Curvature of spacetime

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The curvature of spacetime is what gives rise to the "fictitious forces" that are perceived as gravity under General Relativity.

It's necessary to define the terms carefully here. A "fictitious force" is a perceived force on a body in the absence of any visible agent pushing on it. Examples are the centrifugal force and the Coriolis force. These forces all have the property that they are exactly proportional to the object's mass. With careful analysis under Netwtonian/Galilean physics, it's not hard to see why. The "force" arises from the acceleration of the frame of reference. Gravity also satisfies the criterion of being exactly proportional to mass—this is Einstein's "equivalence principle".

When the physical system is formulated as "spacetime", that is, something with spatial and time coordinates, the acceleration can be seen to be just an aspect of the curvature of the coordinate system. In cases like the simple centrifugal or Coriolis force, the curvature of the coordinate system can be "transformed away" by analyzing the situation in a non-accelerating frame of reference. For example, if one stands outside of an amusement park ride, one can easily see the coordinate curvature effects that cause people on the ride to perceive a force. This has nothing to do with relativity. It arises in pure Netwtonian/Galilean physics.

This "transforming away" is similar to the difference between Cartesian and polar coordinates on a flat sheet of paper. The Cartesian coordinate system is flat, and the polar system is curved. But they are both coordinate systems on a flat sheet of paper. However, there are surfaces ("curved surfaces"), like a sphere, that have no flat coordinate system at all. In spacetime, this is equivalent to something that has no flat coordinate systems, and hence gives rise to fictitious forces that can't be transformed away. This is what gives rise to gravity, a fictitious force that, unlike the centrifugal or Coriolis force, can't be transformed away. That is, there is no place where one can stand and see gravity as just a case of people being pushed on by a floor that is accelerating upward.

This is the curvature of spacetime. The Einstein Field Equations explain how that curvature arises from the presence of massive objects like the Earth or the Sun.

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