# Understanding non-inertial reference frames in CM and SR.

• dipole

#### dipole

Sorry if this is a frequent topic, but I think it's interesting and probably worth repeating if it's come up often before. Let me lay out my question by example (and if anything is inaccurate feel free to point it out), and then I'll summarize at the end.

It's easy to show in Classical Mechanics and Special Relativity that all inertial reference frames are equivalent and equally valid. In classical mechanics, if you perform a Galilean Transformation you'll find that that the Lagrangian is invariant (assuming you properly include any induced Electric or Magnetic Fields), and in SR we know well that Lorentz Transformations leave the laws of physics invariant. This is a basic symmetry of nature that we can equally well look at a problem from multiple inertial frames of reference, and they are all equally good. It doesn't matter weather you consider the particle to be at rest w.r.t. some chosen coordinate system, or moving w.r.t. a coordinate system, everything works out in the end so that everyone agreed on what happened.

As soon as you start introducing non-inertial frames of reference (rotating frames or linearly accelerating) these symmetries disappear. You no longer have the freedom to choose which you consider to be an accelerated frame or not, you only have the inertial frames, and the non-inertial frame, and you must distinguish from the two. In Classical Mechanics this is done in an almost ad-hoc way by introduction of the so-called fictitious forces within the non-inertial frame.

If I am in a rotating frame of reference I will experience a Coriolis Force and Centrifugal force which are not present in inertial frames. I can't think of myself as being in a stationary frame, and the rest of the universe rotating. For example, if I'm sitting on a rotating table, and I look at two balls which are outside the rotating frame, laying on an adjacent table which isn't rotating, them from my point of view the other table appears to be rotating about my axis of rotation. However, there is no inclination for the balls on the other table to want to fly away from that axis of rotation, and if I were to try and think of myself as stationary, and the other frame of reference as the rotating one, I'd surely make some very bizarre and untrue predictions

There is clearly something different about non-inertial reference frames from inertial ones, and this is what my question is about. Can you understand why non-inertial reference frames are so different within the context of Classical Mechanics and Special Relativity, or is it simply a postulate of the two theories that this is true, by experience?

If not, then I assume the answer lies somewhere within General Relativity. I've tried reading up about Mach's Principle and the Equivalence Principle, and how they relate to this, but for some reason I can't seem to extract a clear answer to my question, which means either my question is not well-defined or there is some uncertainty about all of this still.

It's just an innate property of Newtonian mechanics and SR that states of uniform velocity are relative whereas states of acceleration/rotation are absolute; one can easily operationally locally detect acceleration/rotation unambiguously whereas uniform velocity carries with it an ambiguity from frame to frame in the context of these theories. When one applies the frameworks of said theories (such as in the manner in which you have done with the Coriolis force experiment) this is what one finds.

Note also that this does not change when one goes to GR. Furthermore, GR is not a Machian theory (in the sense originally intended by Mach) for the aforementioned reason. The equivalence principle isn't really related to this; all it says is that locally one can transform away the gravitational field.

If I am in a rotating frame of reference I will experience a Coriolis Force and Centrifugal force which are not present in inertial frames. I can't think of myself as being in a stationary frame, and the rest of the universe rotating.

(A)

For example, if I'm sitting on a rotating table, and I look at two balls which are outside the rotating frame, laying on an adjacent table which isn't rotating, them from my point of view the other table appears to be rotating about my axis of rotation.

(B)

However, there is no inclination for the balls on the other table to want to fly away from that axis of rotation, and if I were to try and think of myself as stationary, and the other frame of reference as the rotating one, I'd surely make some very bizarre and untrue predictions

In (A) and (B) you confused a rotating frame with a rotating object. If you introduce a reference frame, then you cannot just say "it ends here". It goes all the way to "the edge of the Universe" (ignoring GR's manifold-related complications here). So the "rest of the Universe" is also in the rotating frame.

You can introduce a rotating frame even if there is no rotating object anywhere, and then suddenly everything, as seen from that frame, will move in weird trajectories, in full accord with the Coriolis and centrifugal forces - even though these same trajectories are rectilinear as seen from some inertial frame.