WOLFRAM|DEMONSTRATIONS PROJECT

Gravitation versus Curved Spacetime

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attracting mass M
energy E
trajectory progress
According to Newton's law of universal gravitation, two masses
M
and
m
attract one another with a force varying as the inverse square of the distance between them:
F=-
GMm
2
r
, where
G
is Newton's constant of gravitation. Orbits of attracting masses, including Kepler's laws of planetary motion, can be calculated on the basis of this force law. The left-hand graphic shows some possible trajectories of a "test mass"
m
, with
m≪M,
around a stationary mass
M
. The trajectories, shown as red curves, depend on the central mass
M
and the energy
E
of the test mass. When the test mass moves more slowly than the escape velocity, it spirals into the center. At higher energies, a stable orbit becomes possible in a progression of conic sections: circle, ellipse, parabola and hyperbola. (Hyperbolic orbits are not included here.)
Einstein's general theory of relativity gives a completely different picture of gravitation. It is not a force, per se, but rather a consequence of the curvature of spacetime. As John Wheeler said, matter tells spacetime how to curve while spacetime tells matter how to move. The right-hand graphic is a simplified representation of the curvature of spacetime caused by the mass
M
. This has been likened to a cannonball warping a mattress. The test mass then moves along a geodesic path in curved spacetime, which reduces to a straight line in the absence of curvature.