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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 ...
Some definitions and concepts from classical mechanics do carry over to SR, such as force as the time derivative of momentum (Newton's second law), the work done by a particle as the line integral of force exerted on the particle along a path, and power as the time derivative of work done. However, there are a number of significant ...
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 ...
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.
The momentum is equal to the energy multiplied by the velocity divided by c 2. As such, the Newtonian mass of an object, which is the ratio of the momentum to the velocity for slow velocities, is equal to E/c 2.
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To find the form of the momentum and energy as a function of velocity, it can be noted that the four-velocity, which is proportional to (,), is the only four-vector associated with the particle's motion, so that if there is a conserved four-momentum (,), it must be proportional to this vector.
"Any physical law which can be expressed in tensor notation in SR has exactly the same form in a locally inertial frame of a curved spacetime." The 4-gradient commas (,) in SR are simply changed to covariant derivative semi-colons (;) in GR, with the connection between the two using Christoffel symbols. This is known in relativity physics as ...