|
In the first part of this article, up until the section on general relativity, the words and expressions are used with their daily life meaning in mind, and they should be read as such.
Definition
If an observer mistakenly assumes an object is kept stationary while in fact it is being accelerated he will automatically assume that the force he can measure is opposing another force, instead of recognizing that inertia is manifesting itself in response to the acceleration. The assumed force is called a fictitious force (also known as an apparent force, fictional force, imaginary force, or pseudo force)
Inertia
In order to discuss the concept of fictitious force, the concept inertia needs to be discussed first.
Inertia opposes any change of velocity. Unlike drag it does not relate to the amount of velocity itself, but to the rate of change of the velocity. The unit of inertia is mass: more mass equals more inertia. Since inertia manifests itself only when there is actual acceleration due to a force, it would not add information to include the manifestation of inertia in diagrams representing forces.
There is the intuitive notion that if two forces that are acting on an object are balanced then the object remains stationary. This is what happens when an object resting on a floor is firmly attached, or if a lot of friction would have to be overcome to cause the object to slide. If you try to push that object, the friction will push back, and the object doesn't move.
Inertia does not prevent change of velocity, as it depends on change of velocity. When an electric car designed to regain energy on decelerating is swithched to braking, the manifestation of inertia is driving the generators, charging the electric car's battery system. Inevitably, the car comes to a halt.
If a car is accelerating hard, then the passenger feels himself 'pushed into his seat'.
The seat is the first to push: as movement begins, inertia manifests itself. The manifestation of inertia does not prevent the acceleration. The amount of acceleration of the passenger is determined by the acceleration of the car.
Natural motion
If a car is turning a sharp corner, the tyres 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.
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. The centripetal force is first to push (or pull): as the observer is deviated from a straight line, inertia manifests itself. The inertia does not prevent the centripetal force from maintaining the circular motion.
If the observer is unaware the disk is rotating (or if he chooses to ignore that possibility), he finds that in order to maintain his position he needs to provide some force directed towards the middle of the disk. Since he chooses to assume he is maintaining a stationary position, he assumes that he is counteracting a force. He mistakenly reverses cause and effect.
The observer mistakenly assumes a force pulling him away from the middle of the disk that he must subsequently oppose to maintain position. (If the rotating disk is in space, the observer would need a space-suit with magnetic boots.) The force that is supposedly pulling him away from the middle of the disk is the fictitious force called centrifugal force.
This supposedly centrifugal force is very different from the centripetal force that maintains the circular motion. A centripetal force is transmitted by contact. The observer on the rotating disk who is standing up needs grip. If the centrifugal force would actually exist it would be a force that is the same for all particles in the body, so you cannot feel it and there is no instrument that can measure it, you can only measure the effort to oppose it.
Another example of a fictitious force is the coriolis force
It is apparent that on Earth gravity fits the description of fictitious force rather well. Really, you can only measure the effort to oppose gravity, and like the fictitious forces it is exactly proportional to the mass of an object.
Relativistic dynamics
Newtonian theory describes gravity as an inverse square law of force. In newtonian theory there is no prospect of finding out how this force is transmitted. 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. The properties of the mediator of gravitational interaction are unique. General relativity shows what exactly the parallels and differences are between gravity and the fictitious forces.
Thought experiments
Einstein used thought experiments to explore the logical consequences of certain hypotheses.
Imagine a very large amount of adjacent conveyer belts, the kind of conveyer belts that are used in large buildings for public transport. Imagine each conveyer belt is moving faster than the adjacent belt. Moving from conveyer belt to conveyer belt an observer is accelerating step by step. Compared to adjacent belts the observer is shifting his relativistic time dilation and length contraction at every step.
If an observer is on a rotating disk, divided in concentric corridors, and he is moving from corridor to corridor, then he has for every distance r to the middle of the disk a different velocity. Moving from corridor to corridor the observer is shifting his time dilation and length contraction at every step.
Einstein decided to follow the logic through and he assumed that an observer who is moving up and down from floor to floor on the surface of a planet is shifting his time dilation and length contraction all the time. For every distance r to the center of the planet a different Lorentz frame should be valid.
So, if gravity deforms the geometry of 4-dimensional space-time in such a way that there is stronger time dilation and space contraction close to a gravitating body than further away, then to stand on the surface of a planet is fundamentally indistinguishable from being accelerated with respect to the surrounding space-time.
This is called the principle of equivalence.
In general relativity space-time geometry is part of the physical world, with physical properties. For example, according to general relativity, when two masses are orbiting each other they lose angular momentum due to radiation of gravitational waves. The energy is not lost, the energy is conserved, it is being carried away by a propagating deformation of the geometry of space-time.
Two distinct kinds of acceleration
Space-time is transparant to relative velocity, and deformed space-time has the same transparancy, but it also imparts acceleration on objects moving through it. This acceleration is without manifestation of inertia
According to general relativity, planets in orbit around a sun stay in their orbit because the space-time they are moving through is imparting acceleration on them. Their resultant motion is a combination of the acceleration that is imparted on them, and their relative velocity. (This is why light is deflected by gravity, but by a much smaller angle than anything else, light moves faster than anything else.) The two suns of a double star system stay in orbit around the common center of mass because the space-time they are moving through is deformed by the other sun of the system. The deformation of space-time around a gravitating body is spherically symmetrical.
We now suddenly have two distinct kinds of acceleration. That is one of the fundamental differences between the everyday notion of acceleration, and what is described by general relativity: two different kinds of acceleration.
When an object is in free fall straight towards the center of a planet, then at each moment the object is not accelerating with respect to the local volume of space-time. To be in free fall is locally indistinguishable from floating in space-time far away from any gravitating body. Any accelerometer will not measure anything when it is in free fall. Satellites in orbit around a planet are in free fall, with a lot of velocity perpendicular to the direction of fall, so they never reach the surface.
Criterium for measurement
For now the two kinds of acceleration will be distinguished by using accelerometer measurements as criterium. If an observer uses mechanical force, for example the thrust of rocket engines to accelerate, then the observer is accelerating with respect to the surrounding space-time and the accelerometers will give a reading.
In summary:
- There is acceleration due to moving through deformed space-time geometry, this acceleration is without manifestation of inertia, and it can only be observed to take place by looking at it from a sufficiently large distance, when considering a sufficiently large volume of space.
- There is acceleration due to a mechanical force, this is acceleration with respect to the surrounding space-time.
The nature of gravity
Now the stage is set to get an understanding why gravity fits the description of a fictitious force rather well:
- Unless you are maintaining grip with something that opposes it, you don't feel anything.
- Like the fictitious forces it is exacly proportional to the mass of an object.
- On the surface of a planet, you - quite understandibly - think that there is a gravitational force, and that in response to that the surface is providing an opposing force, maintaining your position.
According to general relativity, the surface of the planet is accelerating you with respect to local space-time. In response to that inertia manifests itself.
With the fictitious forces, once the observer releases his grip (for example if the observer in space gripping a rotating disk with electromagnetic boots switches the electromagnets of) his motion will from that moment on remain unaccelerated motion, seen from all perspectives, from local to universal. Once contact has been broken, the observer still has a large velocity, but it won't increase.
If the observer is in a region of space-time where space-time is imparting acceleration then if the observer releases his grip, he may be unaccelerating in a very local volume of space, but in all larger perspectives his velocity keeps increasing, and it's towards something that is much, much heavier than he is.
| 
