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Given that in general for a closed system with generalized coordinates q i and canonical momenta p i, [3] = =, = =, it is immediate (recalling x 0 = ct, x 1 = x, x 2 = y, x 3 = z and x 0 = −x 0, x 1 = x 1, x 2 = x 2, x 3 = x 3 in the present metric convention) that = = (,) is a covariant four-vector with the three-vector part being the ...
and orbital 3-angular momentum = of a point-like particle are combined into a four-dimensional bivector in terms of the 4-position X and the 4-momentum P of the particle: [8] [9] = where ∧ denotes the exterior product. This tensor is additive: the total angular momentum of a system is the sum of the angular momentum tensors for each ...
In order to find out the transformation of three-acceleration, one has to differentiate the spatial coordinates and ′ of the Lorentz transformation with respect to and ′, from which the transformation of three-velocity (also called velocity-addition formula) between and ′ follows, and eventually by another differentiation with respect to and ′ the transformation of three-acceleration ...
The energy and momentum of an object measured in two inertial frames in energy–momentum space – the yellow frame measures E and p while the blue frame measures E ′ and p ′. The green arrow is the four-momentum P of an object with length proportional to its rest mass m 0.
For reference and background, two closely related forms of angular momentum are given. In classical mechanics, the orbital angular momentum of a particle with instantaneous three-dimensional position vector x = (x, y, z) and momentum vector p = (p x, p y, p z), is defined as the axial vector = which has three components, that are systematically given by cyclic permutations of Cartesian ...
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The 4-divergence of the stress–energy tensor as the conserved Noether current associated with spacetime translations, gives four conservation laws in SR: [4]: 101–106 The conservation of energy (temporal direction) and the conservation of linear momentum (3 separate spatial directions).
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