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Let’s assume we have 2 different observers. Observer 1 sits in space and observer 2 sits in a space lab which is in a free fall state toward the Earth. We further assume that observer 2 in the space lab does not have any information regarding its surrounding. (e.g Space lab has no windows and observer 2 does not know that there is planet around him which causes any gravitational force).

  1. Is there anyway that observer 2 can figure out whether any frame attached to the space lab can be regarded as an inertial or non-inertial frame? (According to the equivalence principle this should not be possible. Is this true?)
  2. Lets assume there is a 1kg ball inside the space lab which is also free falling together with the space lab? Observer 1 (with its true inertial frame) can measure the acceleration of the ball and deduce that 9.8Newton force is acting on the ball. On the other hand, observer 2 measures the acceleration of the ball as 0, and hence, deduces that the net force acting on the ball is 0. As the observer 2 has no way of knowing that space lab is not an inertial frame, he will not have any doubt about his force measurement. Does this mean that definition of force is dependent on the frame of reference?

Update after lesnik's answer:

  1. Is it possible to define an experiment inside the space lab which proves the observer 2 that he is in fact accelerating? From my understanding, as long as it is assumed that gravity field is uniform, all 3 laws of Newton hold in this free falling space lab without requiring any fictitious force definition. In this case, why cannot we clasify this free falling body as an inertial frame of reference in Newtonian Context?
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  • $\begingroup$ Posted answers good. Just one point.. you talk about observer 1 sitting in space, presumably you mean near the earth and able to watch the observer2 fall past in spacelab capsule. However your obs1 is going to need a floor to stop him falling or be moving sideways at 10 km/ sec in orbit. You need to be clear when posing question. Any floor will be pushing him up and he may think he is being accelerated by a rocket. So now your query goes the other way. Equivalence principle of Einstein. $\endgroup$
    – blanci
    Commented Sep 19, 2020 at 23:20

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Looks like you understand the situation already and only need slight confirmation/clarification.

Let's start with simple case. There is an inertial frame of reference, there are some objects, there are forces acting between objects. Objects are moving around, accelerated by the forces.

In any other inertial frame of reference, the forces between objects are the same.

Next case is more complicated. Consider the same set of objects in some non-inertial frame of reference. You can't just use usual Newton laws any more because you are not in an inertial frame of reference! But it may be very convenient for your purposes to use this frame, no matter is it inertial or not. F.e you describe what's going on inside a space station orbiting Earth. Using inertial frame of reference would be very inconvenient for this purpose!

It turns out that it is possible to use Newton's laws even in this situation. You only have to add some additional, "fictional" forces to the equations.

You switch to inertial frame of reference - these additional forces disappear. You switch back - the forces are back. Very real, you can feel how this force pushes you into the chair in airplane. So, the existence of force depends on frame of reference if there is no restriction that frames of reference must be inertial.

And sometimes it's not possible to decide, if your frame of reference is inertial or not. There is no way to find out if the force acting on all the objects in a room is a gravitational force produced by some real planet or is it a "fictional" force existing only because the room is accelerating.

Bottom line: forces do not change if you switch between inertial frames of reference, forces may change if you switch between non-inertial frames, sometimes it's tricky to find you if your frame of reference is inertial or not.

Update to part 3 of question

No, observer 2 can't make an experiment which would help him to decide if he is staying still or accelerating in gravity field. (As long as he stays inside his small box!) But we can't classify his frame of reference as inertial for the following reason.

In Newtonian Mechanics (and even in Special Relativity Theory) the frame of reference is "global". Inertial frame of reference is inertial everywhere, not just in some area around the observer. Which is obviously not true for observer's 2 lab.

General Relativity Theory is much more complicated. As far as I understand there are no inertial frames of reference at all. You can choose coordinates such that space-time is flat in the vicinity of some point (which means the frame of reference would be inertial in this area), but you can not choose coordinates so that space-time is flat everywhere. But I would rather not go deeper. I am not good enough in this area, sorry.

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  • $\begingroup$ There is still one point that bothers me. All 3 Newton's laws are valid in the free falling space lab. In other words, as far as I understand, the observer in space lab does not need to define any fictitious force to describe any motion he observes in the space lab. Therefore, from his point of view, he is in an inertial frame and the net force acting on the ball is 0. Is there any way that you can convince observer 2 in space lab is wrong? $\endgroup$
    – tantuni
    Commented Aug 11, 2017 at 9:43
  • $\begingroup$ Observer 2 ignores the fact that Earth attracts all the objects inside his lab. There is an excuse: it's not easy for him to find out if there is an Earth around. But still he ignores this fact. From the Newtonian mechanics point of view he just got lucky: ignored the gravitational field, did not account for inertial force, but these two errors compensated each other and he got correct results. $\endgroup$
    – lesnik
    Commented Aug 11, 2017 at 10:06
  • $\begingroup$ So in GR you can consider any freely falling observer as being in a local inertial frame. And @lesnik has it right on GR, you can not choose a global frame that is flat unless the spacetime has no curvature, i.e., it's a purely Minkowski spacetime. $\endgroup$
    – Bob Bee
    Commented Aug 13, 2017 at 3:06

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