A Black Hole is a Faraday Cage

John Reid

Back in 2017,  Geoff Hudson posted A Challenge to Dark Matter in which he highlighted some of the the shortcomings of the Dark Matter proposition in accounting for the form and evolution of galaxies. He pondered where the angular momentum goes when black holes collide and introduced us to the idea of the gravitomagnetic field.

The idea of gravitoelectromagnetism or GEM is that there is an almost exact correspondence between the field equations of electricity and magnetism and those of gravity. Just as the electrostatic force between two charged bodies is proportional to the product of their charges and inversely proportional to the square of the distance between them (Coulomb’s Law), the gravitational force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them (Newton’s Law). Almost, because while two like charges repel one another, two masses attract one another leading to sign changes in the relevant equations.

But the really neat idea is that there is a second force called gravitomagnetism which corresponds to magnetism. Just as a magnetic field arises when charges move in a circuit as an electric current, a gravitomagnetic field arises when masses rotate. The effect is very small because the square of the velocity of light occurs in the denominator. The gravitomagnetic field causes a gyroscope to precess (corresponding to Larmor Precession).  The minute gravitomagnetic field of the rotating earth was measured by six gyroscopes on board Gravity Probe B in 2011 and found to be -37.2 +/- 7.2 milliarcseconds per year, within 5% of the predicted value.

Just as an electric current or a changing electrical field gives rise to magnetism, a changing magnetic field gives rise to an electromotive force; a process known as electromagnetic  induction. Hence gravitomagnetic induction also exists whereby a body experiences a force due to a changing gravitomagnetic field. The problem is that a gravitomagnetic field can only change if the angular momentum of the rotating bodies which gave rise to it changes. Angular momentum is a conserved quantity and can only change locally if there is a corresponding change in the opposite sense in some other locality. This does not augur well for gravitomagnetic induction which is  small because of the conservation of angular momentum.

Or is it?

Do black holes have angular momentum? Yes, they do: The Kerr metric is a generalisation to a rotating body of the Schwarzschild metric, discovered by Karl Schwarzschild in 1915, which described the geometry of spacetime around an uncharged, spherically-symmetric, and non-rotating body.  The exact solution for an uncharged, rotating black-hole, the Kerr metric, remained unsolved until 1963, when it was discovered by Roy Kerr.

Can the angular momentum of a black hole change? It is commonly assumed that it can and that it monotonically increases as the black hole accretes matter and so accretes mass and angular momentum.

Can the changing angular momentum of a massive object be observed remotely? Every time a body accretes angular momentum the surrounding gravitomagnetic field will change proportionately. However this does not happen instantaneously; a gravitational wave front carrying the information about the new distribution of angular momentum will be propagated outward at the speed of light.

Can the changing angular momentum of a black hole be observed? In the case of a black hole this outward propagating wave cannot get past the event horizon. The new gravitomagnetic field must remain trapped inside the event horizon. The event horizon acts as a perfect Faraday Cage from which no radiation can escape, not even gravitational waves.

The net effect is that a black hole removes angular momentum from the rest of the universe and traps it inside the event horizon.

Outside the event horizon angular momentum is no longer conserved.

Black holes eat angular momentum.

This has significant implications for cosmology, particularly for galactic evolution. It is likely that massive or supermassive black holes are present at the centre of every galaxy. As the black hole accretes matter it also removes angular momentum. Removal of angular momentum from near the centre will give rise to a changing gravitomagnetic field exerting forces on material elsewhere in the galaxy. This gravitomagnetic induction may have a similar effect to the putative dark matter. Furthermore it can be modelled based on reasonable assumptions about galactic black hole accretion rates without requiring the ad hoc assumption of dark matter.

Viewed in this way, a galaxy is the gravitomagnetic equivalent of a transformer. A changing “primary current circuit ” of matter rotating around the central black hole gives rise to a “secondary current circuit” elsewhere in the galaxy. To pursue the analogy, the changing secondary current circuit should cause a “back EMF” to be set up in the primary circuit. This analogy is very simplistic. Extensive modelling needs to be done to explore the myriad possible interactions of this “gravitomagnetic plasma”. It may well be that explanations of other features will come out of such modelling: the tendency to form spiral arms, the formation of accretions discs, the dynamics of barred galaxies and so forth.

Gravitomagnetic induction brings about an increase in angular momentum in the secondary circuit “out of nowhere”. Could it be that this increase precisely compensates the angular momentum lost to the black hole?

Maxwell’s equations and the gravito-electromagnetic or “GEM” equations are compared in the following table:

Comparison of GEM waves and gamma rays from the same source by the LIGO team showed that the two speeds of propagation are equal.