onsdag 17 februari 2016

Gravitation, Motion, Programming with Codea and Zeno's Paradox

                                     Fresh reprinting of the world at each new time instant.  

The experience with the programming platform Codea which I am reporting on Matematik-IT  connects to an earlier view exposed on The World as Computation on the connection between matter and gravitation described by Newton's gravitational equation
  • $\Delta\phi =\rho$,            (1)
where $\phi (x,t)$ is gravitational potential and $\rho (x,t)$ mass density depending on a Euclidean space coordinate $x$ and a time coordinate $t$, and $\Delta$ is the Laplacian differential operator. 

The conventional way of viewing (1) is to think of mass density $\rho$ as primordial which generates a gravitational potential $\phi$, and corresponding gravitational force $\nabla\phi$, through the integral
  • $\phi (x,t) =\frac{1}{4\pi}\int\frac{\rho (y,t)\, dy}{\vert x-y\vert}$      (2)
with apparent (instant) action at distance expressed by the global nature the integral. The trouble with this view is that the physics of (instant) action at distance, supposedly being transmitted by graviton particles (appearently traveling at infinite speed), is still completely open despite centuries of deep thinking by deeply thinking physicists. The idea of matter density as primordial may come from primitive thinking that what we can see/touch, must come first and what we cannot see/touch, must be secondary.

But there is another way of thinking, maybe less primitive, which is to view instead gravitational potential $\phi (x,t)$ as primordial and mass density
  • $\rho (x,t) =\Delta\phi (x,t)$      (3)
as being produced by the local operation of differentiation. With this view, there is no action at distance to explain.  What asks for understanding is the physics of (3).

Viewing the world as computation then connects to the way Codea works with the screen being freshly redrawn 60 times a second from the code under function draw()....end, without storing anything previously written on the screen.  This is the same way we perceive the world with our eyes with a fresh image at each new instant.

With this perspective we can think of (3) as the computer code which at each instant draws the world in a new configuration with new mass density $\rho (x,t)$  from a gravitational potential $\phi (x,t)$ which is changing in time according Newtonian mechanics.

In the same way as motion is exhibited by Codea by a sequence of fresh images coded under function draw()...end, the motion of matter we perceive would be a result of fresh reprinting of the world at each new instant according to the code (3).

The alternative view suggests a solution to Zeno's paradox, still unsolved after 2500 years, with the arrow being reprinted at each new instant in time giving the appearance of motion as change of position with time. Think of that!

Note that it Einstein's equations in nearly flat Minkowski space-time reduces to a wave equation variant of Newton's equation of the form
  • $-\frac{1}{c^2}\frac{\partial^2h_{\alpha\beta}}{\partial t^2}+\Delta h_{\alpha\beta} =\frac{16\pi G}{c^4}T_{\alpha\beta}$,
where $h_{\alpha\beta}$ is metric perturbation and $T_{\alpha\beta}$ stress-energy. This equation allows waves traveling at the speed $c$ of light as "ripples of the fabric of space-time" in the common mysterious jargon of relativity theory, but the presence of the factor $c^4$ make such waves vanishingly weak. In short, there seems to be little reason to expect a wave equation variant of (1) to have physical significance, which can be seen as support of the alternative view.

söndag 14 februari 2016

Gravitational Waves in Newtonian Mechanics?

We consider a cosmological model in the form of a compressible gas under interaction by gravitation: Find $(\phi ,m,e)$ depending on a space-time coordinate $(x,t)$, such that for all $(x,t)$
  • $\Delta\dot\phi + \nabla\cdot m =0$
  • $\dot m +\nabla\cdot (um) +\nabla p + \nabla\phi =0$
  • $\dot e +\nabla\cdot (ue) +p\nabla\cdot u =0$,       
where $\rho = \Delta \phi$ is matter density with $\phi$ gravitational potential, $u=\frac{m}{\rho}$ is matter velocity with $m$ momentum, $p=(\gamma -1)e$ is pressure with $\gamma > 1$ a gas constant and $e$ is internal (heat) energy, and the dot denotes differentiation with respect to time, see Many-Minds Relativity 20.3.

In the case $u=0$ and $\nabla p=0$, this model reduces to
  • $\Delta\dot\phi + \nabla\cdot m =0$,                      (1a)
  • $\dot m +\nabla\phi =0$,                                (1b)     
from which follows by differentiating the first equation with respect to time:
  • $\Delta\ddot\phi + \nabla\cdot\dot m =0$,
  • $\dot m +\nabla\phi =0$,      
that is
  • $\Delta\ddot\phi + \Delta\phi =0$,
and thus
  • $\ddot\phi + \phi =0$,
which expresses a periodic oscillation of $\phi (x,t)$ in time.

Alternatively, eliminating instead $\phi$ in (1), gives the following wave equation in $m$:
  • $\ddot m + \nabla\Delta^{-1}\nabla\cdot m=0$. 
A universe in the form of a compressible gas thus allows a periodically varying gravitational potential $\phi (x,t)$ with corresponding periodic mass density $\rho (x,t)=\Delta\phi (x,t)$, if we extend the model to include mass density varying from positive to negative mass, along with periodic momentum $m$.  The effect of such a locally in space oscillating gravitational potential may at distance be perceived as a varying gravitational field and and thus as a gravitational wave. In this context, recall the recent post on dark energy and dark matter where you can follow the expansion a gravitational wave originating from a local oscillation of gravitational potential:

Newtonian mechanics extended to include both positive and negative mass,  thus appears to allow gravitational waves.  The conclusion that gravitational waves somehow require Einstein's general theory of relativity, may be drawn too quickly. Maybe Newton's theory extended to negative mass is enough.

fredag 12 februari 2016

Absurdity of Modern Physics: LIGO Gravitational Wave Detection as Ill-posed Problem.

                                    Simple mathematical modelling of gravitational waves.  

The BIG NEWS from the physics community is the announced experimental detection of gravitational waves:
  • Long ago, deep in space, two massive black holes—the ultrastrong gravitational fields left behind by gigantic stars that collapsed to infinitesimal points—slowly drew together. The stellar ghosts spiraled ever closer, until, about 1.3 billion years ago, they whirled about each other at half the speed of light and finally merged. The collision sent a shudder through the universe: ripples in the fabric of space and time called gravitational waves. Five months ago, they washed past Earth. And, for the first time, physicists detected the waves, fulfilling a 4-decade quest and opening new eyes on the heavens.
  • LIGO researchers sensed a wave that stretched space making the entire Earth expand and contract by 1/100,000 of a nanometer, about the width of an atomic nucleus.  
  •  The oscillation emerged at a frequency of 35 cycles per second, or Hertz, and sped up to 250 Hz before disappearing 0.25 seconds later.
  • Comparison with computer simulations reveals that the wave came from two objects 29 and 36 times as massive as the sun spiraling to within 210 kilometers of each other before merging.
  • The collision produced an astounding, invisible explosion. Modeling shows that the final black hole totals 62 solar masses—3 solar masses less than the sum of the initial black holes. The missing mass vanished in gravitational radiation—a conversion of mass to energy that makes an atomic bomb look like a spark. “For a tenth of a second [the collision] shines brighter than all of the stars in all the galaxies,” Allen says. “But only in gravitational waves.”
  • “It will win a Nobel Prize,” says Marc Kamionkowski, a theorist at Johns Hopkins University in Baltimore, Maryland.
We understand that an expansion-contraction of the Earth of the size of an atom nucleus diameter as an effect of a "ripple in the fabric of space-time" was detected during 0.25 seconds, and from this observation the conclusion is drawn by computer simulations and modelling that this extremely minute effect as a "ripple in the fabric of space-time", was the result of a very specific extremely gigantic invisible explosion 1.3 billion light years away shining brighter than all stars in all galaxies for 0.25 seconds in the form of gravitational waves. 

We see a combination of a biggest possible cause/input and a smallest possible effect/output in a certain mathematical model. The conclusion comes from using this mathematical model in inverse form, where a smallest possible signal is used to identify a biggest possible origin of the signal. 

This means that the mathematical model in inverse form is extremely ill-posed and as such cannot be used to draw conclusions. To do so requires that all alternative explanations of the zero signal can be eliminated, and it is then not enough to just say that no other explanation immediately suggest themselves, that is to draw conclusions from ignorance with the precision of the conclusions increasing as the ignorance or stupidity grows.