Equivalence Principle

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The Principle of Equivalence is generally defined as being the principle that a body's inertial mass and gravitational mass are different aspects of the same underlying property. A weaker version of the principle is that a body's inertial and gravitational masses are proportional.

The immediate result of the principle of equivalence is that if the gravitational mass of a body (which persuades it to change its motion in a gravitational field) and the body's inertial mass (which resists that change in motion) always have exactly the same proportion, then all bodies immersed in the same simple background gravitational field should fall at the same rate, giving Eotvos' Principle.

A further assumption that gravitational and inertial effects are interchangeable then leads to the idea that inertial and gravitational descriptions are fully interchangeable, giving Mach's Principle and a general theory of relativity.

Duality of gravitation and inertia

In the "bucket" thought-experiment in Newton's Principia, we are invited to consider the case of a spinning bucketful of water that has reached equilibrium.

Considered in the frame in which the background stars are fixed, we see that the surface of the water is curved – the components of the rotating body of water are attempting to travel in inertial straight lines, but are being prevented by the walls of the bucket, causing the water to ride up the bucket sides. According to Newton, we can take this identifiable curvature of the water's surface as demonstrating that the bucket is "really" rotating, in the sense that the consequences of it motion are real for all observers, and cannot be transformed away with an arbitrary choice of coordinate systems.
We can say that within this interpretation, the curve of the water surface is due to its inertial mass.

However, Mach pointed out an alternative explanation:

Considered in the frame in which the bucket is stationary and the background stars rotate, we are forced to agree that the surface of the water is curved, but we can no longer blame the effect on inertia, as nothing in the immediate vicinity of the experiment is moving. The inertial explanation can only be used here is we say that our own experiences are invalid, and that the experiences and interpretation of a different ("nonrotating") observer, with a different state of motion, override ours. However, if we require that physics be seen to be consistent in our own frame ("observerspace"), we can point out that our instruments seem to report the existence of an outward-pointing radial gravitational field associated with the rotation of the background shell of surrounding stars - the outward force increases if the star-shell seems to rotate faster, and the field is aligned to point directly away from the central axis around which the shell rotates. It is this field which pulls the water away from the centre of the bucket and causes it to press harder "downhill" against the surrounding bucket walls,

We can say that within this interpretation, the curve of the water surface is due to its gravitational mass.