Hyperbolic anisotropy of vacuum
- From: "Just G. Waller" <wallermax@xxxxxxxxxxx>
- Date: Wed, 14 Feb 2007 21:58:42 +0000 (UTC)
A photon is emitted from source A to receiver B, in perfect vacuum,
with a constant distance r between them, and without gravitational
influences, so they are regarded as inertial systems. It is assumed
that the emitted photon will follow all posible paths from A to B,
even those that aren't straight lines. The measurement of that
photon in B is related, according to Quantum Electrodynamics, to the
resultant sum of probability amplitude contributions of all the
possible paths that photon could have been followed. From this
assumption we can see that the paths which take less time are those
which contribute the most to the final probability amplitude. But,
this is only an interpretation, and although it is a successful one,
others are also possible.
Let us replace vacuum with a different medium, between A and B. Now,
it is easier to understand why a large path means larger spent time.
The photon must intersect a larger amount of atoms in that path, and
it must be retransmitted by means of spontaneous emission in each
atom
step. We can easily express the extra distance dr = r' - r, for a
path
r' as function of a time delay dt, by
dr = r(Exp(dt/t_m) - 1),
where t_m is a timelife, characteristic of the transition
for that medium, and r is the shortest path between A and B.
Noting that dt is a function (statistics) of the number of
atoms
of the medium involved in the retransmission of that
photon, by means of the spontaneous emission mechanism.
So, we actually get
r' = r Exp(dt/t_m)
If the medium is now the vacuum, and there is a speed v between A and
B
then, we state the identity
dt/t_m = v/c,
so the shortest path r now becomes r' = r Exp(v/c),
which means it becomes anisotropic for a
non zero v. With the signature convention, v < 0 if
A and B are approaching, and v > 0 if they are moving
away.
If we want to know the time t_a a photon will take to propagate
through path r in the vaccum (in proper time of A), we get
t_a = r'/c = (r/c)Exp(v/c)
But, it results that, from the proper time of B, that photon emitted
by A towards B will take
t_b = (r/c)Exp(-v/c)
A round trip (a photon emitted by A, reflected off B and
returned to A) will take a delay dT, in proper time of A,
dT = t_a + t_b - 2r/c,
dT = (2r/c)(cosh(v/c) - 1)
Once we have fixed the hyperbolic anisotropy of vacuum, for a non-
zero
speed v between source and receiver, we can easily predict several
known
effects. For example,
Sagnac effect:
In a closed circular loop of length r, rotating at tangent
speed v, we would get a delay dT,
dT = t_a - t_b ,
where t_a is total time for photons traveling
in the direction of rotation, and t_b is that
in the opposite direction.
Therefore, the hyperbolic anisotropic interpretation
predicts a delay of
dT = (2r/c)sinh(v/c)
Compared with the standard equation for a Sganac effect delay, dT_s,
in the same Sagnac interferometer,
dT_s = 2rv/(c^2 - v^2),
we see that they match for a first order approximation. For
non-relativistic speed v<<c, it becomes
dT_s = 2rv/c^2
The Taylor expansion series for sinh(v/c) is
sinh(v/c) = v/c + (v/c)^3/6 + (v/c)^5/120 + ... +
(v/c)^(2n+1)/(2n+1)! + ...
So, the first order approximation yields
dT = (2r/c)(v/c) = 2rv/c^2
dT = dT_s
Noting that a second order approximation would involve
the term (v/c)^3/6 , which would be within the noise
measurement, for non-relatistic v. And, it is already
well known that a Sagnac effect is of first order of v/c,
for non-relativistic v.
.
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