Some of the basic concepts of general relativity can be outlined outside the relativistic domain. In particular, the idea that mass–energy generates curvature in space and that curvature affects the motion of masses can be illustrated in a Newtonian setting. We use circular orbits as our prototype. This has the advantage that we know the kinetics of circular orbits. This allows us to calculate curvature of orbits in space directly and compare the results with dynamical forces.
The equivalence of gravitational and inertial mass A unique feature of the gravitational force is that all massive objects accelerate in the same manner in a gravitational field. This is often expressed as "The gravitational mass is equal to the inertial mass." This allows us to think of gravity as a curvature of spacetime.
Test for flatness in spacetime If initially parallel paths of two particles on nearby geodesics remain parallel within some accuracy, then spacetime is flat to within that accuracy. [Ref. 2, p. 30]
Two nearby particles in a radial gravitational field
Newtonian mechanics for circular orbits
The geodesic and field equations for circular orbits Consider the situation in which there are two particles in nearby circular polar orbits of the Earth at radius r {\displaystyle r} and speed v {\displaystyle v} . Since the orbits are circular, the gravitational force on the particles must equal the centripetal force,
v 2 r = G M r 2 {\displaystyle {v^{2} \over r}={GM \over r^{2}}}
where G is the gravitational constant and M {\displaystyle M} is the mass of the earth. The particles execute simple harmonic motion about the earth and with respect to each other. They are at their maximum distance from each other as they cross the equator. Their trajectories intersect at the poles. From Newton's Law of Gravitation the separation vector h {\displaystyle \mathbf {h} } can be shown to be given by the "geodesic equation"
d 2 h d τ 2 + R h = 0 {\displaystyle {d^{2}\mathbf {h} \over d\tau ^{2}}+R\mathbf {h} =0}
where R = 1 r 2 v 2 c 2 {\displaystyle R={1 \over r^{2}}{v^{2} \over c^{2}}} is the curvature of the trajectory and τ = c t {\displaystyle \tau =ct} is the speed of light c times the time. The curvature of the trajectory is generated by the mass of the earth M {\displaystyle M} . This is represented by the "field equation"
R = G M r 3 {\displaystyle R={GM \over {r^{3}}}}
In this example, the field equation is simply a statement of the Newtonian concept that centripetal force is equal to gravitational force for circular orbits. We refer to this expression as a field equation in order to highlight the similarities with the Einstein field equation. This equation is in a much different form than Gauss's law, which is the usual characterization of the field equation in Newtonian mechanics.
Relationship between curvature and mass density Mass can be written in terms of the average mass density ρ ( r ) {\displaystyle \rho (r)} inside a sphere of radius r {\displaystyle r} by the expression
M = 4 π ρ ( r ) r 3 3 {\displaystyle M={4\pi \rho (r)r^{3} \over 3}} . The field equation becomes
R = 4 π G 3 ρ ( r ) {\displaystyle R={4\pi G \over {3}}\rho (r)} . The curvature of the particle trajectories is proportional to mass density.
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