**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

**
**

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 **F*** _{I
}*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 **v**_{0}
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 **F**_{I}.
In Figure (2b) **v**_{0}
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 S_{1}
is at rest, but not to the space R_{2},
in respect to which the body S_{2}
is at rest. But the privileged space R_{1}
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 S_{1}
and S_{2}

The only
satisfactory answer must be that the physical system consisting of bodies S_{1}
and S_{2}
reveals within itself no imaginable cause to which the differing behavior of S_{1}
and S_{2}
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 S_{1}
and S_{2}
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 S_{1}
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 S_{1}
and S_{2}.
They take over the role of the factious cause R_{1}.
Of all imaginable spaces R_{1},
R_{2},
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
*d***p**/*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 Pres*s, (1999), page
6-7

INERTIAL
FRAMES AND MACH’S PRINCIPLE We
have just deduced in a rather cumbersome fashion the familiar fact **F**= *m***a**
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**= *m***a**
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**= *m***a** 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**= *m***a**
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**
= *m***a** + *m***g**

**F**
= *m***a** + *m***W**´(**W**´**r**)
– 2*m*(**v**´**W**)
+ (*d***W**/*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*/(4p*e*_{0}) ® *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 g

_{ik}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.

**References:**

[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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