Gravity Probe B and TGD

Gravity Probe B experiment tests the predictions of General Relativity related to gravimagnetism. Only the abstract of the talk C. W. Francis Everitt summarizing the results is available when I am writing this. Here is a slightly reformatted abstract of the talk.

The NASA Gravity Probe B (GP-B) orbiting gyroscope test of General Relativity, launched from Vandenberg Air Force Base on 20 April, 2004, tests two consequences of Einstein's theory:

1. the predicted 6.6 arc-s/year geodetic effect due to the motion of the gyroscope through the curved space-time around the Earth;

2. the predicted 0.041 arc-s/year frame-dragging effect due to the rotating Earth.

The mission has required the development of cryogenic gyroscopes with drift-rates 7 orders of magnitude better than the best inertial navigation gyroscopes. These and other essential technologies, for an instrument which once launched must work perfectly, have come into being as the result of an intensive collaboration between Stanford physicists and engineers, NASA and industry. GP-B entered its science phase on August 27, 2004 and completed data collection on September 29, 2005. Analysis of the data has been in continuing progress during and since the mission. This paper will describe the main features and challenges of the experiment and announce the first results.

The Confrontation between General Relativity and Experiment gives an excellent summary of various test of GRT. The predictions tested by GP-B relate to gravitomagnetic effects. The field equations of general relativity in post-Newtonian approximation with a choice of a preferred frame can in good approximation (g_ij=-δ_ij) be written in a form highly reminiscent of Maxwell's equestions with g_tt component of metric defining the counterpart of the scalar potential giving rise to gravito-electric field and g_ti the counterpart of vector potential giving rise to the gravitomagnetic field.

Rotating body generates a gravitomagnetic field so that bodies moving in the gravitomagnetic field of a rotating body experience the analog of Lorentz force and gyroscope suffers a precession similar to that suffered by a magnetic dipole in magnetic field (Thirring-Lense efffect or frame-drag). Besides this there is geodetic precession due to the motion of the gyroscope in the gravito-electric field present even in the case of non-rotating source which might be perhaps understood in terms of gravito-Faraday law. Both these effects are tested by GP-B.

In the following something general about how TGD and GRT differs and also something about the predictions of TGD concerning GP-B experiment.

1. TGD and GRT

Consider first basic differences between TGD and GRT.

1. In TGD local Lorentz invariance is replaced by exact Poincare invariance at the level of the imbedding space H= M⁴× CP₂. Hence one can use unique global Minkowski coordinates for the space-time sheets and gets rids of the problems related to the physical identification of the preferred coordinate system.

2. General coordinate invariance holds true in both TGD and GRT.

3. The basic difference between GRT and TGD is that in TGD framework gravitational field is induced from the metric of the imbedding space. One important cosmological implication is that the imbeddings of the Robertson-Walker metric for which the gravitational mass density is critical or overcritical fail after some value of cosmic time. Also classical gauge potentials are induced from the spinor connection of H so that the geometrization applies to all classical fields. Very strong constraints between fundamental interactions at the classical level are implied since CP₂ are the fundamental dynamical variables at the level of macroscopic space-time.

4. Equivalence Principle holds in TGD only in a weak form in the sense that gravitational energy momentum currents (rather than tensor) are not identical with inertial energy momentum currents. Inertial four-momentum currents are conserved but not gravitational ones. This explains the non-conservation of gravitational mass in cosmological time scales. At the more fundamental parton level (light-like 3-surfaces to which an almost-topological QFT is assigned) inertial four-momentum can be regarded as the time-average of the non-conserved gravitational four-momentum so that equivalence principle would hold in average sense. The non-conservation of gravitational four-momentum relates very closely to particle massivation.

There are excellent reasons to expect that Maxwellian picture holds true in a good accuracy if one uses Minkowski coordinates for the space-time surface. In fact, TGD allows a static solutions with 2-D CP₂ projection for which the prerequisites of the Maxwellian interpretation are satisfied (the deviations of the spatial components g_ij of the induced metric from -δ_ij are negligible).

Schwartschild and Reissner-Norströom metric allow imbeddings as 4-D surfaces in H but Kerr metric assigned to rotating systems probably not. If this is indeed the case, the gravimagnetic field of a rotating object in TGD Universe cannot be identical with the exact prediction of GRT but could be so in the Post-Newtonian approximation. Scalar and vector potential correspond to four field quantities and the number of CP₂ coordinates is four. Imbedding as vacuum extremals with 2-D CP₂ projection guarantees automatically the consistency with the field equations but requires the orthogonality of gravito-electric and -magnetic fields. This holds true in post-Newtonian approximation in the situation considered.

This raises the possibility that apart from restrictions caused by the failure of the global imbedding at short distances one can imbed Post-Newtonian approximations into H in the approximation g_ij=-δ_ij. If so, the predictions for Thirring-Lense effect would not differ measurably from those of GRT. The predictions for the geodesic precession involving only scalar potential would be identical in any case.

The imbeddability in the post-Newtonian approximation is however questionable if one assumes vacuum extremal property and small deformations of Schwartschild metric indeed predict a gravitomagnetic field differing from the dipole form.

3. Simplest candidate for the metric of a rotating star

The simplest situation for the metric of rotating start is obtained by assuming that vacuum extremal imbeddable to M⁴ × S², where S² is the geodesic sphere of CP₂ with vanishing homological charge and induce Kähler form. Use coordinates Θ,Φ for S² and spherical coordinates (t,r,θ,φ) in space-time identifiable as M⁴ spherical coordinates apart from scaling and r-dependent shift in the time coordinate.

1. For Schartschild metric one has Φ= ωt

   and

   u= sin(Θ)= f(r),

   f is fixed highly uniquely by the imbedding of Schwartschild metric and asymptotically one must have

   u =u₀ + C/r

   in order to obtain g_tt= 1-2GM/r (=1+Φ_gr) behavior for the induced metric.

2. The small deformation giving rise to the gravitomagnetic field and metric of rotating star is given by

   Φ = ωt+nφ

   There is obvious analogy with the phase of Schödinger amplitude for angular momentum eigenstate with L_z=n which conforms with the quantum classical correspondence.

3. The non-vanishing component of A^g is proportional to gravitational potential Φ_gr

   A^g_φ= g_tφ = (n/ω)Φ_gr.

4. A little calculation gives for the magnitude of B_g^θ from the curl of A^g the expression

   B_g^θ= (n/ω)× (1/sin(θ)× 2GM/r³.

   In the plane θ=π/2 one has dipole field and the value of n is fixed by the value of angular momentum of star.

5. Quantization of angular momentum is obtained for a given value of ω. This becomes clear by comparing the field with dipole field in θ= π/2 plane. Note that GJ, where J is angular momentum, takes the role of magnetic moment in B_g (see this). ω appears as a free parameter analogous to energy in the imbedding and means that the unit of angular momentum varies. In TGD framework this could be interpreted in terms of dynamical Planck constant having in the most general case any rational value but having a spectrum of number theoretically preferred values. Dark matter is interpreted as phases with large value of Planck constant which means possibility of macroscopic quantum coherence even in astrophysical length scales. Dark matter would induce quantum like effects on visible matter. For instance, the periodicity of small n states might be visible as patterns of visible matter with discrete rotational symmetry (could this relate to strange goings on in Saturn?).

4. Comparison with the dipole field

The simplest candidate for the gravitomagnetic field differs in many respects from a dipole field.

1. Gravitomagnetic field has 1/r³ dependence so that the distance dependence is same as in GRT.

2. Gravitomagnetic flux flows along z-axis in opposite directions at different sides of z=0 plane and emanates from z-axis radially and flows along spherical surface. Hence the radial component of B_g would vanish whereas for the dipole field it would be proportional to cos(θ).

3. The dependence on the angle θ of spherical coordinates is 1/sin(θ) (this conforms with the radial flux from z-axis whereas for the dipole field the magnitude of B^θ_g has the dependence sin(θ). At z=0 plane the magnitude and direction coincide with those of the dipole field so that satellites moving at the gravitomagnetic equator would not distinguish between GRT and TGD since also the radial component of B_g vanishes here.

4. For other orbits effects would be non-trivial and in the vicinity of the flux tube formally arbitrarily large effects are predicted because of 1/sin(θ) behavior whereas GRT predicts sin(θ) behavior. Therefore TGD could be tested using satellites near gravito-magnetic North pole.

5. The strong gravimagnetic field near poles causes gravi-magnetic Lorentz force and could be responsible for the formation of jets emanating from black hole like structures and for galactic jets. This additional force might have also played some role in the formation of planetary systems and the plane in which planets move might correspond to the plane θ=π/2, where gravimagnetic force has no component orthogonal to the plane. Same applies in the case of galaxies.

5. Consistency with the model for the asymptotic state of star

In TGD framework natural candidates for the asymptotic states of the star are solutions of field equations for which gravitational four-momentum is locally conserved. Vacuum extremals must therefore satisfy the field equations resulting from the variation of Einstein's action (possibly with cosmological constant) with respect to the induced metric. Quite remarkably, the solution representing asymptotic state of the star is necessarily rotating (see this).

The proposed picture is consistent with the model of the asymptotic state of star. Also the magnetic parts of ordinary gauge fields have essentially similar behavior. This is actually obvious since CP₂ coordinates are fundamental dynamical variables and the field line topologies of induced gauge fields and induced metric are therefore very closely related.

As already discussed, the physicists M. Tajmar and C. J. Matos and their collaborators working in ESA (European Satellite Agency) have made an amazing claim of having detected strong gravimagnetism with gravimagnetic field having a magnitude which is about 20 orders of magnitude higher than predicted by General Relativity. Hence there are some reasons to think that gravimagnetic fields might have a surprise in store.

Addition: Lubos Motl's blog tells that the error bars are still twice the size of the predicted frame-dragging effect. Already this information would have killed TGD inspired (strongly so) model unless the satellite had been at equator!

For details and background see the chapter TGD and GRT.