Let's ignore GR(scalar) and I am wondering why do we need to model Newtonian gravitational field using vectors? I can understand electromagnetism because of Lorentz force (right hand rule) but what about gravitational field it just the difference in strength at each point in space! Could there be some problems that can only be solved using vector field for gravity? Maybe I should use temperature (scalar) as a better example to compare Newtonian gravitational field ;D
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The way gravity comes into the framework of Newtonian mechanics is as a force, i.e., it has a direction from the get-go. So, it has to be a vector. More directly, as mentioned elsewhere, the gravitational field at the North pole and the South pole are roughly of the same magnitude but they are still different vis-à-vis their direction.
Of course, since gravitational force is a conservative force, one can also describe it using a potential formulation where the gravitational potential is simply a scalar which varies from point to point only in its magnitude. However, the physically observable aspect of this scalar gravitational potential is the force that it exerts on a particle. This force would depend on the gradient of the scalar potential, not the value of the scalar potential. This gradient is, of course, a vector, namely, the gravitational field.
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The force of gravity does not just vary in strength from location to location, but also in direction. A person standing at the South Pole and another person standing at the North Pole (on a boat) experience the same strength of gravity, but the forces are in opposite directions. So, you need vectors to best describe gravity.
Mark H
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