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The principle of general covariance from the Equivalence Principle (EEP) tells us that there is no way in principle to locally distinguish between an inertial acceleration and the effects of a gravitational field. When general covariance is expressed in tensorial form we find that if the equation holds true in an inertial frame that it must hold true in a non-inertial frame of reference.

My question is, if the above is true, is a gravitation field just an illusion? Why do we have to consider a gravitational field at all. Why not think of gravity as the effect of the bending of space and time, with gravitation revealed as an illusion, a by-product of geometry. I feel it is the presence of matter that warps space around it, giving us the illusion that there is a gravitation force pulling on neighboring objects.

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marked as duplicate by StephenG, Aaron Stevens, John Rennie general-relativity Dec 7 at 8:57

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What you've just described is Einstein's Theory of General Relativity. It proposes treating gravity not as a field (as Newton had done), but instead as a geometric property of space and time. The idea is that objects with mass bend space and time, drawing in objects around them. The theory has held up incredibly well for over 100 years.

The issue is that the theory of general relativity is not compatible with quantum field theory, which is used to describe all the other fundamental interactions of our universe.

These two theories have proven difficult to reconcile. Quantum mechanics is relevant in extremely small scale systems (think atoms). Since gravity is extremely weak and mass is quite spread out, gravity is only really significant on a very large scale (think stars, galaxies). Accordingly, we've built QFT on the assumption that gravity does not play a role in atomic interactions.

A very interesting area of further study is black holes. The centre of a black hole is so dense (maybe infinitely), that as an object approaches, gravity is far stronger than quantum forces, and the object is torn into its components. This can not be described by GR, and must be described using QFT.

Unfortunately, we can't measure forces inside a black hole as by definition information can not exit a black hole. Also, any attempt to incorporate gravity into the standard model has failed as we are unable to quantise gravity.

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If you call every effect of relavistic observation as an illusion, than yes, it is. It is an illusion like magnetic fields, which are just relativistic electrical fields. In this case you should check geodesics equiation for the movement: https://en.wikipedia.org/wiki/Geodesics_in_general_relativity

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