in the Lorentz transformations was not arrived at independently of the speed of light. This is explained here:

http://www.bartleby.com/173/7.html
http://www.bartleby.com/173/11.html

However, the outcome of that derivation is that "c plays the part of an unattainable limiting velocity", as explained in the same book:

http://www.bartleby.com/173/12.html

Cheers,
Harald

"However, the speed of propagation of electromagnetic waves and light in vacuum was found to be a constant, c."

I read that there is a a limit to the precision with with we have measured the speed of light, and that it might be slightly less. And if it were slightly less, it would not change the value of c.

You are saying that is not possible? You are saying that the speed of light *must* be equal to c? That contradicts bcrowell's FAQ and what Pengwuino just said. Darn. It was making perfect sense.

 

[Shawakaze]

Interesting discussion. According to my knowledge all massles particles travel at c, which includes photons, gluons, W & Z bosons and gravitons(theoretically). C is the causality constant if I'm permitted to regard it that way, it is the focus of the epr paradox. Ftl particles simply do not follow causlity in the usual classical sense.c is just the limiting speed, light is not special at all.

 

[Shawakaze]

If the speed of light does not equal c, is it correct to read e=mc^2 as "energy equals mass times the speed of light squared"?

Is the point of the c in that equation to say that the amount of energy a certain amount of mass equals depends on the speed of light, or is the point to say it depends on the maximum speed of cause and effect?

Ok this has really caught my attention. Ok here is a crazy thought I'm offering u, its just a suggestion. Suppose ultimately all matter was made up of massless "stuff". And something else gave it a measure of inertia, "mass". Then if u wanted to calculate the energy of a bunch of that "stuff", how would u calculate for it?
Would u use e=mc^2? Just a crazy thought remember!

 

[harrylin]

"However, the speed of propagation of electromagnetic waves and light in vacuum was found to be a constant, c."

I wrote: "The assumption that this is exactly the case is part of the light postulate".

I read that there is a a limit to the precision with with we have measured the speed of light, and that it might be slightly less. And if it were slightly less, it would not change the value of c.

You are saying that is not possible? You are saying that the speed of light *must* be equal to c? That contradicts bcrowell's FAQ and what Pengwuino just said. Darn. It was making perfect sense.

No, I did not say such a thing. In order to clarify this, I summarized the history and referred you to a book by Einstein to give you a reference that you asked for.

So, Pengwuino just said that c is the speed of light: the speed of light is assumed to be a constant, and its speed is designated with the symbol c. The definition of c is based on Maxwell's hypothesis that the speed of light is a constant, and this was taken as postulate for special relativity. Thus c is defined as the speed of light, and in relativity it has also taken the meaning of limit speed (you asked for the book, did you now read the chapters to which I provided the links?).

If the speed of light would turn out to be very slightly less than the limit speed of nature (and probably not a constant), this would impose a redefinition of c in the way bcrowell described.

PS, as an afterthought: c being the speed of light is itself already a secondary meaning, if I'm not mistaken. Originally c stood apparently for a property of vacuum. Maxwell and others proposed that the speed of light is equal to this vacuum constant.
- http://en.wikisource.org/wiki/On_Physical_Lines_of_Force

Thus the speed of light was regarded to be a consequence of the electromagnetic properties of space. So, even if the speed of light would turn out to be very slightly less than that, this first meaning of c would remain - together with the additional meaning of speed limit, which may be regarded as a consequence of the properties of space.

 

[homology]

Jiminey this is interesting! I had just read, a few weeks back, that if the photon has non-vanishing mass we lose gauge invariance (and thus charge conservation). Also, I think this would alter the 1/r dependence of the Coulomb potential. (I'm thinking of Zee's derivation in his QFT in a nutshell book).

The consequences of such a detail are amazing!

 

[harrylin]

Jiminey this is interesting! I had just read, a few weeks back, that if the photon has non-vanishing mass we lose gauge invariance (and thus charge conservation). Also, I think this would alter the 1/r dependence of the Coulomb potential. (I'm thinking of Zee's derivation in his QFT in a nutshell book).

The consequences of such a detail are amazing!

I don't think so... where did you read that?

 

[bcrowell]

Interesting discussion. According to my knowledge all massles particles travel at c, which includes photons, gluons, W & Z bosons and gravitons(theoretically).

The W and Z are not massless.

So I wonder what a more accurate, hopefully a perfectly accurate, way of reading e=mc^2 is.

Energy equals mass times what squared? And the speed limit might not apply to the quantum world, right? How c is defined should incorporate that fact, and also the fact that there might be particles already traveling faster than c.

One way of reading it: "Energy equals mass multiplied by the square of the maximum speed of cause and effect."
Another way of reading it: "Energy equals mass" (in an approproate systems of units where c=1).

 

[homology]

I don't think so... where did you read that?

Read what? That photon mass violates gauge invariance and charge conservation? Proca mass would get you some hits via google. I read about it first in my EM book (by Brau) where when setting up the lagrangian he includes a term quadratic in the 4-potential, just for kicks. That breaks gauge invariance which is the symmetry giving charge conservation. The coefficient of the term $$A_{\mu}A^{\mu}$$ ends up related to the mass of a photon. There have a been a number of papers over the years investigating such consequences as this and also the effects of the electric force. I was just reading one last night:

Experimental limits on the Photon Mass and Cosmic Magnetic Vector Potential, by Roderic Lakes, Phys. Rev. Lett, Vol. 80, no.9 1826-1829

But check out Zee's quantum field theory book, its chapter I.4

 

[harrylin]

Read what? That photon mass violates gauge invariance and charge conservation? Proca mass would get you some hits via google. I read about it first in my EM book (by Brau) where when setting up the lagrangian he includes a term quadratic in the 4-potential, just for kicks. That breaks gauge invariance which is the symmetry giving charge conservation. The coefficient of the term $$A_{\mu}A^{\mu}$$ ends up related to the mass of a photon. There have a been a number of papers over the years investigating such consequences as this and also the effects of the electric force. I was just reading one last night:

Experimental limits on the Photon Mass and Cosmic Magnetic Vector Potential, by Roderic Lakes, Phys. Rev. Lett, Vol. 80, no.9 1826-1829

But check out Zee's quantum field theory book, its chapter I.4

Thanks, I didn't know about Proca! Now, following your suggestion I stumbled on an overview article in http://iopscience.iop.org/0034-4885/68/1/R02, which states:

"Phase invariance (U(1) invariance) is lost in Proca theory, but the Lorentz gauge is automatically held, and this is indispensable to charge conservation, i.e. the Lorentz condition becomes a condition of consistency for the Proca field."

Doesn't that mean that charge is conserved?

Harald

 

[yuiop]

I wonder why no one has mentioned the local nature of this maximum speed limit? For example in cosmological terms over vast distances, objects are said to be receding from us at greater than the speed of light, but locally the maximum relative speed is always c (if using a sensible coordinate system).

 

[bcrowell]

I wonder why no one has mentioned the local nature of this maximum speed limit? For example in cosmological terms over vast distances, objects are said to be receding from us at greater than the speed of light, but locally the maximum relative speed is always c (if using a sensible coordinate system).

IMO the right way to look at this is that in GR, there is no unique way to define the relative velocity of objects that are far apart. Since velocity is only defined locally, there is clearly no way we could have a speed limit that was more than local. Distant galaxies can be said to be receding from us at >c. Distant galaxies can also be said to be at rest relative to us. There is no meaningful way to decide which of these is true.

 

[harrylin]

I wonder why no one has mentioned the local nature of this maximum speed limit? For example in cosmological terms over vast distances, objects are said to be receding from us at greater than the speed of light, but locally the maximum relative speed is always c (if using a sensible coordinate system).

I was thinking about mentioning that in post #16 which logically leads to such a remark (the properties of space are local), but left it out as it's apparently not the focus of the discussion here. "Why can't you go faster than the speed of light?" is, as I understood it, a local question anyway!

 

[CosmicVoyager]

I wonder why no one has mentioned the local nature of this maximum speed limit? For example in cosmological terms over vast distances, objects are said to be receding from us at greater than the speed of light, but locally the maximum relative speed is always c (if using a sensible coordinate system).

My understanding is...

They are not actually receding. That is, they are not moving *through* space. Space is moving carrying them with it.

 

[jambaugh]

A better understanding of c, is not so much as the speed limit but as a unit conversion factor. The E=mc^2 equation is an identity "mass is energy" plus a unit conversion.
1kg = c^2 joules.

In that context, also note that "speed" is just a slope in space-time and unitless when working in common units. I find this helps to understand the "speed limit" as a geometric phenomenon instead of a dynamic one.

 

[CosmicVoyager]

A better understanding of c, is not so much as the speed limit but as a unit conversion factor. The E=mc^2 equation is an identity "mass is energy" plus a unit conversion.
1kg = c^2 joules.

In that context, also note that "speed" is just a slope in space-time and unitless when working in common units. I find this helps to understand the "speed limit" as a geometric phenomenon instead of a dynamic one.

This is excellent. I think this is headed in the right direction. Conceptally, the important form of the equation e=mc^2 is c=squarroot(m/e). The real question is why is the speed limit of the universe equal to the square root of the mass of any object divided by it's energy. I get the feeling that it must be, by the very definitions of what mass and energy are. I don't see it yet though. I think this is very close. Something to do with the fact that energy is motion. I will think about it. If anyone can illustrate why that must be so, please do.

The square root of mass divided by energy. Hmm...

 

[homology]

Thanks, I didn't know about Proca! Now, following your suggestion I stumbled on an overview article in http://iopscience.iop.org/0034-4885/68/1/R02, which states:

"Phase invariance (U(1) invariance) is lost in Proca theory, but the Lorentz gauge is automatically held, and this is indispensable to charge conservation, i.e. the Lorentz condition becomes a condition of consistency for the Proca field."

Doesn't that mean that charge is conserved?

Harald

Gosh Harald, I don't know. When you look at the Lagrangian its covariant whether or not you have the term $$A_{\mu}A^{\mu}$$ and so I never saw how that gave us anything than the fact that we know Maxwell's equations are covariant. What I've seen so far is that if you perform a gauge invariance and you want the action to be invariant you have to invoke charge conservation. But that trick doesn't work if you have the term with the two A's. I've always wanted to look at it a bit deeper, but its not clear to me (from just a quick glance over the paper) how charge conservation follows from the Lorentz invariance. I wonder if we should start a new thread on this?

P.S. that is, start a thread on the consequences of a nonzero photon mass (and this and other articles). It could be a fun way to explore the consequences.

 

[jambaugh]

...Conceptally, the important form of the equation e=mc^2 is c=squarroot(m/e).

No, I disagree. The important form is the more general form (in c=1 units e.g. time in seconds distance in light-seconds):
$$m^2 = e^2 - p\cdot p$$
Mass is proper energy, i.e. the energy of a particle as seen by a co-moving observer.
Viewed another way it is the magnitude of the energy-momentum 4-vector.

I think you're looking for the wrong thing by looking for the "why" of a speed limit. At such a fundamental level, the universe simply is as it is. There are no alternatives to compare as if a choice were to be made. This is distinct from e.g. the question of say why a cannon ball lands in a particular spot. One can speak of the dependence on changeable prior factors.

With e.g. existence of speed limit, one can at best, utilize that limit as a basis for unifying spatial and temporal units of measurement... then describe other phenomena in that context. One can incorporate how this condition of nature relates to others, e.g. relativity of moving observers, conservation of certain quantities and not of others, symmetries of systems and the nature of their dynamic evolution.

One can also investigate the reasonableness of certain questions, for example "why isn't c bigger or smaller?" is meaningless when one realizes we define the value relative to our arbitrary choices of units for measuring distance and duration. One may ask "can c vary over space, and depend on direction?" and realize that to make the question meaningful you must assume a geometry. The variation and directional dependence of c then becomes equivalent (in the context of fixed c) to a question of the variability of the geometry . . . Voilà! one has the theory of General Relativity.

Look at it this way. To describe "where" some point is in the plane one must pick an origin point, consider it fixed and describe "where" relative to that origin. Likewise we pick c to be a fixed point of kinematic description and describe "what's happening" relative to that. With this understanding, c=1 is the best "value" to pick when we are not trying to build bridges or coordinate meetings.

Finally, that having been said. There is a way to attempt to explain the "why" question. There are really only a few choices for 3 space + 1 time structure if space itself is isotropic. These are enumerated by the thee choices of "orthogonal" groups on 4 dimensions which contain the group of rotations in the three spatial dimensions. These are SO(4) (the simple extension to rotations in 4 space-like dimensions, ISO(3) the group of velocity transformations in Galilean relativity involving rotating velocities and adding them as vectors, and finally SO(3,1) the Lorentz group which we now believe is correct. If SO(4) was the group then we would be able to rotate around so we move backwards in time and we would be able to create causal paradoxes. SO(3,1) and ISO(3) do not allow this. Of the two SO(3,1) (implying a speed limit) is stable under small perturbations of its defining relations. If we leave the "choice" to random chance then SO(3,1) implying our relativistic Minkowski space-time structure is infinitely "more probable".

There are few scientific questions which can be answered universally in the affirmative. This is one of those. We can positively show that empirical evidence eliminates all but a locally Minkowski space-time structure.

 

[_PJ_]

Really it's best to just abandon c as a measurement of 'speed' altogether.

Since "speed" is displacement/time, both of which are subject to lorenzian contraction, and time itself is not a scientific concept, since it cannot be defined nor readily proven.

c is easier understood as the ratio of Energy/mass.

When relativistic motion requires the consideration of the amaximum speed, it can be understood as the "greatest possible distance traversed in the shortest possible time".
Now there is nothing to prevent faster than light speeds. In fact, there's every reason to believe that such is a feature of the universe. What IS impossible, is accelerating (by definition, accelerating positively or negatively - so including decelerating) across that 'barrier'. Also impossible to accelerate bosons. However, this would imply an infinitie answre to "greatest possible distance traversed in the shortest possible time". One could resolve this by substituting in the Planck Time, but since that itself iss derivative from c, it is not valid to use in this way.

So, let's forget E=mc^2, and consider the source, concentrating only on distance, speed and time.:

s= _________________________
\/(v^2*t^2)-(t^2*(v^2-c^2)

Einstein proved that light (and therefore all bosons) travel along null-lines. Once s=0, all results for v=c show that distance along the same path are equivalent, implying an infinite speed through spacetime, while at a finite velocity v=c.

This is WHY relativity shows c as a limit.

 

[Sam Gralla]

Proca theory is presented confusingly in almost every reference, and your quote,

"Phase invariance (U(1) invariance) is lost in Proca theory, but the Lorentz gauge is automatically held, and this is indispensable to charge conservation, i.e. the Lorentz condition becomes a condition of consistency for the Proca field."

is a prime example. The logic here is totally backwards and in fact I would say simply incorrect. In standard Maxwell theory charge conservation is guaranteed--you cannot find solutions to Maxwell's equations with non-conserved sources. (This can be viewed as a consequence of the gauge symmetry, if you like.) In proca theory charge conservation is not guaranteed by the equations--you can perfectly well find solutions with non-conserved source. However, if you demand that charge be conserved, then you find that the "Lorentz gauge condition" holds. (I put it in quotes because there is no notion of gauge in proca theory. The condition is just an extra equation that comes out if you demand charge conservation.) You can see how the above quote has the logic backwards. Also I think it is crazy to call the Lorentz gauge condition a "consistency condition" for the field. The "Lorentz gauge condition" enforces charge conservation, but there is nothing inconsistent about not conserving charge! As far as I know there are no consistency problems with the proca equations, with or without the "Lorentz gauge condition".

 

[Demystifier]

Proca theory is presented confusingly in almost every reference, and your quote,

"Phase invariance (U(1) invariance) is lost in Proca theory, but the Lorentz gauge is automatically held, and this is indispensable to charge conservation, i.e. the Lorentz condition becomes a condition of consistency for the Proca field."

is a prime example. The logic here is totally backwards and in fact I would say simply incorrect. In standard Maxwell theory charge conservation is guaranteed--you cannot find solutions to Maxwell's equations with non-conserved sources. (This can be viewed as a consequence of the gauge symmetry, if you like.) In proca theory charge conservation is not guaranteed by the equations--you can perfectly well find solutions with non-conserved source. However, if you demand that charge be conserved, then you find that the "Lorentz gauge condition" holds. (I put it in quotes because there is no notion of gauge in proca theory. The condition is just an extra equation that comes out if you demand charge conservation.) You can see how the above quote has the logic backwards. Also I think it is crazy to call the Lorentz gauge condition a "consistency condition" for the field. The "Lorentz gauge condition" enforces charge conservation, but there is nothing inconsistent about not conserving charge! As far as I know there are no consistency problems with the proca equations, with or without the "Lorentz gauge condition".

After seeing your excellent points on another topic, I came to see your other comments on the Forum (and this one is the only at the moment). Again, your comments are excellent. You are new here, but I hope you will stay here for a longer time a contribute a lot.

 

[homology]

Proca theory is presented confusingly in almost every reference, and your quote,
.

Thanks for clearing that up. I've only just recently even heard of the Proca mass and have started reading some papers on it. That being said I'm a total noob.

Welcome to PF forums :)

 

[Sam Gralla]

Thanks for the welcoming comments, demystifier and homology. I stumbled across these forums a few days ago. They seem well moderated and very active, and it's a great idea to have such a place where people can discuss physics. So, I thought I'd join in.

 

[Neandethal00]

I think I probably am being a bit confusing. So let's make it concrete. If we find the photon has a non-zero mass, then the speed of light would not be the same as the maximum speed limit. The number 'c' in E = mc^2, would be the maximum speed limit, not the speed of light because as far as I know, you can derive special relativity independently of electrodynamics.

First of all, one unwritten universal law is 'everything in this universe has a limit', so does the speed of anything. But the question is 'how do we know maximum speed limit is c?' It is just 'one theory' that tells us c is the highest speed of things. A theory.

If we use E=mc², and assume c is the highest speed of any object, doesn't it also mean 'the rest energy of mass m is twice its maximum kinetic energy?'