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I understand from a non-rotating falling reference frame that that gravity is fictitious since there is no apparent force acting on the falling observer. However, from a rotating frame of reference due to gravity (e.g. or orbits) distant objects may seem to be orbiting around the observer faster than $c$. So why can't we say that this frame is non inertial since this reference frame indicates that distant objects seem to be travelling faster than light even though from a non rotating reference frame its not.

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    $\begingroup$ It seems to me that the title of your question is not well matched to the content of your question. $\endgroup$ – Andrew Steane Apr 12 at 13:21
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This question is about the distinction between the concept of inertial motion and the actual layout of the matter in the universe.

First of, the internal dynamics of a small body under no external force apart from gravity are the same as those of a similar body situated in empty space far from any gravitating object. Therefore the state of motion of such a body earns the name 'inertial' irrespective of how that body may be moving relative to other things such as the distant stars.

Next, be careful about your example with orbits. A frame in Earth orbit, for example, is not rotating except by a tiny amount owing to de Sitter precession (unless someone made it rotate at some other rate, but then it wouldn't be an inertial frame). Having said that, even that tiny rotation would be enough to result in faster-than-light relative motion (in the transverse direction) between that frame and distant objects, according to some reasonably sensible definition of relative motion when objects are far apart. The way most people familiar with general relativity argue here is that the concept of relative velocity for things not at the same event in spacetime is rather hard to define in a useful way in general relativity. You are right that there are plenty of situations where a pair of inertial frames separated by distances on the order of many lightyears might be said to have a relative velocity many times the speed of light. But the 'relative velocity' invoked in such statements does not have any direct impact on any physical effect.

Relative velocity between two entities located at the same event and moving past one another is, on the other hand, a well-defined four-vector.

Finally, by observing the distant stars it is certainly possible to pick out one local frame from others. Here in the solar system, for example, there is just one state of motion which makes the cosmic microwave background radiation look isotropic to first approximation (to be precise, to have no dipole term). When we say that different inertial frames have some sort of equivalence between them, we are not denying this. The equivalence is to do with the internal dynamics for systems at rest relative to one frame or another. But among all these frames, you can certainly pick out one and say 'this is the one in which the motion of the universe on the largest scale is the simplest'. It doesn't mean that other local inertial frames are not inertial.

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A rotating frame orbiting a planet under gravity does not qualify as an inertial frame because, although in free fall, it does not satisfy the definition of an inertial frame, most simply expressed as

  • An inertial reference frame is one in which inertial bodies remain at rest or in uniform motion.

It is implicit in this definition that an inertial frame is local, in time as well as space. The size of an inertial frame ddepends upon the accuracy of measurement, but it is worth noting that a second in time corresponds to a light-second in distance. In terms of normal timescales, local refers to quite short intervals of time.

Tidal forces become apparent in the orbiting frame within a fairly short interval on ordinary time scales, showing that it is not an inertial frame.

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