Velocity and acceleration in 4D
Recognising other physical quantities as tensors also simplifies their transformation laws. First note that the velocity four-vector In physics, in particular in special relativity and general relativity, the four-velocity of an object is a four-vector that replaces classical velocity (a three-dimensional vector). It is chosen in such a way that the velocity of light is a constant as measured in every inertial reference frame Uμ is given by
Recognising this, we can turn the awkward looking law about composition of velocities into a simple statement about transforming the velocity four-vector of one particle from one frame to another. Uμ also has an invariant form:
So all velocity four-vectors have a magnitude of c. This is an expression of the fact that there is no such thing as being at coordinate rest in relativity: at the least, you are always moving forward through time. The acceleration 4-vector and γu is the Lorentz factor for the speed u. A dot above a variable indicates a derivative with respect to the coordinate time in a given reference frame, not the proper time τ is given by . Given this, differentiating the above equation by τ produces
So in relativity, the acceleration four-vector and the velocity four-vector are orthogonal.
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