As far as I understand some of the readings, Gravity does not exist in real terms. It's only a way of modeling motion as we see. Einstein for example explained the motion without having to have Gravity. Is this true?
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Physics does not answer existential problems. It gathers data and observations and models them with mathematical equations and functions, and then can explain the data with the model and predict new observations. This has been going on for centuries, and what we see if we study the history of physics is that there are regions of validity for the mathematical models, regions in the variables where the models are valid within measurement errors and regions where they fail
The Newtonian gravitational model is valid for our everyday experiences and experiments with sizes of the order of kilograms, meters, and seconds. Our whole civilizations are built and maintained on the accuracy of the Newtonian model. Thus to ask if gravity exists is a bit like asking "do I exist?", which, as far as experience goes, is a philosophical question.
Now in the region where the unit of length is light years and the unit of mass sun masses, deviations from Newtonian mechanics have been measured validating the General Relativity model. This does not invalidate the Newtonian model in its region of validity since for the low values of the quantities involved, the predictions of the GR model are indistinguishable from the Newtonian model within measurement error.
In conclusion, if we exist, gravity exists, as gravity is a definition in the simplest mathematial model that describes mechanics in human dimensions.
Gravity definitely does exist. Einstein did not provide a model without gravity, he simply found a new way to think about it. Gravity, as we experience it, is a consequence of bodies moving along so-called geodesics, which can in simple terms described as "shortest paths through spacetime". The effect of gravity now arises if there is motion along geodesics through curved spacetime: from our point of view, this looks like an acceleration. This is not an explanation without gravity, but according to general relativity, this is the mechanism behind gravity. By no means should one conclude that gravity does not exist.
Apart from this, the existence of gravity is certainly verified by observation (be it in everyday life or complicated experiment). More specifically, general relativity has proven to be an excellent description of gravitational phenomena in many experiments.
It's only a way of modeling motion as we see.
You can think in that way about General Relativity, but physicists strongly believe that there exists theory of quantum gravity, which means that there exists a corresponding fundamental particle called the graviton.
The most developed examples of quantum gravity theories are String Theory and Loop Quantum Gravity. If experiments prove any quantum gravity theory and find the graviton then gravity would "exist in real terms" as you said.