Abstract
Gravitational waves cannot escape a black hole, which acts like a Faraday cage for both electromagnetic and gravitational waves. Thus, when a massive object is accreted by a black hole, its gravitomagnetic field ceases to be detectable outside the event horizon. This, in turn, gives rise to an accretion shock, a step-function shock wave propagating outward from the accretion event as the gravitomagntic field disappears. Unlike the gravitational waves detected by LIGO, it is the gravitomagnetic component of the accretion shock which is linearly polarised. It affects the angular momentum of any matter it encounters, redistributing angular momentum otherwise lost to the black hole. A background radiation of black hole accretion shocks should be detectable with an array of ring laser gyros.
Introduction
The Lense-Thirring Effect whereby a gyroscope precesses near a rotating massive object has been known since the early days of Relativity . It is due to the gravitomagnetic field generated by the rotating object and is the gravitational analogue to the magnetic field. Gyroscope precession is the gravitational analogue of Larmor Precession. The idea of a gravitational analogue to Maxwell’s electromagnetic field equations was first proposed well before General Relativity. Its existence wa recently confirmed by Gravity Probe B .
The so called GEM (for “gravitoelectromagnetic”) field equations relate g , the acceleration due to gravity (corresponding to the electrostatic field) and the gravitomagnetic field, Ω , (corresponding to the magnetic field) , to the mass density (corresponding to the charge density) and to the mass current density (corresponding to the charge current density). Constants include the gravitational permittivity, ϵg = − 1/4πG, (where G is the gravitational constant), which is negative, reflecting the fact that masses attract one another whereas like electric charges repel one another.
Just as Maxwell’s equations lead directly to electromagnetic waves without the need for a propagating medium, so too do the GEM equations. The speed of gravitational waves can be derived from the GEM equations , just as the speed of light can be derived from Maxwell’s equations. It was recently demonstrated , that gravitational waves travel with the speed of light. It follows that, in empty space, GEM waves are propagated in precisely the same way and at the same speed as EM waves.
The generation of shocks by black holes
The observed acceleration of particles within the polar jets of black holes has been attributed to EM shocks. Given the homomorphism between the EM and GEM equations it seems likely that GEM shocks also play a role.
While the tensor calculus of General Relativity provided a powerful tool for investigating the properties of the space-time continuum, it has been strangely recalcitrant regarding the formation of shocks and other turbulent phenomena. Turbulence associated with a continuum, such as vortex streets and breaking waves in a liquid, originates at its boundaries. It may be productive to allow for similar, turbulence-generating boundaries of the space-time continuum. One such boundary is the event horizon of a black hole. This assumption puts such turbulence beyond the reach of the tensor calculus of General Relativity, just as the turbulence of breaking surface gravity waves lies beyond the reach of the Navier-Stokes equations of fluid dynamics.
It does not, however, put it beyond the reach of physics. While relativistic methods may not be sufficiently robust to deal with turbulence, a combination of physical intuition and Maxwellian field theory may prove more useful. The event horizon of a black hole has the properties of a one-way, Faraday cage in that Maxwellian fields emanating from physical events in the interior are undetectable outside the cage. Thus when a tidally disrupted star is accreted by a black hole, the EM and GEM fields of the newly accreted material cannot penetrate the event horizon and must vanish from the observable universe. The EM and GEM fields which existed prior to accretion must therefore collapse. Such a collapse takes the form of a shock wave which originates at the event horizon and propagates outwards with the speed of light.
Note that Green’s Theorem implies that, unlike the magnetic and gravitomagnetic fields, the electrostatic and gravitational far-fields of the system are unaffected by accretion, since the total charge and mass inside a spherical envelope remain constant during the accretion process.
The decay function, ∂J/∂t , describes the rate at which mass current density is removed from the accretion disc when matter falls beneath the event horizon. It is as if a virtual ring current of similar magnitude and opposite direction were instantaneously created immediately outside the event horizon. Spatially, to first order, the field change ΔΩ has the same form of the magnetic field generated by a wire loop and given by the Biot-Savart Law, i.e. the field of a magnetic dipole. However it also resembles a current pulse in an annular antenna, the far field of which is determined by the rules of Fraunhoffer diffraction and approximated by the Fourier Transform of the ring current “antenna” shape. Such an antenna has two opposed, narrow lobes normal to its plane. This may account for the creation and excitation of the relativistic jets emanating from black holes normal to the plane of the accretion disc.
The ΔΩ shock persists for the time it takes for the ring current, J, to fall through the event horizon and disappear. The gravitomagnetic shock so generated propagates outward from the black hole. When it encounters a fluid such as interstellar gas or plasma, matter will be accelerated, i.e. the shock loses energy by increasing the vorticity or angular momentum of any material it encounters. This has the effect of carrying angular momentum far afield. Such a mechanism may play a role in galactic evolution by redistributing angular momentum outwards from the galactic centre; dark matter may be unneccessary.
Since the advent of gravitational wave detectors such as LIGO we are accustomed to regarding gravitational waves as deterministic, coherent and linearly polarized g-waves, as indeed they are when generated as gravitational radiation by celestial objects rapidly orbiting a common centre of mass. On the other hand, the ubiquity of accretion shocks in a galaxy must also lead to an incoherent gravitational background radiation taking the form of a series of random uncorrelated impulses or shot noise. The magnitude of each impulse is a delta function convoluted with the decay function discuused above. Furthermore, it is not the g-wave which is linearly polarized but the Ω-wave, the gravitomagnetic component, because the Ω-wave is forced by ∇ × J, which is perpendicular to the plane of the accreted ring current.
Experiment and Observation
Dimensional analysis reveals that Ω has units of T − 1 and is angular frequency with which a gyroscope or similar inertial sensor is observed to rotate relative to the fixed stars. Gravity Probe B is an example. Likewise ∂Ω/∂t is the change in angular frequency gravitomagnetically induced in a rotating body by a changing gravitomagnetic field.
Recent observation of small variations in pulsar frequencies are cited as evidence of the existence of a gravitational wave background and a number of mechanisms were proposed by them, for example, changing propagation times between pulsar and observer. Real changes in pulsar spin rates induced by accretion shocks are another possibility.
High precision, terrestrial observations of the Lense-Thirring Effect using ring laser gyroscopes are being undertaken in Italy under the GINGER project (for “Gyroscopes in General Relativity”). GINGERINO, a square ring laser prototype, with high sensitivity in the frequency band of fractions of a Herz, has been in operation since 2017 . When fully operational, the project will have sufficient resolution to resolve the Lense-Thirring Effect of the earth’s rotation at the earth’s surface, but, to date, long term variations have precluded this. These are attributed to temperature dependent disturbances coming from the laser .
This low frequency noise in the GINGERINO data may not be instrumental in nature. It could, for example, be due to the accretion shocks discussed here. A simple experiment could resolve the two sources, viz.: by comparing the output of a number of independent ring laser gyros (RLGs), each driven by its own laser. Low frequency noise due to accretion shocks would be identical for each RLG whereas instrumental noise in each RLG would be statistically independent of the others. By averaging the output from a number of identically oriented RLGs, instrumental noise would decrease inversely with the number of detectors while the Lense-Thirring and accretion shock signals would remain undiminished. RLGs are a commercially available navigational tool. A number of off-the-shelf RLGs may turn out to be no more expensive than the high precision RLGs currently used by GINGERINO and would have much better low frequency noise characteristics.
Journal Version
A .pdf file of the relevant paper, suitable for peer-review and which includes equations and bibliography is available on-line and may be downloaded from: Reid24.
The paper was rejected by the journal editor and is published here for the first time. It was not peer reviewed.