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Centrifugal Forces and Black Holes

by John G. Cramer

Alternate View Column AV-55
Keywords: gravitation centrifugal force inversion black hole lightlight orbit

Published in the November-1992 issue of Analog Science Fiction & Fact Magazine;
This column was written and submitted 4/7/92 and is copyrighted ©1992 by John G. Cramer.
All rights reserved. No part may be reproduced in any form without
the explicit permission of the author.

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This column is about forces and pseudo-forces. In particular, it is about the surprising thing that happens to centrifugal force in a very strong gravitational field, as recently discovered in the formalism of general relativity by M. A. Abramowicz and his co-workers. Everyone, of course, knows about centrifugal force. It's the force that pushes you away from the center of the turn when your car goes around a tight corner. If you tie a rock to a string and swing it in a circle, it's the force that pulls the string tight and slings the rock away if the string breaks. It's the force that pulls the water out of your laundry when the washing machine is on the spin cycle. And it is the force that balances the pull of gravity and holds the earth in its orbit around the sun.

What perhaps you did not know is that centrifugal force is, strictly speaking, not a force at all. It is a pseudo-force. It is, in a sense, an illusion produced by changing coordinate systems. This is a slippery idea which will require some explanation. So to understand why centrifugal force is not a true physical force, let's start by considering what the true forces of nature are.

There are four fundamental forces, the strong and weak interactions that govern the behavior of quarks, nuclei and radioactive decay; the electromagnetic force which governs light, electricity, and chemical reactions; and the gravitational force which keeps us stuck to the surface of our planet and governs the behavior of planets, stars, and galaxies. Of these forces, gravity is by far the weakest, and the strong interaction, as its name suggests, is the strongest.

Each of these fundamental forces has its own "mediating particle," a particle that is pitched back and forth in a quantum-level game of catch between interacting particles. For the strong interaction the mediating particle is one of the eight varieties of gluon; in the weak interaction it is one of the three weakons (the Zo, W+ and W-). The mediating particle of the electromagnetic interaction is the photon, the basic quantum of light discovered by Planck and Einstein. The mediating particle for gravity is the graviton, the basic quantum of the gravitational radiation (gravity waves). Gravity waves, however, have only been observed rather indirectly as they steal energy form a binary neutron star, and the graviton itself is a theoretical construct which may never be observed directly.

The four forces each have a characteristic distance or "range" over which they act. The weak force has the shortest range, about 10-18 meters (.0004 of the diameter of a proton). The strong interaction's range is about 10-15 meters. Both forces drop rapidly to zero when interacting particles are separated by a distance greater than the range of the force. Effectively, the strong and weak forces switch off outside their range.

Electromagnetism and gravity have an infinite range, and so they never switch off. Both forces fall off as 1/r2 with distance, diminishing with distance but never going to zero. Because gravity extends over large distances and because it accumulates in strength with contributions that always add, ultimately when enough gravitating matter is present the effects of gravitational attraction can overwhelm those of the stronger forces. The supernova and the neutron star are examples of systems dominated by gravity, in which the actions of the electromagnetic, strong, and weak forces are nullified or even made to work backwards under the influence of very strong gravity.

With this background, we can see why it is not correct to call centrifugal force a force. It has no mediating particle. It has no range. It has no place setting at nature's table of the forces because it is an illusion, a pseudo-force that is the result of inertia. Newton taught us that mass, once moving in some direction with some velocity, will continue to move in a straight line in the same direction with the same velocity unless some external force bends its path, speeds it up, or slows it down. This is inertia, the underlying principle that produces the centrifugal force effect.

If you are in a car that turns a corner, it is the action of inertia that causes your body to try to continue in a straight line. Forces exerted by the seat, the seat belt, the doors, etc. are required to push your body in the new direction. Thus you have the illusion, because your point of view in a reference frame that is being accelerated in a circle, that some external force is pushing you away from the center of the turn. There is no such force. What you actually experience is inertia trying to continue your motion in a straight line while the interior of the car around you is moving in the circular path of the turn.

It was the subtle connection between inertia, centrifugal force and gravity that Albert Einstein called the Principle of Equivalence and used to establish an entry point for developing the general theory of relativity, the present standard theory of gravity. Think of the equivalence principle as a TV quiz show. The contestant is anesthetized and placed in a sealed but well equipped and environmentally controlled box. When he regains consciousness, he is asked to determine on the basis of any measurements he can perform inside the box whether (a) he is at rest on the earth in a 1 g gravitational field, (b) in gravity-free space and being accelerated in a straight line with an acceleration of 1 g, or (c) moving at a constant and very high speed along a circular path of very large radius so that the centrifugal force is 1 g.

Einstein's working hypothesis, as embodied in the equivalence principle, is that the contestant's task in this game would be hopeless. There is no possible physical measurement that he could perform within the box that would distinguish between these three conditions. The effects of gravity, as they are experienced at rest on the earth (condition a) are indistinguishable from the effects of inertia as experienced in linear acceleration (condition b) or in circular motion of sufficiently large radius (condition c). This, Einstein said, is because mass distorts the geometry of space itself to produce gravity, which at its root is just another manifestation of inertia. Gravity and inertia are the same.

Now we will consider what happens to centrifugal force in a strong gravitational field. The centrifugal force is present when a mass moves at a uniform speed in a circle, but it is absent when the same mass moves at the same speed in a straight line. But what, in a universe where space itself can be curved, do we mean by a straight line? Einstein's answer was to use a beam of light as the surveying instrument that determined which lines are straight lines: whatever path light takes is what we mean by a straight line.

In gravity-free space this is obvious. Near a massive star or a black hole, where light itself can be strongly bent or even trapped by gravity, it is not. With G as Newton's gravitational constant (the strength of the gravitational force), a black hole of mass m has a characteristic radius rs = 2Gm/c2 called the Schwarzschild radius at which the escape velocity equals c, the velocity of light. This means that at rs light itself cannot escape that gravitational pull of the black hole. At a radius half-again as big of rc = (3/2)rs light will be so bent by gravity that it will travel in a circular orbit around the black hole. At just this radius rc, space is bent so that the "straight line" path traveled by light becomes a perfect circular orbit.

M. A. Abramowicz and his co-workers have recently investigated the effects of centrifugal force in the vicinity of rc. They have found a remarkable result: inside radius of rc the centrifugal force acts backwards , pointing toward the center of turning. In other words, instead of the centrifugal force slinging an orbiting object outward, it will push the object inward.

A standard Keplerian orbit in a weak gravitational field can be thought of as delicate balance between the force of gravity which pulls the orbiting object toward the center of attraction, and the centrifugal pseudo-force which pushes the orbiting object outward. If the centrifugal force changes sign and points inward, there can be no stable orbits. Any matter that wanders inside the zone bounded by rc will immediately fall further inward.

This result is not so surprising when we consider that at the radius rc light travels in a closed circular path. This is the equivalent, for general relativity, of a straight line and a mass traveling along the path of the light beam, at whatever velocity, should experience no centrifugal force, any more than it would in traveling in a normal straight line in gravity-free space. It therefore becomes reasonable that when we venture even deeper into the gravity well the curvature of space will increase even more, with the result that a mass traveling in a smaller circular orbit experiences a negative centrifugal force.

The significance of this result relates to the problem of observing black holes. The black hole, being intrinsically a trap for light, cannot in itself be observed optically. However, particularly when the black hole is one member of a binary star system, there is expected to be an accretion disk, a very hot disk made of gas and dust that is in the process of releasing its gravitational energy as heat as it falls into the black hole. This material becomes so hot that it provides a bright beacon in the ultraviolet and x-ray regions of the electromagnetic spectrum pointing to the locations of binary systems with black holes.

Previous computer simulations of the behavior of the accretion disk have indicated very peculiar behavior inside a radius of rc. Very peculiar distributions of momentum and angular momentum were calculated. Long ellipsoids of gas became spherical and the direction of flow mysteriously reversed as the viscous gas fell into the black hole. With this new insight provided by the reversal of the centrifugal force, this behavior becomes clearer. Inside rc the same forces that had stretched the gas into an ellipsoid force it toward a spherical shape.

The radius rc (which is 50% larger than the Schwarzschild radius) is of special significance in the dynamics of balck holes because it is within this boundary that the black hole becomes a matter-eating vortex with no stable orbit possible. We can ask, in the spirit of Larry Niven and Bob Forward who have conjured up heroes that orbit their ships close to neutron stars and black holes, whether it is possible to venture within the zone bounded by rc.

The answer, given a sufficiently powerful engine, is yes. The hero would have to point his engine always inward and at right angles to the direction of motion, balancing with engine thrust both the gravitational attraction of the black hole and the now-reversed centrifugal force that would be pulling him inward. If he could manage this and at the same time avoid being disintegrated by the tidal forces near the black hole, he might live to tell the tale. And that's a tale that I'd like to read somewhere, sometime ...

References:

Centrifugal Force near Black Holes:
B. Allen, Nature 347,615 (1990);
M. A. Abramowicz and A. R. Prasanna, Mon. Not. R. Astr. Soc. 245, 720 (1990);
M. A. Abramowicz and J.C. Miller, Mon. Not. R. Astr. Soc. 245, 729 (1990);
M. A. Abramowicz, Mon. Not. R. Astr. Soc. 245, 733 (1990).


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