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what is the difference in the the way in which gravitaion is described in newtonian, relativistic and quantum mechanics?
Galilean Gravitation
Galileo dropped objects from the Leaning Tower of Pisa to demonstrate that objects fall at the same rate regardless of mass. These results were in contradiction to the ancient Aristotelian conception of gravity, which stated that the speed of an object's fall was proportional to its weight.
Galileo also described a mathematical formula for the distance an object will fall in a given time, though his description was geometric rather than algebraic.His formula, that the distance fallen is proportional to the square of the elapsed time, or d ? t^{2}, remains valid for many approximations today.
However, Galileo's theory that the acceleration of gravity is independent on an object's height above the ground could not stand the test of time.
Newtonian Gravitation
Whereas Galileo described gravitation in terms of motion, Newton described it in terms of a Force. Gravitation itself went from the natural tendency of objects to fall to a vector field theory in which the force between two objects was inversely proportional to the square of the distance between them.
Newton's vector equation for gravitation states that each particle in the universe attracts another particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them or:
Where:
 F is the gravitational force vector.
 m_{1} and m_{2} are the object's masses.
 r is the vector that separates the objects' centers of mass.
 is the unit vector of r;
 G is the gravitatiopnal constant
Another way to express this equation, that is more convenient for programming and expansion into power series, is:
Near the surface of the Earth, where the difference in gravitational acceleration at different heights is very small, Newtonian gravitation "reduces" to, or is mathematically equivalent to within a known margin of error, Galilean gravitation. That is, both Galilean gravitation and Newtonian gravitation can reasonably describe the motion of falling apples and artillery shells. But further from the surface of the Earth, the errors in Galileo's approximation become too great, and Newton's model is required.
Newton's theory of gravitation stood, refined and reformulated, but essentially unchanged, for more than two centuries.
Relativistic Gravitation
Galileo described gravity in terms of motion, and Newton described gravity in terms of force, Einstein described gravity in terms of inertial motion in curved spacetime, such curvature being a function of the matter and fields in a given region.
Just as the Newtonian theory reduced to Galilean gravitation near the surface of the Earth, general relativity gravitation reduces to Newtonian gravity where gravitational fields are relatively weak. Only when gravitational fields are strong, such as the field around the sun within the orbit of Mercury do Newton's equations fail to produce a good approximation of reality.
The Possibility of Quantum Gravitation
Physicists do not universally agree that a quantum theory of gravity is inevitable, or even possible. Einstein's theory of gravitation described gravity in a way that is so fundamentally different from the other interactions in nature that many physicists no longer gravity to be a force at all. Rather, under the Einsteinian interpretation, gravitation is an inherent property of spacetime and has no analogue at the quantum level at all. Other physicists believe that the Einsteinian interpretation is merely a useful largescale approximation of an interaction which we have not yet detected on the subatomic scale.
To date, no satisfactory theory of quantum gravity has been advanced, but research continues.