"The Alternate View" columns of 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 withoutthe 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 Z^{o}, 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/r ^{2}** 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
**r _{s} = 2Gm/c^{2 }**called the Schwarzschild radius at
which the escape velocity equals

M. A. Abramowicz and his co-workers have recently investigated the effects of
centrifugal force in the vicinity of **r _{c}**. They have found a
remarkable result: inside radius of

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 **r _{c}** will immediately fall further inward.

This result is not so surprising when we consider that at the radius
**r _{c}** 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 **r _{c}**. 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

The radius **r _{c}** (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

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).

*This page was created by John G. Cramer
on 7/12/96.*