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The local conservation of non-gravitational linear momentum and energy in a free-falling reference frame is expressed by the vanishing of the covariant divergence of the stress–energy tensor. Another important conserved quantity, discovered in studies of the celestial mechanics of astronomical bodies, is the Laplace–Runge–Lenz vector.
The inertial trajectories of particles and radiation in the resulting geometry are then calculated using the geodesic equation. As well as implying local energy–momentum conservation, the EFE reduce to Newton's law of gravitation in the limit of a weak gravitational field and velocities that are much less than the speed of light. [4]
In this example, the time derivative of q is the velocity, and so the first Hamilton equation means that the particle's velocity equals the derivative of its kinetic energy with respect to its momentum.
In special and general relativity, there is a local law for the conservation of energy–momentum. It can be succinctly expressed by the tensor equation: ; = This illustrates the rule of thumb that 'partial derivatives go to covariant derivatives'.
Conservation of momentum is a mathematical consequence of the homogeneity (shift symmetry) of space (position in space is the canonical conjugate quantity to momentum). That is, conservation of momentum is a consequence of the fact that the laws of physics do not depend on position; this is a special case of Noether's theorem. [25] For systems ...
However, the Einstein field equation is not the only equation which satisfies the three conditions: [6] Resemble but generalize Newton–Poisson gravitational equation; Apply to all coordinate systems, and; Guarantee local covariant conservation of energy–momentum for any metric tensor.
A local conservation law is usually expressed mathematically as a continuity equation, a partial differential equation which gives a relation between the amount of the quantity and the "transport" of that quantity. It states that the amount of the conserved quantity at a point or within a volume can only change by the amount of the quantity ...
This equation is called the Cauchy momentum equation and describes the non-relativistic momentum conservation of any continuum that conserves mass. σ is a rank two symmetric tensor given by its covariant components. In orthogonal coordinates in three dimensions it is represented as the 3 × 3 matrix: