IntroductionFreal = m a
People moving in an accelerated reference frame observe phenomena which, to observers "outside" that frame of reference, do not exist. For such an outside observer, the phenomena are not real, but are a mirror-image illusion caused by the acceleration of the reference frame. Since the phenomena are not real, no force is needed to explain them. But for the observer in the accelerated reference frame, the illusory motions usually appear to be perfectly real. This leads to the supposition, by the accelerated observer, that some kind of force is responsible for the motions. Since, from the viewpoint of the non-accelerated observer, neither the motion nor the force actually exist, such forces are referred to as a fictitious force or, after the French mathematician who first proposed such a way of reconciling the differing viewpoints, a d'Alembert force.
As an example, consider the motion of the sky -- that is, the motion of the stars in the sky westward each day, which is caused by our eastward motion around our axis of rotation. To an observer outside the Earth, it appears that the stars are essentially immobile, or "fixed" in space, since their actual motions are very small compared to their vast distances. But to an observer on the ground, the stars seem to circle around the sky, in uniform circular motion, which should require a force, pointing toward the center of the circular motion, called a centripetal force. If the ground-based observer supposed that there really was such a force, the outside observer would call that a fictitious force. (Of course, when people thought that the stars really did move in this way, they also thought that it was natural for them to move that way, so no force was required to cause their motion.)
Inertial and Accelerated Reference Frames
To understand fictitious forces, we need to define inertial reference frames, and accelerated reference frames.
An inertial reference frame is one in which Newton's First Law of Motion, the Law of Inertia, holds true. In such a frame, objects which have no force acting on them move in straight lines, with constant speed. Any deviation from such a motion is presumed to be due to a force whose magnitude and direction can be deduced by applying Newton's Second Law of Motion, the Force Law, to observations of that deviation. An observer who is at rest is in an inertial reference frame, and observers who are not at rest, but are moving with uniform (unchanging, straight-line) motion are also in inertial reference frames.
An accelerated reference frame is one in which the Law of Inertia does not hold true; that is, one in which an object which has no real force acting on it appears to have a force acting on it, because the observer sees it moving with an acceleration which is a mirror image of his reference frame's acceleration. Perhaps surprisingly, since we assign such importance to Newton's Laws in general, and the Law of Inertia in particular, such reference frames are very common in everyday life. Any acceleration, whether a change in speed, a change of direction (such as, in the example above, our motion around the Earth's axis of rotation), or both, puts observers who share in that acceleration in an accelerated reference frame.
Motion in a linearly accelerated reference frame
As an example, consider a car which is accelerating from a fixed start. Prior to its acceleration, it and its contents are at rest, whether observed by someone standing outside the car, or by someone sitting inside it. At that time the observers share an inertial frame of reference (since they are not moving relative to each other), and would agree that according to the Law of Inertia and the observed lack of motion, no net force is acting on the car or its contents.
Once the car starts up, however, things change. The observer who is outside the car remains in the inertial frame of reference, and interprets the acceleration of the car and its contents as being due to a forward force, which accelerates them in the direction of the car's motion. The observer inside the car, however, tends to use the car as his reference frame, just the same as if it were still at rest, and since it is accelerating, will perceive things differently from the observer who really is at rest.
Consider something which is sitting on the dashboard of the car, and as the car moves forward, is "thrown off" the dashboard, toward the rear of the car. Presuming there is no net force acting on the object, the outside observer will see it as remaining at rest, while the car moves forward. Its apparent backward motion, as seen by someone sitting in the car, is merely a mirror image of the car's forward motion. However, to a person in the car, the object appears to accelerate toward them, as though some backward force is acting on it. In fact, the person in the car can feel that there is some kind of backward force acting on not only that object, but all other objects in the car, because they feel themselves being thrown backward against their seat, as well.
Of course, they are not being thrown backward, at all. The car is moving forward, and according to Newton's Force Law, must push them forward, with a force equal to their mass times the car's forward acceleration, if they are to move forward with it. Or at least, that's how the outside observer would interpret the forward motion of the person in the car. But according to Newton's Law of Action and Reaction, every force (or "action") causes an opposing force (or "reaction") which acts in the opposite direction, on the opposite object, and is equal in magnitude to the original force. So, as the car pushes the person inside the car forward, a "reaction" force is created, with which the person pushes the car backward. This mutual pair of forces, one pushing forward on one object (the person), and the other backward on the other object (the car), is what the person inside the car feels. And as long as they can detach themselves from their immediate surroundings, and mentally evaluate things in the same way as the outside observer, they will conclude that the forward force is the real one, and the backward force is the reaction to the forward force.
However, it is very difficult to separate one's mind from one's senses; and the senses of the person in the car very clearly tell them that they are being thrown backward, against the seat, and that of the action/reaction pair of forces, the backward force is the real one, and the forward force is the reaction force.
In other words, the person who is not in the accelerated reference frame of the car will see it as moving forward, pushing forward on objects moving with the car, and objects which seem to fly backwards as the car runs into them, have no real force acting on them. But the person who is in the accelerated reference frame of the car will view the car as an inertial reference frame (which it is not), and by doing so, will imagine that there are backward forces, which the outside observer would consider fictitious forces, to explain the apparent backward force acting on all objects in the car.
What is true for the accelerated car is true for all accelerations. In accelerated reference frames, observers outside the accelerated reference frame, in inertial reference frames, will perceive that the acceleration of the frame requires forces which push objects and people inside the frame in the direction of its acceleration; but for persons inside the accelerated reference frame, there is a tendency to treat the accelerated reference frame as though it were an inertial reference frame, thereby "creating" a backward acceleration which is a mirror image of the reference frame's forward acceleration, and as a result, "fictitious" forces which are a mirror image of the forces causing the real acceleration.
How to recognize fictitious forces / accelerated reference frames
So, suppose we observe things moving in a way which suggests that forces are acting on them. How can we tell whether we are in an inertial reference frame, and the forces we observe are real, or whether we are in an accelerated reference frame, and the forces we observe are "fictitious"? The answer lies in the Force Law, which specifies that the acceleration which an object receives depends not only on the force acting on it, but also on its own mass, or inertia, or resistance to a change in its motion. For real forces, there is no guarantee that the force acting on the object and the mass of the object will produce a particular acceleration. The acceleration could be large or small, for a given force, depending upon how much inertia the object has. For fictitious forces, all objects seem to be accelerated in the same way, which means that the force acting on the objects seems to be directly proportional to the inertia of the object. Things that have little inertia seem to have little force acting on them, and things that have a lot of inertia seem to have a lot of force acting on them.
In mathematical terms, real forces Freal acting on different masses m will produce varying accelerations a, according to the rule
where a could have any value, while fictitious forces Ffictitious acting on different masses will all produce the same acceleration aconstant, according to the ruleFfictitious = m aconstant
So if we see different objects moving with different accelerations, at least some of those accelerations, and the forces causing them, must be real; whereas if we see different objects moving with the same acceleration, the accelerations are probably not real, but a mirror-image of an acceleration of the reference frame, and forces required to explain them are "fictitious".W = m g
Is Gravity A Fictitious Force?
There is one situation in which we see accelerations which are the same, and presume that the forces causing those accelerations are real -- namely, when we see objects falling toward the Earth, under the influence of gravity. No matter what the objects are made of, or how big or small they are, all objects fall under the influence of gravity with exactly the same acceleration, the acceleration of gravity. Solids, liquids, gases, green cheese, moonbeams, fairy dust and horsefeathers would all fall the same under the influence of gravity, as would sand grains, pebbles, boulders, mountains, moons and planets. Because of this we write the Force Law in a special way for the force of gravity, replacing the force F with the object's weight W, and the acceleration a with the acceleration of gravity, g, thusly
The fact that gravity, like fictitious forces, involves a constant acceleration, makes us wonder whether gravity could be a fictitious force. It's hard to imagine that anything so pervasive and seemingly real could be "fictitious", but the forces experienced by the person in the accelerated car feel real, and are presumably fictitious. Is there some way that we could create the phenomenon of gravity, without the force?
There is indeed such a way. Suppose that you were in a rocket ship, headed upwards at the acceleration of gravity, so that anything not attached to the ship seems to "fall" with a mirror image of that upward acceleration. Then every such object would fall toward the back of the ship, at the acceleration of gravity, and trying to stop such a fall would require a force, in the direction of the acceleration, proportional to the object's mass, which would be equal to, and appear to be, its real weight.
Of course, we can't explain gravity in that way, as that would require every part of the Earth to be accelerating upward and outward, which would make the Earth bigger and bigger, which is not observed. So the simplest explanation is to assume that, peculiar though it may be, gravity -- although a perfectly real force -- acts as though it is a fictitious force. No other real force is known to act in this way, but perhaps gravity is "special", and it is merely a coincidence that it looks like a fictitious force.
The strange and in some ways disturbing answer to this supposition is that the phenomenon of gravity (the fact that things fall, and have weight) is real, but the force of gravity, as described by Newton, is not a real force, but a fictitious force. According to Einstein's General Theory of Relativity, gravity is a curvature of space-time such that in the future, things are closer together than they are now, even if they are moving in straight, parallel lines, with no force between them. For in curved space-time, there is no such thing as a straight line, but instead, only curved lines, called geodesics, which are the straightest possible paths in curved space-time, but are always and inexorably curved. (for now, see the chapter on black holes and general relativity in the text for a more detailed discussion) And since curved paths, in our experience, require some centripetal force to create them, the motion of things along geodesics seems to require some force to explain the acceleration observed, as a result of that curvature.
So we see things falling, with an acceleration which we call the acceleration of gravity, and thinking that we live in a straight-line, uniformly moving or stationary inertial reference frame, we attribute that acceleration to a force, the force of gravity. Whereas in reality, objects falling toward the Earth are moving along geodesic paths, with no acceleration, and according to a modified version of the Law of Inertia (objects which are at rest tend to remain at rest, and objects which are moving tend to move along geodesic paths with uniform motion, unless some force acts on them), have no force acting on them. They fall simply because the curved space-time near the Earth makes it natural for them to be closer to us in the future, than they are now.
But if no force is required to make them fall, why do they seem to have weight, when we hold them? In Newton's physics, the weight we perceive is a direct measure of the force of gravity acting on them. How can they have weight, if there is no force acting on them?
The answer is, that if we are holding them, there is a force acting on them, namely the upward force we are exerting, which keeps them from doing what they are supposed to do, namely fall. And of course that upward force on the object we're holding creates a reaction force, which is downward and equal to the upward force we are exerting, and that is what we are observing, when we observe that something has weight. In fact, even when we perceive our own weight, that is what we are observing. Whatever is keeping us from falling is pushing us upward, with a force equal to our "weight", and what we think is our weight, pushing downward, is just a reaction to that upward push. If there weren't anything pushing us upward -- if, for example, we were to jump off the top of the D building -- we would feel no upward force, and would therefore feel "weightless". We would presume that we weren't really weightless, because the ground is rapidly rising to meet us, but that isn't because we have weight, but because it is natural, in the absence of a force preventing it, for us to fall toward the Earth, along a space-time geodesic. (splat conveying the distinct notion that there's some force around here, somewhere, Einstein's Theory or not)
(more to follow ASAP)
Motion in a rotationally accelerated reference frame: centripetal / centrifugal forces
(point to be made -- the centripetal force is always the real one, and the centrifugal force is always the fictitious one)
Application to the Earth / Coriolis forces
(for now, see Coriolis Forces for a discussion of this topic)