How inertial frame of reference differs from non-inertial frame?

We know that we can't say whether we are at rest or uniformly moving if we're in a einstein cage..but if the same medium is accelerating/decelerating can we being inside(and can't see outside) claim abt state of cage..?I ve read that a non-inertial can be converted to inertial by incorporating a fictious force..?? Does this fictious force has any orgin.If yes,where this origin disappears when cage stops accelerating.?If there are 2 bodies inside a cage which is accelerating uniformly,does 1 body see other body in rest..?

Dale
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You can tell if your cage is inertial or not simply by using an accelerometer comoving with the cage. If it reads 0 then the cage is inertial.

well..I remember a example stating why non-inertial frames can be recognised being inside.When the person inside a cage tosses a coin,if the coin falls back in this hand then he's in inertial frame else if it falls right ahead of him or behind he's in decelerating or accelerating frame..but why does this phenomenon occur,why can't the coin in same way as inertial frame fall back into his hand by consuming the acceleration in the air also while tossed

Dale
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If the cage is inertial then the coin won't fall back into his hand, it will continue to move in a straight line at a constant speed. That is Newtons first law.

HallsofIvy
Homework Helper
That is assuming no external forces. Am I in an "inertial coordinate system" sitting here in front of my computer? If I were to toss a coin in the air, it would come back to my hand!

Any body inside a inertial frame is a source of non-inertiality because the body is held intact by forces..does that mean there's NO INERTIAL FRAME in known physical applications.?

A.T.
Am I in an "inertial coordinate system" sitting here in front of my computer?
According to GR, no.
If I were to toss a coin in the air, it would come back to my hand!

Dale
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Any body inside a inertial frame is a source of non-inertiality because the body is held intact by forces..does that mean there's NO INERTIAL FRAME in known physical applications.?
As long as the net real force is zero an object may be subject to multiple external forces and still be inertial. Similarly with the various sub-parts of an extended body.

HallsofIvy
Homework Helper
Ah. So a room stationary on the surface of the earth is "non-inertial" and a room falling with non-zero accleration due to gravity is "inertial".

DrGreg
Gold Member
Ah. So a room stationary on the surface of the earth is "non-inertial" and a room falling with non-zero acceleration due to gravity is "inertial".
Yes, although to be precise and avoid ambiguity it would be better to say "non-zero acceleration relative to the Earth's surface". A free-falling room has zero "proper acceleration" i.e. acceleration relative to a local inertial observer.

A.T.
Ah. So a room stationary on the surface of the earth is "non-inertial" and a room falling with non-zero accleration due to gravity is "inertial".
Yes. In GR gravity is an inertial force which appears only in non-inertial frames. An answer to the OPs question valid for both: Newton and GR is:

Inertial frames are those where all inertial forces disappear (all Newtons Laws hold).

The difference is only the classification of gravity as a interaction or inertial force respectively.

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Matterwave
Gold Member
The coin trick has to assume that the coin is not charged, or that there is no external electro-magnetic field (or both). Only gravity gets to modify what you call "inertial" or not.

My question is still not answered completely.! Whats use of fictious force in this context..and how to convert a non-inertial frame to inertial frame.and does a body appear stationary when watched by a object in its accelerating frame.

My question is still not answered completely.! Whats use of fictious force in this context..and how to convert a non-inertial frame to inertial frame.and does a body appear stationary when watched by a object in its accelerating frame.

You cannot convert a non inertial frame to an inertial frame. The proper force experienced in a non inertial frame cannot be transformed away. We can sometimes use the concept of an "instantaneous co-moving inertial reference frame" to study non inertial frames. For example let us say we have a rocket accelerating with a constant proper acceleration of one g. In an initial reference frame S, the velocity of the rocket is zero. At a later time the rocket is momentarily at rest in inertial reference frame S' which is moving at 0.6c relative to S. At an even later time the rocket is momentarily at rest in an inertial reference frame S'' moving at 0.8c relative to S and so on. In inertial reference frames, S, S' and S'' the rocket is always measured to have an acceleration of g when the rocket is momentarily at rest, but in frame S for example, the initial acceleration is g but progressively the acceleration gets slower as the velocity of the rocket increases relative to S.

Now if we have object accelerating with constant proper acceleration then there exists a non inertial reference frame, in which the object will appear stationary. If we have an inertial object that is accelerating from the point of view of a non inertial reference frame, then there is also exists an inertial reference frame in which the object appears to be stationary. This is trivially obvious. However, if we have a non inertial object experiencing proper acceleration, then there is no inertial reference frame where the object appears stationary for more than an instant. Conversely, if we have a inertial object with coordinate acceleration as measured in a non inertial reference frame, then there is no non inertial reference frame where the object appears stationary for more than an instant.

DrGreg
In relativity this is taken account of by the difference between a coordinate derivative and a covariant derivative. Newton's law in a non-inertial frame becomes$$F^\alpha = \frac{\mbox{D} P^\alpha}{\mbox{D} \tau} = \frac{\mbox{d} P^\alpha}{\mbox{d} \tau} + \Gamma^\alpha_{\beta\gamma} U^\beta P^\gamma$$The term involving $\Gamma^\alpha_{\beta\gamma}$ is the "fictitious force" term and becomes zero in an inertial frame.