Proof Minkowski Spacetime is Poorly Conceived

Discussion in 'Alternative Theories' started by danshawen, Apr 21, 2016.

  1. The God Valued Senior Member

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    This is a thread where total non sense is being perpetuated by both the sides. And the problem with Danshaven is that he is trying to substitute a bigger nonsense with an existing well established nonsense. Odds are against him.

    Spacetime as a mathematical tool is fine, but it is meaningless and inexpressible in reality.

    space as light travel time is as meaningless and as inexpressible in reality as spacetime, infact a bit more ridiculous as no maths can be derived out of it.
     
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  3. danshawen Valued Senior Member

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    It took a very long time and considerable help from people here (many of them now opposed to the idea they helped create) to come up with that "bigger nonsense".

    There is a reason the neutron is unstable outside of a nucleus or proximity to other protons or neutrons, but indefinitely stable inside of atomic structure. This nonsense explains that.

    There is a reason that a beam of photons between mirrors in a laser cavity share the inertia of the electrons and atoms that confine it to that space, and it is the same reason that elementary particles of matter and atomic structure itself has inertia. Inertia derives of energy being bound, by whatever means. But in the case of bound energy, this can only happen because propagating energy is bound by other propagating energy, and this can only occur in a means of propagation that is by its very nature, FTL, with both a tangential and a radial propagation component of a minimum of two photons. Time dilation that varies in value from the centers of fundamental particles to the outer edge is a key component of this structure and is not currently supported by either QM or Relativity. This is principally the reason both quantum spin and entanglement remain bizarre adjuncts to the math that supports our understanding of particle physics. This is the reason particle physicists believe there is a disconnect between the most recent discoveries of the particle that supports the mechanism of inertia but does not support or provide a deeper understanding of relativity's principle of equivalence. It is a means for understanding how the "space" or light travel time outside of the bound energy that is matter may also support rotational and/or linear inertia with a gravitational field that directs falling objects to the exact geometric centers of planets, because the spin zero particle that does that is able to pass through other such bosons and matter all the way to the centers of planets with less difficulty than a neutrino, to which the same mechanism also imparts a minuscule amount of inertial mass. The "divine hand" of gravitation goes away permanently. The Pope probably won't be very happy about that.

    There is a reason that rest mass of particles seem to increase with increasing energies or relative velocity in a given direction, and that no matter how much energy you push it with, it can never exceed the speed of light. And that more energy can be added in any other of an infinitude of directions without causing a fundamental particle of matter to release all of its bound energy into unbound energy, until or unless it collides with an equivalent mass of antimatter. And there is a reason that all the while the mass is increasing, so does time dilation. If the internal rate of propagation of energy for bound particles was only the speed of light, this would not be possible without releasing all of the energy of matter when you pushed it close enough to that limit. The matter in the universe at cosmological distances would simply fly apart. It would be very hard not to observe such a process. The night sky would be very bright indeed.

    In a universe of energy transfer events including both bound and unbound energy, the speed of light is an invariant and universal constant in all inertial reference frames, but it is not the only invariant state of motion. The other one is the domain of the bound energy that is matter. The rest frame quite literally derives of the relativistic vector sum of +c and -c in all inertial reference frames in any state of motion. Without these TWO motion invariants, as well as the spin invariant (quantum spin of zero), an understanding of physics itself remains locked in the timeless, inertialess, static geometry of Euclid and Pythagorus.

    Anything that moves, moves relative to something else. As soon as it moves, the time dilation it experiences is different from the rate found anywhere else in the universe. Simultaneity, even if it exists, does not persist long enough outside of the domain of quantum entanglement for a 19th century mathematician to make assertions about anything not restricted to discussions of absolute space and/or time, and was certainly no excuse to reinstate it by means of an artificially invariant interval that equivocates the invariance of the speed of light with absolute time.
     
    Last edited: May 4, 2016
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  5. rpenner Fully Wired Valued Senior Member

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    Quoting my summary of the philosophy of physics:
    A physical theory is a useful, precise, communicable framework for predicting the behavior of a wide variety of related observable phenomena.​

    This definition leaves space-time, specifically the geometry of space-time, rather better off than you indicate.
     
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  7. el es Registered Senior Member

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    It seems that for danshawen, energy transfer events occur in zero space and zero time. Geometry and math are of no avail.

    Outside observers are forbidden.

    An outside observer would require the geometry of space and time to formulate a mathematical theory.
     
  8. el es Registered Senior Member

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    duplicate
     
  9. danshawen Valued Senior Member

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    Science always works better if all of us don't completely agree. I have made my case. I'm sure, it isn't the last word.
     
  10. przyk squishy Valued Senior Member

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    Then why did you say the invariant spacetime interval was unjustified before? You can't say it's completely unjustified and then go "Oh, I knew that all along" when I show you a partial justification.

    Then you can't have any issue with the invariance of the spacetime interval.

    Lorentz transformations are linear, so they also apply to coordinate intervals. For instance, a Lorentz boost along the \(\mathrm{x}\) axis would relate coordinate intervals in different reference frames by

    \(\begin{eqnarray} \Delta t' &=& \gamma ( \Delta t - \tfrac{v}{c^{2}} \Delta x ) \,, \\ \Delta x' &=& \gamma ( \Delta x - v \Delta t ) \,, \\ \Delta y' &=& \Delta y \,, \\ \Delta z' &=& \Delta z \,. \end{eqnarray}\)​

    Substituting this into the spacetime interval, you get

    \(\begin{eqnarray} c^{2} \Delta t'^{2} - \Delta x'^{2} - \Delta y'^{2} - \Delta z'^{2} &=& c^{2} \gamma^{2} \bigl( \Delta t - \tfrac{v}{c^{2}} \Delta x \bigr)^{2} - \gamma^{2} \bigl( \Delta x - v \Delta t \bigr)^{2} - \Delta y^{2} - \Delta z^{2} \,, \\ &=& \gamma^{2} \Bigl( c^{2} \Delta t^{2} - 2 v \Delta t \Delta x + \tfrac{v^{2}}{c^{2}} \Delta x^{2} \Bigr) - \gamma^{2} \Bigl( \Delta x^{2} - 2 v \Delta x \Delta t + v^{2} \Delta t^{2} \Bigr) - \Delta y^{2} - \Delta z^{2} \,, \\ &=& \gamma^{2} \bigl( 1 - \tfrac{v^{2}}{c^{2}} \bigr) c^{2} \Delta t^{2} - \gamma^{2} \bigl( 1 - \tfrac{v^{2}}{c^{2}} \bigr) \Delta x^{2} - \Delta y^{2} - \Delta z^{2} \,, \\ &=& c^{2} \Delta t^{2} - \Delta x^{2} - \Delta y^{2} - \Delta z^{2} \,, \end{eqnarray}\)​

    since \(\gamma^{2} = \frac{1}{1 - \tfrac{v^{2}}{c^{2}}}\). So you can't dispute invariance of the spacetime interval without also disputing the Lorentz transformation.

    You keep talking about "Minkowski rotations" as if Minkowski made up some new type of rotation for his geometry. There is no such thing. The only thing anything remotely like a rotation in Minkowski geometry is the Lorentz transformation, which you've just said you're completely fine with.

    Normal (Euclidean) geometry is based around rotations and the invariance of the Euclidean interval \(\Delta x^{2} + \Delta y^{2} + \Delta z^{2}\). Minkowski geometry is a different kind of geometry based around Lorentz transformations instead of rotations and the invariance of the spacetime interval \(c^{2} \Delta t^{2} - \Delta x^{2} - \Delta y^{2} - \Delta z^{2}\) instead of the Euclidean interval.

    Minkowski geometry doesn't answer this or even attempt to do so. That isn't the point of it.

    The simple physical answer is that we measure speed using e.g. physical rulers to measure the distance travelled and clocks to measure time intervals, and these are all affected by relative velocity in such a way that we always measure the speed of light to be invariant. This in turn happens because rulers and clocks are physical systems and the physics governing their structure and evolution are Lorentz invariant.

    For example, if you work out the electromagnetic field around a moving charged particle, you'll find (for instance) that the electric part of the field is length contracted compared to the Coulomb field around a charge at rest. You could do this applying relativity (work out the Coulomb field in the rest frame and then switch to the moving frame), but even if you didn't know relativity you could derive the same result directly from Maxwell's equations. (In fact, this is how it was first done, by Oliver Heaviside in the 1880s.) So it shouldn't be too surprising if a physical ruler bound together by electromagnetic forces becomes length contracted when in relative motion.

    It's invariant with respect to a change of reference frame.

    No, this is a non sequitur. It is not trivial that you can change what you call "at rest" without changing the speed of light. In Galilean relativity, for instance, any reference frame can equally well be considered the one "at rest", but velocities undergo normal velocity addition when you change from one reference frame to another, so that there is no invariant speed. This shows that you can have a relativity principle without an invariant speed of light. We had one long before Einstein.

    This is why Newtonian mechanics and Newton's theory of gravity were modified and replaced in light of special relativity. Newton's theories already had a relativity principle -- Galilean relativity -- built into them. It was just what we now believe is the wrong one.

    You look like you're taking "invariant interval" to mean that space and time intervals are invariant. It doesn't mean that at all. The coordinate intervals \(\Delta t\), \(\Delta x\), \(\Delta y\), and \(\Delta z\) are individually reference-frame dependent and change when you change the reference frame in the way described by a Lorentz transformation, but this always happens in such a way that the combination \(c^{2} \Delta t^{2} - \Delta x^{2} - \Delta y^{2} - \Delta z^{2}\) stays the same. You can see this by taking a hypothetical spacetime interval and seeing how it changes if you change the reference frame using a Lorentz transformation. You find that it doesn't change. This is what I show above in this post.
     
    Last edited: May 4, 2016
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  11. danshawen Valued Senior Member

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    No. It checks out mathematically alright.

    Here is the part that DOESN'T check out:

    c^2Δt^2−Δx^2−Δy^2−Δz^2

    Say you have two interferometer-like events (connected by light travel time) separated in space by the Pythaogorean spatial geometric distance: sqrt(Δx^2+Δy^2+Δz^2).

    Is Δt^2 the same in all inertial reference frames, of is it just possible that delta t in one location is very different from delta t in another location related by Minkowski's interval? Assuming that they are the same implies a Lorentz invariance property for time that does not exist. Lorentz invariance only works for the speed of light, NOT time, and NOT time dilation.

     
  12. paddoboy Valued Senior Member

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    Many experiments suggest otherwise and show spacetime to be far more than just a mathematical tool. GP-B has showed it can be curved/warped and twisted.....aLIGO has showed that in the event of a catastrophic disturbance in spacetime, it will create waves also.
    Spacetime certainly is not a vanilla type physical thing, but that does not make it any the less real, due to the above effects. Spacetime is a dynamical entity and as such real.
    Is an electromagnetic field real in your opinion?
    Is time real? Is space real?
    What is your definition of real?
     
    Last edited: May 4, 2016
  13. przyk squishy Valued Senior Member

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    No, it is frame dependent. (Did you read the end of my last post?)

    No, this is completely wrong. The second postulate from Einstein's 1905 paper says:

    Einstein subsequently derived the Lorentz transformation based on the invariance of \(c\). Now consider the first postulate in light of this:

    This means that all the laws of physics must be Lorentz invariant, i.e., they must remain invariant in form under a Lorentz transformation of the coordinates.
     
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  14. danshawen Valued Senior Member

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    I completely agree with every single quote from Einstein, EXACTLY the way you quoted them. The last one, "all the laws of physics" REMAIN THE SAME in ANY SINGLE INERTIAL REFERENCE FRAME is absolutely, precisely, exactly correct. Inertial reference frames are where, in the ideal case, we do all of our physics. This does not mean that every physical measure of such frames, other than the invariant speed of light, are equal. Time dilation may be different. The twin paradox, which is not really a paradox at all, is the best example.

    What is NOT CORRECT is to expect that TIME DILATION in TWO DIFFERENT INERTIAL REFERENCE FRAMES. will be the same. What is NOT CORRECT is to assume an invariant 4D interval BETWEEN TWO NON COLOCATED SIMULTANEOUS EVENTS, and expect that DELTA T, an interval of time dilation, will be EXACTLY SAME DELTA T IN TWO REFERENCE FRAMES IN RELATIVE MOTION WITH RESPECT TO EACH OTHER.

    As soon as this is posited, the laws of physics that hold in a single reference frame no longer apply because we are now dealing with two reference frames in relative motion. The speed of light, which is invariant will measure the same in both reference frames. Time dilation however is not invariant and in general WILL NOT be the same in both reference frames. Because the interval is defined as if they are the same time interval in both frames, Minkowski's interval based on this assumption is an invalid formulation, and one that goes to some considerable trouble to equivocate absolute spacetime with absolute time dilation contained in the definition of his 4D intervals substituted for the invariant speed of light.

    ALL OF YOUR MATH is perfectly correct in a static universe with no time and no movement. But this is not what those invariant intervals are intended to describe.

    ALL OF MINKOWSKI'S vectors are relativistic, not Euclidean. There are specific rules for how you can and cannot add relativistic distances, all which are all subject to a vector component that shortens in the direction of relative motion. Of course, if both FoRs are at rest, and far from any gravitation, his intervals will work just fine in principle. Big deal. This is no check of the validity of the interval as a universal invariant, mathematically or physically.

    There are specific rules for how you calculate time dilation, and it is different literally everywhere, and depends on proximity to gravitating bodies as well as relative motion. Your boost matrices may take care of these transformations just fine, but they cannot compensate for Minkowski's fundamental error with equivocating delta t time dilations in two different reference frames as though it were a law of physics that they must be the same. They most certainly are not, with the sole exception of the case where both reference frames are at rest with respect to each other. It simply does not work that way physically, even if the math for the rest case may seem sound enough. That would be a trivial and a superficial test of the validity of interval invariance, and in most dynamic cases, they will most certainly fail that test.
     
    Last edited: May 5, 2016
  15. przyk squishy Valued Senior Member

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    I did not say that every physical measure is invariant. I said that the laws of physics must be invariant in form under Lorentz transformations, which are the transformations that relate the coordinates of inertial reference frames in relativity. Einstein, for example, checked explicitly that Maxwell's theory of electromagnetism is Lorentz invariant in the same 1905 paper in which he introduced special relativity.

    I only brought this up because you said that Lorentz invariance applies only to the speed of light. That's wrong, and a claim like this makes me think that you don't understand what "Lorentz invariance" actually means. I wasn't saying anything about distance or time here.

    I never said this.

    Why do you keep denying the opposite of what I said? Invariance of the spacetime interval does not mean that \(\Delta t\) is invariant. I've told you this three times now.
     
  16. The God Valued Senior Member

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    Perfect, no dispute, but that does not make spacetime real.
    The observable phenomena can be said to be real, spacetime is a tool to describe these relevant phenomena. To describe some real observation, you may need mathematical tools and thats what spacetime is. Just because the concept of spacetime is used in describing gravity etc, it does not make it real.


    You may take web help to learn the definition of 'real'.

    A physical thing is a physical thing....nothing like vanilla type or exotic candy type....Got it ?
     
  17. The God Valued Senior Member

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    I urged you to put your theory clearly, but el es is doing the honors, why not you. after all its your baby ?

    There are some questions..

    1. What is bound energy ?
    2. What is unbound energy ?
    3. What is the difference between two and what is the similarity between two ?
    4. The bound energy is kept bound by what mechanism ?
    5. What is bound particle ?
    6. What is unbound particle ?
    7. Pleas explain below statement of your theory ?
    'propagating energy is bound by other propagating energy'
    What if there is speed mismatch ?
    8. Why light speed is invariant in your theory ?
    9. Why time dilation is different in radial direction of a fundamental particle ?
    10. Take a fundamental particle of radius r (if we can define a particle that way), then what is the time dilation at r/2 from center ?
    11. Take 'n' fundamental particles, lump them (either bosonic or fermionic) and now please tell the time dilationa spect of such lump ?
    12. What is a fundamental particle in your theory ?
    13. Why have you used the word seem ib below statement ?
    rest mass of particles seem to increase with increasing energies or relative velocity
    14. Anything that moves, moves relative to something else.
    Thats not your idea ?
    15. Why the time dilation should increase as the mass increases ? Mass can be increased by lumpting together or by increase speed, so why time should dilate ?
    16. You can't chuck Minkowski out by saying...whatever means. Can you ?
    Inertia derives of energy being bound, by whatever means.


    Many more......
     
    Last edited: May 5, 2016
  18. paddoboy Valued Senior Member

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    Please Register or Log in to view the hidden image!

    At least you still maintain some humor...nice to see my friend.
    But you failed to answer the question is an electromagnetic field real?
    Space is real....it separates all the matter energy in the universe.
    Time is real as it stops everything from happening together.
    If we had neither, we would not be here.
    Spacetime is a unified multi-dimensional framework within which it is possible to locate events and describe them in terms of spatial coordinates and time.Spacetime is also a valid concept that follows from the observation that the speed of light is invariant, i.e. it does not vary with the motion of the emitter or the observer. Spacetime allows a description of reality that is common for all observers in the universe, regardless of their relative motion. Intervals of space and time considered separately are not the same for all observers.
    Space time as a dynamical object is most certainly real....
    I was going to give you a rest from links But I also like to help out whenever necessary.
    https://einstein.stanford.edu/content/relativity/q411.html

    https://einstein.stanford.edu/content/relativity/q411.html
    Got it?

    Please Register or Log in to view the hidden image!

     
  19. danshawen Valued Senior Member

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    The follow wing laws of physics are invariant under Lorentz transformation:

    1. The invariance of the speed of light as applied to linear propagation in a vacuum

    1.5 The invariance of the rest frame for matter tracing linear trajectories at velocities <c, which is the frame of reference which makes item 1. above possible. Even the invariant speed of light is invariant RELATIVE or measured with respect to something else. The rest frame, not an interval, is that something else.

    2. Rest mass of matter (measured in the inertial rest frame in any state of motion)

    3. Conservation of energy for relativistic projectiles with linear trajectories. This includes a combination of both rest mass and relativistic linear momentum, which includes any increase in mass / energy due to added momentum in a given direction.

    Are you with me so far?
     
  20. danshawen Valued Senior Member

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    Now about the Lorentz transformation:

    This is a shining gem of mathematical physics. Without it, Special Relativity itself would not even have been possible.

    I have spent more time probably than half of the physics PhDs in the world studying it because of an unfortunate incident involving my freshman physics professor and his own derivation of it.

    I'm not going to derive it again. That' is counterproductive and a waste of time and energy. I concede that it is correct in every way, in all its glory.

    It contains an awful lot of TIME INSTANTS, which is to say, NOT INTERVALS of time, but discrete INSTANTS for time, and also discrete position origins and coordinates for any and all measurements used.

    x, x', t, and t' are used to represent the stationary (unprimed) and the moving (primed) frames of reference, and when things start moving, you get difference terms for position and difference terms for time. The important thing to note about it is that TIME AND POSITION TERMS ARE NOT COMBINED.

    At no point in the derivation of the Lorentz transformations is anything said about MASSES (rest mass or otherwise), or about TIME DILATION that may be present at one of the position coordinates that is different than the time dilation at another. In this way, the Lorentz transformations too are an incomplete description of physical reality, but their success in describing and extending the applicable physics pretty much speaks for itself.
     
  21. danshawen Valued Senior Member

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    It means that the entire interval is invariant. What I said is, THE INTERVAL IS NOT INVARIANT. THE DELTA T IS THE REASON IT IS NOT INVARIANT.

    What Minkowski is saying is, that for for ANY TWO SIMULTANEOUS EVENTS viewed from ANY INERTIAL REFERENCE FRAME IN ANY STATE OF RELATIVE MOTION, THE 4D INTERVAL IS INVARIANT (THE SAME, MEASURED BY TWO DIFFERENT OBSERVERS).

    LIAR. Minkowski's derivation is not nearly as clean as the Lorentz transformations. He deliberately attempts to combine terms of time and space into a single invariant interval. This is wrong on many levels.

    Minkowski uses DELTA T, in other words, t2 - t1 time intervals to create his invariant intervals, NOT INSTANTS of time the way the Lorentz transformations successfully apply.

    These intervals take no account of:

    1) Whether these intervals between simultaneous events in spacetime happen to be in relative motion when they occur
    2) Whether these intervals between simultaneous events in spacetime happen to be at different distances from gravitating bodies, or whether or not one of those events is simply located nearer to a larger rest mass.

    The DELTA T is going to be problematic in any event, because time dilation is already predicted by the Lorentz transformations, and he literally takes no account of it in computing his DELTA Ts. To Minkowski, DELTA T over here is EXACTLY THE SAME time dilation factor as DELTA T over yonder. It isn't so, no matter where you measure them. His interval does not have the same standing, mathematically or physically, that the Lorentz transformations do.

    THAT is why his 4D Intervals are not invariant in the same way that the speed of light is.

    In actual fact, time dilation also increases with increasing mass. The example of protons and neutrons remaining stable in atomic nuclei when neutrons decay outside of them is the best example I can give. Time dilation is why.
     
  22. danshawen Valued Senior Member

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    The Lorentz transformation of the laws of physics mentioned three posts ago actually do not apply to the laws of physics pertaining to quantum spin or the propagation of energy inside of particles of matter. This is because the Lorentz transformations only apply to linear propagation of energy or matter, <c. That is manifestly NOT what is going on inside of the bound energy we call matter.

    Rotational modes of energy propagation contain both a tangential and a radial mode of propagation not covered by the Lorentz transformations. There are different laws of physics anywhere v > c. Relativity will not be obeyed, but evidently there is still some means of equivalent time dilation that is occurring. Otherwise, matter would be much less persistent in terms of decay into unbound energy.
     
  23. danshawen Valued Senior Member

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    It's not "my baby". I told you it was developed here, by dozens of contributors. It was.

    Bound energy is matter or antimatter.
    Unbound energy would be anything from long wave radiation to energies of 250 GeV. Not talking about particles or hadrons here; just photons.

    Bound energy appears outwardly to be "at rest" or moving with a constant velocity <c in its natural state. Rest mass (energy) is invariant.
    Unbound energy travels at c in a vacuum relative to any bound energy in any state of motion. The speed of light is invariant.

    Energy and matter (unbound and bound energy) are the same. E=mc^2

    There's no such thing as an "unbound" particle. That would be energy. Particles and waves are equivalent in this theory, as in QM.

    It's not my theory. It is not an alternative theory. It is a compilation of physics other than Minkowski's, which work together very well.

    I don't know how propagating energy can be bound by other propagating energy. I only understand that it can.

    Outside of particles of bound energy, time dilation does not have a direction. Inside, it is less at the geometrical centers of bound particles of energy than it is at their outer shells. Bound energy rotates internally at velocities > c. The latter may be part of the definition of time, but it is not all. If it was, matter itself would be as frozen in place and static, and without time or inertia, as Euclidean geometry.

    I cannot predict time dilation at a given radius within bound energy, other than at the centers, and at the rim. At the rim, I believe it to be c^2, but this is a guess. This is all new physics. The rotational inertia bound in matter may be composited of as many or more modes of rotation as there are for electrons, as has been directly observed in hydrogen atoms. There may be no physical experiments possible in order to decompose smaller fundamental particles, but understanding that there is fine structure there and how fast it propagates is the first step in a new direction of scientific inquiry.

    The only spin zero boson is the Higgs boson. This is as significant a physical find as the revelation of the laws of linear inertia. It's a beginning. Fermions like electrons have quantum entangled paired half integer spins and are the best known example of quantum entanglement. As such, ENTANGLEMENT IS EVERYWHERE. So get used to it.

    "Anything that moves does so relative to something else" is the foundation of Special Relativity. I did not originate it.

    Why should time dilation increase as mass increases? This, in a perfect world, SHOULD HAVE BEEN the connection that Minkowski made, but missed by a mile. It most certainly does, and it also makes perfect sense, in terms of conservation of energy for a start. More mass means more inertia. More inertia means more time dilation. It is possible, this is new, but if someone had done a paper already (likely), I'm not claiming credit. I'm not claiming credit for anything. I have said that I am freely offering any of this to be used by any of the contributors here in any manner they wish, including publishing about it. That offer, subject to copyright limitations set by Xenoforums, is still open. I don't know that I have the right to offer unlimited or fair use. Possibly not.

    "Inertia derives of energy being bound, by whatever means." is an idea developed and promoted by a laser physicist John Macken over 10 years before I found the idea. John has his own website "Only Spacetime", and dozens of laser patents to his credit.

    Minkowski and his attempts at justifying his role in the education of Einstein and Hilbert has been an ugly blemish on relativity physics since his death from a burst appendix in 1908. It is not in the same league either with Lorentz or the work of Minkowski's less favored student Einstein. Space isn't just covariant with time. Space IS light travel time in all three dimensions, in every direction, combined with FTL rotation / propagation of energy inside of the bound energy we call matter (or antimatter). There is no inertia, other than that possessed of bound energy, in the universe of energy transfer events.

    I think that answers most of the inquiries in your short list.
     
    Last edited: May 5, 2016

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