Inertial Force

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In general relativity inertial forces are identical in nature to gravitational forces. For a complete precise mathematical treatment of inertial forces see gravitational forces.

Consider a frame of reference S (X, Y, Z) that is at rest in an inertial frame of reference in flat spacetime. Let S’ (x, y, z) be a frame of reference which is uniformly accelerating in a straight line parallel to the Y/y axes. Let the spatial axes of S’ remain parallel to the corresponding spatial axes of S.  Let there also be a several particles at rest in S. The relationship between the two components of the velocity between the two coordinate systems in Newtonian mechanics by

The relationships between the velocity is given by

The relationships between the accelerations is given by

From Eq. (3) we see that if the Y-components of the acceleration of all the particles as reckoned in the inertial frame S is zero then the y-components of all the particles will acceleration in the -y direction in S’ having the value of ay = -a for each and every particle.  This will also be true if the particles are moving with constant velocity in S. Thus it is easy to see that the particle will be reckoned by observers in S’ to be accelerating in the –y direction with all the y-components of acceleration of each particle having the value to a as shown in Figure 1 below



It is obvious that the rate of acceleration of the particle as reckoned by observers in S’ that the acceleration of the particles in S will have the same acceleration of all the other particles, irrespective of the mass of each test particle. If one were to hold a particle such that it was at rest in S’ then it would require a force whose direction is in the +y direction. The magnitude of this force is defined as the weight of the particle as reckoned by observers in frame S’.
     Consider now, instead of S’ accelerating linearly with respect to S, that S’ is rotating at a constant angular velocity
w about the Z-axis of S. The XY-plane coincides with the xy-plane while the origins of S and S’ coincide. It can be shown that the inertial force FI on a particle as reckoned by an observer in the rotating frame is given by

where r and v are quantities measured in S’. There are no forces acting in the frame of reference S. The first term on the right hand side represents the Centrifugal force and the second term represents the Coriolis force [1]. These forces are called inertial forces. Two important items that can be pointed out at this time are (1) each term is proportional to the particle’s inertial mass and (2) the Coriolis term is velocity dependant. See Figure 2 below

Figure (1a) shows a particle that has an initial velocity v0 in the radial direction. The total inertial force in this case does not point in the radial direction but rather in the direction shown in the figure labeled FI. In Figure (2b) v0 is in the tangential direction. The deflection due to this force is shown in the example below in Figure 3

In Newtonian physics the laws of physics were defined with respect to inertial frames of references. However Einstein changed all that with general relativity. In the article The Foundation of the General Theory of Relativity, [2] Einstein presented the following argument

In classical mechanics, and no less in special relativity or relativity, there is an inherent epistemological defect which was, perhaps for the first time, clearly pointed out by Ernst Mach. We will elucidate it by the following example: - Two fluid bodies of the same size and nature hover freely in space at so great a distance from each other and from all other masses that only those gravitational forces need be taken into account which arise from the interaction of different parts of the same body. Let the distance between the two bodies be invariable, and in neither of the bodies let there be any relative movements of the parts with respect to one another. But let either mass, as judged by an observer at rest relatively to the other mass, rotate with constant angular velocity about a line joining the masses. This is a verifiable relative motion of the two bodies. Now let us imagine that each of the bodies has been surveyed by means of measuring instruments at rest relatively to itself, and let the surface of S1 prove to be a sphere, and that of S2 an ellipsoid of revolution.

A picture of what Einstein describes is shown in Figure 4 below

Einstein continues,

Thereupon we put the question – What is the reason for the difference in the two bodies? No answer can be admitted as epistemologically satisfactory, unless the reason given is an observable fact of experience. The law of causality has not the significance of a statement as to the world of experience, except when observable facts ultimately appear as causes and effects.
     Newtonian mechanics does not give a satisfactory answer to this question. It pronounces as follows: - The laws of mechanics apply to space R
1, in respect to which the body S1 is at rest, but not to the space R2, in respect to which the body S2 is at rest. But the privileged space R1 of Galileo, thus introduced, is a merely factitious cause, and not a thing that can be observed. It is therefore clear that Newton’s mechanics does not really satisfy the requirement of causality in the case under consideration, but only apparently does so, since it makes the factitious cause R1 responsible observational difference in the bodies S1 and S2
     The only satisfactory answer must be that the physical system consisting of bodies S
1 and S2 reveals within itself no imaginable cause to which the differing behavior of S1 and S2 can be referred. The cause must therefore lie outside this system. We have to take it that the general laws of motion, which in particular determine the shapes of S1 and S2, must be such that the mechanical behavior of S1 and S2 is partly conditioned, in quite essential respects, by distant masses which we have not included in the system under consideration. These distant masses and their motion relative to S1 and S2 must then be regarded as the seat of the causes (which must be susceptible to observation) of the different behavior of the two bodies S1 and S2. They take over the role of the factious cause R1. Of all imaginable spaces R1, R2, etc., in any kind of motion relatively to one another, there is none which we may look upon as privileged a prori without reviving the above-mentioned epistemological objection. The laws of physics much be of such a nature that they apply to systems of reference in any kind of motion. Along this road we arrive at an extension of the postulate of relativity.

We therefore see that in general relativity there are no special frames, such as inertial frames, which require special attention be paid to them.

Definition - Inertial force: When the motion of the reference system generates a force (defined as the time rate of change of momentum, i.e. f º dp/dt, which has to be, as measured in that system, we call that force an inertial force.

From Gravitation, by Misner, Thorne and Wheeler, Box 6.1, page 164

A tourist in a powered interplanetary rocket feels "gravity." Can a physicist by local effects convince him that this "gravity" is bogus? Never, says Einstein's principle of the local equivalence of gravity and accelerations. But then the physicist will make no errors if he deludes himself treating true gravity as a local illusion caused by acceleration. Under this delusion, he barges ahead and solves gravitational problems by using special relativity: if he is clever enough to divide every problem into a network of local questions, each solvable under such a delusion, then he can work out all influences of any gravitational field. Only three basic principles are invoked: special-relativity physics, the equivalence principle, and the local nature of physics. They are simple and clear. To apply them, however, imposes a double task: (1) take spacetime apart into locally flat pieces (where the principles are valid), and (2) put the pieces together into a comprehensible picture. To undertake this dissection and reconstruction, to see curved dynamic spacetime inescapably take form, and to see the consequences for physics: that is general relativity.

From Introducing Einstein's Relativity, by Ray D'Inverno, Oxord/Clarendon Press, (1992) page 122  

Notice that all inertial forces have the mass as a constant of proportionality in them. The status of inertial forces is again a controversial one. One school of thought describes them as apparent or fictitious which arise in non-inertial frames of reference (and which can be eliminated mathematically by putting the terms back on the right hand side). We shall adopt the attitude that if you judge them by their effects then they are very real forces. [Author gives examples]



Albert Einstein -That the relation of gravity to inertia was the motivation for general relativity is expressed in an article Einstein wrote which appeared in the February 17, 1921 issue of Nature [28]

Can gravitation and inertia be identical? This question leads directly to the General Theory of Relativity. Is it not possible for me to regard the earth as free from rotation, if I conceive of the centrifugal force, which acts on all bodies at rest relatively to the earth, as being a "real" gravitational field of gravitation, or part of such a field? If this idea can be carried out, then we shall have proved in very truth the identity of gravitation and inertia. For the same property which is regarded as inertia from the point of view of a system not taking part of the rotation can be interpreted as gravitation when considered with respect to a system that shares this rotation. According to Newton, this interpretation is impossible, because in Newton's theory there is no "real" field of the "Coriolis-field" type. But perhaps Newton's law of field could be replaced by another that fits in with the field which holds with respect to a "rotating" system of co-ordinates? My conviction of the identity of inertial and gravitational mass aroused within me the feeling of absolute confidence in the correctness of this interpretation.

A.P. French - Inertial force is defined as the force on a body that results solely from observing the motion of the body from a non-inertial frame of reference. This in addressed in Newtonian Mechanics, A.P. French, The M.I.T. Introductory Physics Series, W.W. Norton Pub. , (1971) , page 499. After describing the inertial force as seen from an accelerating frame of reference French writes

From the standpoint of an observer in the accelerating frame, the inertial force is actually present. If one took steps to keep an object "at rest" in S', by tying it down with springs, these springs would be observed to elongate or contract in such a way as to provide a counteracting force to balance the inertial force. To describe such force as "fictitious" is therefore somewhat misleading. One would like to have some convenient label that distinguishes inertial forces from forces that arise from true physical interactions, and the term "psuedo-force" is often used. Even this, however, does not do justice to such forces experienced by someone who is actually in the accelerating frame of reference. Probably the original, strictly technical name, "inertial force," which is free of any questionable overtones, remains the best description.

Cornelius Lanczos - The subject of inertial force is also addressed in The Variational Principles of Mechanics - 4th Ed., Cornelius Lanczos, Dover Pub., page 98.

Whenever the motion of the reference system generates a force which has to be added to the relative force of inertia I’, measured in that system, we call that force an “apparent force.” The name is well chosen, inasmuch as that force does not exist in the absolute system. The name is misleading, however, if it is interpreted as a force which is not as “real” as any given physical force. In the moving reference system the apparent force is a perfectly real force, which is not distinguishable in its nature from any other impressed force. Let us suppose that the observer is not aware of the fact that his reference system is in accelerated motion. Then purely mechanical observations cannot reveal to him that fact.

John A. Peacock - From Cosmological Physics, John A. Peacock, Cambridge University Press, (1999), page 6-7

INERTIAL FRAMES AND MACH’S PRINCIPLE  We have just deduced in a rather cumbersome fashion the familiar fact F= ma only applies in inertial frames of reference. What exactly are these? There is a well-known circularity in Newtonian mechanics, in that inertial frames are effectively defined as being those sets of observers for whom F= ma applies. The circularity is only broken by supplying some independent information about F – for example, the Lorentz force F = q(E + v´B) in the case of a charged particle. This leaves us in a rather unsatisfactory situation: F= ma is really only a statement about cause and effect, so the existence of non-inertial frames comes down to saying that there can be a motion with no apparent cause. Now, it is well known that F= ma can be made to apply in all frames if certain ‘fictitious’ forces are allowed to operate. In respectively uniformly accelerating and rotating frames, we could write

F = ma + mg

F = ma + mW´(W´r) – 2m(v´W) + (dW/dt)´r

The fact that these ‘forces’ have simple expressions is tantalizing: it suggests that they should have a direct explanation, rather than taking the Newtonian view that they arise from an incorrect choice of reference frame. The relativist’s attitude will be that if our physical laws are correct, they should account for what observers see from any arbitrary view – however perverse.

            The mystery of inertial frames is deepened by a fact of which Newton was well aware, but did not explain: an inertial frame is one in which the bulk matter in the universe is at rest. This observation was taken up in 1872 by Ernst Mach. He argued that since the acceleration of particles can only be measured relative to other matter in the universe, the existence of inertia for a particle must depend on the existence of other matter. This idea has become known as Mach’s principle, and was a strong influence on Einstein in formulating general relativity. In fact Mach’s idea ended up very much in conflict with Einstein’s eventual theory – most crucially, the rest mass of a particle is a relativistic invariant, independent  of the gravitational environment in which a particle finds itself. However, controversy still arises in debating whether general relativity is a truly ‘Machian’ theory – i.e. one in which the rest frame of the large-scale matter distribution is inevitably an inertial one (…)

            A hint at an answer to this question comes by returning to the expression for the inertial forces. The most satisfactory outcome would be to dispose of the notion of inertial frames altogether, and to find a direct physical mechanism for generating ‘fictitious’ forces. Following this route in fact leads us to conclude that Newtonian gravitation cannot be correct, and that inertial forces can be affectively attributed to gravitational radiation. Since we cannot at this stage give a relativistic argument, consider the analogy with electromagnetism … It is highly plausible that something similar goes on in the generation of inertial forces via gravity, and we can guess the magnitude by letting e/(4pe0) ® Gm. This argument was proposed by Dennis Sciama, and is known as inertial induction. …. Thus, it does seem qualitatively valid to think of inertial forces as arising from gravitational radiation. Apart from being a startling different view of what is going on in non-inertial frames , this argument also sheds light on Mach’s principle: for a symmetric universe, inertial forces clearly vanish in the average rest frame of the matter distribution. Frames in constant relative motion are allowed because (in this analogy) a uniformly moving charge does not radiate.

            It is not worth trying to make this calculation more precise, as the approach is not really even close to being a correct relativistic treatment. Nevertheless, it does illustrate very well the prime characteristic of relativistic thought: we must be able to explain what we see from any point of view.


            It may seem that we have actually returned to something like the Newtonian viewpoint: gravitation is merely an artifact of looking at things from the 'wrong' point of view. This is really not so; rather, the important aspects of gravitation are not so much first order effects as second order tidal forces: They cannot be transformed away and are the true signature of gravitating mass. However, it is certainly true in one sense to say that gravity is not a 'real' force, the gravitational acceleration is not derived from a 4-force and transforms differently.

Albert Einstein, in a letter to Lincoln Barnett (1948), wrote

The concepts of Physics have always been geometrical concepts and I cannot see why the gik field would be called more geometrical than f.i. the electro-magnetic field or the distance between bodies in Newtonian Mechanics. The notion probably comes from the fact that the mathematical origin of the  is the Gaussian-Riemann theory of the metrical continuum which we are wont to look at as part of a geometry.


[1] A Coriolis Tutorial, by James F. Price, Woods Hole Oceanographic Institution, January 10, 2006.
[2] The Foundation of the General Theory of Relativity, by Albert Einstein, Annalen der Physik, 49, 1916.

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