Assumption of Relativity Activated

 13 July 02:33   

    Returning to the appearance of a wave, we anon see that

    :$phi\; =\; mathbf\; cdot\; mathbf\; -\; omega\; t\; =$

     mathbf cdot mathbf - ( omega /c)(ct)

     = underline cdot underline .

    Thus, a bunched way to address a beachcomber is

    $A(\; underline\; )\; =\; sin\; (\; underline\; cdot\; underline\; )\; .$(6.8)

    Since $underline$ is accepted to be a four-vector and back the appearance of a beachcomber is accepted to be a scalar absolute of advertence frame, it follows that $underline$ is absolutely a four-vector rather than just a set of numbers. Thus, the aboveboard of the breadth of the beachcomber four-vector haveto aswell be a scalar absolute of advertence frame

    :$underline\; cdot\; underline\; =$

     mathbf cdot mathbf - omega^2 / c^2 = const .

    

    Let us analysis absolutely what this means. As this amount shows, we can boldness a position four-vector x into apparatus in two altered advertence frames, e.g (X,T) and (X′,T′), but these are just altered means of autograph the aforementioned vector.

    This is absolutely the aforementioned as the way a three-vector has altered apparatus in a rotated frame.

    Similarly, just as a three-vector has the aforementioned magnititude in all frames, so does a 4-vector; i.e,

    :$X^2-c^2\; T^2\; =\; X^\; -\; c^2\; T^$

    Applying this to the beachcomber four-vector, we infer that

    :$k^2-c^\; omega^2\; =\; k^\; -\; c^\; omega^$

    where the unprimed and abreast ethics of k and ω accredit to the apparatus of the beachcomber four-vector in two altered advertence frames.

    Up to now, this altercation applies to any wave. However, after-effects can be disconnected into two categories, those for which a appropriate advertence anatomy exists, and those for which there is no such appropriate frame.

    As an archetype of the former, complete after-effects attending simplest in the advertence anatomy in which the gas accustomed the complete is stationary. The aforementioned is true of ablaze breeding through a actual average with an basis of refraction not according to unity. In both cases the acceleration of the beachcomber is the aforementioned in all admonition alone in the anatomy in which the actual average is stationary.

    If there is no actual medium, then there is no actual way of award a appropriate anatomy so the after-effects haveto abatement into the additional category. This includes all after-effects in a vacuum, such as light.

    In this case the afterward altercation can be made. An eyewitness affective with account to after-effects of abundance ω and beachcomber amount k sees after-effects of abundance ω′ and beachcomber amount k′. If the eyewitness can acquaint in any way that they appear from a antecedent affective with account to them, then they can use this to analyze a appropriate anatomy for those waves, so the after-effects haveto attending just like ones from a anchored antecedent of abundance ω

    This armament us to achieve that for such waves

    :$omega^2=\; c^2\; k^2\; +\; mu^2\; ,$

    where μ is a constant. All after-effects in a exhaustion haveto accept this form, a abundant added belted best than in classical physics.