When physics is geometry: a new proof for general relativity through geometric interpretation of Mössbauer rotor experiment. Celebration of the 100th anniversary of general relativity by Christian Co

Dear Ch,

I think I know how to do this k_1 + k_2 within a single treatment of the metric. We go back to the metric

ds^2 = Adt^2 - 2ωr^2dφdt - dr^2 - r^2dφ^2

for A = 1 - (ωr)^2. Now divide this entire equation by dt so that

(ds/dt)^2 = A - 2ωr^2(dφ/dt) - (dr/dt)^2 - r^2(dφ/dt)^2

This is the gamma factor for the system with ds/dt = 1/γ. The Lorentz gamma factor is then approximately after binomial theorem etc

γ =~ 1 + ½[(ωr)^2 + v_r^2 + 2ωr^2Ω + r^2Ω^2],

where v_r is the radial velocity and Ω is the angular velocity of the particle.

The particle is a photon and in the experimental set up it is set in a radial direction. The standard gamma factor for a particle does not pertain to a photon, but we are using that here with the idea that v_r is the speed of light or very close to that. We might consider the beam of photons to be just a highly relativistic beam of electrons, where these behave approximate to massless particles. We now consider that the photons are constrained to remain in a type of photon guide or fiber optic. This means the photons have an angular velocity component to them with

c^2 = v_r^2 + (ωr)^2.

The radial part we just write as c or unity as an approximation and we substitute this into the gamma factor, here modified to account for a photon that makes this different from the standard definition, to get

γ =~ ½ + 2(ωr)^2

This is then used to compute the time dilation dτ = γdt. We let dt -- > dr and the time dilation integrate along the radial direction

τ = ∫^R(½ + 2(ωr)^2)dr = R/2 + (2/3)ω^2R^3 = R(1/2 + (2/3)v^2R).

The R/2 can be eliminated if we just consider the difference in gamma factors as the relevant factor so that δγ = 2(ωr)^2 is the relevant factor to compute redshift factors.

Cheers LC

Dear Christian,

Thank you for reading my essay and glad that you liked it. I went through that link that you posted and it was very interesting, I also need to go through your essay very thoroughly because I have not gotten the right results for the gravity as I have for the others.

Indeed the exotic results that I am getting is even surprising to me and I am not too sure what to make of them, but I do have a general idea. I did rate your essay and I may have some questions for you later.

Thanks

Dear Christian,

A rotating laboratory is not "freely falling", i.e. subject to only gravitational effects. Put otherwise, a body "at rest" in a rotating frame (i.e. with constant spatial coordinates) is not following a geodesic. That is why it isn't an inertial frame.You are not separating linear acceleration from rotation, which are quite different. The Strong Equivalence Principle is confined comparisons between effects in various circumstances, none of which include rotating labs or rotating frames. (The physical effects of rotation were, of course, whole point of Newton's bucket experiment, which gets the same explanation in relativity as it does for Newton: rotation is an objective feature of some motions.)

One application of the principle is this: experiments done in a lab "at rest in a constant gravitational field" (to a good approximation, a *non rotating* lab on the surface of the Earth, but that is only approximate) will display the same phenomena as a *linearly accelerating* lab in flat space-time. That gives the "bending of light". Another is the (approximate) equivalence of a non-accelerating lab in flat space-time to a *non rotating* lab in "free fall" on Earth (again this is only to first order since the field on Earth is not constant). There is no principle equating a rotating lab with a non-rotating lab in any gravitational field. Phenomena in a rotating lab will not even be spatially isotropic (referring to the rotating coordinates).

It is kind of odd that you say I cited MTW, I only brought it up because you cited it as one source for the application of the Equivalence Principle to rotational situations. Since you seem to concede that they nowhere make such a claim, perhaps you should remove that citation. Now you say you do not trust textbooks. If so, then don't cite them, especially when they do not make the claim you are trying to establish.

There is a reason that the SLAC was built as a linear accelerator, rather than a closed circuit like the LHC, and that reason has to do with the difference between linear acceleration and non-linear acceleration. Your claim that rotation is the same as linear acceleration is not accurate. And no Equivalence Principle, including the one you cite above, equates rotating labs or rotating frames to non-rotating labs or non-rotating frames.

Regards,

Tim

The equivalence principle simply states that freely falling frames are equivalent to a purely inertial frame independent of gravity. Similarly a frame that is accelerated and one on the surface of a gravitating body are equivalent:

we ... assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system. -- Einstein, Albert (2003). The Meaning of Relativity.

Departures come about for two reasons. The weak EP is a form of Galileo's principle, and it says the motion of a particle in a gravity field is the same as on an accelerated frame. This insures Galileo's observation on the independence of mass of a body with respect to its motion. With the weak equivalence principle the main departure is due to tidal forces and the radial direction of gravity. so the WEP requires that the size of the frame in a gravity field, say the dimensions of a lab sitting on the surface of a gravitating body, be very small relative to the dimensions of the gravitating body. The Einstein EP (EEP) says that any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime. This is the inertial idea of being in a freely falling frame, such as the infamous elevator. Again if this frame falls through a region of a gravity field so that tidal forces are apparent there are departures. The strong EP (SEP) says that the velocity of the frame relative to any outside frame, such as a distant coordinate system, is not a determinant of the measured physics on that local frame. There is again locality of measurements required to eliminate tidal forces. This means that gravitational physics is purely geometric. This is "strong" because it makes reference to regions of spacetime that are removed from any local frame.

When it comes to the rotating frame and the EP, we have certain stipulations that are required. Certainly for the WEP and EEP we require that the dimensions of any local frame be small. This holds for the SEP as well, but we have another stipulation that physics in the lab frame be independent of motion relative to the outside world. This does not happen with the rotating frame. One clear departure is the Coriolis acceleration 2ωxv, which in the rotating frame is rather apparent if there is some motion of a particle relative to the rotating frame. An observer on a frame which observes motion of a freely moving particle as cycles or circles, with no central gravitating body present, suspects then that they are on a rotating frame. As a result the additional caveat for the SEP with rotating frames is that the motion of a particle not under any local force in that frame must have a small velocity v

Dear Christian,

An interesting essay, offering a simple and elegant resolution to a puzzling experimental problem. Very nice work.

Do you know this paper by Bini et al.? http://arxiv.org/abs/gr-qc/0106013

Best wishes and good luck in the contest!

Christine

Hi Christian,

I always look forward to your essays, because I know I can expect a maximum of theoretical prediction and numerical result, with a minimum of blah,blah, blah ...

It is evidenced that relativists do not often get due respect in this forum -- I hope you are an exception. The time synchronization issue does indeed cross boundaries of classical and quantum gravity, and rotation is key to the geometry of time synchronized systems.

I hope you get a chance to visit my essay where, in part, I examine rotation in the complex plane (Euler's geometric interpretation of C) that may have the potential to join Hilbert space quantum dynamics to the continuous functions of classical spacetime.

All best,

Tom

Dear Christian,

I enjoyed your essay, and I agree with you that "General relativity is [...] the best example showing that Mathematics is Truth instead of Trick." And what is the best way to show once more this, than to confirm its validity by explaining the deviation in the Mössbauer rotor experiment, deviation which was even considered an evidence against general relativity's adequacy. That's a good way to answer the contest's question, and in the same time to celebrate 100 years of general relativity. Congratulations!

Best wishes,

Cristi Stoica

Dear Christian,

Your essay as always is a rich educational resource, and it's a perfect sentiment for the 100th celebration of General Relativity - a strong and independent proof which reveals the full geometric interpretation of gravity. You provide an excellent analysis of the Mossbauer effect - one I haven't seen before, and your geometric interpretation of time dilation and clock synchronization are absolutely enlightening! You spell out specific pivotal ideas and back them up with technical rigor and lucid experimental evidence; I thoroughly thank you for this solid approach. I also appreciated the application to GPS systems at the end of your essay, a field I briefly consulted in many years ago. A class act contribution to this topic and the forum, I give it the highest rating.

My essay also brings out how changing the mathematical representation can educe quite amazing revelations in physical explanation, and discusses General Relativity's geometric interpretation as a key feature. Furthermore, it mentions the geometric effect on the Turing machine, and uses relativity theory in understanding the multiverse explanation of self referentially induced superposition. Please take a moment to read my essay and rate it as well,

Thanks, Steve

Dr. Corda,

I also was pleased to see someone celebrate the centennial of General Relativity and had read your abstract immediately. I was not surprised to recognize that I'm not equipped to venture in without a guide and browsing for information confirmed that. The salient feature appears to be the disjunct between SR and GR, and the lack of a unified field resolution that would make GR definitive of a discrete non-zero point particle applicable to QM.

Though a novice to GR, and only dragged kicking and screaming by the persistence of Thomas Ray to an understanding of it being a selection of interconnecting maths that operate more like a computational device than 'a theory' which is its own co-ordinate system, I was finally disabused of what is probably a common niave assumption that GR is an extension of SR. And while SR is perhaps the most experimentally confirmed theory to date, the time metric remains under constant challenge.

At issue is the observer experience commonly illustrated by Einstein's epiphany of riding a beam of light and time 'stops'. It is mathematically complete but is based on the metric that the speed of time is one second per second at relative rest and ,Lorentz fashion, is zero second per second at light velocity. If we look at that in reverse, gravitation is the negative acceleration linearly from 1 sec/sec at c, down to relative rest at 0 sec/sec, and perhaps analogous to Unruh's 'uneven flow of time'.

If we look at SR from a paradigm that time in a stationary frame appears to stop in the experience of an observer in the light velocity frame, because light velocity is equivalent to the limit rate to which time can extend, then both observers can proceed through time at their respective gravitational rate. Energy is mass existing at light velocity but doesn't have to extend spatially to infinity. It remains mathematically complete, and suggests that 0 sec/sec @c | | 1 sec/sec @ 0 , is a 5th dimension.

I think it was Fitzgerald whom remarked that the finite speed of light is 'astonishingly slow'. Coming down from instantaneous at infinity (?), yeah, he's right. It's like a stone. Thank-you for daring to push the limits in such a public forum, and giving feast for thought. Sincerely, jrc

OOPS! drat...

at end of paragraph 4 should read ' 1 sec/sec @ c || 0 sec/sec @ 0 '

jrc

Christian,

Great essay. You maintain an exceptional standard. I also agree the wisdom of your approach, which I also use, not attacking Einstein for incompleteness or flaws but helping complete it and improve the understanding. I also found your writing very clear and concise. I think it should be another top essay (in the community scoring at least - certainly for me) but I hope may also now seem more palatable to those more theoretically entrenched.

You should be aware of the other commonalities, but this year I also identify and analyze the consequences of the great mathematical self-deception underlying QM and hampering unification. i.e. I show we CAN trick ourselves by carelessly 'abusing' mathematics in application. I do hope you'll read and comment as I'm sure you'll like and agree it and also maybe find it helpful

Very well done and thank you for a different vista on the reality we agree on.

Best of luck

Peter

Christian,

Great opportunity to celebrate GR and Einstein's 100th anniversary. You provide a cogent explanation of a strong and independent proof which reveals the full geometric interpretation of gravity.

Indeed, Math is truth rather than trick. I have done modelling and simulation and always valued the power of math to lend understanding to operation of weapon systems in aerospace and solve practical problems. I also see its need in bringing us forward in quantum biology, a better understanding of the universe's beginnings (LHC) and the origins of life (DNA). These are sort of my proofs.

Jim