Sunday, February 06, 2005

Cahill is not using Galilean Relativity in his Ether Drift Claims

In order to calibrate what Cahill and the Catanians are claiming, let's review the "mainstream position" circa 1955 in the text book "Classical Electricity and Magnetism" by Panofsky & Phillips Ch 15 "The Experimental Basis of the Theory of Special Relativity".

15-1 presents the equations of the linear Galilean boosts along the x-axis between inertial frames

x' = x - vt
y' = y
z' = z
t' = t i.e. absolute Newtonian time

The Newtonian point particle mechanical laws are form-invariant ("covariant") under this transformation boost connecting two global inertial frames. This is a group unto itself if we do not change direction of the boost. So are the Lorentz boost transformations in a fixed direction in space, but if we change the direction of the boost, if I recall correctly, we need the spatial rotations to keep the group property - that is we need the full O(1,3) in that case? The Lorentz boosts alone in all directions are not a group. The space rotations of the Lorentz group O(1,3) are a subgroup O(3), but not the boosts unto themselves? I need to check that. I mean is O(3) a normal subgroup of O(1,3) so that the quotient of cosets of O(3) in O(1,3) is a quotient group O(1,3)/O(3)?

Back to P&P:

"unless all sources of force are known, an inertial frame is not strictly definable" in Galilean relativity p. 273

The Maxwell free wave equation is not form-invariant under linear the Galilean relativity boosts.

Therefore, if Galilean relativity were true we would need a preferred frame of absolute rest in which Maxwell's wave equation has its canonical form e.g. in 1 space-dimension for simplicity

A,ct,ct - A,x,x = 0

A is the Maxwell vector potential (piece of U(1) internal local gauge compensation connection field in modern parlance)

Note that

x' = x - vt

1 = x,x' - vt,x'

A,x' = (A,x)(x,x') + (A,t)(t,x')

(t,x') = v^-1(x,x' - 1)

as v -> 0, x,x' -> 1 i.e. 0/0 indeterminate that we consistently define as 0 since v -> 0 is the identity map.

A,x' = (A,x)(x,x') + (A,t)v^-1(x,x' - 1)

Note that since the transformation is linear x,x',x' = 0 etc.

A,x',x' = (A,x,x)(x,x')^2 + (A,t,t)v^-2(x,x' - 1)^2

At't' = At,t

If we say the primed frame is the absolute rest "ether frame" with the canonical form for the wave equation, then

c^-2A,t',t' - A,x',x' --> c^-2A,t,t - (A,x,x)(x,x')^2 + (A,t,t)v^-2(x,x' - 1)^2

which is not form-invariant under these Galilean transformations.

That is, Maxwell's free radiation EM theory is not covariant (form-invariant) under the Galilean boost group in one fixed space direction.

Note, that this kind of Galilean ether frame of absolute rest is not at all what Cahill and the Catanians are talking about! They use the Lorentz group O(1,3) from the beginning. Their claim has nothing at all to do with 15.1 & 15.2 in P&P! It's important not to get confused about that!

The O(1,3) rest frame is a spontaneous broken vacuum symmetry in which the field equations remain O(1,3) covariant! The broken symmetry is in the solution not in the dynamics!

When P&P write, on the basis of the Galilean transformations in eqs. (15-6) to (15-10), not the Lorentz transformations that Cahill uses, for an individual run, rotating the Michelson interferometer by 90 degrees then and there, not averaging over an ensemble of such runs when Earth is in widely different parts of its elliptical orbit round the Sun: "On rotating the apparatus through 90 degrees, we should expect the interference pattern to shift by N fringes where

N ~ [(L1 + L2)/(wavelength)](v/c)^2 (15-10) p. 276

where v is the absolute speed of the interferometer here with respect to the above hypothetical Galilean ether.

P&P's (15-10) is not same as Cahill's (2) on p.4 derived using the Lorentz group and assuming gas index of refraction of light along the paths n =/= 1 i.e. Cahill's n(n^2 - 1) factor not at all in P&P's analysis. Therefore, P&P's Table 15-1 "Trials of the Michelson-Morley Experiment" and P&P's rejection of Miller's claims have no direct bearing on what Cahill is claiming! It's Apples and Oranges.

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