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Scale Relativity

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Title: Scale Relativity  
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Subject: Laurent Nottale, RST model, Jackiw–Teitelboim gravity, Liouville gravity, IR/UV mixing
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Scale Relativity

Scale relativity is a theory of space-time initially developed by Laurent Nottale, researcher at Paris Observatory, near Paris. It is an extension of the concept of relativity found in special relativity and general relativity to physical scales (time, length, energy, or momentum scales). Scale relativity means that scales always depend on the scale used as a reference. As a consequence, the fundamental laws of physics cannot involve scales themselves because their values are arbitrary choices. Another important consequence is that the geometry of space itself becomes scale dependent by relaxing the hypothesis of differentiability. While differentiable trajectories found in standard physics are automatically scale invariant, it is the main insight of the theory that also certain non-differentiable trajectories (which explicitly depend on the scale of the observer) can be scale invariant and new mathematical tools are developed to model such trajectories.Scale invariance is closely related to the self-similarity observed in fractals. One of the claimed successes of the theory is that the laws of quantum mechanics, like the Schroedinger equation, can be derived directly from a continuous space in which coordinate curves are generally non-differentiable through the use of a new complex total derivative with respect to time in the Euler–Lagrange equation. The mass involved in these laws is of pure geometrical origin and no particle is assumed to derive them.[1]

Galilean scale relativity

Motivating observations

Two everyday observations are, that if we look at an object at a very small distance, say through a microscope, then even the slightest movement of this object will appear very fast; if on the other hand we look up to the sky and follow the movement of a jumbo-jet we sometimes wonder why it doesn't fall down, because from this distance it appears to be almost standing still.
Is this a pure subjective perception? The passengers in the jet will say that the clouds rushing by prove that the plane is moving fast, whereas the earth below is nearly standing still. And if the 'object' under the microscope were an ant that just woke up from coma, it would observe itself moving - relatively to the surface it is bounded to - with merely a few centimeters per minute.
This is reminiscent of the situation where one walks inside a train. One observes oneself walking rather slowly, while an observer outside will add the velocity of the train to the walking speed, and say that the person inside the train is walking fast relatively to the ground. A similar observation led Galileo to formulate a relativity principle of motion. Likewise the former observations led Nottale to formulate scale relativity.

Mathematical formulation

A sharpened pencil in extreme perspective. Note the shallow depth of field.
Angular velocity \omega describes how much around the circle d \theta something moves per time change dt

While Galilean relativity of motion can be expressed by differences:

v = v_2 - v_1 = (v_2 - v_0)-(v_1 - v_0)

The relativity of scales can be expressed by ratios:

\rho = \frac{ x_2 }{ x_1 } = \left( \tfrac{x_2}{x_0} \right) \diagup \left( \tfrac{x_1}{x_0}\right)

This can be derived by taking visual perspective into account, which is the phenomena that as objects become more distant, they appear smaller, because their angular diameter (visual angle) decreases. Then an observer on the ground sees v ' as the tangential velocity v (observed inside the plane) scaled down by the ratio of r'/r, that's between radius r (distance to the observed movement) and r ' (distance to the projective plane, e.g. 'the window').

v'(r,r ')= \frac{r '}{r } v \quad \Leftrightarrow \quad \rho = \frac{ r ' }{ r } = \frac{ v ' }{ v } = \left( \tfrac{r ' \theta}{t} \right) \diagup \left( \tfrac{r \theta}{t}\right)

The last term follows from (*)
Now the difference between a jet flying in a circle from its own perspective, i.e. a rotation that our intuition would favor as a real movement, and a perspectively projected rotation conceived by a far away observer, is that

  • in the real case the angular velocity \omega is constant, while the tangential velocity depends on the radius

v \;(r)= \omega r = \frac{\theta}{t} \; r (*)

  • in the projective case the tangential velocity v (the speed the passengers observe) is constant, while here the angular velocity (a.k.a. angular frequency) depends reciprocal on radius r

\omega(r)= \frac{1 }{r} v

That is a jet flying through your garden (small r) will have a much higher angular velocity, than one that is 'hanging' in the sky. Now what if r goes to zero? Then the angular velocity would become infinite for any non-zero v > 0 (ultraviolet catastrophe). This is the analog to infinite speed in Galilean Relativity, when one rides on a train that is riding on a train, ... ad infinitum. This leads to (Lorentzian) scale relativity that is analogous to special relativity.

(Lorentzian) Scale Relativity principle

The scale relativity extends to scales the reasoning made by Einstein on speeds in special relativity: just like a constant speed c= \frac {1} {\sqrt{\varepsilon_0\mu_0}} in Maxwell's equations, which does not appear to depend on the speed of the observer, suggests that the law of combination of speeds must preserve this invariant, similarly, the appearance of a constant length \ell_P = \sqrt { \frac {\hbar G} {c^3} } in Schrödinger's equation suggests that the law of combination of scales must preserve this invariant. In other words, just like c is a physical speed limit, \ell_P is a physical length limit.

Predictions and retrodictions

Scale relativity made a number of true predictions, as well as a number of retrodictions, both in cosmology and at small scale, including:

  • Prediction of the location of exoplanets [1]
  • Explanation of some observed large-scale structures [2]
  • Relation between mass and charge of the electron [3]

See also


  1. ^ Nottale, Laurent (1998). "Scale Relativity and Schrodinger's Equation".  

External links

  • Laurent Nottale's site
  • The original 1992 scale relativity article
  • List of Papers Downloadable from the site
  • Derivation of the postulates of quantum mechanics from the first principles of scale relativity last article
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