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Using Einsteins ''Hole Argument'' to Give Photons Mass Watch

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    Abstract




    Following Susskinds lectures led to an important equation when concerning the primordial boson field into the stage of symmetry breaking without adding anything trivial. It's almost definitely a question he raised to his audience and yet no doubt considered it with depth. In work, I'll show a possible way to get around this scalar field problem of how to add mass.




    1.








    A symmetry can be understood from the most simplest langrangian term








    \partial\phi^{*} \partial \phi








    Suppose we had a transformation








    \partial \phi' \rightarrow e^{i\theta}\phi(x)








    In this transformation, the derivatives of \phi' are concerned only with \phi. This transformation will look like








    \partial \phi'(x) = e^{i \theta} \partial \phi(x)








    \partial \phi'^{*}(x) = e^{-i \theta} \partial \phi^{*}(x)








    Therefore, when you multiply \partial \phi ' with \partial \phi ^{*} the e^{i\theta} and e^{-i\theta} will cancel out because that is how you would compute them with their conjugates. Thus








    In other words, our field \theta is constant, and does not require the same derivatives as our boson. In return, we would just get








    \partial \phi^{*}\partial \phi








    and viola! This was the most simplest demonstration of a symmetry conserving field. An even simpler demonstration would be:








    \frac{d(x +C)}{dt}








    is in fact simply the same as








    \frac{dx}{dt}








    this means that the equations where symmetry existed and what we have in these symmetries are extra constants always remain the same, just as our Gauge example above.








    But what if \theta also was a function of position \theta (x)? Well let's see shall we.








    \partial \phi' = (\partial \phi + i \phi \frac{\partial \theta}{\partial x})e^{i \theta}








    and our conjugate would be








    \partial \phi'^{*} = (\partial \phi^{*} - i \phi^{*} \frac{\partial \theta}{\partial x})e^{-i \theta}








    Multiplying the two, we need to factorize it








    (\partial \phi + i\phi\frac{\partial \theta}{\partial x})(\partial \phi^{*} - i\phi^{*}\frac{\partial \theta}{\partial x})








    which gives








    =\partial \phi^{*} \partial \phi + i(\phi \partial \phi^{*} - \phi^{*} \partial \phi)\frac{\partial \theta}{\partial x} + \phi^{*} \phi (\frac{\partial \theta}{\partial x})^2








    The reason, again why this equation turned out to be such a mess was because our \theta-field now depended on position, so the derivatives of our massless boson field also included it.




    Now Susskinds asks...




    ''is there any obvious way to give mass to this field, a question he has no doubt asked many, for there does not appear to be obvious ways. In my work, ''The Fully Relativistic Spacescape,'' http://www.thestudentroom.co.uk/show....php?t=2488277 ... that Einstein discovered that giving physical meaning to space coordinates involved taking not only their worldlines into perspective but also their physical interactions.


    This was mathematically the same thing as ''dragging the metric tensor'' under deiffeomorphism invariance, which is like a Lorentz invariance, except it applies to the universe holistically.


    Now, a bunch of photons or whatever energy radiated from big bang, must be related to giving physical meaning back to massless bosons occupying space. Instead of Higgs Boson you could take General Relativity and meld it into Machian relativity and restore physical points in space where we consider none.


    As I wrote in previous work:






    =\partial \phi^{*} \partial \phi + i(\phi \partial \phi^{*} - \phi^{*} \partial \phi)\frac{\partial \theta}{\partial x} + \phi^{*} \phi (\frac{\partial \theta}{\partial x})^2








    The reason, again why this equation turned out to be such a mess was because our \theta-field now depended on position, so the derivatives of our massless boson field also included it. The massless boson field could attain the mass by the same holistic approach melding GR with Machian relatvity, dragging the metric of the position-dependent boson field would be done so using diffeomorphism invariance - No longer does the system have an appearance of masslessness, GR + Machian relativity solves it by tackling what Einstein called his ''hole argument.'' Definitely a pun... Einstein was heavily unfluenced by Mach.
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