Gravito-magnetic field distributions

Initially we assume a constant-speed spinning mass source for the gravitomagnetic field that is a ring (annulus) of finite radius a and infinitesimal thickness. The detector is assumed to be a fiber-optic gyroscope (FOG) oriented with axis of symmetry parallel with that of the source, but otherwise arbitrarily offset. In the event that the FOG is rotated at constant rate about its axis of symmetry, the gyroscope measures a phase-difference between two counter-propagating EM modes of the fiber that increases linearly with time, corresponding to the Sagnac effect. The FOGs will respond similarly in the presence of a sufficiently strong gravito-magnetic field, the latter derived from a linear approximation to a Lens-Thirring metric. Anthony Lasenby has confirmed that to good approximation the FOG phase-shift measures the gravito-magnetic flux threading the fiber. Accordingly, one can use the results of EM theory for a current ring to predict the gravito-magnetic field distribution and corresponding FOG response. 

In general, that theory gives no simple analytic expression for the field intensity. We can define far-field as the domain in which the dipole approximation for the source is trustworthy. Generally, we expect this to be the case when the angle subtended by the source at the detector, and, for good measure, the angle subtended by the detector at the source, are both small, though it is possible that these constraints require some refinement. In the far-field case one can give closed expressions and therefore relatively easily obtain plots of the field intensity.

In the MT setup and in the IASA replication however, we cannot use the far-field approximation; the dimensions of the fiber path (which, by the way, is not quite a circle) is of the same order as the radius of the spinning mass. And the distance between these two is relatively small. The following plots of gravitomagnetic flux and therefore FOG phase-shift-rate have been obtained numerically from the expression given here. The scaling for the flux is in arbitrary units - the MT results are alleged to be around 10¹⁸ greater than that predicted by GR for the source mass in question. All distances are normalized with respect to the radius of the spinning source mass. The path of light in the FOG detector has been assumed to be circular.

		In this plot the detector is in plane with the source but at arbitrary horizontal (radial) offset. The detector is exactly coincident with the source when the horizontal displacement is zero. This is the radius at which the signal is maximally positive. Actually, in this approximation, the signal at this point is theoretically infinite. Of course in practice the signal will be rendered finite due to the fact that the source is radially distributed (ie with non-zero thickness). (In the plot below the actual value at zero displacement has been numerically truncated.)  At displacement = 1 the center of the detector lies exactly on the path of the annulus of matter. At displacement = 2 the detector lies entirely outside the spinning matter, the two circles just grazing each other. Notice that this is the radius at which the signal is maximally negative. Notice also that the maximum magnitude of the flux reading outside the source is about an order of magnitude less than the flux when the displacement is very close (~ 0.1) to zero.
		This figure show the flux versus radial offset for various non-zero heights - normalized with respect to the source-mass radius. Notice how quickly the signal falls off with height above the source.
		This shows the fall-off with height for the perfectly concentric arrangement. I would expect that this result can be given in closed form.
		This shows the fall-off with height for various offset detector positions. rho = 2 is the maximum magnitude for the detector in-plane but exterior to the mass. At other heights the maximum occurs at a different radial offset.

Below is the 2D contour plot of flux threading the FOG. Click here to open an animated gif (4.63 MB) of this plot in a new window.

For anyone interested, here are the links to the Maple code used to generate these plots:

Possible experimental test

Using these field calculations it is possible to estimate the maximum magnitude signal external to the existing MT dewar as a ratio with the signal currently obtained inside his dewar. The former is obtained when the detector is in-plane with, exterior to, and otherwise and as close as possible to the spinning mass. Observation of the associated sign reversal is proposed by IASA/HSC as a significant test of the hypothesis that the 'MT effect' is frame-dragging-related - at least in so far as it confirms a minimal expectation of a gravitomagnetic dipole field distribution. A summary of the relevant geometry and consequent numerical predictions is here.

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