Coriolis field
Encyclopedia
In theoretical physics
a Coriolis field is one the apparent gravitational fields felt by a rotating or forcibly-accelerated
body, together with the centrifugal field and the Euler field.
is the angular velocity vector of the rotating frame, 
the speed of a test particle used to meassure the field, and using the expression of the acceleration in a rotating reference frame
, it is known that the acceleration of the particle in the rotating frame is:
the Coriolis force is assumed to be the fictitious force that compensates the second term:
Where
denotes the linear momentum. It can be seen that for any object, the coriolis force over it is proportional to its momentum vector. As vectorial product can be expressed in a tensorial way using the Hodge dual
of
:
This matrix can be seen as a constant tensor field
, defined in the whole space, that will yield coriolis forces when multiplied by momentum vectors.
, this “apparent”, “fictitious” or “pseudo-gravitational” field effect can be treated as genuine.
As an example, when an object is set down on a rotating children’s roundabout, it is seen to slide away from the centre of the roundabout. In the non-rotating frame of reference, the outward motion is a consequence of the object’s inertial mass and the object's tendency to continue moving in a straight line. However, in the rotating frame as a reference, the object is pulled outwards by a radial gravitational field
caused by the relative rotation of the outside universe. In that view, the (outward) motion is instead a consequence of its gravitational mass.
This dual description is used to unify the ideas of inertial and gravitational mass under general theories of relativity, and to explain why an object's inertial mass and gravitational mass are proportional in classical theory. In these descriptions, the distinction is purely a matter of convenience; inertial and gravitational mass are different ways of describing the same behaviour.
effect and not just a mathematical artifact is justified by Machian
theory. It notes that evidence of the field's existence is not only visible to the rotating observer; its distortion is also visible and verifiable for non-rotating onlookers. Thus, the relative rotation of the roundabout and universe masses creates a real physical distortion in spacetime that is visible to all observers (see: Kerr black hole, frame-dragging
, light-dragging effects
). The physical consequences of rotation experienced by the rotating-frame observer can be said to be “smudged into” the non-rotating observer’s physics. The Coriolis field can thus be said to have a genuine existence; it is expressed in the intrinsic curvature of the region and cannot be made to vanish with a convenient mathematical change of coordinate system. The forces and effects are mutual–the roundabout observer feels the outside universe pulling more strongly along the rotation plane, and pulling matter around, and (to a far lesser extent) the mass of the rotating roundabout creates a stronger inward pull and pulls matter around with it as well.
In this way, general theories of relativity are supposed to also eliminate the strict distinction between inertial and noninertial frames
. If we take an inertial observer in flat spacetime and have them observe a rotating disc, the existence of the rotating mass means that spacetime is no longer flat, and that the concept of rotation is now subject to the democratic principle
.
This elimination of the concept of the inertial frame was initially described by Einstein as one of the great successes of his general theory of relativity.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
a Coriolis field is one the apparent gravitational fields felt by a rotating or forcibly-accelerated
Acceleration
In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...
body, together with the centrifugal field and the Euler field.
Mathematical expression
Being is the angular velocity vector of the rotating frame, the speed of a test particle used to meassure the field, and using the expression of the acceleration in a rotating reference frameRotating reference frame
A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth. A rotating frame of reference is a special case of a non-inertial reference...
, it is known that the acceleration of the particle in the rotating frame is:
the Coriolis force is assumed to be the fictitious force that compensates the second term:
Where
denotes the linear momentum. It can be seen that for any object, the coriolis force over it is proportional to its momentum vector. As vectorial product can be expressed in a tensorial way using the Hodge dualHodge dual
In mathematics, the Hodge star operator or Hodge dual is a significant linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented inner product space.-Dimensions and algebra:...
of
:This matrix can be seen as a constant tensor field
Tensor field
In mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...
, defined in the whole space, that will yield coriolis forces when multiplied by momentum vectors.
Mach's view
In a theory that conforms to some versions of Mach's principleMach's principle
In theoretical physics, particularly in discussions of gravitation theories, Mach's principle is the name given by Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach....
, this “apparent”, “fictitious” or “pseudo-gravitational” field effect can be treated as genuine.
As an example, when an object is set down on a rotating children’s roundabout, it is seen to slide away from the centre of the roundabout. In the non-rotating frame of reference, the outward motion is a consequence of the object’s inertial mass and the object's tendency to continue moving in a straight line. However, in the rotating frame as a reference, the object is pulled outwards by a radial gravitational field
Gravitational field
The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...
caused by the relative rotation of the outside universe. In that view, the (outward) motion is instead a consequence of its gravitational mass.
This dual description is used to unify the ideas of inertial and gravitational mass under general theories of relativity, and to explain why an object's inertial mass and gravitational mass are proportional in classical theory. In these descriptions, the distinction is purely a matter of convenience; inertial and gravitational mass are different ways of describing the same behaviour.
Is it real?
Support for the idea that the Coriolis field is a real physicalNature
Nature, in the broadest sense, is equivalent to the natural world, physical world, or material world. "Nature" refers to the phenomena of the physical world, and also to life in general...
effect and not just a mathematical artifact is justified by Machian
Ernst Mach
Ernst Mach was an Austrian physicist and philosopher, noted for his contributions to physics such as the Mach number and the study of shock waves...
theory. It notes that evidence of the field's existence is not only visible to the rotating observer; its distortion is also visible and verifiable for non-rotating onlookers. Thus, the relative rotation of the roundabout and universe masses creates a real physical distortion in spacetime that is visible to all observers (see: Kerr black hole, frame-dragging
Frame-dragging
Einstein's general theory of relativity predicts that non-static, stationary mass-energy distributions affect spacetime in a peculiar way giving rise to a phenomenon usually known as frame-dragging...
, light-dragging effects
Light-dragging effects
In physics, there are several situations in which the motion of matter might be said to drag light.Under special relativity's simplified model it is assumed that these light-dragging effects do not happen, and that the speed of light is independent of the speed of a body's motion...
). The physical consequences of rotation experienced by the rotating-frame observer can be said to be “smudged into” the non-rotating observer’s physics. The Coriolis field can thus be said to have a genuine existence; it is expressed in the intrinsic curvature of the region and cannot be made to vanish with a convenient mathematical change of coordinate system. The forces and effects are mutual–the roundabout observer feels the outside universe pulling more strongly along the rotation plane, and pulling matter around, and (to a far lesser extent) the mass of the rotating roundabout creates a stronger inward pull and pulls matter around with it as well.
In this way, general theories of relativity are supposed to also eliminate the strict distinction between inertial and noninertial frames
Non-inertial reference frame
A non-inertial reference frame is a frame of reference that is under acceleration. The laws of physics in such a frame do not take on their most simple form, as required by the theory of special relativity...
. If we take an inertial observer in flat spacetime and have them observe a rotating disc, the existence of the rotating mass means that spacetime is no longer flat, and that the concept of rotation is now subject to the democratic principle
Democratic principle
In the context of General Relativity, the democratic principle allows quick, order-of-magnitude calculations for the strength of gravitomagnetic effects such as frame-dragging...
.
This elimination of the concept of the inertial frame was initially described by Einstein as one of the great successes of his general theory of relativity.
See also
- Classical theories of gravitationClassical theories of gravitationThe current Gold Standard Theory of Gravitation is the general theory of relativity. This is a classical, relativistic field theory of gravitation...
- Coriolis effectCoriolis effectIn physics, the Coriolis effect is a deflection of moving objects when they are viewed in a rotating reference frame. In a reference frame with clockwise rotation, the deflection is to the left of the motion of the object; in one with counter-clockwise rotation, the deflection is to the right...
- Equivalence principleEquivalence principleIn the physics of general relativity, the equivalence principle is any of several related concepts dealing with the equivalence of gravitational and inertial mass, and to Albert Einstein's assertion that the gravitational "force" as experienced locally while standing on a massive body is actually...
- General relativityGeneral relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
- GravitomagnetismGravitomagnetismGravitomagnetism , refers to a set of formal analogies between Maxwell's field equations and an approximation, valid under certain conditions, to the Einstein field equations for general relativity. The most common version of GEM is valid only far from isolated sources, and for slowly moving test...
- Mach's principleMach's principleIn theoretical physics, particularly in discussions of gravitation theories, Mach's principle is the name given by Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach....
- Newton's bucket