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Suppose I am in an arbitrary reference frame and i want to know if it is inertial or not. So I decide to do a test, i take a particle of mass $ m $ and let go of it. I observe that the particle is accelerating backwards with an acceleration $ a $. Therefore, i can conclude from this two things:

1st: I am in an non-inertial reference frame.

2nd: It could be that someone is 'tricking' me. It might be the case that I am standing still and there is a uniform field that interacts with my particle and exerts on it a force $ -ma \hat{i} $ on it.

Is there a way to distinguish between the two ?

Further thoughts

If i have complete knowledge of the forces that are exerted on the particle that i can easily deduce whether i am in an inertial reference frame or not as I can simply check Newton's second law and see if it agrees with observation. For example, in the first case, it is:

$$ \vec{F} = m \vec {a} \Rightarrow \vec{0}=m\vec{a} \Rightarrow \vec{a} = \vec{0} $$ ($\vec{F}=\vec{0} $ as i 'let go of the particle of mas $m$') That results contradicts what I observe and therefore the second Law of Newton does not hold and by definition I am not an inertial reference frame.

For the second case, we know that: $$ \vec{F}=m\vec{a} \Rightarrow -ma\hat{i}=m\vec{a} \Rightarrow \vec{a}=-a\hat{i} $$ This time, Newton's second law agrees with observation and therefore I am an inertial reference frame.

The conclusion is that: If we have complete knowledge on the forces we know the kind of reference frame we're in. The thing is that in reality we are never able of determining the exact forces that are exerted on a particle.

So how can we really know whether we are in an inertial reference frame or not ?

Thank you in advance for your answers.

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It is in fact not possible to distinguish between these two situations. This observation is called the equivalence principle. It is the basis of general relativity.

Note that this principle is stated with regards to gravity, not any generic interaction, because the only field that will impart the same acceleration to everything inside it, is a uniform gravitational field.

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You are on the right track. First one studies acceleration under various forces and one eventually arrives at a definition of inertial mass. After that one can notice whether the local forces you are experiencing are all proportional to inertial mass, with the same proportionality constant. If they are then you can go into free fall and the forces will go away.

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