Sagnac

In the case of a space separation of the laser beams (i.e. if the atomic velocity is perpendicular to the direction of the laser beams), the interferometer is in a Mach-Zehnder configuration. Then, the interferometer is also sensitive to rotations, as in the Sagnac geometry (Sagnac, 1913) for light interferometers. For a Sagnac loop enclosing area A, a rotation Q, produces a phase shift ... 

Figure 2. Diagram of the atomic Sagnac interferometer at Yale (Gustavson et al., 2000). Individual signals from the outputs of the two interferometers (gray lines), and difference of the two signals corresponding to a pure rotation signal (black line) versus rotation rate.

The high sensitivity of atomic Sagnac interferometers to rotation rates will enable HYPER to measure the modulation of the precession due to the Lense-Thirring effect while the satellite orbits around the Earth. In a Sun-synchronous, circular orbit at 700 km altitude, HYPER will detect how the direction of the Earth s drag varies over the course of the near-polar orbit as a function of the latitudinal position 9 ... 

Cotton, and Sagnac to the little Curie home, which was thus transformed into an intimate academy. There the master, with his customary simplicity, explained his ideas to us and deigned to discuss ours.. . . 74). 

In a rotating interferometer, fringe shifts have been observed between light beams that propagate parallel and antiparallel with the direction of rotation [4]. This Sagnac effect requires an unconventional explanation. 

The group space of 0(3) is doubly connected (i.e., non-simply connected) and can therefore support an Aharonov-Bohm effect (Section V), which is described by a physical inhomogeneous term produced by a rotation in the internal gauge space of 0(3) [24]. The existence of the Aharonov-Bohm effect is therefore clear evidence for an extended electrodynamics such as 0(3) electrodynamics, as argued already. A great deal more evidence is reviewed in this article in favor of 0(3) over U(l). For example, it is shown that the Sagnac effect [25] can be described accurately with 0(3), while U(l) fails completely to describe it. 

The Sagnac effect cannot be described by U(l) electrodynamics [4,43] because of the invariance of the U(l) phase factor under motion reversal symmetry (T) ... 

The explanation of interferometric effects in U(l) electrodynamics is in general self-inconsistent, and sometimes, as in the Sagnac effect, nonexistent. In this... 

In U(l) electrodynamics in free space, there are only transverse components of the vector potential, so the integral (158) vanishes. It follows that the area integral in Eq. (157) also vanishes, and so the U(l) phase factor cannot be used to describe interferometry. For example, it cannot be used to describe the Sagnac effect. The latter result is consistent with the fact that the Maxwell-Heaviside and d Alembert equations are invariant under T, which generates the clockwise... 

C) Sagnac loop from the counterclockwise (A) loop [17]. It follows that the phase difference observed with platform at rest in the Sagnac effect [47] cannot be described by U(l) electrodynamics. This result is also consistent with the fact that the traditional phase of U(l) electrodynamics is invariant under T as discussed already in Section (III). The same result applies for the Michelson-Gale experiment [48], which is a Sagnac effect. 

The Sagnac effect with a platform at rest [47] is explained as the phase factor ... 

This is an expression for the observed phase difference with the platform at rest in the Sagnac experiment [47] it is a rotation in the internal gauge space. In U(l) electrodynamics, there is no phase difference when the platform is at rest, as discussed already. 

When the platform is rotated in the Sagnac effect, there is an additional rotation in the internal gauge space described by... 

This result is true for all matter waves and also in the Michelson-Gale experiment, where it has been measured to a precision of one part in 1023 [49]. Hasselbach et al. [51] have demonstrated it in electron waves. We have therefore shown that the electrodynamic and kinematic explanation of the Sagnac effect gives the same result in a structured vacuum described by 0(3) gauge group symmetry. 

The Maxwell-Heaviside theory of electrodynamics has no explanation for the Sagnac effect [4] because its phase is invariant under 7 as argued already, and because the equations are invariant to rotation in the vacuum. The d Alembert wave equation of U(l) electrodynamics is also 7 -invariant. One of the most telling pieces of evidence against the validity of the U(l) electrodynamics was given experimentally by Pegram [54] who discovered a little known [4] cross-relation between magnetic and electric fields in the vacuum that is denied by Lorentz transformation. 

It can be shown straightforwardly, as follows, that there is no holonomy difference if the phase factor (154) is applied to the problem of the Sagnac effect with U(l) covariant derivatives. In other words, the Dirac phase factor [4] of U(l) electrodynamics does not describe the Sagnac effect. For C and A loops, consider the boundary... 

The Sagnac effect caused by the rotating platform is therefore due to a rotation in the internal gauge space ((1),(2),(3)), which results in the frequency shift in Eq. (171). The frequency shift is experimentally the same to an observer on and off the platform and is independent of the shape of the area Ar. The holonomy difference (172) derived theoretically depends only on the magnitudes and ff, and these scalars are frame-invariant, as observed experimentally. There is no shape specified for the area Ar in the theory, and only its scalar magnitude enters into Eq. (172), again in agreement with experiment. 

The Lehnert equations are a great improvement over the Maxwell-Heaviside equations [45,49] but are unable to describe phenomena such as the Sagnac effect and interferometry [42], for which an 0(3) internal gauge space symmetry is needed. 

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