Non-inertial reference frame

A non-inertial reference frame (also known as an accelerated reference frame^[1]) is a frame of reference that undergoes acceleration with respect to an inertial frame.^[2] An accelerometer at rest in a non-inertial frame will, in general, detect a non-zero acceleration. While the laws of motion are the same in all inertial frames, in non-inertial frames, they vary from frame to frame depending on the acceleration.^[3][4]

In classical mechanics it is often possible to explain the motion of bodies in non-inertial reference frames by introducing additional fictitious forces (also called inertial forces, pseudo-forces^[5] and d'Alembert forces) to Newton's second law. Common examples of this include the Coriolis force and the centrifugal force. In general, the expression for any fictitious force can be derived from the acceleration of the non-inertial frame.^[6] As stated by Goodman and Warner, "One might say that F = ma holds in any coordinate system provided the term 'force' is redefined to include the so-called 'reversed effective forces' or 'inertia forces'."^[7]

In the theory of general relativity, the curvature of spacetime causes frames to be locally inertial, but globally non-inertial. Due to the non-Euclidean geometry of curved space-time, there are no global inertial reference frames in general relativity. More specifically, the fictitious force which appears in general relativity is the force of gravity.

Avoiding fictitious forces in calculations

See also: Inertial frame of reference and Fictitious force

In flat spacetime, the use of non-inertial frames can be avoided if desired. Measurements with respect to non-inertial reference frames can always be transformed to an inertial frame, incorporating directly the acceleration of the non-inertial frame as that acceleration as seen from the inertial frame.^[8] This approach avoids use of fictitious forces (it is based on an inertial frame, where fictitious forces are absent, by definition) but it may be less convenient from an intuitive, observational, and even a calculational viewpoint.^[9] As pointed out by Ryder for the case of rotating frames as used in meteorology:^[10]

A simple way of dealing with this problem is, of course, to transform all coordinates to an inertial system. This is, however, sometimes inconvenient. Suppose, for example, we wish to calculate the movement of air masses in the earth's atmosphere due to pressure gradients. We need the results relative to the rotating frame, the earth, so it is better to stay within this coordinate system if possible. This can be achieved by introducing fictitious (or "non-existent") forces which enable us to apply Newton's Laws of Motion in the same way as in an inertial frame.

— Peter Ryder, Classical Mechanics, pp. 78-79

Detection of a non-inertial frame: need for fictitious forces

That a given frame is non-inertial can be detected by its need for fictitious forces to explain observed motions.^{[11][12][13][14][15]} For example, the rotation of the Earth can be observed using a Foucault pendulum.^[16] The rotation of the Earth seemingly causes the pendulum to change its plane of oscillation because the surroundings of the pendulum move with the Earth. As seen from an Earth-bound (non-inertial) frame of reference, the explanation of this apparent change in orientation requires the introduction of the fictitious Coriolis force.

Another famous example is that of the tension in the string between two spheres rotating about each other.^[17][18] In that case, prediction of the measured tension in the string based upon the motion of the spheres as observed from a rotating reference frame requires the rotating observers to introduce a fictitious centrifugal force.

In this connection, it may be noted that a change in coordinate system, for example, from Cartesian to polar, if implemented without any change in relative motion, does not cause the appearance of fictitious forces, despite the fact that the form of the laws of motion varies from one type of curvilinear coordinate system to another.

Fictitious forces in curvilinear coordinates

See also: Mechanics of planar particle motion

A different use of the term "fictitious force" often is used in curvilinear coordinates, particularly polar coordinates. To avoid confusion, this distracting ambiguity in terminologies is pointed out here. These so-called "forces" are non-zero in all frames of reference, inertial or non-inertial, and do not transform as vectors under rotations and translations of the coordinates (as all Newtonian forces do, fictitious or otherwise).

This incompatible use of the term "fictitious force" is unrelated to non-inertial frames. These so-called "forces" are defined by determining the acceleration of a particle within the curvilinear coordinate system, and then separating the simple double-time derivatives of coordinates from the remaining terms. These remaining terms then are called "fictitious forces". More careful usage calls these terms "generalized fictitious forces" to indicate their connection to the generalized coordinates of Lagrangian mechanics. The application of Lagrangian methods to polar coordinates can be found here.

Relativistic point of view

Frames and flat spacetime

Further information: Proper reference frame (flat spacetime)

If a region of spacetime is declared to be Euclidean, and effectively free from obvious gravitational fields, then if an accelerated coordinate system is overlaid onto the same region, it can be said that a uniform fictitious field exists in the accelerated frame (we reserve the word gravitational for the case in which a mass is involved). An object accelerated to be stationary in the accelerated frame will "feel" the presence of the field, and they will also be able to see environmental matter with inertial states of motion (stars, galaxies, etc.) to be apparently falling "downwards" in the field along curved trajectories as if the field is real.

In frame-based descriptions, this supposed field can be made to appear or disappear by switching between "accelerated" and "inertial" coordinate systems.

More advanced descriptions

As the situation is modeled in finer detail, using the general principle of relativity, the concept of a frame-dependent gravitational field becomes less realistic. In these Machian models, the accelerated body can agree that the apparent gravitational field is associated with the motion of the background matter, but can also claim that the motion of the material as if there is a gravitational field, causes the gravitational field - the accelerating background matter "drags light". Similarly, a background observer can argue that the forced acceleration of the mass causes an apparent gravitational field in the region between it and the environmental material (the accelerated mass also "drags light"). This "mutual" effect, and the ability of an accelerated mass to warp lightbeam geometry and lightbeam-based coordinate systems, is referred to as frame-dragging.

Frame-dragging removes the usual distinction between accelerated frames (which show gravitational effects) and inertial frames (where the geometry is supposedly free from gravitational fields). When a forcibly-accelerated body physically "drags" a coordinate system, the problem becomes an exercise in warped spacetime for all observers.

See also

• Rotating reference frame
• Fictitious force
• Centrifugal force
• Coriolis effect
• Inertial frame of reference
• Free motion equation

References and notes

1. "Accelerated Reference Frames". Retrieved 2023-09-06.
2. Emil Tocaci, Clive William Kilmister (1984). Relativistic Mechanics, Time, and Inertia. Springer. p. 251. ISBN 90-277-1769-9.
3. Wolfgang Rindler (1977). Essential Relativity. Birkhäuser. p. 25. ISBN 3-540-07970-X.
4. Ludwik Marian Celnikier (1993). Basics of Space Flight. Atlantica Séguier Frontières. p. 286. ISBN 2-86332-132-3.
5. Harald Iro (2002). A Modern Approach to Classical Mechanics. World Scientific. p. 180. ISBN 981-238-213-5.
6. Albert Shadowitz (1988). Special relativity (Reprint of 1968 ed.). Courier Dover Publications. p. 4. ISBN 0-486-65743-4.
7. Lawrence E. Goodman & William H. Warner (2001). Dynamics (Reprint of 1963 ed.). Courier Dover Publications. p. 358. ISBN 0-486-42006-X.
8. M. Alonso & E.J. Finn (1992). Fundamental university physics. Addison-Wesley. ISBN 0-201-56518-8.^{[permanent dead link]}
9. "The inertial frame equations have to account for V_Ω and this very large centripetal force explicitly, and yet our interest is almost always the small relative motion of the atmosphere and ocean, V' , since it is the relative motion that transports heat and mass over the Earth. … To say it a little differently—it is the relative velocity that we measure when [we] observe from Earth’s surface, and it is the relative velocity that we seek for most any practical purposes." MIT essays by James F. Price, Woods Hole Oceanographic Institution (2006). See in particular §4.3, p. 34 in the Coriolis lecture
10. Peter Ryder (2007). Classical Mechanics. Aachen Shaker. pp. 78–79. ISBN 978-3-8322-6003-3.
11. Raymond A. Serway (1990). Physics for scientists & engineers (3rd ed.). Saunders College Publishing. p. 135. ISBN 0-03-031358-9.
12. V. I. Arnol'd (1989). Mathematical Methods of Classical Mechanics. Springer. p. 129. ISBN 978-0-387-96890-2.
13. Milton A. Rothman (1989). Discovering the Natural Laws: The Experimental Basis of Physics. Courier Dover Publications. p. 23. ISBN 0-486-26178-6. reference laws of physics.
14. Sidney Borowitz & Lawrence A. Bornstein (1968). A Contemporary View of Elementary Physics. McGraw-Hill. p. 138. ASIN B000GQB02A.
15. Leonard Meirovitch (2004). Methods of analytical Dynamics (Reprint of 1970 ed.). Courier Dover Publications. p. 4. ISBN 0-486-43239-4.
16. Giuliano Toraldo di Francia (1981). The Investigation of the Physical World. CUP Archive. p. 115. ISBN 0-521-29925-X.
17. Louis N. Hand, Janet D. Finch (1998). Analytical Mechanics. Cambridge University Press. p. 324. ISBN 0-521-57572-9.
18. I. Bernard Cohen, George Edwin Smith (2002). The Cambridge companion to Newton. Cambridge University Press. p. 43. ISBN 0-521-65696-6.