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Inertia and gravity seem to be two quite different phenomena. However, as shown by somebody careless dropping lead balls off a badly built tower in Pisa, inertia and gravity are proportional.So, somehow they must be related. Relativity has it that they both deform space-time, but that does not explain much. Does anybody have an idea?

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.

My point, exactly. However, could we develop such a basic principle, we might be able to advance our understanding of Cosmos. At this stage even some wild idea would be worth considering.

It IS easier to say than to do. But without ambition, you go nowhere. With ambition, well maybe

Professional scientists tend to work with projects with high probability of success, even though the result might not expand our understanding significantly.

Well, it is true that generalizing is ALLWAYS, without exception, dangerous. However, I work with scientists (ok, no physicists) all the time. Most are interested in their career and their next paper, only (generalizing again).

Professional scientists tend to work with projects with high probability of success, even though the result might not expand our understanding significantly. To do the opposite is a sure way to put a promising career behind you.

Moreover, often amateurs, not burdened by ruling paradigms, are better thinking out of the box. Think of Albert Einstein who did the most revolutionary thinking before he turned professional scientist.

It is the path that requires no force to travel.

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.

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]

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 "pseudo-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.

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.

It is possible to launch a rocket from the earth in September to arrive in October to the point in space that the earth will reach only in December. Supposing it had enough fuel for the required acceleration. It would be a shorted path that the earth orbit, but that path would not be a "free lunch".

Think of Albert Einstein who did the most revolutionary thinking before he turned professional scientist.

That's not quite true. It's the path that requires only the gravitational force to travel. In general relativity the gravitational force is interpreted to be an inertial force.

In the first place we're talking not about the shortest path but an extremal path. And the magnitude of the spacetime interval between two points on a worldline in spacetime is not determined by the Euclidean metric but by the Minkowski metric. E.g. consider a photon which travels in flat spacetime between point A and point B. Suppose the spatial distance between A and B is d. The magnitude of the spacetime interval is 0. That's an example of how different Euclidean geometry is from Minkowski geometry.

I agree that it is useful for calculation to take gravity as a (inertial) force.

Generally speaking, all geodesical paths in the spacetime are inertial movements if we understand inertia as absence of forces.

By the way, my path in the spacetime, being in this couch, is not a geodesical one, so it is not inertial.

I was answering a remark from yor_on about curved space, not spacetime.

In a flat spacetime, the Minkovski geometry applies, and the geodesicals are straight lines. that correlates well to uniform retinear inertia movement.

What exactly do you mean by "Minkowski geometry applies"?

Einstein was more interested in what made sense theoretically and not what made things easy to calculate.

You neglected to mention that all its necessary that all non-inertial forces be zero. It’s fine to have inertial forces acting. And remember that inertial forces being “real” was a very important part of Einstein’s thinking that led him to GR.