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Author
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Topic: Gravity = Statistics?
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chimp
Member
Member # 333
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posted 09. July 2003 12:35
Is gravity a statistical probability distribution?
Yes, the force called gravity is actually geometry, non-Euclidean geometry, where spacetime becomes anisotropic and inhomogeneous in the presence of mass-energy.
Then the question becomes "what is space?" "What is time?"
Space is relational. Time is a process.
Heisenberg Uncertainty:
DxDp >= hbar/2
The relation becomes totally "chaotic" below the Planck length. So, space could be described as a self similar relation which is generated by the quantum foam, and forms Penrose's "spin networks".
The curvature of spacetime could be represented as a Gaussian distribution? If mathematics only is an approximation of reality, then the mathematics of probability corresponds "exactly" with reality.
The Riemann tensor explains how a tangent vector, parallel translated around a tiny parallellogram is changed. So, to say that spacetime is "curved" means how much a tangent vector changes during parallel transport around a loop. Parallel transport is the translation of an infinitesimal tangent vector along a geodesic.
The probability distribution should agree exactly with the Riemann tensor of Einstein's relativity.
Is the universe a closed system? The million dollar question ![[Smile]](smile.gif)
Russell E. Rierson
analog57@yahoo.com
[ 11. July 2003, 03:28: Message edited by: Russell E. Rierson ]
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Claire
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Member # 725
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posted 11. July 2003 00:35
An infinite tangent vector along a geodesic, that's an interesting proposition! What constitutes an infinity (in) the acse of an aproximation of reality? or what constitutes finitenes in the case of the accuracy in a particular of a function for an inference of the same reality as a parallel with the same vector along the same geodesic? The inference of the distribution and the geodisic might appear infinite, is it really though. I think we could go to towards mirror vectors even and the mind/brain maybe, maybe not. What about transposing them.
Claire
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chimp
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Member # 333
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posted 11. July 2003 03:23
A geodesic is the shortest distance between two points on a curved surface. An object in free fall traces out a geodesic. Parallel translation with tangent vectors along a curved manifold is an abstraction of course. It is difficult to imagine actual "infinitesimal tangent vectors" following imaginary paths on spacelike hypersurfaces.
Infinitesimals are mathematical objects, less than finite but greater than zero. It seems to me that symmetry groups resolve the infinitesimals paradox quite nicely!
Worldlines seem much more "realistic" than imaginary paths and tangent vectors. The coordinate independence of GR, is very appealing.
Richard Feynman's sum over histories-path integral, gives a particle's four dimensional worldline, from point A to point B. The principle of least action applies and energy is conserved. It appears that the universe has laws to maximize efficiency.
Thought experiment:
An object in "free fall" is basically equivalent to an inertial reference frame, if the object is relatively small. Two clocks are synchronized at the top of a tower. Also, there are more clocks affixed along regular intervals from the bottom, to the top OF the tower.
When the clock is dropped from the tower, it will be accelerated at 9.8 meters/sec^2. Yet, since it is in free fall it will be equivalent to a rest frame. As its velocity continues to increase in its fall, it will have a relativistic time dilation, t1:
t1/sqrt[1-(v/c)^2]
Where "v" is the instantaneous velocity at any one "instant", approximately ![[Wink]](wink.gif)
The clocks affixed to the tower will have the approximate gravitational time dilation, t2:
t2/sqrt[1-2GM/((c^2)*r)]
G is Newton's universal gravitational constant. M is mass of the perfectly spherical planet that the tower is standing on. r is the radius of the perfectly spherical planet and c is the speed of light in vacuum.
As the clock falls next to the tower, at each instant that the falling clock passes a clock affixed to the tower, a third observer would observe the two clocks to be ticking at the same rate.
The clock at the bottom of the tower will be the slowest, since it is in the stronger part of the gravitational field. When the falling clock reaches this bottom level, the two clocks should have the "same" time dilation, if the falling clock was dropped from rest, with no additional accelerating forces.
Energy is conserved.
t1/sqrt[1-(v/c)^2] = t2/sqrt[1-2GM/((c^2)*r)] ?
[ 11. July 2003, 03:40: Message edited by: Russell E. Rierson ]
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chimp
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Member # 333
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posted 12. July 2003 02:42
Theoretical physicist Richard Feynman derived the "sum over histories" interpretation of quantum mechanics, where a system does not have a single history, but it has every possible history, and each history has its own probability amplitude. For example, an electron travels from point A to point B by every possible route at once. Each possible route or "path" corresponds to a history.
http://www.maths.usyd.edu.au:8000/u/hughl/PI.html
The amplitude for each history defines the probability of that particular path being followed. The number involves the "action" associated with the history-path, which seems to determine that the path taken, will be the history closest to the "classical" trajectory, in accordance with the law of conservation of energy.
Stephen Hawking explains that when we apply the Feynman sum over histories to particles moving in a background of spacetime, we must also include histories in which the particle travels faster than light and backwards in time.
I propose that these histories and worldlines are fundamental, or elementary waves-distributions, that are ripples in a basic substrate of stochastic noise - chaos. Waves that are in phase, travelling in opposition to each other, constructively interfere with each other and are at resonance. The resonating probability waves are what we can call the "collapse of the wave function" of the Copenhagen interpretation.
The waves that are out of phase destructively interfere with each other and form the basis of the stochastic noise and quantum fluctuations, which have been empirically verified through the "Casmir effect".
A system's history or "worldline" is a resonating four dimensional entity!
Time antisymmetry and reverse time translation of probability amplitudes is given by the equations of special relativity:
t = t'/sqrt[1-B^2]
t = -t'/sqrt[1-B^2]
(t*L) = (-L*t)
Strings and branes could actually be resonating waveforms ![[Eek!]](eek.gif)
Since general relativity is a background independent theory, spacetime must also have its own probability density wavefunctions and sum over histories. Distributed identity. A stratification of probability density functions for relational space-time.
Russ
[ 12. July 2003, 02:44: Message edited by: Russell E. Rierson ]
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chimp
Member
Member # 333
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posted 17. July 2003 01:21
A clock is dropped from a high tower. Along the length of the tower, at fixed intervals, are more clocks. As the clock continues its downward descent, it is accelerated at one "g", ~ 9.8 m/sec^2 .
tower......free fall
[Ct0]<--->[Cf0]
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[Ct1]<--->[Cf1]
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[Ct2]<--->[Cf2]
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[Ctn]<--->[Cfn]
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At the top of the tower, at the "instant" of release, both clocks t0
and f0 are ticking at the same "rate".
During the fall, the free falling clock wil pass a clock on the tower
at one "instant". Does the rate of the two clocks correspond at the
insant? Does the rate of [Ct1] = [Cf1] ? Does the rate of [Ctn] =
[Cfn] ?
approximately?
Does T/sqrt[1-(v/c)^2] = T/sqrt[1-2GM/(c^2)*r] ?
I posted my clock-tower thought experiment to different places on the internet. Someone who wishes to remain anonymous sent the answer below:
Answer:
quote:
On Tue, 15 Jul 2003, Russell E. Rierson wrote:
> A clock is dropped from a high tower. Along the
length of the tower,
> at fixed intervals, are more clocks. As the clock
continues its
> downward descent, it is accelerated at one "g", ~
9.8 m/sec^2 .
[snip]
> At the top of the tower, at the "instant" of
release, both clocks t0
> and f0 are ticking at the same "rate".
If you are asking about weak-field gtr, this is true
automatically if all
the clocks are "ideal clocks".
(An ideal clock is just one which is not subject to
instabilities or
affected by acceleration, temperature, etc., as any
real clock would be.)
> During the fall, the free falling clock wil pass a
clock on the tower
> at one "instant". Does the rate of the two clocks
correspond at the
> insant?
For a moment forget all but two clocks, one falling
and the other held
motionless in a uniform gravitational field. You are
apparently asking
about "the relative rates of these clocks at the
instant when the falling
clock passes the static one". When you pose the
question in this way,
perhaps it is easier to see that the answer depends
upon precisely how you
are comparing the "rates" of the two clocks!
Here is a suggestion for refining your question.
Instead of a falling
clock and a stationary clock, assume you have two
labs, each equipped with
an ideal clock. The first lab is at the origin.
There is a uniform
gravitational field in the z direction (this only
makes sense in the
weak-field approximation to gtr, incidentally). The
second lab is
initially at the origin (the second clock is initially
synchronized with
the first clock, for example by the usual "Einstein
synchronization
procedure", which is valid in any static spacetime)
and at time t = 0 is
dropped and begins to fall in the -z direction. An
observer in the second
lab keeps a laser beam aimed at an observer in the
first lab throughout
the experiment. Question: how does the frequency of
the laser as observed
by the observer in the first lab change over time (by
his own clock)?
Anon.
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marco
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Member # 883
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posted 20. August 2003 18:44
I am not so sure about your statement that time is a process. Couldn't time be an illusion of the human mind to comprehend and navigate in the ever present different states of the universe.
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Evan
Member
Member # 164
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posted 20. August 2003 22:50
Russell offers an interesting link to a short history of path integrals. At the end of this article, the author mentions how Feynman's ideas have been broadened to other complex probabilistic histories.
I have mentioned several times that if Dembski's ideas concerning the explanatory filter are ever going to be applied to evolutionary events at the genetic level, some version of the path integral idea will need to be developed and quantitated.
Evolutionary change involves literally billions of possible genetic changes taking places in thousands, millions, or billions of organisms over some large number of generations. The overall “path integral” probability of all the possible paths from the genetic structure of creature A to creature B, with a probability associated with reproductive fitness attached to each path, is the measure that the EF wishes to use to establish design - if this probability is below some agreed upon lower bound, Dembski wishes to declare design.
Several points stand out here:
1) This approach will be very difficult to quantitize in the real world. However, Feynman spent years, working in the context of other scientists thinking about the same issues, before he succeeded. ID theorists need to start working on how this can be done.
2) The approach needs to clearly work in situations where design is not expected. Take a situation where a simple organism, when exposed to a certain environmental stress, regularly undergoes a certain "microevolutionary" change. Any "path integral theory of genetic change" will need to produce probabilities close to one for this situation. This is where the ID theorist should start - the easy cases where we pretty much know what we can expect to happen.
3) Last, such a theory clearly needs to take into account a huge number of events. (Feynman's actually takes in a infinite number.) This is in contradistinction to all estimates that I have ever seen offered, where one event only is considered - the event by which all the components parts for B come together randomly and instantaneously.
There is a huge difference between this simplistic "one-event" hypothesis of random assemblage and a true theory of genetic changes analogous to Feynman's path integral method .
As Dave Barry once said, "You never can tell when the inevitable is going to happen." Design theorists need to develop the "path integral" type tools that will let us truly distinguish between the virtually inevitable from the vastly improbable. Until such quantitative methods are developed and tested, declarations of design will remain nothing but intuitive speculation.
I know this post is off-topic: it is certainly not about gravity. But Russell's link mentioning the use of path integrals to other stochastic phenomena provides the connection, I hope.
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