Fictitious force
The term fictitious force refers to a manifestation of inertia. Certain other names for "fictitious force" are apparent force, fictional force, imaginary force, pseudo force, or inertial force.
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Acceleration in a straight line
The most straightforward example is what is felt when a car is accelererating hard. The passenger feels himself 'pushed into the seat'. To change the velocitiy of an object - in this case the passenger - a force must be exerted. It is only after the springs of the back of the carseat have been compressed sufficiently that the springs exert sufficient force to accelerate the passenger as hard as the car is being accelerated. So the passenger feels himself being 'pushed into the seat'; the springs in the carseat transmit the force of the car engine to the passenger.
By general agreement, the mechanical force exerted by the engine of the car is called a force, and the manifestation of inertia, the 'being pushed into the seat', is not categorized as a force. Often the manifestation of inertia is called apparent force or fictitious force. In physics there is the general agreement to define the concept of force in such a way that one of the properties of force is that if two opposing forces are equally strong then there will be no change (of velocity). If the manifestation of inertia would be categorized as a force, then that would lead to an inconsistency in the framework of definitions.
When an electric car designed to regain energy on decellerating is switched to braking, the manifestation of inertia drives the generators, recharging the battery system. Manifestation of inertia is always involved when there is conversion of kinetic energy to potential energy and vice versa, and brakes heating up on braking is conversion of kinetic energy to heat. Whenever kinetic enerty is converted to another form of energy manifestation of inertia is doing work. Other than manifestation of inertia only forces can do work.
A principle difference between manifestaton of inertia and force is that manifestation of inertia occurs only if there is actual change of velocity due to a force being exerted. Manifestation of inertia occurs in response to a force being exerted. As long as no force is being exerted inertia is 'ready for action' in a matter of speach, but it only becomes noticable (and measurable) when a force is being exerted.
When a car is accelerating this acceleration is being opposed by inertia, but this opposition does not prevent the acceleration. Conversely, if the car brakes it is inertia that necessitates that the tires need a lot of grip on the road. Because of inertia a force is required for decelleration, but the manifestation of inertia does not prevent the decelleration. Whether a particular change of velocity is seen as acceleration of decelleration only depends on an arbitrary choice of inertial coordinate system. Only amount of change of velocity is unambiguous in physics. The amount of change of velocity determines the magnitude of the manifestation of inertia.
Circular motion
If a car is turning a sharp corner, the tires of the car need to have enough grip to provide the necessary force directed towards the inside of the curve. If they have no grip at all the car will continue to move in a straight line. Natural motion is motion in a straight line, with constant velocity. This is called uniform motion. Uniform motion is motion without manifestation of inertia. To make an object deviate from unifom motion, a force must be exerted. When turning a corner, a car is being accelerated, in a direction perpendicular to the direction of the velocity.
An observer on a rotating disk is in motion, but it is not uniform motion, it is circular motion and in order to maintain a circular motion a centripetal force must be provided. In the case of human scale setups, this centripetal force is provided in the form of mechanical force: a cord under tension, or friction between the surface of the disk an the object (or observer) on the disk. As this force, directed to the center of rotaton, deviates the observer from moving in a straight line, inertia manifests itself in the opposite, outward direction. The manifestation of inertia does not prevent the centripetal force from maintaining the circular motion.
The centrifugal manifestation of inertia is in itself not measurable, it's not a mechanical force. The only way that the magnitude of the centrifugal manifestation of inertia can be measured is to measure how much centripetal force is required to maintain the circular motion.
Manifestation of inertia, trajectories, and work
In the case of motion that is not linear, manifestation of inertia can do work, just like in the case of linear motion. Take some weights in your hands, and sit down in a chair that can easily swivel. Stretch your arms outward, and start yourself rotating. Then pull in your arms. In order to pull in the weights some extra force must be exerted, above the amount of force that is necessary for maintaining a constant diameter of circular motion. So when the weights are being pulled in, work is being done and the rotational kinetic energy goes up. (In this example the angular momentum does not change, but the rotational kinetic energy does!). When the weights are allowed to 'swing out' again the centrifugal manifestation of inertia is doing work.
Examples discussing motion almost always present a circular trajectory, in order to simplify the explanation. This makes it seem as though centrifugal force is allways perpendicular to the direction of motion. But whenever the trajectory is elliptical or in any other way non-circular the manifestation of inertia is doing work. In the real world trajectories are always somewhat non-circular; mathematically perfect circular trajectories exist only in theory.
In general, a force is perpendicular to the direction of motion if and only if the motion is uniform circular motion; it is a property of circular motion, rather than a property of any particular force.
In the case of the coriolis force: when an object is following a uniform circular trajectory no work is being done. Whenever the motion is not circular energy is converted from one form to another, for example potential energy to kinetic energy; work is being done.
Conclusion
Whenever the expression 'fictitious force' is encountered in a text, it must be kept in mind that it refers to manifestation of inertia. Manifestation of inertia is (1) real physics, (2) by general agreement not to be categorized as a force.
This concludes the article on fictitious force. The next part is an in-depth discussion of fictitious force in the context of general relativity. In general relativity inertia is seen the same way as in newtonian dynamics so in general relativity and in newtonian dynamics fictitious force is seen in the same way.
In-depth discussion: Relativistic dynamics
Newtonian theory describes gravity as an inverse square law of force. In newtonian theory there is no prospect of finding out more about the properties of the mediator of gravity. With the introduction of general relativity Einstein moved the theory of gravity to a deeper level. General relativity describes the nature of the mediator of gravity in detail. General relativity shows what exactly the parallels and differences are between gravity and the fictitious forces.
Gravity and inertia
In general relativity, the description of gravity and the description of inertia are unified. Both seem to involve an action-at-a-distance. What physical interaction determines the path of a planet orbiting a sun? And what physical interaction determines how much a celestial body that is rotating bulges at the equator? Why is the amount of bulging that is observed invariably observed to be proportional to the angular velocity with respect to the universe as a whole? The 'how much to bulge' question seems to imply that there must be some sort of action-at-a-distance between distant matter, perhaps all of the matter in the universe, and the rotating body, providing the information that is necessary to determine the amount of equatorial bulging uniquely.
In general relativity it is assumed that space-time geometry itself is the mediator of all that. In general relativity space-time geometry determines inertia, and gravity is mediated by deformation the space-time geometry. There is no need to assume that gravity is mediated by something that is present in space, it is mediated by the very fabric of space-time itself. In general relativity the concept of space-time geometry and the concept of a gravito-inertial field are considered to be one and the same thing.
The gravito-inertial field is transparant to velocity, all uniform velocities are indistinguishable, but whenever there is change of velocity there is interaction with the gravito-inertial field, with the gravito-inertial field opposing (but never preventing) the change of velocity.
Geodesics
When matter is moving through non-deformed space-time (in other words: straight space-time), it follows paths that are the same as euclidean straight lines. These lines are universally straight, meaning that when they are viewed from a distance they are still observed to be straight, they are straight lines universally. Objects that are moving through straight space-time will move in straight lines and with constant velocity: that is the free motion in straight space-time geometry. When there is a change of velocity due to a force being exerted then inertia manifests itself.
When matter is free-moving through deformed space-time it follows geodesics. Geodesics are paths with no manifestion of inertia. When a force is deviating an object from moving along its proper geodesic, inertia manifests itself. In curved space-time, the concept of an inertial frame of reference is definable only locally, because of the deformation of space-time geometry by matter. Matter is concentrated in Suns and Planets and so on, and in the neighbourhood of these gravitating bodies there is spherically symmetrical deformation of space-time geometry. An important part of the deformation of space-time geometry is alteration of the rate of time. An object free falling towards the center of gravity of a gravitating mass is moving along its proper geodesic. The local frame of reference, co-moving with that free falling object, is a local inertial frame of reference.
Geodesics include all free fall trajectories; not only motion straight towards a center of gravity: all trajectories that have both a component straight towards (or away from) a center of gravity, and a component perpendicular to that are geodesics.
By nature space-time geometry tends to be straight. Further away from a gravitating body the deforming stress is distributed over a larger area, and the space-time geometry is closer to straight space-time geometry there.
The deformation of space-time geometry around a gravitating body is spherically symmetrical. However, according to geometrodynamics, rotating bodies do cause a rotation of space-time around them. This rotation is extremely small, the space probe Gravity Probe B, orbiting Earth, is expected to measure an accumulated rotation of space-time geometry (with respect to the universe as a whole) of 42 milliarcseconds over the course of a year.
What interaction determines the equatorial bulging of rotating celestial bodies?
Celestial bodies (in general: all rotating bodies) are interacting with the local gravito-inertial field, and this interaction with local space-time geometyr influences the shape of the rotating body. It is extremely rare for space-time geometry to rotate significantly with respect to the universe as a whole, therefore 'non-rotating frame of reference' is often defined as a frame of reference that doesn't rotate with respect to the universe as a whole.
The principle assumption of general relativity is that all laws of physics relate in the same way to local space-time geometry. You can do a local measurement with a mechanical instrument like a gyroscope, or you can measure with an interferometric instrument, the instruments will agree on how much (or how little) the local inertial frame of reference rotates with respect to the universe as a whole. No counter-examples of this assertion of general relativity have been encountered in physics.
Centrifugal manifestation of inertia is what arises when an object is maintained in circular motion in straight space-time. Centrifugal manifestation of inertia cannot accelerate an object; as soon as the centripetal force ceases, the centrifugal manifestation of inertia ceases, and the object continues in uniform motion. Centrifugal manifestation occurs when centripetal acceleration interacts with straight space-time geometry, and undeformed space-time geometry mediates inertia only. manifestation of inertia is not categorized as a force.
Whether gravity accelerates depends on the scale of the perspective when looking at it. When only considering a small volume of space in a gravitational field, co-moving with a free falling object, a volume of space in which the tidal forces are effectively zero, then there is no acceleration in that volume of space. When looking at it from a perspective that envelopes the entire planet that is the source of the gravitaton, then the object is accelerating towards the common center of mass. Obviously, the largest perspective available is the one to base decisions on.