Equivalence principle & nonlinearity

[Thrice]

How does nonlinearity of the gravitational field equations follow from the equivalence principle? I remember hearing a handwaving example of this & I'm interested in more details.

 

[MeJennifer]

How does nonlinearity of the gravitational field equations follow from the equivalence principle? I remember hearing a handwaving example of this & I'm interested in more details.

The closer a collection of masses come together the less the sum of their inertial masses becomes. Hence, to maintain equivalence the total gravitational mass has to couple to this increasing "defecit".

 

[Entropia]

The equivalence principle is a fundamental concept in physics that states that the effects of gravity are indistinguishable from the effects of acceleration. This means that an observer in a uniform gravitational field cannot tell whether they are being accelerated or in a stationary frame in a gravitational field. This principle has been confirmed by numerous experiments and is a cornerstone of Einstein's theory of general relativity.

One consequence of the equivalence principle is that the gravitational field equations must be nonlinear. This can be understood by considering a simple example. Imagine an observer standing on a scale in a stationary elevator on the surface of the Earth. The scale will read their true weight, which is the force of gravity acting on their mass. Now, imagine the same observer in an elevator that is accelerating upwards at the same rate as the gravitational field on Earth. In this case, the observer will also read their true weight on the scale. This is because the acceleration of the elevator is canceling out the effects of the gravitational force.

However, if the gravitational field equations were linear, the observer would not read their true weight in the accelerating elevator. Instead, the scale would read a different value depending on the strength of the gravitational field. This would violate the equivalence principle, as the observer would be able to distinguish between being in a stationary gravitational field and being in an accelerating frame.

Therefore, the nonlinearity of the gravitational field equations is a direct consequence of the equivalence principle. It ensures that the effects of gravity and acceleration are always indistinguishable, as observed in experiments. This is just one example of how the concept of equivalence plays a crucial role in understanding the behavior of gravity and the universe as a whole.