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The classic example of a non-relativistic spacetime is the spacetime of Galileo and Newton. It is the spacetime of everyday "common sense". [1] Galilean/Newtonian spacetime assumes that space is Euclidean (i.e. "flat"), and that time has a constant rate of passage that is independent of the state of motion of an observer, or indeed of anything external.
The definition of the non-relativistic gravitational fields provides the answer to this question, and thereby describes the image of the metric tensor in Newtonian physics. These fields are not strictly non-relativistic. Rather, they apply to the non-relativistic (or post-Newtonian) limit of GR.
The physical content of these different formulations is the same, but they provide different insights and facilitate different types of calculations. While the term "Newtonian mechanics" is sometimes used as a synonym for non-relativistic classical physics, it can also refer to a particular formalism based on Newton's laws of motion.
From a non-relativistic point of view this should have been the same as the previous result. This example demonstrates how the proper time equation is used, even though the Earth rotates and hence is not spherically symmetric as assumed by the Schwarzschild solution. To describe the effects of rotation more accurately the Kerr metric may be used.
For example, for a speed of 10 km/s (22,000 mph) the correction to the non-relativistic kinetic energy is 0.0417 J/kg (on a non-relativistic kinetic energy of 50 MJ/kg) and for a speed of 100 km/s it is 417 J/kg (on a non-relativistic kinetic energy of 5 GJ/kg). The relativistic relation between kinetic energy and momentum is given by
Relativistic Hamiltonians are analogous to those of non-relativistic QM in the following respect; there are terms including rest mass and interaction terms with externally applied fields, similar to the classical potential energy term, as well as momentum terms like the classical kinetic energy term.
Relative velocities between two particles in classical mechanics. The figure shows two objects A and B moving at constant velocity. The equations of motion are: = +, = +, where the subscript i refers to the initial displacement (at time t equal to zero).
For example, the centrifugal force that appears to emanate from the axis of rotation in a rotating frame increases with distance from the axis. All observers agree on the real forces, F; only non-inertial observers need fictitious forces. The laws of physics in the inertial frame are simpler because unnecessary forces are not present.