gravity - Can we determine an absolute frame of reference taking into account general relativity?

Given that acceleration induces measurable physical effects, would it be correct to say that there should be an absolute inertial frame of reference? I know that one cannot distinguish a priori between acceleration and gravitational effects, but there should be a determined distribution of mass in the universe, and assuming it is known, its effects should be able to be subtracted to deduce 'absolute' acceleration. Is this incorrect?

Answer

The problem is that to determine the distribution of mass in the universe you need to choose a coordinate system that you're going to be using for measuring the positions of all those masses. The trouble is that you are free to choose whatever coordinate system you want to make this measurement. There is no absolute coordinate system for measuring the mass distribution. Your choice of coordinate system will determine how much of any acceleration you measure is inertial and how much is gravitational.

The four-acceleration is given by:

$$ A^\alpha = \frac{\mathrm d^2x^\alpha}{\mathrm d\tau^2} + \Gamma^\alpha{}_{\mu\nu}U^\mu U^\nu $$

and speaking rather loosely the first term on the right is the inertial acceleration and the second term is the gravitational acceleration. The problem is that while the four-acceleration is a tensor the two terms on the right are not. It is always possible to choose a coordinate system that makes the inertial acceleration zero - in fact this is simply the rest frame of the accelerating object. Likewise it's always possible to choose coordinates that make the Christoffel symbols, $\Gamma^\alpha{}_{\mu\nu}$, equal to zero - these are the normal coordinates.

This is the equivalence principle in action. While the four-acceleration is a tensor, and therefore a coordinate independent object, the two terms on the right can be interchanged by a choice of coordinates making the acceleration look purely inertial, purely gravitational, or some combination of the two just by changing coordinates.

Since there is no absolute coordinate system for measuring the mass distribution there is no absolute coordinate system for measuring the inertial acceleration. The two types of acceleration are fundamentally indistinguishable.