Is it always possible to determine whether or not one is accelerating? Consider the following two situations: A: You wake up in an elevator that is in free fall in a gravitational field.
B: You wake up in an elevator that is floating in a vacuum.
Is it possible to distinguish between these two situations?
It seems to me that Newton's second law formulated in a local coordinate system would look the same whether or not situation A or B is the truth: If P is an object in the elevator (which we denote by E), Newton's second law would be:
$$F_P = m_P\ddot{x}_P \Rightarrow$$
$$(-m_Pg + F'_P) = m_P(\ddot{x}_E+\ddot{x}_{P\backslash E})$$
where $F'_P$ is the force applied to the object in addition to whatever gravitational force is present, $x$ denotes position relative to an inertial frame, and $x_{P\backslash E}$ is the position of the object relative to the elevator.
Since the elevator is in free fall,
$$F_E = m_E\ddot{x}_E \Rightarrow -m_Eg = m_E\ddot{x}_E \Rightarrow \ddot{x}_E = -g$$
Thus, the equation of motion for the object P becomes
$$F'_P = m_P\ddot{x}_{P\backslash E}$$
But this would be true regardless of the value of $g$! Or am I missing something? If not, is there any other way of knowing whether or not situation A or B is the truth?
A: You might want to have a look at Accelerating Frame and Gravity as it is closely related though not an exact duplicate. The answer is that there is locally no way to tell the difference between your two cases $A$ and $B$. In fact this is enshrined in Einstein's equivalence principle.
Note the emphasis on the local equivalence. Gravitational fields are never completely uniform so for a freely falling observer there are always some tidal forces acting, and if your elevator is big enough you could tell the difference between the two cases. Suppose you place test particles near the walls of the elevator and initially at rest with respect to yourself. The tidal forces mean that in some directions those test particles would accelerate towards you and in other directions they would accelerate away from you. Measuring this acceleration would allow you to determine that you're falling in a gravitational field rather than floating in space.
A: Well, the conventional answer is that you cannot distinguish the two situations. Einstein's General Relativity is based upon not being able to distinguish between the two situations.
However - if you were falling towards a planet or star etc individual objects in free fall would appear to move towards each other because the gravitational field is not uniform - they are falling towards a central point - the centre of mass.
