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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.
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.
The arclength parameter is called proper time and usually denoted τ. The length of M is called the proper time of the particle. If the worldline M is a line segment, then the particle is said to be in free fall. [1]: 62–63 A world line traces out the path of a single point in spacetime.
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.
Rindler chart, for = in equation (), plotted on a Minkowski diagram.The dashed lines are the Rindler horizons. The worldline of a body in hyperbolic motion having constant proper acceleration in the -direction as a function of proper time and rapidity can be given by [16]
Figure 2–8. The invariant hyperbola comprises the points that can be reached from the origin in a fixed proper time by clocks traveling at different speeds. Fig. 2-8 illustrates the invariant hyperbola for all events that can be reached from the origin in a proper time of 5 meters (approximately 1.67 × 10 −8 s). Different world lines ...
On this usage, comoving and proper distances are numerically equal at the current age of the universe, but will differ in the past and in the future; if the comoving distance to a galaxy is denoted , the proper distance () at an arbitrary time is simply given by = where () is the scale factor (e.g. Davis & Lineweaver 2004). [2]