A Derivation of the Speed of Light Postulate by Armin Nikkhah Shirazi

Being able to see my abstract as any other potentially interested reader will see it here has caused me to notice that it could easily give the impression that my derivation violates Occam's razor (i.e. one can reasonably ask why it is necessary to use three axioms to derive one postulate) and therefore be dismissed altogether.

I would like to take this space to explain why this impression is false. If one invests the time to read the paper, one will find that the derivation leads not just to the speed of light postulate, but to a framework that is equivalent to SR, with one exception: An unambiguous explanation for the speed of light postulate is built in that is completely absent in the standard formulation of SR, and this explanation leads to a definite experimental prediction which would be unexpected in the standard form. Therefore this derivation is not simply a tautological chain of reasoning presented from a novel starting point (which would violate Occam's razor), but really a "deeper" framework, provided the experimental prediction fails to be falsified (because the prediction is that of a null-result, it can never be be confirmed with absolute certainty).

Armin,

You are the first person who has welcomed some criticism that shows the strength of your character.

I think the lack of comments you have received is due to mathematical derivations. A real physics argument requires the reader to work through the derivations and is time consuming. Maybe our essays would have been discussed more if we avoided mathematics and wrote about the applications of LQG, string theory and the multiverse on human consciousness. Maybe we were mistaken and this contest had nothing to do with time.

Reading your essay reminded me of the Einstein-Lorentz clock I did find a video that explains it well:

http://www.learner.org/resources/series42.html#

It is episode 42, "The Lorentz Transformation" about 11:45 into the video. The series is very 80s and is an introduction to physics. You do have to register to watch the videos but it is free and requires only an email address.

The "photon clock" is a photon that is reflected between two mirrors. The clock is put in motion and an observer at rest views the path of the photon as the hypotenuse of a right triangle. Gamma is then derived using the Pythagorean theorem. Axiom I can be derived by the same argument. If you are using a similar Lorentz derivation it would seem C is fixed because the two mirrors are a fixed distance apart. Indeed you do need a proof for the "triangle becoming a line". I would start with the law of cosines and see where it took me.

In equation (1) you wrote V as the magnitude of a constant velocity. I think you were too careful here. V can be a vector because V^2 is the scalar (dot) product which gives a scalar. Similarly (1) can be rewritten so Vt is Vt^2 and a scalar product too.

Axiom II seems like a stretch but if it is true then Axiom III is possible, it looks identical to the Zeroth law of thermodynamics.

I found some mathematical steps hard to follow because my relativity is rusty. I had to remember that

dTau / dt = dt / gamma.

Photons are described in two extreme branches of physics, special relativity and quantum mechanics. Feynman does a great job explaining the photon and its properties:

http://vega.org.uk/video/subseries/8

Overall I found your essay very creative. I'm impressed that a double major working a midnight shift had the energy to write such a creative essay. I like that you are both mathematically and musically gifted like Einstein and Pythagoras. Good luck on your finals.

B^2

Hi Brian,

Thank you for your input and suggestions.

If I am not mistaken, Einstein himself used something like the photon clock to derive the Lorentz Transformations. The photon clock argument assumes the speed of light postulate and therefore I could not use it in my derivation. However, once the speed of light postulate is derived, then, it appears to me, one could use the photon clock argument to derive the Lorentz transformations.

My main goal with axiom I was just to find a way to isolate the mathematics of special relativity from the speed of light postulate so that it can be clearly examined by itself. I admit that I am still not certain whether I succeeded because if the alternative possibility given by the triangle argument contains some internal logical inconsistency, then axiom I would already by itself imply the postulate. I have spent many hours trying to find a proof either way, but so far without success.

You have put your finger also on another technical point which has given me some (mild) headaches when you pointed out that v can be considered a vector. I agree with you in general, of course, but I was not sure whether the quantity (c^2-(v_tau)^2)^1/2 can be given a spatial direction since v_tau does not point in any direction in space (it points in the "direction" of the spacetime metric).

Axiom II is an extraordinary claim which requires extraordinary evidence. I certainly do not blame you for being skeptical, as you should be. The onus is on me to show that this idea merits serious consideration.

One way to do this is to come up with an experimental test that allows it to be falsified. Unfortunately, the test suggested in my paper is with today's technology extremely difficult to perform because the relatively low energy density of photons coupled with the weakness of the gravitational force makes it a challenge to distinguish any actual effects from a null-result. I have discussed with one faculty member here who is an expert on LIGO (the interferometer which is designed to detect gravitational waves) whether it might be possible to use it for such an experiment (e.g. have laser beam directed in the vicinity of the mirror and watch for evidence of a gravitational field). He seemed at least to find the idea interesting but was not very optimistic about its feasibility. Of course, a definite determination of its feasibiliy requires me to do the hard calculation (which is a challenge for me because I have not yet taken GR, but I intend over the next year to learn how to do it on my own).

Another way is to show that it can also explain other unsolved fundamental problems in physics and, suffice it to say, I am working on that as well with some encouraging results.

It is not an accident that axiom III looks like the 0th law, it was in fact inspired by it.

A small correction, dTau/dt=1/gamma.

I have watched the Feynman videos twice each and also read the book QED-the strange theory of light and matter, which is essentially a transcript of them. I am fascinated by this interpretation of quantum theory and I strongly believe that that it is possible to connect it to my second axiom.

Finally, thank you very much for taking the time to make some detailed comments, and for your compliments.

To put what I am doing in a more grounded perspective, I work pretty much full time so I can only take 1 or 2 classes each term. Also, I had already written an earlier (and less refined) version of this paper as a term paper for my philosophy of space and time class.

Armin

Armin,

I was in the same situation of working full time while taking physics and math classes. Physics and non-physics classes should be weighted differently. Being a full time student in physics is like drinking from a fire hose.

Is the connection between axiom II and quantum the many worlds interpretation?

I'm not so certain that light doesn't have a gravitational field. We know from GR that light is influenced by gravity. Newton tells us that every action has an equal and opposite reaction. The rest mass of a photon is zero, however, the photon does have energy and momentum according to

P = h/lambda

E = hv

My QED and GR is not strong enough to tackle this problem head on but I have thought about it a lot too.

Armin,

An intriguing and well constructed essay. I think it is interesting how you explore the issue of light having zero spacetime interval.

Regards,

Paul

Brian,

Yes, I agree that physics classes should have different weight.

wrt your question about axiom II and MWI, in a word, no.

Thank you for raising this issue. I think that conservation of momentum presents the toughest challenge to my idea.

Newton's third law generally does not hold in classical Electrodynamics. If you consider, for instance, two charges q1 and q2 approaching a common point, the magnetic force of q1 on q2 is generally not equal and opposite to that of q2 on q1. But momentum is still conserved because we consider the fields themselves to carry and "store" momentum.

I would respond to your question then as follows: Yes, I agree that if the gravitational field of photons is zero then it is true that N3 is violated. But I would like to raise the possibility that overall momentum may still be conserved if one takes into account the overall consequences of the interaction.

Let me give a concrete scenario: Suppose a photon comes in from a distant star, and is deflected by the sun's gravitational field in such a way that it is absorbed by an electron on the surface of the earth (call it m_e), whereas in the absence of the gravitational field it would have been absorbed by an electron on the surface of mars (call it m_m) (say). Here the classical language of trajectories is once again shorthand for saying that the gravitational field changed the wave function of the photon so that it collapsed at the location of m_e instead of m_m.

As a result of the absorption, the electron goes in an excited state, which will be marked by *. So we assume we have with the gravitational field the configuration (m_e*, m_m) and without the field the configuration (m_e, m_m*). The binding energy in the excited state is higher, and since it is associated with an entity that ages, it also increases the gravitational field of the electron.

Of course, these are ridiculously small effects, but the point is that the action of the Sun's gravitational field on the photon, while by my idea not being met with an equal and opposite reaction, nevertheless resulted in a slight reconfiguration of gravitational fields in its vicinity.

Let me be the first to point out that this is a rather unsatisfactory argument.

The main problem I see is, how do we know that the reconfiguration occurs in exactly such a way as to conserve momentum overall? It seems to me highly non-trivial to prove that it does or does not. Also, what if the photon is not absorbed by anything within the solar system but just keeps on traveling? Even if the eventual reconfiguration does conserve momentum, we would have to wait until the photon is absorbed by something (unless we consider the alteration in the photon's "direction" (classically) or wave function as a result of the gravitational field itself as a sort of "momentum storage").

So I don't have a good answer. I can neither show that my idea does result in momentum conservation nor does it seem to me that it can be clearly shown that it violates it. In my view the best way to settle this is to just attempt to measure the gravitational field of electromagnetic radiation, but of course that is not easy either.

There is another major challenge that one could raise against my idea, and fortunately (for me) I have a much better response to that. One may ask whether my idea does not violate the principle of equivalence.

The argument, as I understand it, goes as follows:

From SR we know that inertial mass is equivalent to Energy. The principle of equivalence says that inertial mass is equal to gravitational mass. Therefore, Energy is equivalent to gravitational mass and photons should produce gravitational fields (i.e. spacetime curvature).

Now I confess that I have never seen this argument stated exactly in this way, because every GR text I have consulted simply treated the energy density of massless particles and that of massive particles as it pertains to their inclusion in the Energy momentum tensor exactly the same without ever explicitly providing a justification for doing so (as if to imply that this should be obvious).

So because of that I am not 100% certain that the argument above is the actual relativistic argument for considering electromagnetic radiation to produce gravitational fields. If it is something else, I would be most grateful to anyone who would point it out to me.

But, given my caveat, let us suppose that it really is the reason why one may believe that because of the principle of equivalence, photons should produce

gravitational fields.

This argument contains a subtle but elementary logical error, and to highlight it, let me express the argument in the form of a syllogism:

premise 1: m_i c^2=E_i for all m_i

premise 2: m_i=m_g for all m_i and all m_g

Conclusion: m_g c^2=E_g

m_i is inertial mass

c^2 is the speed of light squared

E_i is the energy equivalent to inertial mass

m_g is gravitational mass

E_g is the energy equivalent to gravitational mass (in the sense of producing spacetime curvature)

the conclusion does not follow because it makes an extra assumption not contained in the premises, namely that E_i=E_g.

If photons do not produce gravitational fields, then the logically correct form of this syllogism is

premise 1: m_i c^2=E_i for all m_i

premise 2: m_i=m_g for all m_i and all m_g

Conclusion: m_g c^2=E_i

If photons do produce gravitational fields, then the logically correct form of this syllogism is

premise 1: m_i c^2=E for all E_i and E_g

premise 2: m_i=m_g for all m_i and all m_g

Conclusion: m_g c^2=E

The bottom line is that the equivalence principle in the standard form m_i=m_g has nothing to say about whether photons produce gravitational fields or not. If one wishes to extend the empirically established domain of the equivalence principle to E_i=E_g, then one must do the experiment.

Sorry for the length, any feedback is appreciated.

Armin

Hey Armin,

I have not replied sooner because I have been thinking a lot about this thread and I like where it is going. Your post has given me an idea for an experiment, but first I have one correction.

The full relativistic energy equation is:

E^2 = (M_o)^2 C^4 P^2 C^2

The rest mass of the photon is zero so the first term above drops out reducing the equation to

E = PC

The energy of the photon is its' momentum multiplied by the speed of light, this comes from the quantum mechanics equations:

E = h (frequency) and P = h / (wavelength)

and another familiar equation for waves:

C = (wavelength)(frequency)

After reading your post I began thinking about light, nuclear fusion in stars, as well as "types" of mass. Fusing elements creates both photons, with only relativistic mass, and neutrinos that have a very small inertial mass. The sum of the energy of the nuclear product, with the photons, plus the neutrinos gives the initial energy of the nuclear reactants, E_i = (M_i)C^2.

Before I read your post I thought if photons were affected by gravity, then they should also have a gravitational field. After thinking through it I convinced myself that only neutrinos produced gravitational fields not photons. The speed with which I switched my belief has left me exhausted; so I'm going to rest and sit on the fence. I tried to think of an experiment that would settle my fence sitting problem; let me know what you think.

When a photon is created it does not immediately punch out of the star and soar through the cosmos. The photon bounces from atom to atom before it reaches the surface of the star. The reason is the photon interacts with the electromagnetic force and the star is incredibly dense; this limits the photon's time as a free particle. The photon bouncing in the star would be analogous to a random walk. A constant stream of neutrinos traveling near the speed of light penetrate through the earth because they rarely interact with matter. This makes sense because neutrinos interact with the nuclear forces but they do not interact with the electromagnetic force. Therefore, part of the nuclear reactants' mass escapes the star quickly and the other part is significantly delayed; this should produce some physical effects.

Assume astronomers could measure the gravitational lensing of a ray of light (laser, starlight, etc) around a star for an extended period of time. If the rate of nuclear fusion and neutrino emission were known, then any deviation in the light ray's geodesic indicates a change in the star's gravitational field. It would shed light (pun intended) on how gravitational fields depend on the "type" of mass required to produce them. There are several MAJOR experimental drawbacks.

Neutrinos are hard to detect and this limits our understanding on neutrino flux. Knowing the exact rate of conversion of matter into energy would also be difficult. I believe the hardest part would be measuring the small deviation in the light ray's geodesic. In other words, there are many large uncertainties and the experiment requires almost absolute precision.

No doubt my knowledge of astrophysics could benefit from more study; I'm not sure if this has been thought of before or is just ludicrous. It could have been thought of before because I do read a lot of physics. I often forget where I learned the physics, but fortunately I remember some of the material. I may have read this and it just sat in my subconscious because it seems too clever.

Respect,

B^2

P.S. Only reply if you are done studying for your physics finals. I know how the fun physics can distract from the stressful physics, but the stressful physics is what people care about. If you only have humanities finals remaining I expect a prompt reply :)

Hi Brian,

I have no more finals as of today, so I can once again delve into this issue.

Your statements about the relationship between mass and Energy are correct. I am not sure, but it seems to me that you made them in reference to my equivalence principle argument. If so, I should have perhaps pointed out that I was referring to inertial and gravitational rest mass.

This point can sometimes cause confusion because E=mc^2 actually refers to two distinct equations, one involving rest mass and its equivalent rest energy, and the other involving rest mass multiplied by the gamma factor and total energy. Let m_0 be the rest mass and m=gamma *m_0, then

E=(m_0^2 c^4+p^2c^2)^1/2

=(m_0^2 c^4+gamma^2 m_0^2 v^2 c^2)^1/2

=m_0 c^2 (1+gamma^2 (v^2/c^2))^1/2

=m_0 c^2 (1+gamma^2 beta^2)^1/2

=m_0 c^2 (1 +gamma^2 (1-gamma^-2))^1/2

=m_0 c^2 (1 +gamma^2-1)^1/2

=m_0 c^2 gamma

=mc^2

I wrote out all the intermediate steps because skipping steps when it is a pain to read the equations might not be such a great idea. If I misunderstood your point and you already knew this, I apologize, and ask that you tell me to which statement of my previous post you were referring.

I did make a careless statement when I said that the "binding energy is higher" when referring to the photon absorption scenario in my previous post. Since the mass of two particles when bound together is less than the sum of their separate masses, binding energy is negative, hence in that sense I should have actually said that the binding energy is less negative. The net effect, though, that an excited state of a system of bound masses should produce a stronger gravitational field than that of a system of less excited masses should still hold, though.

Also, when I mentioned that perhaps a change in "direction" of the photon might be considered a form of "momentum storage" I was referring to the fact that the direction is given by the wave vector, which is proportional to the photon's momentum. So I was thinking about something analogous to the magnetic force, which can only change the direction of a particle but never accelerate it in its direction of motion.

Interestingly, neutrino oscillations can be regarded as evidence for a finite neutrino mass by a simple argument very similar in spirit to the existence paradox: If neutrinos oscillate, then they must observe themselves to change in their proper frame, therefore they age, therefore they must travel at v

The last part of my previous post got somehow cut off, so here is the rest:

...therefore they must travel at v

Hmmm,

it seems that this program does not like the "less than" sign, so here is yet another attempt to complete my post:

...therefore they must travel at v less than c, therefore they have mass.

I think you have an interesting idea there; if I understood you correctly you say that if the reaction products of the same fusion process exit a start at very different times, then this produce observable consequences in the star's gravitational field.

Let me think about this, I'll get back with you.

Armin

Provocative paper and discussion! Very insightful ideas. I think your idea of existence in another continuum is intuitively very satisfying and has great potential. Here are a few questions and comments.

Armin, you state that photons exist in another continuum. You discuss "interactions" as events seperated in space by interval ct. This demonstrates a clear relationship between the two continua which is still difficult to characterize by your axioms. What is the intuitive nature of this relationship?

Also, your mention of the effects of gravitons on photons then assumes that gravitons also exist in the same continuum as photons. If these two "particles" do exist together then we would expect intuitively to see an effect of photons on gravitons. (I thought such an interaction has already been postulated from general relativity by Hawkins, etc? This would be most relevant at the beginning of the universe when photons are postulated to exist in "high density" without matter. I must check.) The lensing of light by massive bodies suggests a spatial location for this interaction. How can this be explained if photons do not exist over any point in space where this interaction would occur?

Lastly, what is the resultant intuitive explanation for the continuum of relative v from 0 to c (max) as related to c itself? This seems to me an equivalence principle in itself that begs an understanding.

Daniel

PS - Thanks for the acknowledgement!

An updated version of this paper which contains some minor corrections can be found at

http://deepblue.lib.umich.edu/handle/2027.42/83152

To anyone who is interested,

I think part of the reason this paper has been essentially ignored the last 3 years is because it must seem hard to see how one of the central assumptions, the one pertaining to the ontological status of photons, fits in with everything we know. recently posted a paper which gives a conceptual discussion of the framework which purports to help 'understand' what the formalism of quantum mechanics tells us about reality, and one of the important aspects of this is that the same assumption about photons is validated from a completely different perspective.

The paper is entitled "A Novel Way of `Understanding' Quantum Mechanics" and can be found either at Deep Blue, the University of Michigan repository at

http://hdl.handle.net/2027.42/86651

or at

http://www.vixra.org/abs/1110.0005

I appreciate any feedback. Thanks,

Armin