|
|
| Line 1: |
Line 1: |
| {{third-party|date=September 2012}}
| | Andrew Simcox is the name his parents gave him and he totally enjoys this name. Her family members life in Ohio. Invoicing is my occupation. It's not a typical factor but what she likes performing is to perform domino but she doesn't have the time recently.<br><br>Take a look at my web-site - good psychic ([http://srncomm.com/blog/2014/08/25/relieve-that-stress-find-a-new-hobby/ http://srncomm.com/]) |
| | |
| '''Scale relativity''' is a theory of [[space-time]] initially developed by [[Laurent Nottale]], working at the French observatory of Meudon, near Paris. It is an extension of the concept of [[Theory of relativity|relativity]] found in [[special relativity]] and [[general relativity]] to physical [[scale (spatial)|scale]]s (time, length, energy, or momentum scales). If scales in nature are always relative, an absolute scale cannot exist. As a consequence, fundamental physical laws need to be [[scale invariant]]. While [[differential]] [[trajectories]] found in standard physics are automatically scale invariant, it is the main insight of the theory that also certain non-differential trajectories (which explicitly depend on the scale of the observer) can be scale invariant and new tools are developed to treat such trajectories. One of the claimed successes of the theory is that the laws of [[quantum mechanics]], like the [[Schroedinger equation]], can be derived directly from the assumption that space-time itself is non-differential{{Clarify|date=January 2012}} and scale invariant. [[Scale invariance]] is closely related to the [[self-similarity]] observed in [[fractals]].
| |
| | |
| ==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 [[Motion (physics)|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. <br>
| |
| 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. <br>
| |
| 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===
| |
| | |
| [[Image:Sharpened Pencil.jpg|thumb|right|250px|A sharpened [[pencil]] in extreme perspective. Note the shallow [[depth of field]].]]
| |
| [[Image:Angularvelocity.png|thumb|250px|Angular velocity <math>\omega</math> describes how much around the circle d<math> \theta</math> something moves per time change dt]]
| |
| | |
| While [[Galilean relativity]] of ''motion'' can be expressed by differences:
| |
| | |
| <math> v = v_2 - v_1 = (v_2 - v_0)-(v_1 - v_0)</math>
| |
| | |
| The relativity of ''scales'' can be expressed by ratios:
| |
| | |
| <math> \rho = \frac{ x_2 }{ x_1 } = \left( \tfrac{x_2}{x_0} \right) \diagup \left( \tfrac{x_1}{x_0}\right) </math>
| |
| | |
| This can be derived by taking [[Perspective (visual)|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').
| |
| | |
| <math> 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)</math>
| |
| | |
| The last term follows from '''(*)''' <br>
| |
| 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]] <math> \omega</math> is constant, while the ''tangential velocity'' depends on the radius
| |
| <math> v \;(r)= \omega r = \frac{\theta}{t} \; r</math> '''(*)'''
| |
| * 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
| |
| <math>\omega(r)= \frac{1 }{r} v</math>
| |
| | |
| 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 <math>c= \frac {1} {\sqrt{\varepsilon_0\mu_0}}</math> 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 <math>\ell_P = \sqrt { \frac {\hbar G} {c^3} }</math> in [[Schrödinger's equation]] suggests that the law of combination of scales must preserve this invariant. In other words, just like <math>c</math> is a physical speed limit, <math>\ell_P</math> is a physical length limit.
| |
| | |
| ==Predictions and retrodictions==
| |
| Scale relativity made a number of true predictions, as well as a number of [[retrodiction]]s, both in cosmology and at small scale, including:
| |
| | |
| * Prediction of the location of exoplanets [http://luth2.obspm.fr/~luthier/nottale/ukresult.htm]
| |
| * Explanation of some{{which|date=January 2014}} observed [[large-scale structure]]s [http://luth2.obspm.fr/~luthier/nottale/DaRochaNottaleL.pdf]
| |
| * Relation between mass and charge of the electron [http://luth2.obspm.fr/~luthier/nottale/ukmachar.htm]
| |
| | |
| ==See also==
| |
| * [[Causal dynamical triangulation]]
| |
| * [[Conformal group]]
| |
| * [[Doubly special relativity]]
| |
| * [[Fractal cosmology]]
| |
| * [[Fractals]]
| |
| * [[Olbers' paradox]]
| |
| * [[Perspective (graphical)]]
| |
| * [[Scale invariance]]
| |
| | |
| ==External links==
| |
| *[http://luth.obspm.fr/~luthier/nottale/ Laurent Nottale's site]
| |
| *[http://luth.obspm.fr/~luthier/nottale/arIJMP2.pdf The original 1992 scale relativity article]
| |
| *[http://luth.obspm.fr/~luthier/nottale/ukdownlo.htm List of Papers Downloadable from the site]
| |
| *[http://arxiv.org/abs/0711.2418 Derivation of the postulates of quantum mechanics from the first principles of scale relativity] last article
| |
| | |
| [[Category:Quantum mechanics]]
| |
| [[Category:Theory of relativity]]
| |
Andrew Simcox is the name his parents gave him and he totally enjoys this name. Her family members life in Ohio. Invoicing is my occupation. It's not a typical factor but what she likes performing is to perform domino but she doesn't have the time recently.
Take a look at my web-site - good psychic (http://srncomm.com/)