Math Imitating Life

These are going to just some comments and references for a future article.

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Here is something by Albert Einstein to wonder about.

Relativity, The Special and General Theory

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In the first section of this book, Einstein makes the following points:

  1. Truth is relative to one’s experience. 
  2. If the experience is limited then one’s “truth” is limited.
  3. Euclidean Geometry is based on limited experience. 
  4. How do we consider a straight line? How do we consider distance?
  5. We assume locations in space to exist as if on a rigid body.

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Points from Section II:

  1. Let’s use a “rigid” Cartesian Coordinate System, and a rigid body as a unit, to describe positions in space.
  2. But, since optical observations are involved, let’s also take into account the properties of the propagation of light in determining the measurements.

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Points from Section III:

  1. Regarding “motion in space,” there is no such thing as an independently existing trajectory but only a trajectory relative to a particular body of reference. 
  2. The finiteness of the velocity of propagation of light would influence the perception of change in position with time.

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Points from Section IV:

  1. A system of co-ordinates of which the state of motion is such that the law of inertia holds relative to it is called a “Galileian system of co-ordinates.” 
  2. We cannot use a system of coordinates rigidly attached to earth, because, in that system of coordinates, stars would appear to be moving in a circle in violation of the law of inertia.
  3. We assume a Galileian systems of co-ordinates, which are rigid but not attached to earth. 

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Points from Section V:

  1. Principle of Relativity: Natural phenomena run their course according to exactly the same general laws with respect to all Galileian co-ordinate systems that are translating uniformly, and not rotating or acceleratingrelative to each other.
  2. We may assume, as our body of reference, a Galileian coordinate system K0, in which natural laws are capable of being formulated in a particularly simple manner.
  3. We may then assume K0 to be “absolutely at rest,” and all other Galileian systems K “in motion.”
  4. The “motion” of Galileian systems K shall contribute to their diminished simplicity, or increased complexity, of the formulation of natural laws.

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Points from Section VI:

  1. The theorem of the addition of velocities employed in classical mechanics, as we shall see, does not hold in reality.

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Points from Section VII:

  1. The velocity of light of any wave-length is constant in vacuum. It does not depend on the motion of the body emitting the light.
  2. The theorem of the addition of velocities employed in classical mechanics does not hold for the velocity of light, which remains the same with respect to all bodies of reference.
  3. This result comes into conflict with the principle of relativity set forth in Section V, because it would appear that different laws of propagation of light must necessarily hold for different coordinate systems.
  4. It seems that the principle of relativity must be rejected because the theoretical investigations into electromagnetic phenomena, leads conclusively to the constancy of the velocity of light in vacuo.
  5. In reality there is not the least incompatibility between the principle of relativity and the law of propagation of light, as shown in the special theory of relativity, by an analysis of the physical conceptions of time and space.

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Points from Section VIII:

  1. “Time” should be defined as a measure in the immediate vicinity (in space) of the event.

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Points from Section IX:

  1. Events that are simultaneous with reference to one co-ordinate system are not necessarily simultaneous with reference to another co-ordinate system because of the finite velocity of light.
  2. Every reference-body (co-ordinate system) has its own particular time.

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Points from Section X:

  1. Owing to the consideration of simultaneity, the measure of the same distance from two different coordinate systems, which are in relative motion to each other, is not necessarily the same .

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There is no loss of energy in a pure standing wave. In other words, the energy would be conserved. We may then look at this physical universe as an example of pure standing wave, because energy is conserved in this universe.

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The most fundamental phenomenon in this universe seems to be a back-and-forth motion around a reference value. Call it a vibration that creates waves; but this phenomenon seems to underlie all other phenomena.

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Light seems to be tied to the vacuum of the space itself rather than to anything existing in that space. In other words, light does not seem to travel relative to anything in space. The velocity of light is the same regardless of the motion of the frame of reference that is used.

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The velocity of light is finite . Einstein used this fact to query the very basis of perception on cosmic scales.

At atomic scales, light seems to condense as standing waves instead of reflect. It seems to be the fractal iteration of this condensation that appears as electrons, protons, neutrons, etc. This makes perception at atomic scales questionable too. Our perception seems to be limited to the middle band.

The frequency and wave-length of light corresponds in such a way that the velocity of light always remains the same. That is the basis of confirming that the universe is expanding using the redshift of the Doppler effect. We apply the same argument to the shift in the pitch of the sound of whistle of a passing train. This brings up the following questions:

(1) The velocity of sound is constant with respect to a medium, such as air. Would we perceive the same shift in the pitch of sound if the air is not all displaced and moved around  by the passing train?

(2) The velocity of light is constant in space. Can we treat space as a “medium” made up of not the usual material, but dark matter perhaps!

(3) Does this dark matter get disturbed by passing light, similar to the way air gets disturbed by the passing train?

(4) How does this dark matter appear to behave at atomic dimensions?

Note: This seems to be looking at the old ether theory again.

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Counting goes up to infinity; then that collection of infinity may be regarded as “one.” For example, Infinity of fundamental particles may take the shape of an apple. We may then count apples. Not all apples will have the same number of fundamental particles, but each would be regarded as a unit apple.

http://io9.com/5809689/a-brief-introduction-to-infinity

The counting to infinity may be repeated with this new “one.”  This procedure may continue without limit. This procedure may be reversed without limit also.

This raises the question, “Do we have a rigid relationship between the “numbers” used to account for the atomic phenomena and the numbers used to account for the cosmic phenomena?”

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The following is the basic assumption underlying Euclidean Geometry, which was pointed out by Einstein:

Euclidean geometry assumes that points, directions and distances behave as if they are associated with a rigid body. We are conditioned to think this way because the rigid body of earth provides our frame of reference.

But the question remains, “Does the “fabric” of space behaves like a rigid body, and if not, then do the laws of Euclidean Geometry still apply?”

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Here is a very interesting comment on mechanical brain and free will by Alan Turing (AMT/B/5 Image 5):

http://www.turingarchive.org/viewer/?id=459&title=5

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[To be continued…]

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Comments

  • vinaire  On February 2, 2012 at 7:49 PM

    From Luminiferous aether

    “Later, Maxwell’s equations showed that light is an electromagnetic wave. The apparent need for a propagation medium for such Hertzian waves can be seen by the fact that they consist of perpendicular electric (E) and magnetic (B or H) waves. The E waves consist of undulating dipolar electric fields, and all such dipoles appeared to require separated and opposite electric charges. Electric charge is an inextricable property of matter, so it appeared that some form of matter was required to provide the alternating current that would seem to have to exist at any point along the propagation path of the wave. Propagation of waves in a true vacuum would imply the existence of electric fields without associated electric charge, or of electric charge without associated matter. Albeit compatible with Maxwell’s equations, electromagnetic induction of electric fields could not be demonstrated in vacuum, because all methods of detecting electric fields required electrically charged matter.

    “In addition, Maxwell’s equations required that all electromagnetic waves in vacuum propagate at a fixed speed, c. As this can only occur in one reference frame in Newtonian physics (see Galilean-Newtonian relativity), the aether was hypothesized as the absolute and unique frame of reference in which Maxwell’s equations hold. That is, the aether must be “still” universally, otherwise c would vary along with any variations that might occur in its supportive medium. Maxwell himself proposed several mechanical models of aether based on wheels and gears, and George FitzGerald even constructed a working model of one of them. These models had to agree with the fact that the electromagnetic waves are transverse but never longitudinal.

    “Nevertheless, by this point the mechanical qualities of the aether had become more and more magical: it had to be a fluid in order to fill space, but one that was millions of times more rigid than steel in order to support the high frequencies of light waves. It also had to be massless and without viscosity, otherwise it would visibly affect the orbits of planets. Additionally it appeared it had to be completely transparent, non-dispersive, incompressible, and continuous at a very small scale.”

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    The data above seems to be quite useful as a starting point to analyze and understand what space really is.

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    • Chris Thompson  On February 2, 2012 at 8:14 PM

      I agree. And because the qualities described are of a substance which does not “exist” in an orthodox sense, you may call it unknowable. (sort of joke but not really)

      So we are stuck with something called the neti-neti of space which exists and yet is unlike anything else except for more of itself which isn’t like anything.

      I am not joking when I say that neti neti helps give me relief from the apparent inconsistencies of this discussion of space. And I think that if we support one another including anyone desiring to join in to follow or comment upon this thread and if we are mindful to continue to look I believe that we could make some meaningful progress toward understanding what space really is. And what electromagnetism really is. And what mass really is.

    • vinaire  On February 2, 2012 at 9:54 PM

      Here are some comments on the quote above;

      (1) There is no aether filling the space. There is only space.
      (2) Space is not material. Therefore, “motion” is not there as part of space. Space may be considered being “still” universally.
      (3) Mass seems to be concentrated at points within space that are relatively very far apart. These points stay at stable distances from each other. This shows rigidity. So, space may be considered rigid enough to not allow longitudinal waves, and to be able to support high frequencies of light waves.
      (4) Space, among these points, is non-material, without mass and viscosity.
      (5) Space is completely transparent, non-dispersive, incompressible, and continuous at a very small scale, except where the “point masses” are.

      These above are the properties of space. What happens to space near the “point masses” is not being considered here. Thus, away from the point masses, space is capable of supporting electromagnetic waves.

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    • vinaire  On February 2, 2012 at 10:01 PM

      The quote above seems to imply that electric field is capable of existing in space without associated electical charge. The implications of this needs to be investigated further.

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    • vinaire  On February 2, 2012 at 10:15 PM

      Space may be hypothesized as the absolute and unique frame of reference in which Maxwell’s equations hold. This will ensure that all electromagnetic waves in vacuum propagate at a fixed speed, c. Again the implications of this needs to be investigated further.

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  • vinaire  On February 2, 2012 at 10:36 PM

    From Luminiferous aether

    Maxwell wrote in Encyclopædia Britannica: “Aethers were invented for the planets to swim in, to constitute electric atmospheres and magnetic effluvia, to convey sensations from one part of our bodies to another, and so on, until all space had been filled three or four times over with aethers…. The only aether which has survived is that which was invented by Huygens to explain the propagation of light.”

    Let’s look at Huygen’s explanation here.

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    • vinaire  On February 3, 2012 at 11:53 AM

      A simple example of the operation of the [Huygen’s] principle can be seen when two rooms are connected by an open doorway and a sound is produced in a remote corner of one of them. A person in the other room will hear the sound as if it originated at the doorway. As far as the second room is concerned, the vibrating air in the doorway is the source of the sound.

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    • vinaire  On February 3, 2012 at 12:05 PM

      Huygens’ principle can be seen as a consequence of the isotropy of space – all directions in space are equal. Any disturbance created in a sufficiently small region of isotropic space (or in an isotropic medium) propagates from that region in all radial directions. The waves created by this disturbance, in turn, create disturbances in other regions, and so on. The superposition of all the waves results in the observed pattern of wave propagation.
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      Space is isotropic for there is no inherent reason for any bias in any direction.

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  • Chris Thompson  On February 3, 2012 at 2:57 AM

    REASON FOR CONSTANT SPEED OF LIGHT FOR ALL FRAMES OF REFERENCE: Light is emitted in packages that I will refer to as a wave-pulse. This wave-pulse package may contain wave-trains called “wellenzug.” However, wave-pulses need not necessarily be thought of as more than a single oscillation. This is not my idea, but conventional QM.

    Light pulses outward from an emitter and propagates at 186,000 miles/second. Because the pulses are nearly instantaneous and because the pulses have no particular mass, and because they move at the speed of light they have momentum. But because they are never at rest they have no particular inertia. Clumsy language that – pointing to more understanding required.
    Regardless, though an emission of light may seem to “stream” or “shine,” in fact it pulses and these emissions pulse instantaneously from the exact point in space where the emission occurs. They have momentum in the direction in which they are emitted but none in the direction of the motion of the emitter. See?

    ***Shifts in wavelength of light are NOT caused by motion of the emitter. Rather, they are caused by any motion of the observer with respect to the emitted wave-pulse of light — after the fact of its emission.***

    http://arxiv.org/pdf/0803.2596.pdf “How Long Is a Photon”

    • Chris Thompson  On February 3, 2012 at 1:58 PM

      *** This is a mistake on my part due to looking too closely at a single photon emission.***

      Please see a better statement below…

  • Chris Thompson  On February 3, 2012 at 2:15 PM

    It is a postulate that the speed of light is constant for all frames of reference but the reason why this should be so is heretofore unknown. However, I want to postulate a model whereby the reason for this can be known. My reasoning is supported by the quantum jump of photon emission shown in QM.

    Stated better:
    1. Emissions of light occur in pulses rather than streaming.
    2. The pulses are of short duration.
    3. The single electromagnetic pulse is of such very short duration that it emanates from a single point in space-time.
    4. Rapid and consecutive emanations or wave pulses create the following illusions:
    A. Motion
    B. Color shift by stringing out the pulses (red-shift) by moving away or compressing the pulses (blue-shift) by moving closer.
    5. The medium of space-time is “expanding” by quantum jumps and thus the wave-pulse point of emanation is moving away from any observer for purposes of cosmology.
    6. It may be found that all motion occurs in quantum jumps rather than a smooth and gradual motion. This is at the quantum level and would only be visible to a human observer or his machinery through the lens of mathematics.
    7. This postulated motion and wave-pulsing by quantum jumps means that every quantum moment – possibly Planck Second – the universe is holding rigidly perfectly still.
    8. Thus sources of wave-pulses of electromagnetic radiation (light) are never in motion at the moment of emanation.

  • vinaire  On February 3, 2012 at 7:10 PM

    Here are some more postulated properties of space:

    (1) Space permeates all point of material bodies.
    (2) Space must give rise to electric and magnetic phenomena.
    (3) Space must determine the propagation velocity of their effects.
    (4) The speed of light and all other propagating effects are determined by space at the relevant location.
    (5) This is analogous to the way that gaseous, liquid and solid media affect the propagation of sound waves.
    (6) Space may be considered the overall reference frame for the universe and thus velocities are all absolute relative to its “rest frame.”

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  • vinaire  On February 4, 2012 at 7:44 AM

    Robert B. Laughlin, Nobel Laureate in Physics, endowed chair in physics, Stanford University, had this to say about ether in contemporary theoretical physics:

    “It is ironic that Einstein’s most creative work, the general theory of relativity, should boil down to conceptualizing space as a medium when his original premise [in special relativity] was that no such medium existed . . .

    “The word ‘ether’ has extremely negative connotations in theoretical physics because of its past association with opposition to relativity. This is unfortunate because, stripped of these connotations, it rather nicely captures the way most physicists actually think about the vacuum. . . . Relativity actually says nothing about the existence or nonexistence of matter pervading the universe, only that any such matter must have relativistic symmetry.

    “It turns out that such matter exists. About the time relativity was becoming accepted, studies of radioactivity began showing that the empty vacuum of space had spectroscopic structure similar to that of ordinary quantum solids and fluids. Subsequent studies with large particle accelerators have now led us to understand that space is more like a piece of window glass than ideal Newtonian emptiness. It is filled with ‘stuff’ that is normally transparent but can be made visible by hitting it sufficiently hard to knock out a part. The modern concept of the vacuum of space, confirmed every day by experiment, is a relativistic ether. But we do not call it this because it is taboo.”

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    • Chris Thompson  On February 4, 2012 at 1:26 PM

      Vinaire,

      This is what I am finding as well. My sense is that the idea that there are vast portions or in fact any portions of the universe which are void and contain “nothing” is as counter-intuitive to me as thinking that I will make manifest what is “unknowable.”

      It is no leap of logic to me that space is composed of something, and also that – – that something can be manipulated.

  • vinaire  On February 4, 2012 at 7:49 AM

    From Wikipedia:

    Quantum mechanics can be used to describe spacetime as being “bitty” at extremely small scales, fluctuating and generating particle pairs that appear and disappear incredibly quickly. Instead of being “smooth”, the vacuum is described as looking like “quantum foam”. It has been suggested that this seething mass of virtual particles may be the equivalent in modern physics of a particulate aether.

    Can the electric and magnetic fields of electromagnetic waves be looked upon as a “particle pair”?

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    • Chris Thompson  On February 4, 2012 at 1:52 PM

      “particle paired” as in your comment about potential vs kinetic?

      • vinaire  On February 4, 2012 at 2:15 PM

        Yes.

        This goes back to the binary nature of this universe.

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  • vinaire  On February 4, 2012 at 9:27 AM

    This article, Electromagnetism, provides a wonderful introduction. Here are a few viewpoints worth noticing.

    (1) The interaction between electromagnetic force and the momentum of the electrons determine interactions between the electrons of neighboring atoms. This governs the processes involved in Chemistry.

    (2) Electrons are bound by electromagnetic wave mechanics into orbitals around atomic nuclei to form atoms, which are the building blocks of molecules.

    (3) It seems that the electric field stores the potential, which is expressed through the magnetic field as force. The electromagnetic wave is like a “potential-actual” wave. This is analogous to how potential and kinetic energy switches back and forth in a pendulum.

    “Light and other electromagnetic waves take the form of quantized, self-propagating oscillatory electromagnetic field disturbances called photons. Different frequencies of oscillation give rise to the different forms of electromagnetic radiation, from radio waves at the lowest frequencies, to visible light at intermediate frequencies, to gamma rays at the highest frequencies.”

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  • vinaire  On February 4, 2012 at 12:02 PM

    I guess, my current question is, “Why is the speed of light a constant?”

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    • Chris Thompson  On February 4, 2012 at 1:38 PM

      I am a little obsessed with this question. I understand you to ask it ambiguously.

      Please ask it again to clear up what you want to know: Why is there a constant?, or why is it constant?, or why is it constant for all frames of reference?, or why does EMR represent that constant?, or why is the constant speed 186K mi/sec?, or other phrasing . . . ?

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