Why does light have invarient speed?

[rbj]

I'd argue though that the above is only true under the premise that c is max and constancy to anyone and everyone. It does not prove, nor explain, why c (or its special case, light speed in vacuum) is constancy. And the only proof we have is by experiment/example (and perhaps Maxwell's equation, which is up to interpretation and subject to the question of whether it is complete). If neither of the observers can claim that he’s solely being stationary, why must we assume both of them are right?

well, evidently you didn't get it. nothing that a little repetition cannot fix:

actually, it's the other way around; YOU need to justify why any two inertial observers would have different laws of physics applying to them. what would be the mechanism for why one inertial observer would have one set of physical laws applying to what he observes and the other inertial observer would have a different set of physical laws applying to what she observes. the postulate of SR is that we can tell no difference between the two different constant velocity frames of reference because there is nothing different about them just because they do not have the same constant velocity. from the POV of the first observer, he is stationary and it's the second observer who is moving. from the POV of the second observer, she is stationary and it's the first observer who is moving. what tangible reason would you prefer one over the other?

if there is no meaningful difference [between two different observers, neither whom are accelerated], if both observers have equal claim to being stationary (and it's the other guy who is moving), then the laws of physics (particularly Maxwell's Equations) have to be exactly the same for both of them, both in a qualitative sense, and in a quantitative sense. both observers have the same permittivity of free space (\epsilon_0) and permealbility of free space (\mu_0). (why should one get values lower or higher than the other?) so when they apply Maxwell's equations to this E&M wave (of this flashlight beam), they both will see that this changing E field is causing a changing B field which, in turn, is causing a changing E field which is causing a changing B field, etc. and for both observers, the laws of physics governing the interactions of electomagnetic fields (Maxwell's Eqs. and the parameters \epsilon_0 and \mu_0) are the same. (as well as all other physical law.) then it turns out, when either observer solves Maxwell's Equations for this case, they both get a propagating wave with wave speed of:

c = \sqrt{\frac{1}{\epsilon_0 \mu_0}} [/itex]<br /> <br /> <b>but that&#039;s the same for both observers!</b> (even though they both are moving relative to the other.) there is no reason one should solve the Maxwell&#039;s equations and get a different <i>c</i> than the other (because they both have the same \epsilon_0 and \mu_0)! even if the two are looking at the very same beam of light.<br /> <br /> so, let&#039;s try again. did you read it? the onus is on <b>you</b>, not me nor Einstein (nor anyone else who accepts SR) to indicate to us <i>why</i> you would prefer one inertial frame of reference over the other. if you cannot justify such a preference, then the onus is you <b>you</b> to &quot;prove&quot; that the laws of physics is different in the two unaccelerated frames of reference. why should anyone think that they are?<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> You (and we all) say, well, otherwise Maxwell’s equation (and SR) would be broken. This to me, is a circular argument that does not address the deeper question. </div> </div> </blockquote><br /> no, not SR. where did you get that? the issue is if Maxwell&#039;s equation<b>s</b> (there are 4 of them) are in any way different for the two different inertial observers. are they? what thin little shred of evidence or reason would you think that they are? again, it&#039;s up to <b><i><u>YOU</u></i></b>, who suggests otherwise, to say why. not the other way around. this onus of reason or proof, of making a case, lies with you, not me (nor Einstein).<br /> <br /> this is the basis of SR: that we can find no reason to believe that one inertial observer is any more preferred than any other inertial observer. that is the postulate and it wouldn&#039;t be true if there really was an aether out there that sets the standard for what is truly stationary and what is truly moving. but we have no reason to believe that such an absolute frame of reference exists. there is nothing in our everyday experience, nor in scientific experiments to suggest otherwise. what is the logic in believing something to exist (outside of a trancendental faith, which is okay to have, but not okay to apply to science) when there is no reason nor evidence to believe that such exists? the onus is on <b>you</b> to prove the contrary, not on me.<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> On the other hand, if we can prove mathematically, without assuming Einstein&#039;s relativity principle (constancy of c and relativity) that c must exist and has to be constancy, we likely would be able to show that light speed in vacuum satisfies the proof (i.e., is a special case of c) and consequently derive SR. To me, locality principle seems to be capable of doing this. I believe that the locality principle assumes something more fundamental than constancy of c , and thus would give us deeper insight about relativity. </div> </div> </blockquote><br /> i think you need to look into the obvious insight before looking into the deep. <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> It is ‘the’ speed without mathematical proof, though SR shows us that if we take this for granted we can come to lots of useful and consistent conclusions. </div> </div> </blockquote><br /> no, it&#039;s the <b>only</b> consistent conclusions. if the postulate of SR was not true, we would be <b>forced</b> to come to inconsistent conclusions. how does one justify an understanding of reality with inconsistent conclusions when there exists another understanding that is consistent?<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> This would bring up another interesting question. Is vacuum really “nothing”? </div> </div> </blockquote><br /> there is &quot;space&quot;. but, unless you can find evidence that there is something else in there (and the M-M experiment tried to, with negative results), the onus is really on you to show that there is the slightest reason to believe that there actually <b>is</b> something else there. outside of religious faith (which i definitely do not denigrate) there is no reason to believe in something that has absolutely no measurable properties of existence.<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Is there such a thing that is truly empty where *nothing* can travel on/by/within/through it? And what would be the equivalent of photon for magnetic field, gravitational field, or quantum fields where fundamental particles interact within? Is there any theoretical or experimental proof so far that “graviton” exists, has zero rest mass, travels at c to each one and everyone? </div> </div> </blockquote><br /> i don&#039;t think anyone is saying that gravitons exist as a matter of fact (as they would say that photons exist). like strings and branes, it&#039;s a hypothesis.<br /> <br /> <blockquote data-attributes="" data-quote="mdeng" data-source="post: 1561725" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> mdeng said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> Einstein’s relativity theory does not preclude ether. </div> </div> </blockquote><br /> actually, it does preclude aether, if you define it as the medium of which light (E&amp;M) and/or the perturbations of other interaction propagate in.

 

[mdeng]

well, evidently you didn't get it. nothing that a little repetition cannot fix:

actually, it's the other way around; YOU need to justify why any two inertial observers would have different laws of physics applying to them.

I am not sure I follow you. Why is the burden on me to show/justify this when I am merely asking why light travels at a constant c (in vacuum) to anyone and everyone? You seem to be equating my question about constancy of light to about why one should prefer one inertia frame over another when observing light. Well, I don't think the two questions are the same, even though they may be related or even equivalent to some extent. The reason is that the assumption of constancy of light speed is the foundation of SR. Without it, I'm not even sure how one can argue about why's wrong to have preferential treatment of one inertial frame over another.

what would be the mechanism for why one inertial observer would have one set of physical laws applying to what he observes and the other inertial observer would have a different set of physical laws applying to what she observes.

SR itself does *not* prove there must exist constancy c. On the contrary, it assumes so and is built on top of it. And without this assumption, I am not sure how SR could claim what you stated above. Maxwell's equation may have predicted this (or simply happens to agree with it), but it's not explain why light must behave this way. IMO, constancy c must be the entailment of something beyond SR, not something proven by SR.

I don't have any answer as to what this "something" is, nor why or why not we'd prefer one inertia frame over the other, other than "that is what SR (together with its assumption) says". I am not saying SR is wrong, and in that sense I agree with you that we should not prefer so and so. But I'd like to ask what dictates light to behave this way? How could we arrive at the same conclusion as SR by starting with something more fundamental without first assuming constancy c or talking about light? I hope questions like these may one day lead to an answer.

this is the basis of SR: that we can find no reason to believe that one inertial observer is any more preferred than any other inertial observer. that is the postulate and it wouldn't be true if there really was an aether out there that sets the standard for what is truly stationary and what is truly moving. ... the onus is on you to prove the contrary, not on me.

While the SR postulate may very well be true (I believe is), we can't prove it merely by asking whether anybody is able to disprove it. Nevertheless, the intention of my original question is not whether SR is wrong, but what its ultimate foundation is which may replace its postulate with something more fundamental. I am just not satisfied with postulates, no matter how consistent it may seem to be. :)

there is "space". but, unless you can find evidence that there is something else in there (and the M-M experiment tried to, with negative results), the onus is really on you to show that there is the slightest reason to believe that there actually is something else there. outside of religious faith (which i definitely do not denigrate) there is no reason to believe in something that has absolutely no measurable properties of existence.

I don't have any clue or suggestion to offer. As I said, it's more a philosophical question at this stage, just like what the smallest particle is.

i don't think anyone is saying that gravitons exist as a matter of fact (as they would say that photons exist). like strings and branes, it's a hypothesis.

I'd believe that if GR is right, then graviton must exist and travels at c because GR says (assumes) gravitational wave has the property of constancy c.

 

[sas3]

As we approach speed of light, time slows down next to stand-still (and space contracts by an infinite factor). However, I don’t think this can apply to speed = c because 1/sqrt(1-v^2/c^2) is undefined. Intuitively, how can we have time frozen and yet "travel" (i.e., how can we experience "change" without time)? Maybe, when time is frozen, we must be at anywhere and everywhere. That is, there is no longer a concept of "space" because there is no longer a concept of distance.

I forgot about the “space contracts by an infinite factor” part.

Thanks

 

[mdeng]

I have never done what you say; I have only talked about rapidity and I showed that you could travel along all the universe in (almost) zero time of your clock. The fact rapidity has nothing to do with space and time should have been clear by the way I have defined it, so it's not very useful to say that it couldn't be called "speed" because "you're taking the distance traveled as measured in one reference frame, and dividing it by the time as measured in another reference frame": the idea is just to avoid defining it in terms of space and time.

With this idea, the fact light's speed cannot be reached is transformed in the idea that light's "speed" = rapidity is actually infinite, but s/t doesn't result as infinite but finite, because space and time are not mutually independent, excepting at v << c.

By the way: rapidity R = arctgh(v/c).

I think what you are saying is that although R is related to space and time (because by R's definition, it is related to v and thus to space and time by which the particular v is calculated), it avoids the dependencies between space and time. I read that rapidity has the property that relative rapidities are additive. But how is the "v" (and hence R, or vice versa) here decided anyway, as in your example? ‘v’ to which inertial frame, and by which frame's time?

 

[lightarrow]

I think what you are saying is that although R is related to space and time (because by R's definition, it is related to v and thus to space and time by which the particular v is calculated), it avoids the dependencies between space and time. I read that rapidity has the property that relative rapidities are additive.

Quite simple to show. If you want to add

v_1
and

v_2

the formula is

V = \frac{v_1+v_2}{1+v_1v_2/c^2}

as you know. I write it this way:

V/c = \frac{v_1/c+v_2/c}{1+v_1v_2/c^2}

By definition:

V/c = tangh\ R

v_1/c = tangh\ R_1

v_2/c = tangh\ R_2

so

tangh\ R\ =\ \frac{v_1/c+v_2/c}{1+v_1v_2/c^2}\ =\ \frac{tangh\ R_1+tangh\ R_2}{1+tangh\ R_1\ tangh\ R_2}\ =\ tangh(R_1+R_2)

which clearly means

R = R_1+R_2.

But how is the "v" (and hence R, or vice versa) here decided anyway, as in your example? ‘v’ to which inertial frame, and by which frame's time?

When you talk about the velocity of an object you are referring to a specific ref frame, exactly as in Newtonian mechanics; frame's time is not relevant here.

 

[rbj]

I am not sure I follow you.

i guess you don't.

Why is the burden on me to show/justify this when I am merely asking why light travels at a constant c (in vacuum) to anyone and everyone?

it's because of the fact that the laws of physics, both qualitatively and quantitatively are the same for observers that are both inertial that c is the same for both observers. otherwise, you have two observers that are in the same situation, an inertial frame of reference, and they both perform the same experiment and get measurably different results. if c = 1/\sqrt{\epsilon_0 \mu_0} was measurably different, then at least one observer either gets a different law of physics, at least quantitatively with a different c or \epsilon_0 or \mu_0. you have to tell us why in the world would two observers, in equivalent situations - that is inertial frames of reference, just not the same inertial frame of reference - two observers with equal claim to being the stationary observer, so there is no good reason for those Maxwell's equations to be inaccurate for either one of these observers.

unless you supply us with the good reason. if your "good" reason is that c is different for one observer over the other, that isn't good enough. that's the circular reasoning. it continues to beg the question for why should the laws of physics be different for one inertial observer than the other.

so, yes. the onus continues to be to come up with a reason for why the laws of physics are different for these two observers. if you cannot, the inescapable consequence is that the laws of physics are the same. if the laws of physics are the same, the parameters in those laws are the same. one of those parameters is c.

You seem to be equating my question about constancy of light to about why one should prefer one inertia frame over another when observing light. Well, I don't think the two questions are the same, even though they may be related or even equivalent to some extent.

they are to any extent unless you can tell us why one observer gets one set of laws and the other gets another set.

SR itself does *not* prove there must exist constancy c. On the contrary, it assumes so and is built on top of it. And without this assumption, I am not sure how SR could claim what you stated above.

i only said how, two or three times.

Maxwell's equation may have predicted this (or simply happens to agree with it),

in Maxwell's equations related to magnetism, Ampere's Law and the Biot-Savart Law.

\oint_C \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 \iint_S \mathbf{J} \cdot \mathrm{d}\mathbf{S} = \mu_0 \iint_S \rho \mathbf{v} \cdot \mathrm{d}\mathbf{S}

d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{(\mathbf{J}\, dV) \times \mathbf{\hat r}}{r^2} = \frac{\mu_0}{4\pi} \frac{(\rho \mathbf{v}\, dV) \times \mathbf{\hat r}}{r^2}

\mathbf{F} = q \cdot(\mathbf{E} + \mathbf{v} \times \mathbf{B})

so, with just Maxwell's equations, what value do you use for \mathbf{v} for either the moving charge that is the magnetic source or the moving charge that the \mathbf{B} field is acting on? if i am moving relative to you, do you use the values of velocity that you measure or the values that i measure? why should it be the values that you measure? why are your measures of velocity, that you plug into Maxwell's equations to compare the theoretical result to experiement, be the perferred values over the velocities that i measure in my frame of reference? you have to justify that, and if you can't, it necessarily follows that the parameters used in the expression of physlcal law be the same for you as they are for me.

but it's not explain why light must behave this way. IMO, constancy c must be the entailment of something beyond SR, not something proven by SR.

I don't have any answer as to what this "something" is, nor why or why not we'd prefer one inertia frame over the other,

then, for you to be logically consistent, if you have no reason why one inertial frame is preferred over the other, you have no reason to expect that the theoretical nor measured speed of propagation is experienced differently for the two intertial frames of reference.

other than "that is what SR (together with its assumption) says".

it's an ancillary assumption or a corrollary of the main postulate that all inertial frames of reference have equal claim to being "stationary". if either can say that they are stationary when they do experiments, they should get, as it appears to themselves, the same results within an experimental error (assuming they are equally "good" experiments - that their level of experimental error are equally low).

I am not saying SR is wrong, and in that sense I agree with you that we should not prefer so and so. But I'd like to ask what dictates light to behave this way?

and i said precisely why, given the main postulate of relativity: that it is all relative, not absolute (at least as far as how we experience velocity - acceleration is a quantity that we have an absolute measure of, at least until GR). but i don't think that GR says a word about gravitons, does it? i thought that was an extension of the ideas of quantum mechanics to gravity, in a similar sense of how photons are related to the electromagnetic action.

 

[mdeng]

Quite simple to show. If you want to add

v_1
and

v_2

the formula is

V = \frac{v_1+v_2}{1+v_1v_2/c^2}

as you know. I write it this way:

V/c = \frac{v_1/c+v_2/c}{1+v_1v_2/c^2}

By definition:

V/c = tangh\ R

v_1/c = tangh\ R_1

v_2/c = tangh\ R_2

so

tangh\ R\ =\ \frac{v_1/c+v_2/c}{1+v_1v_2/c^2}\ =\ \frac{tangh\ R_1+tangh\ R_2}{1+tangh\ R_1\ tangh\ R_2}\ =\ tangh(R_1+R_2)

which clearly means

R = R_1+R_2.



When you talk about the velocity of an object you are referring to a specific ref frame, exactly as in Newtonian mechanics; frame's time is not relevant here.

Thanks for the math. Regarding the deductions, does the final result hold for v1 = -v2? It would make the denominator equal 0.

For your last statement, I think it meant to say that the time reference is within the same frame as the speed, except that the unit of the time (and for that matter, distance/space) does not matter (i.e., is irrelevant) .

 

[mdeng]

it's because of the fact that the laws of physics, both qualitatively and quantitatively are the same for observers that are both inertial that c is the same for both observers. otherwise, you have two observers that are in the same situation, an inertial frame of reference, and they both perform the same experiment and get measurably different results.

Is this what is assumed by SR? The *why* about the 'c' part above is my question. I am aware of the "otherwise" part but I don't take that as an explanation or proof of why light has the constancy property. It's a postulate, not a proof, nor an explanation based on more fundamental physics laws.

... there is no good reason for those Maxwell's equations to be inaccurate for either one of these observers.

... unless you supply us with the good reason...

so, yes. the onus continues to be to come up with a reason for why the laws of physics are different for these two observers.

... if you cannot, the inescapable consequence is that the laws of physics are the same. if the laws of physics are the same, the parameters in those laws are the same. one of those parameters is c.

I am pretty sure that you have something to show me which I currently l don't grasp although I'd like to. However, we can't prove a theory, or a manifestation of some truth by observing that there is so far no good reason to think differently. It's plain truth that Einstein had to postulate the constancy of light speed to derive SR. If there is a proof of this postulate, and even more importantly an explanation, in terms of some more fundamental physics phenomenon, we then should be able to remove the postulate from SR and replace it with more basic principles. I don't find that your explanation satisfies this criteria which is what my question is about.

 

[Dale]

However, we can't prove a theory, or a manifestation of some truth by observing that there is so far no good reason to think differently.

Uhh, that's the whole basis of the scientific method.

Take a theory, use it to make a prediction (aka hypothesis) about some experiment, do the experiment, check the results against the hypothesis. If there is agreement then the theory is verified (so far no good reason to think differently), if there is not then the theory is falsified (a good reason to think differently).

Science fundamentally isn't about "manifestations of truth" it is about getting good models of how things work. If you want "truth" and "proof" then you want math or religion, not science. And even in math you need to start with some unproven axioms, and in religion you need unproven tenents, so why should science be denied unproven postulates? Seems rather arbitrary.

 

[mdeng]

Uhh, that's the whole basis of the scientific method.

Take a theory, use it to make a prediction (aka hypothesis) about some experiment, do the experiment, check the results against the hypothesis. If there is agreement then the theory is verified (so far no good reason to think differently), if there is not then the theory is falsified (a good reason to think differently).

Science fundamentally isn't about "manifestations of truth" it is about getting good models of how things work. If you want "truth" and "proof" then you want math or religion, not science. And even in math you need to start with some unproven axioms, and in religion you need unproven tenents, so why should science be denied unproven postulates? Seems rather arbitrary.

Well, it's not a proof, is it? "Proof by example" is not a proof and has often led to fallacies (or in the physics world inability to explain new phenomenons), while it certainly serves to support and strengthen the position of a theory (in the framework where the experiment is carried out).

I agree though that it is not necessarily the goal of physics to seek "manifestations of truth". However, physics does seek to reveal, or to explain, what's behind each and every physics phenomenon. And to this regard, it's an endless search. I'd regard the constancy of light speed as one such phenomenon that we, especially physics, like to find an answer to. This is completely different from saying we must deny an unproven postulate, which is not what I said or implied anyway. One of the best characteristics I like about science is that it is strong enough to allow questions on anything and everything, which ultimately gave science its strength and leads to new discoveries.

 

[mdeng]

Ming,

I repeat ( it's been said enough times in earlier posts) - without the postulate that all local observers measure the same speed of light the laws of physics ( Maxwell and Newton) do not work !

Right. I knew my result must be wrong but did not understand why. Some postings have made me aware that I missed the velocity addition formula in relativity.

 

[Mentz114]

Ming,
I saw that my post was unnecessary which is why I deleted it. I'm glad you're up-to-speed ( so to speak).

M

 

[phyti]

To mdeng:
Is your original question, "why is the measured speed of light the same for all observers?"

 

[mdeng]

To mdeng:
Is your original question, "why is the measured speed of light the same for all observers?"

Yes. I'd like to see what insights the current physics offers on this question.

EDIT: is the locality principle a candidate to explain c? It seems to have the potential.

 

[Dale]

Well, it's not a proof, is it?

Of course not. In science theories are never "proven". They are "verified" or "falsified".

You are asking a fundamentally unscientific question.

 

[mdeng]

Of course not. In science theories are never "proven". They are "verified" or "falsified".

You are asking a fundamentally unscientific question.

I hope/think you are not referring to the question of mine about why light has a constant speed c to anyone and everyone. Surely, an answer will come someday even though it will no doubt invoke some new postulates but they nevertheless would be more fundamental.

 

[mdeng]

Of course not. In science theories are never "proven". They are "verified" or "falsified".

You are asking a fundamentally unscientific question.

i only said how, two or three times.

Found something that more or less reflect your line of thoughts, which I don't really dispute at all, but would still like keep/pose my question. Or maybe I should rephrase my question from 'why' to 'how': how does light travel at constant c to anyone and everyone? :)

(from http://physicsworld.com/cws/article/print/24291)

This led to the final - scientific - stage, which saw the maturation of the human intellect. Physics and astronomy, Comte thought, reached this stage in the 17th century. Human beings ceased to ask why phenomena happened and instead sought to answer how they happened by finding the appropriate laws. The number of such laws tends to decrease as science progresses. Gravitation, for example, was found to unify what had seemed to be myriads of forces into one.

 

[rbj]

To mdeng:
Is your original question, "why is the measured speed of light the same for all observers?"

Yes. I'd like to see what insights the current physics offers on this question.

You are asking a fundamentally unscientific question.

i don't think that it's an unscientific question, but i fail to see how mdeng can accept the broader postulate of relativity, that "any law of nature should be the same at all times; and scientific investigations generally assume that laws of nature are the same regardless of the person measuring them", yet insist that a quantitative parameter of some of those laws can vary and, for some reason, needs yet another postulate to tie it down to a fixed value (at least between inertial observers). i don't get it, and i doubt that mdeng will convince me that I'm the one that's missing something in the logic here. the speed of light is the same for all inertial observers because, as a postulate, the laws of physics are the same. that is more than sufficient. http://en.wikipedia.org/wiki/Principle_of_relativity

the speed of light (or of gravity or whatever the interaction) being invariant between these different inertial observers is actually a corollary of the main postulate. so to ask "why is c the same" when one accepts the postulate that the laws of physics are the same is a logical disconnect.
it's a little similar to an unrelated topic that i had recently with someone else about Linear System Theory in Electrical Engineering or in Linear Algebra in mathematics.

it turns out the most basic postulate of a Linear System:

(1) \mathbf{L} \left\{ x_1(t) + x_2(t) \right\} = \mathbf{L} \left\{ x_1(t) \right\} + \mathbf{L} \left\{ x_2(t) \right\}

which is synonymous with "superposition applies" is sufficient, in and of itself, to establish the following as a corollary:

(2) \mathbf{L} \left\{ k x(t) \right\} = k \mathbf{L} \left\{ x(t) \right\}

for any constant k. at least any rational constant k.

and this can be extended to:

(3) \mathbf{L} \left\{ \sum_m k_m x_m(t) \right\} = \sum_m k_m \mathbf{L} \left\{ x_m(t) \right\}

if we set aside the issue of irrational k, simply because in a physical system with any decent continuity in it (it behaves virtually the same for k=3.1415926535897932384 as it does for k=3.1415926535897932385 and k=\pi is somewhere in between), the Eq. (2) is a direct result of Eq. (1) and, as a postulate, Eq. (1) is enough to say we have a linear system. we do not need both Eqs. (1) and (2), yet some (nearly all) textbooks in Linear System Theory list both (1) and (2) as postulates for when you are dealing with a Linear System. but Eq. (1) is good enough. if Eq. (1) is true, so is Eq. (2) (at least for all rational constants), and then so is (3).
Likewise, if one can accept the postulate that any law of nature is the same for observers in equivalent circumstances regardless of which observer is perceiving or measuring reality subject to such laws, then it follows that the parameter c, which exists in some laws, namely a simple rewritten form of Maxwell's Equations, is the same for each observer.

\nabla \cdot \mathbf{E} = c Z_0 \rho

\nabla \cdot \mathbf{B} = 0

\nabla \times \mathbf{E} = -\frac{1}{c}\frac{\partial \mathbf{B}} {\partial t}

\nabla \times \mathbf{B} = Z_0 \rho \mathbf{v} + \frac{1}{c} \frac{\partial \mathbf{E}} {\partial t}

note here that the B field is scaled, similarly to how cgs does it, so that it is dimensionally the same animal as the E field and that, rather than express it with \epsilon_0 \mu_0, we are expressing the equations in terms of two dependent parameters

Z_0 \equiv \sqrt{\frac{\mu_0}{\epsilon_0}}
and
c \equiv \sqrt{\frac{1}{\mu_0 \epsilon_0}}

forget, for the moment, about the meaning of the parameters c and Z₀ (it turns out later, that as we solve Maxwell's Equations in the context of free space, that the wavespeed comes out to be c and the characteristic impedance of propagation is Z₀). if mdeng accepts that the laws of physics are axiomatically the same between the two inertial observers, then if mdeng accepts that the c above is the speed of propagation of an EM disturbance (this is what you get when you solve the above wave equations in free space), then there is no logical reason that mdeng can deny that the postulate of identical laws for both observers does not imply an identical c. he/she can deny that one follows the other, but it is not logical.

 

[YellowTaxi]

Well what effect would there be if c stretched out and became 6x10^8 m/s
ie a value equal to 2c

Would all laws adapt, or would there be something that 'got left behind' as c increased ?

 

[rbj]

Well what effect would there be if c stretched out and became 6x10^8 m/s
ie a value equal to 2c

Would all laws adapt, or would there be something that 'got left behind' as c increased ?

what's important are the dimensionless constants. if a dimensionless 'constant' changes, we would know the difference. if a dimensionful parameter is believed to change, we don't measure that parameter all by itself. just as one counts tick marks on a ruler when measuring length, or tick marks on a weighing scale when measuring mass, or ticks of a clock when measuring time, all physical experiments really have dimensionless results. assuming we revert the definition of the meter to its pre-1960 definition (when it was the length between two scratch marks on a platinum-iridium bar in France), when we measure the speed of light, we are measuring it against some like-dimensioned standard and it would be that ratio that has fundamentally changed. perhaps it's some other parameter in that ratio that has changed. if no dimensionless fundamental constant has changed, we could not know the difference if some dimensionful constant has changed. (from whos perspective? some god who is unaffected by the change of c?) take a look at the Planck Units article in Wikipedia and/or my way-too-long treatise earlier in this thread or the little argument we had at the cosmology forum (regarding a variable \hbar cosmology). if you measure everything in Planck Units (or some other systems of natural units), then there is no c to vary, the speed of light in vacuo is always 1 Planck Length per Planck Time. if the dimensionless ratios of the number of Planck Lengths per meter or the number of Planck Times per second changed, then something would be noticably different, and while we might be tempted to blame it on a changing c, the salient fact is that this dimensionless ratio changed. whatever the dimensionless ratio that we measured that led us to initially think that c has changed. but we have to worry about the standard that c was measured against, we can't just say that it was c that had changed.

 

[YellowTaxi]

Yes' I'm aware of that type of argument rbj , my question was rather would we be able to tell simply from the other constants that maybe got left behind as c stretched.

I doubt they'd all be carried along with the larger version of c, or are they?

 

[lightarrow]

tangh\ R\ =\ \frac{v_1/c+v_2/c}{1+v_1v_2/c^2}\ =\ \frac{tangh\ R_1+tangh\ R_2}{1+tangh\ R_1\ tangh\ R_2}\ =\ tangh(R_1+R_2)

Thanks for the math. Regarding the deductions, does the final result hold for v1 = -v2? It would make the denominator equal 0.

If

v_1 = -v_2

the denominator is not zero but

1 - v_1^2/c^2.

 

[mdeng]

If

v_1 = -v_2

the denominator is not zero but

1 - v_1^2/c^2.

I meant v_1=-v_2 and |v_1| = c. I guess it still holds.

 

[mdeng]

Hi rbj,

Many thanks for all the effort to answer my question.

I think the disconnection between your reasoning and my question is about the understanding of what role the light speed constancy plays in SR. It seems to me that you think that’s implied by the principle of relativity. However, this would mean that Einstein could do away one of the two of his postulates. Since Einstein did not do that, it then follows (blindly, or out of my laziness) that light speed constancy must be independent of and thus can’t be explained by the principle of relativity. But I am having a second thought.

I am not exactly sure why Einstein had to postulate light speed constancy. One explanation is as stated at http://en.wikipedia.org/wiki/Introduction_to_special_relativity that the postulate is needed to establish Maxwell’s equation in the time-space 4D space. For the lack of knowledge on Minkowski's formula and my rusty math, I don’t know how this point worked out or what Minkowski’s equation postulates.
---
Therefore, by assuming that the universe has four dimensions that are related by Minkowski's formula the speed of light appears as a constant and it does not need to be assumed to be constant as in Einstein's original approach to special relativity. Notice that c is not explicitly required to be the speed of light. It is a consequence of Maxwell's electrodynamics that light travels with c. There is no such requirement inherent in special relativity.
---
Another explanation may have to do with “dependence on definition of units” as stated below from http://en.wikipedia.org/wiki/Status_of_special_relativity. But I am not sure where “but then the invariance of c is non-trivial” would come from.
---
Because of the freedom one has to select how one defines units of length and time in physics, it is possible to make one of the two postulates of relativity a tautological consequence of the definitions, but one cannot do this for both postulates simultaneously because when combined they have consequences which are independent of one's choice of definition of length and time. For instance, if one defines units of length and time in terms of a physical object (e.g. by defining units of time in terms of transitions of a caesium atom, or length in terms of wavelengths of a krypton atom) then it becomes tautological that the law determining that unit of length or time will be the same in all reference frames, but then the invariance of c is non-trivial. Conversely, if one defines units of length and time in such a way that c is necessarily constant, then the second postulate becomes tautological, but the first one does not; for instance, if the length unit is defined in terms of the time unit and a predetermined fixed value of c, then there is no a priori reason why the number of wavelengths of krypton in a unit of length will be the same in all reference frames (or even in all orientations).
---
There is yet another possible explanation at http://en.wikipedia.org/wiki/Status_of_special_relativity.
---
In fact Maxwell's equations combined with the first postulate of special relativity can be used to deduce the second postulate. Actually electromagnetism is greatly simplified by relativity, as magnetism is simply the relativistic effect obtained when the simple law of electrostatics is put into a relativistic Universe.
---
Perhaps Einstein did not want SR to depend on Maxwell’s equation and as such he would be able to show that Maxwell’s equation is a logical consequence of SR. But again, I don’t know how one would come to conclude “magnetism is simply the relativistic effect obtained when the simple law of electrostatics is put into a relativistic Universe.”
So it seems that we are very close to an agreement. Nevertheless, in addition to the mathematical consequence of Maxwell equation plus the principle of relativity, I’d like to know what mechanics is behind light to allow it travel that way, or “how does light travel at an invariant speed to anyone and everyone”? :)

EDIT: Just noticed that the last "explanation" actually is problematic given that Minkowski's formula is needed in addition to SR's 1st postulate to derive 'c' in SR.

 

[Mentz114]

Ming,

thus can’t be explained by the principle of relativity.

You've got completely the wrong way round. It is a postulate of relativity that everyone measures the same speed for light.

The postulate is supported by the fact that the laws of physics require it to avoid contradictions.

Also, is the Wiki really the best source you have ? I must say I find your arguments incomprehensible but I don't think you understand relativity at all.

 

[Dale]

I hope/think you are not referring to the question of mine about why light has a constant speed c to anyone and everyone.

As you noted in other posts the more scientific question is "how". Science answers "how" questions much better than "why" questions. However, that was not what I was referring to in this case.

What I am talking about is this:

Well, it's not a proof, is it? "Proof by example" is not a proof

If you refuse to allow experimental evidence in the answer then you are rejecting the scientific method and therefore you are not asking a scientific question. In fact, your question of this thread appears to be a philosophical or mathematical question.

 

[Dale]

i fail to see how mdeng can accept the broader postulate of relativity, that "any law of nature should be the same at all times; and scientific investigations generally assume that laws of nature are the same regardless of the person measuring them", yet insist that a quantitative parameter of some of those laws can vary and, for some reason, needs yet another postulate to tie it down to a fixed value (at least between inertial observers). i don't get it, and i doubt that mdeng will convince me that I'm the one that's missing something in the logic here.

I tend to agree with you on this point. I have often thought that the first postulate was sufficient and that the second postulate is simply a corolary to the first one. But I have been told by rather reliable sources that I was wrong, they were actually two separate postulates, and I didn't feel strongly enough about it to argue. I have a similar "wrong but not strong" opinion about Newton's first and second laws.

 

[phyti]

If the theory is based on the first and second postulates, then the speed of light 'c' is guaranteed, but the abstract mathematical manipulations will not reveal the 'how' or 'why'. The purpose of the theory is to produce numbers that agree with measurements/perceptions by the observer. To explain 'why', you have to analyze the behaviour in terms of physical processes. There is an answer to this, just as there is for time dilation.

 

[mdeng]

As you noted in other posts the more scientific question is "how". Science answers "how" questions much better than "why" questions. However, that was not what I was referring to in this case.

:) And in the sense of the article I quoted, I agree with you about 'why' vs. 'how'.

What I am talking about is this:If you refuse to allow experimental evidence in the answer then you are rejecting the scientific method and therefore you are not asking a scientific question. In fact, your question of this thread appears to be a philosophical or mathematical question.

My statement was actually meant to refer to my misconception of rbj's reasoning as proving one postulate by another or using SR as an absolute truth. I don't refuse experimental evidence at all. That's what physics and all empirical science are about when seeking truth (or "how" :-).

My original question is about the (physical) mechanism/process, not philosophy/abstract-math.

 

[mdeng]

I tend to agree with you on this point. I have often thought that the first postulate was sufficient and that the second postulate is simply a corolary to the first one. But I have been told by rather reliable sources that I was wrong, they were actually two separate postulates, and I didn't feel strongly enough about it to argue. I have a similar "wrong but not strong" opinion about Newton's first and second laws.

I don't know what your reliable sources are, but what they told you appears to be consistent with what I have read so far (except for some loose introductory articles).