I am trying to use some principles of Richard Bellman ‘dynamic programming’ or ‘optimization‘. [I am unsure if I am successful in a rigorous manner, but hopefully there is insight.]

When a colleague remarked that this was not rigorous, Bellman reportedly responded, “Of course not. It’s not even precise. A good principle should guide the intuition.”

Richard Ernest Bellman, biography
BA, MA Mathematics; Brooklyn, Wisconsin, respectively
“… but in December 1944 he was drafted into the army and assigned (t)o the Manhattan Project in Los Alamos. There he worked on problems in theoretical physics until his discharge in 1946”
PhD Mathematics, Princeton
Faculty: Princeton, Stanford
Researcher RAND where developed ‘dynamic programming’, but “applied practitioners were regarded as distinctly second-class citizens of the mathematical fraternity” in Stuart Dreyfus biography
Faculty: Southern California in Mathematics, Electrical Engineering and Medicine
“621 papers, 41 books and 21 translations of books authored (or co-authored) by Bellman”
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bellman.html

Dynamic programming or optimization
http://en.wikipedia.org/wiki/Dynamic_programming
or
A Tutorial on Dynamic Programming, Michael A. Trick, Mini V, 1997
http://mat.gsia.cmu.edu/classes/dynamic/dynamic.html

Sorry for the spelling and notations errors.
I proofread without glasses before sending.

The last line of post #16 should read:
… Max energy at c*sqrt(2)?

However if Einstein is correst and c is the maximum velocity, then the extrema argument still seems viable.

Moreover, it appears to allow Einstien to have a Max relative velocity of 2c with Max energy at c*sqrt*2)

Indeed you are correct, I did make an error.

Newton used a time integration in dt.

Einstein used the velocity integration in dv.

The historical evolution is clearer to me in cps units:
Momentum p≡g•cm/s
Force dyn≡g•cm/s²
Energy erg≡g•cm²/s²

I do not get it: if you integrate in dv, as you seem to be doing in step 2a, the left-hand side, p, becomes pv, not F. And indeed, it has the units of energy, not force.

Cheers,
T.

There may be other alternatives:
c may be a constant extrema (maximum);
as a subtle difference from a simple constant.

Consider: A possible evolution from p=mv to F=ma to E=mc^2

Slide #10 of 14 on 3 pages from Eve Ostriker [UMD-US], Lecture 4: Newton’s Laws Newton’s second law Newton’s third law …
http://www.astro.umd.edu/~ostriker/ASTR340/lectures/lect4.pdf

1 – Aristotle [“object at rest remains at rest], momentum:

[rho or] p=mv

2 – Newton, force:

a – integrate velocity in momentum F=m*(1/2)*v^2

b – make substitute transformation F=ma

3 – Einstein, energy

a – make substitute transformation F=mc^2

this may have been derived from the omission of extrema notation such as:
b – from extrema notation F=m*MAX(a)=m*MAX(v^2)
where MAX(a)=MAX(v^2) as a->c^2 or v->c

The above ideas patterned after PHYSCLIPS from UNSW, physics, Australia;
Let me consider – Einstein’s equations written in different ways
Newton’s laws: background and limitations
Newton’s laws written in different ways
http://www.physclips.unsw.edu.au/jw/Newton.htm

first of all, a mistake in my post – the rate fit is -0.002 km/y and not m/y, and the uncertainty of the laser measurement is 1m/s and not one mm/s. I will proceed to change the post, but unlike other bloggers around I prefer to clarify first

Second, of course there is no detectability of a change of c over time of the size predicted by Louise, and the methods used to measure c might indeed be such that no change is detectable. My plot was just a fun way to spend five minutes with root on my laptop… “When I hear a list of numbers, I reach for my fitter”. It’s the experimenter’s instinct.

Louise, thanks for your updates…

Cheers all,
T.

Since speed of light c is proportional to the density of the medium and also it is proportional to the motion or the frequency of source, speed of light cannot be measured as an absolute quantity.

If we measure c=c1 at t=t1 and c=c2 at t=t2 and if c1>c2 we cannot say that speed of light is slowing down. We cannot know if we measured absolute c or if the density of the medium increased and we obtained lower c. Sort of an equivalency principle is at work here.

If I flash a flashlight to sun it will take longer to reach the sun than 8 minutes?

Also, if speed of light in rarified medium is changing it will also change in other media. But in order to measure the speed of light you have to pass it through a medium of different density. And if light is slowing down it will be slowing down in all media and measuring of the slowing down will be impossible. (I think Carl gives more sophisticated version of this argument.)

I agree hundred per cent with Louise Riofrio that constancy of speed of light cannot be proved. It can only be kept constant as a conventional unit. (The same argument can be applied to impossibility to observe absolute fundamental particles. A particle can only be defined as fundamental temporarily.)

As I said I am no expert. I am just curious. And I would appreciate comments from experts.

Since c(t) ~ t^{-1/3}, cdot/c = -1/3t

Where t is age of the Universe, on the order of 13.7 billion years. Over the 200-year range of Tommaso’s graph, change in c is:

cdot = c(200 yr)/(41 Gyr) = c/(2 x 10^8)

cdot = 1.5 m/sec over 200 years! That is far too small to measure on the graph, where increments are 5 km/sec.

The odds are in favour of “c change” and here’s why: It is impossible to “prove” experimentally that c is constant, because a more accurate measurement could always prove you foolish.

Carl’s question on the Sun deserves an extraordinary reply, which will appear on my blog within the next 2 days.

If you write E = m c^2, you get that when c was faster, energies (obtained from Einstein’s equivalence principle) were higher. Since the sun gets its energy from converting matter into energy, that means that you got more energy for the same number of atoms converted. Thus, the early sun was hotter than modern calculations would suggest.

One can always choose coordinates so that light speed is constant. The point is that to do this, you have to assume that the geometry of spacetime is not locally flat. This is what Einstein did.

The result of the analysis of the equations of GR with geometric algebra is that one should rewrite GR to work on flat space coordinates. This was done by the Cambridge Geometry group and David Hestenes gave the works his stamp of approval. (I should mention here that Hestenes thinks I’m a complete idiot.) See the paper “Gauge Theory of Gravity with Geometric Calculus” on this page.

For a black hole, the implied flat space coordinates are Painleve, which is the reason I did all that simulation stuff with Painleve coordinates.

Now the above discussion of flat space applies to local coordinates only. Louise’s work applies to the universe cosmologically, it is not at all a local theory. In fact, the Cambridge geometry group does do work on cosmological GR. I haven’t read it to see if they make the cosmos flat or not. My interests are mainly in elementary particles.

As far as computing the change in the speed of light over 100 years, I would think that it would be negligible even if you had the method of measuring it. Rather than see the effect in a light measurement, I would think that it would show up in something like a measurement of the fine structure constant or something like that. And it will be a very small thing.

Louise talks about the changing speed of light as an explanation for changes in the brightness of the sun. I’ll go ahead and see if I can understand that well enough to comment on it, though I wish she’d step up.

good question – I think we could make a much better measurement now, but I believe the laser interferometry used to achieve the mm/s precision of the 1973 measurement cannot be easily matched by other techniques.

Hi Jeff,
yes, it is just (un)healthy speculation, just a curiosity on my part.

Cheers,
T.

When we defining the velocity of light as 299 792 458 m/s, we recognize it, in the “light” of relativity, as a scale factor that allows us to define the meter once we define the second; i.e. the velocity of light, seen as a parameter, and not so much as the velocity of propagation of this or that wave, is emebedded in our description of nature based on relativity (our description of nature is based on the assumption that c is a parameter). That includes quantum mechanics, to the extent one uses Dirac, and astrophysics and cosmology to the extent one uses general relativity.

Any experimental evidence that our most experimentally solid theories, have cracks and that these may indeed be explained as signals for non-constancy of velocity? Or is it all just healthy speculation?

and a discussion
http://www.weburbia.demon.co.uk/physics/speed_of_light.html