@notsosureofit Hypothesis

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The proposition that dispersion caused by an accelerating frame of reference implied an accelerating frame of reference caused by a dispersive cavity resonator. (to 1st order using massless, perfectly conducting cavity)


Starting with the expressions for the frequency of a cylindrical RF cavity [1]:

[math]f = \cfrac{c}{2π}\sqrt{(\cfrac{X}{R})^2+(\cfrac{pπ}{L})^2}[/math]

where "c" is the speed of light in the medium (the speed of light in vacuum divided by the square root of the product of the relative magnetic permeability times the relative electric permittivity of the medium).

For TM modes, X = X[sub m,n] = the n-th zero of the m-th Bessel function. [1,1]=3.83, [0,1]=2.40, [0,2]=5.52 [1,2]=7.02, [2,1]=5.14, [2,2]=8.42, [1,3]=10.17, etc.

and for TE modes, X = X'[subm,n] = the n-th zero of the derivative of the m-th Bessel function. [0,1]=3.83, [1,1]=1.84, [2,1]=3.05, [0,2]=7.02, [1,2]=5.33, [1,3]=8.54, [0,3]=10.17, [2,2]=6.71, etc.


  • Here is a table [2] to 15 digits precision for the roots of the cylindrical Bessel functions X[sub m,n] and for the roots of its derivative X'[subm,n] from m=0 to m=10, and from n=1 to n=5


Rotate the dispersion relation of the cavity into doppler frame to get the Doppler shifts, that is to say, look at the dispersion curve intersections of constant wave number instead of constant frequency.

[math]\Delta f = \cfrac{1}{2f}(\cfrac{c}{2π})^2X^2(\cfrac{1}{Rs^2}-\cfrac{1}{Rb^2})[/math]

This is a cylindrical approximation and could be replaced with a tapered dielectric index of refraction in a cylindrical cavity.

and from there the expression for the acceleration g from:

[math]g = \cfrac{c^2}{L}\cfrac{\Delta f}{f}[/math]

such that:

[math]g = \cfrac{c^2}{2Lf^2}(\cfrac{c}{2π})^2X^2(\cfrac{1}{Rs^2}-\cfrac{1}{Rb^2})[/math]

This is the acceleration at which the dispersion of the tapered cavity is balanced out by the dispersion due to its acceleration.

Using the "weight" of the photon in the accelerated frame from:

[math]W = \cfrac{hf}{c^2}g[/math]

where we make the connection via the Equivalence Principle that the acceleration of a photon seen in the rest frame is that which is balanced out in the accelerated frame. That is, the dispersion of the tapered cavity reduces to zero (along the axis) in that accelerated frame of reference.

Such that:

[math]W = T = \cfrac{h}{L}\Delta f[/math]

We identify that as the thrust per photon:

[math]T = \cfrac{h}{2Lf}(\cfrac{c}{2π})^2X^2(\cfrac{1}{Rs^2}-\cfrac{1}{Rb^2})[/math]

If the number of photons is

[math]\cfrac{P}{hf}(\cfrac{Q}{2πf})[/math]

then the total thrust is

[math]NT = \cfrac{PQ}{4πLf^3}(\cfrac{c}{2π})^2X^2(\cfrac{1}{Rs^2}-\cfrac{1}{Rb^2})[/math]

or

[math]NT = \cfrac{2PQ}{L(2πf)^3}(c X)^2 (\cfrac{1}{Ds^2}-\cfrac{1}{Db^2})[/math]

or

[math]NT=\cfrac{2PQ}{L ω^3}(c X)^2 (\cfrac{1}{Ds^2}-\cfrac{1}{Db^2})[/math]

where Ds = diameter at small end, Db = diameter at big end and ω = angular frequency.

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Proof that scaling dimensions inversely proportional to frequency keeps the thrust invariant:

Suppose that the thrust at frequency f1, and dimensions L1, Ds1, and Db1 is

[math]NT_1 = \cfrac{2PQ}{L_1(2πf_1)^3}(c X)^2 (\cfrac{1}{Ds_1^2}-\cfrac{1}{Db_1^2})[/math]

then, at frequency f2 a multiple of frequency f1

f2 = n f1

where n is any irrational number, and scaling dimensions to be inversely proportional to frequency f2:

L2 = L1/n

Ds2 = Ds1/n

Db2 = Db1/n

and substituting, we get the thrust for frequency f2 to be:

[math]NT_2 = \cfrac{2PQ}{L_1(2πf_1 n)^3}(c X)^2 (\cfrac{1}{(Ds_1/n)^2}-\cfrac{1}{(Db_1/n)^2})[/math]

and since the factor of [math]n^3[/math] occurs both in the numerator and the denominator, it cancels out, leaving

NT2 = NT1

The thrust force is independent of frequency when the diameter and the length of the cavity are both scaled to decrease inversely proportional to increasing frequency, in order to maintain the same mode shape.