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In the book Physics by Resnick, Halliday, and Krane, in chapter 3 there was a statement which confused me.

To test whether a particular frame of reference is an inertial frame, we place a test body at rest in the frame and ascertain that no net force acts on it. If the body does not remain at rest, the frame is not an inertial frame. Similarly, we can put the body (again subject to no net force) in motion at constant velocity; if its velocity changes, either in magnitude or direction, the frame is not an inertial frame. A frame in which these tests are everywhere passed is an inertial frame.

Initially, I didn't seem to have a problem with this reasoning. It always worked. But then I stood upon an example which in my opinion is an exception to the statement.

Consider a person in an elevator which is falling. If we perform the test and place an object inside the elevator, the object doesn't seem to move. This should mean the elevator frame is inertial, which in fact is not, since it is accelerated downwards.

What's the problem here? Why does the test fail in this case? Is there a problem in my reasoning? More importantly, what should be the restatement of the original test if this happens to be an exception?

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3 Answers 3

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The falling elevator's frame is inertial! Congratulations, you've found Einstein's equivalence principle.

In relativity, we have to consider the proper acceleration, which is a 4-vector. Gravitational fields do not impart a proper acceleration, and so are considered to be inertial frames of reference.

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    $\begingroup$ Locally inertial, it's worth pointing out $\endgroup$
    – ajd138
    Jun 4 at 17:02
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    $\begingroup$ Is the core here that because gravity is applied to everything within the frame equally, it is therefore cancelled out of the consideration? Putting it differently, for a suffiently large elevator where gravity affects things near the ceiling differently than things near the floor, does that invalidate the locally inertial nature of the frame? $\endgroup$
    – Flater
    Jun 5 at 5:24
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    $\begingroup$ @Flater Yes, this is what ajd138 meant by their clarifying statement. To be locally inertial means that we can rely on the predictions of special relativity rather than needing to concern ourselves with the effects of gravity. If, however, we consider a sufficiently large region that the metric over that space does not closely match a Minkowski metric, we can no longer consider it inertial. In other words, an ant on an apple thinks the surface is flat. $\endgroup$ Jun 5 at 5:37
  • $\begingroup$ @LoganJ.Fisher Can you please explain how it is related to equivalence principle ? $\endgroup$
    – bubucodex
    Jun 5 at 16:47
  • $\begingroup$ @bubucodex The equivalence principle can be expressed in two main ways. One, that inertial mass is equivalent to gravitational mass. Two, that a particle in a box can't tell the difference between being inside an accelerating rocket or on the surface of a gravitational body. Focusing on that second one, if accelerating up is the same as standing on the surface, then being in flat spacetime must be the same as being in free-fall. Ergo, free-fall is an inertial reference frame. $\endgroup$ Jun 5 at 16:53
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The elevator (or more exactly, a point object within the elevator) is following a geodesic, which is a generalization of the concept of an inertial frame.

The problem you hit in your reasoning is when you said "which in fact is not since it is accelerated downwards". The question to ask is, "How do we know it is accelerated downwards?" You'll find that there is no test the elevator rider can do from within the box that would tell him this$^1$. Only someone outside in a different reference frame will see the elevator accelerate.

A geodesic is a particle's natural path through curved spacetime. It is the path it will travel when no forces except gravity are acting on it. I say "except gravity" but in General Relativity gravity is a fictitious force that seems to occur for some observers due to gradients in spacetime curvature. So we should really just say an object follows a geodesic path "when no forces are acting on it." Geodesics include freefall, orbits, flybys of planets or stars, or even falling into a black hole. In all of these cases, the observer inside a closed elevator will not notice any of the outside physical phenomena, and can do no experiment to distinguish which scenario he is in. The reason objects get destroyed by black holes is the curvature gradient becomes so great that regions of uniform gravity become smaller than the elevator, and structures get ripped apart because different parts of them start to take different geodesic trajectories. But if you had an infinitesimally small elevator, it could fall all the way to the center of a black hole or participate in a binary neutron star merger (orbiting thousands of times per second at nearly lightspeed) and not detect anything different from freefall from inside the elevator box.

$^1$ In a uniform gravitational field.

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The two existing answers are correct, in a more general sense, but they do not actually address what the textbook is talking about. The problem is that the problem you have set up:

If we perform the test and place an object inside the elevator, the object doesn't seem to move,

does not describe a force-free situation! You are assuming that there is a gravitational force holding the object to the floor.* So the reasoning that the falling elevator frame is inertial is incorrect, because you have not correctly "ascertain[ed] that no net force acts on it." If there were truly no force (if, for example, the elevator car were being accelerated nongravitationally in deep space), in the frame moving with the car, you would see the object rise from where it was initially set and apparently accelerate toward the roof.

*The other existing answers point out that, in the more advanced context of general relativity, the gravitational field can be conceived of as something other than a force. (Instead, gravity involves objects moving along geodesics in a curved background spacetime. If a bunch objects are close enough together and are thus subject to the same gravitational field and following essentially the same geodesic, they will not change positions relative to one-another. This is the situation in a falling elevator on Earth.) However, that is not the situation that the book is trying to explain.

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