Undecidability, Fractal Geometry and the Unity of Physics by Tim Palmer

Tim,

What do your diagrams and equations actually represent? Do they represent a world ruled by equations or do they represent a world that is taking logical steps and performing logical analysis? Where does the logical analysis and steps that you are personally taking end? Do the logical steps in Figure 1 represent logical steps you are taking (e.g. to solve a problem or equation) or do the logical steps in Figure 1 represent logical steps the world is taking?

Your essay is interesting.

Reading it made me think.

For example if there was a chaotic state in general relativity, then is it possible that Hausdorff's measure of particle trajectory an relativistic invariant? If it were not so, then there would be an observer for whom the relativist trajectory is non-fractal, but this seems unlikely to me (it's like a change of topology, to change from a chaotic trajectory to a non-chaotic trajectory).

Also for the Bell theorem (or the Einstein-Podolsky-Rosen paradox), it is possible to study the Feynmann diagram for the cross section in the scattering of two polarized Dirac particles (I read today the results in Greiner book) and to obtain the probability of the final state (with elicities). If there are interaction, so gauge bosons, then there is not an instantaneous effects; the collapse of Alice state communicate the state to Bob using the gauge bosons interaction, with the light speed.

Hi Tim,

It is quite a revolutionary program you have embarked on, overthrowing the infinitesimal and subverting the continuum. Your standard of rationality includes its mathematical definition: that any rational quantity can be expressed as a ratio of whole numbers. The conviction that the infinite and the infinitesimal have no place in physics goes well with the idea that appropriate mathematics ought to be involved. Nearly all of the infinities have been expelled from physics.

But there is still the century-old foundational problem with infinity in physics that appropriate mathematics might help resolve. It strikes me as ironic that Hilbert's admonishment about how "the infinite is nowhere to be found in reality" still stands today, considering his name is on the Einstein-Hilbert action which is involved in the unlimited gravitational energy required for inflation. This point is made clear by Paul Steinhardt who, when asked where the energy for inflation comes from, confirmed that it comes from a bottomless supply of gravitational energy. Since inflation requires infinite energy, the theory is inadmissible by Hilbert's standard of rationality, and so is the general theory of relativity which is supposed to deliver that energy.

On the presumption that it is essentially classical Newtonian gravitational potential energy which is apparently the source of the unlimited energy, it should be of interest that a relativistic version of gravitational potential energy can be constructed from a consideration of the composition of relativistic gravitational redshift due to a sphere.

For example, given a test particle of mass m, the classical element of potential energy due to a spherical shell of matter is du = -F(r) dr where F(r) is the force of gravity at radius r. The redshift due to the shell is given by dz = du / mc^2. The total redshift, z, from all shells can be composed relativistically as the product, 1 z = PRODUCT[1 dz] = exp[INTEGRAL dz], using Wikipedia's Pi notation (here "PRODUCT") for the Volterra product integral. The composite redshift due to a complete sphere of mass M, at radius R, is then z = exp(GM/Rc^2) - 1, not the conventional relativistic (1 - 2GM/Rc^2)^{-1/2} - 1, and not the first-order approximation, GMm/Rc^2. The corresponding relativistic gravitational potential energy must have similar exponential form to be consistent with the composition of relativistic gravitational redshift.

Unlike Newtonian potential energy which is negative, relativistic gravitational potential energy is positive, and equal to mc^2 exp(-GM/Rc^2). In the absence of a gravitational field, it is equal to rest energy. Gravitational potential energy is taken from that rest energy, and thus has a finite limit. Newtonian potential energy -GMm/R, is an approximation to mc^2 [exp(-GM/Rc^2) -1] for weak fields. Relativistic gravitational potential energy is an exponential map of the classical potential energy.

Relativistic gravitational potential energy gives an escape velocity sensibly limited to the speed of light, as might be expected from a relativistic theory, whereas this condition is violated in both the classical theory and general relativity. The singularity-free metric corresponding to the escape velocity is the same as Brans-Dicke. In that theory, inertial and gravitational mass differ slightly, by a presently undetectable amount. I suspect this discrepancy could arise from failing to account properly for the exponential nature of gravitational energy.

The original work can be found at the link in the file shells2010dec29.pdf. It has some simple examples to demonstrate the essential concepts. I was not aware of product integrals when it was written in 2010. This derivation of the product integral addresses the issue of normalization, which can be inferred from the physics of the problem. I don't have an essay for this contest, but here is a link to an essay from the last contest that shows some radical consequences of accepting the composite relativistic gravitational redshift.

It seems to me that there might be a way to incorporate these relativistic compositions for gravity into general relativity via the product integral and arrive at the Brans-Dicke metric. I wonder, what would be your intuition on this possibility?

Colin Walker

"... the closed Hilbert Space of quantum mechanics only arises in the singular limit where my finite fractal parameter p is set equal to infinity, and this is an unphysical limit! Hence, rather than complete quantum mechanics, my own view is that, guided by quantum theory, we have to go back to basics ..."

There may you meet the humble effort of an old engineer:

Eckard Blumschein

Hello again Tim,

After reading Lawrence Crowell's paper; I have a greater appreciation for your work, and even moreso that you are able to write so lucidly about it for lay audiences. I am impressed. I will have more questions now, after all that fuel for thought.

Would the correctness of your theory imply that the fabric of spacetime is fractal? This is a feature of several quantum gravity theories, in terms of the microstructure. Does that project onto the large scale structure of the cosmos in your view? Would it surprise you if I said it appears some of your starting assumptions would follow naturally, if my own theory pans out?

Tip of the old iceberg for you.

More later,

Jonathan

That was a satisfying answer Tim...

This speaks to the question of what sort of evidence of your theory would we see in the cosmos that might provide verification or refutation for its veracity. I asked a similar question of Gerard 't Hooft at one point and his answer was similar - that it was too early to tell what the cosmic evidence would be.

The following year at FFP11; he elaborated in his talk about the desirability of and difficulties with obtaining Lorentz invariance in a CA based QG theory, but still no hard predictions about what we would observe (in black hole emissions perhaps) that would distinguish it from the standard.

I've seen or heard several predictions from Loop Quantum Gravity folks about possible signature detections - such as Lorentz invariance violations, comb filtered emissions from black holes, and so on. But I see that each time such a prediction is made, folks will jump on it as excluding a theory if the exact signature predicted is not found. And String Theory folks seemingly refrain from making any hard predictions at all.

All the Best,

Jonathan

Dear Tim,

Great essay, congrats. Wish I had a background in physics to completely understand. Please indulge me if you have time.

1) Are you essentially suggesting that mathematical incomputability/undecidability exists as space-time, emergent from quantum non-linearity (as that is what the wave function seems to suggest, which may in effect be the cause of macroscopic gravity?

2) If something (anything that exists as part of detectable science) is incomplete as a matter of ontology (incomplete in the Godel sense) how could that ontology possibly verify determinism or superdeterminism?

Best,

Jack

(Essay: Misalignment Problem - You may enjoy the amalgamated sleuths section)

Thanks Tim,

I am very glad you are thinking (with the great tools of physics) about the same questions I am.

Re Q2 I think you have grasped my question in your statement "are there experimentally testable consequences of determinism?" Because if Godel's incompleteness manifests physically (space-time & mass) then you could never test determinism because the physical system would have unknowable states that cannot be determined by the system itself. So you couldn't have a determinable system, could you?

Best,

Jack

Tim,

What Bell had in mind (and explicitly expressed so in many interviews) is that, if particles are little machines, then his inequality must be respected. Now, as with any statement regarding the physical world, it tacitly assumes also `common sense'. One can bend this vague notion to an arbitrary extent, but there is a more direct attack on Bell's theorem, which has been staring us in the face for over a century: Particles (and chaotic systems and humans) are not machines! (no new-age stuff)

You are invited to read my essay which is further relevant to your main area of expertise - predicting the behavior of chaotic systems. Ensemble average over initial conditions is probably not the right way to do so.

I finally got to reading your paper. I have been working to get a piece of instrumentation developed meant to go to another planet. In reading this I think what you say is maybe not that different from what I develop.

Your paper drives home the point on using the Blum, Shub, and Smale (BSS) concept of computability. This is an odd concept for it involves complete computation of the reals to infinite precision and where our usual idea of close approximations are not real computations. This is a certain definition of incomputability. Since these I_U fractal subsets for an underlying fractal system are forms of Cantor sets the p-adic number or metric system is used to describe them. As fractal sets are recursively enumerable their complements are what are incomputable in a standard Church-Turing sense. Since this fractal is really defined in a set of such, there is a set of p-adic numbers or metrics and by Matiyasevich this is not globally computable. By this there is no principal ideal for the entire set or equivalently a single algorithm for all possible Diophantine equations. This is the approach I take with my FQXi paper. As a result, at this time I am relatively disposed to your concept here.

This meaning to incomputability in the BBS system is different, but not that out of line with the standard Church-Godel-Turing understanding. We can see that determinism is not always computable. The Busy Beaver algorithm of Rado has the first five numbers 0, 1, 4, 6, 13, but beyond that things become tough. The 6th is thought to be 4098, though not proven as yet. The 7th is a number greater than 1.29Г--10^{865}. It is not possible to compute higher Busy Beaver numbers. The failure to do so is a form of the Berry paradox or undecidability. The Busy Beaver is then a sort of model idea of a strange attractor with the exponential separation of differing initial conditions for two systems.

We have for coherent states, a general form of laser states of light, the occurrence of states of the form |p, qвџ© that have both symplectic and Riemannian geometry. My mind is pondering what connection this concept of incomputability has to coherent states. The occurrence of Riemannian geometry for spacetime, particularly if spacetime is a large N entanglement or condensate of states, and an underlying quantum geometry may be ordered as such. Einstein in his Annus Mirabilus proposed that states of light have blackbody or Boltzmann thermal distributions with a coherent set of states in his coefficients. This may really describe quantum gravitation as well.

Dear Tim Palmer,

If you are a physicist rather than an inflexible mathematician, you may hopefully be in position and ready to answer my question:

While I know, "a closed interval is an interval that includes all of its limit points", I guess there might be a fundamental point-based alternative to the "dot-based" mathematics from Dirichlet up to Heine and Borel. Given real numbers constitute Euclidean points, isn't then a discrimination between closed and bounded only justified for rational numbers? Isn't it logically impossible to include a single real number? Is the notion limit point really reasonable?

Well, this request relates to my own essay rather than directly to your essay. I am just curious.

Sincerely,

Eckard Blumschein

Hello,

I liked a lot your general essay. Several ideas are very relevant about the links between this quantum mechanics and this GR. I consider personally in my model of spherisation a gravitational coded aether sent from the central cosmological sphere made of finite series of spheres, I tell me that we have a deeper logic than only our relativity and these photons like main essence. This space, vacuum seems more than we can imagine. I have shared your essay on Facebook because it is one of my favorites, regards