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A four-vector A is a vector with a "timelike" component and three "spacelike" components, and can be written in various equivalent notations: [3] = (,,,) = + + + = + = where A α is the magnitude component and E α is the basis vector component; note that both are necessary to make a vector, and that when A α is seen alone, it refers strictly to the components of the vector.
If the object has mass, so that its speed is necessarily less than the speed of light, the world line may be parametrized by the proper time of the object. The four-velocity is the rate of change of four-position with respect to the proper time along the curve. The velocity, in contrast, is the rate of change of the position in (three ...
The proper time interval between two events on a world line is the change in proper time, which is independent of coordinates, and is a Lorentz scalar. [1] The interval is the quantity of interest, since proper time itself is fixed only up to an arbitrary additive constant, namely the setting of the clock at some event along the world line.
The four-force is defined as the rate of change in the four-momentum of a particle with respect to the particle's proper time.Hence,: =. For a particle of constant invariant mass >, the four-momentum is given by the relation =, where = (,) is the four-velocity.
In differential geometry, the four-gradient (or 4-gradient) is the four-vector analogue of the gradient from vector calculus. In special relativity and in quantum mechanics , the four-gradient is used to define the properties and relations between the various physical four-vectors and tensors .
An accelerating particle's 4-vector acceleration is the derivative with respect to proper time of its 4-velocity. This is not a difficult situation to handle. Accelerating frames require that one understand the concept of a momentarily comoving reference frame (MCRF), which is to say, a frame traveling at the same instantaneous velocity of a ...
In non-inertial coordinates, which include accelerated coordinates in special relativity and all coordinates in general relativity, the acceleration four-vector is related to the four-velocity through an absolute derivative with respect to proper time.
Log-log plot of γ (blue), v/c (cyan), and η (yellow) versus proper velocity w/c (i.e. momentum p/mc).Note that w/c is tracked by v/c at low speeds and by γ at high speeds. The dashed red curve is γ − 1 (kinetic energy K/mc 2), while the dashed magenta curve is the relativistic Doppler factor.