Strange and subtle gravitational effects

You may think you understand gravitation, at least conceptually, but to encourage you to check our other posts explaining different aspects of gravity, let us show you the strangest effects of this interaction.

Beginning with gravitoelectromagnetism, also known as linearized gravity, if two wheels spin on a common axis, their mutual gravitational attraction will be greater if they spin in opposite directions than in the same direction. In gravitoelectromagnetism, mass currents of moving or rotating masses generate a gravitomagnetic field, analogous to the magnetic field from electric currents, but the sign of the force law differs from electromagnetism. Thus, parallel mass currents repel gravitomagnetically, while antiparallel mass currents attract gravitomagnetically. For counter-rotating wheels, the gravitomagnetic interaction is attractive, slightly increasing the net gravitational attraction. In General Relativity, this effect is better described in terms of frame dragging.

Another consequence of gravitoelectromagnetism is that, for an object constrained in an equatorial Earth’s circular path but not in freefall, it weighs more if orbiting anti-spinward, and less if orbiting spinward. The spinning of Earth generates itself a gravitomagnetic field, and test masses moving through that field experience a velocity-dependent force analogous to the magnetic Lorentz force. Motion with the spin is assisted, while against the spin is resisted. With an anti-spinward motion, the gravitomagnetic force acts downward, adding to that of gravity. In General Relativity, this is explained again by frame-dragging, but frame dragging will neither accelerate nor slow down the object in either direction, it is not a “viscosity”. Consequently, a stationary plumb-bob suspended over the rotating Earth will hang vertically and will not lean to one side. But if it starts to fall, induction will push it in the spinward direction.

In principle, gravitoelectromagnetic drag forces can accelerate an object without the object experiencing any stresses or g-forces that could break it apart. An example is a flexible or fluid toroidal mass undergoing minor axis rotational acceleration, that is, rotating on itself like a “smoke ring”. It will tend to pull matter through the throat due to rotational frame dragging. The absence of stresses happens because every component of the object is affected equally by the gravitational drag force and as a whole, and the object does not “feel” any force.

Another gravitoelectromagnetic effect is called the Mashhoon–Cohen–Stodolsky clock effect, by which a clock co-rotating with a spinning mass and another clock counter-rotating, both at the same radius, will accumulate different proper times and tick differently. This time delay is also a consequence of frame dragging in General Relativity.

Moving to General Relativity, there are also counter-intuitive effects. One of them is the not yet confirmed prediction of gravitational memory: test masses can remain permanently displaced after the passing of a gravitational wave. One would expect that after a gravitational wave passes through a system of masses, their relative positions do not change, but General Relativity predicts otherwise, even in its linear approximation (not gravitoelectromagnetism).

But there is one effect that Einstein considered necessary in a complete theory of gravity, which is not satisfied by General Relativity or gravitoelectromagnetism. In 1912, Einstein published a short note describing how his relativistic theory of gravity, still under development, already hinted towards a fulfilment of Ernst Mach’s ideas about the origin of inertia. We explained Mach’s principle in another post. Einstein’s proposal citing Mach was essentially that the inertia of a body increases when masses are present in its neighborhood. These masses were idealized by Einstein as a massive shell, and he asked whether a force acts on the central body if an acceleration is imparted on the shell. Such an apparent positive result was carried on after his General Theory of Relativity, and included as the first of three consequences Einstein expected from Mach’s principle and in his opinion, were fulfilled by his theory: 1. The inertia of a body must increase when ponderable masses are piled up in its neighborhood. 2. A body must experience an accelerating force when neighboring masses are accelerated, and, the force must be in the same direction as the acceleration. 3. A rotating hollow body must generate inside of itself a “Coriolis field” which deflects moving bodies in the sense of the rotation, and a radial centrifugal field as well. The last two were indeed fulfilled by General Relativity by frame dragging, one of them being the Lens-Thirring effect. But Carl Brans showed in 1961 that the first one wasn’t, and that it was a mere coordinate effect.

We now know that galaxy dynamics are in conflict with General Relativity if dark matter doesn’t exist. Given the patterns and regularities found in galaxy rotation curves, which seem to arise from a modified law of gravity rather than invisible mass distribution, Modified Newtonian Dynamics was developed based on these regularities. We explained MOND in our previous post, and how it relates to Mach’s principle as a modified inertia theory in which, at low accelerations, the inertia of a body is decreased. But low accelerations can be trivially related to being far away from other masses, and it looks like MOND could have an origin in Mach’s consequence of “The inertia of a body must decrease when ponderable masses are removed from its neighborhood”. Another possibility is that MOND has an origin in an Einstein’s more complete statement about such a consequence of Mach’s principle: “It must be demanded that the inertial resistance of a body could be increased by having unaccelerated inertial masses arranged in its vicinity; and this increase of the inertial resistance must disappear again if these masses accelerate along with the body”. If one considers an expanding universe, most of the matter in the observable universe is accelerating away from a stationary body at approximately c*H0 (10^-10 m/s^2). If the body happens to have an acceleration of this magnitude, he won’t be accelerating with respect to some part of the universe, and following Mach, its inertia should be decreased.