The Length of A Position Vector
In special relativity the location of a particle in 4-dimensional spacetime is given by its world line
where is the position in 3-dimensional space of the particle, is the velocity in 3-dimensional space and is the speed of light.
The "length" of the vector is a Lorentz scalar and is given by
where is c times the proper time as measured by a clock in the rest frame of the particle and the metric is given by
This is a time-like metric. Often the Minkowski metric is used in which the signs of the ones are reversed.
This is a space-like metric. In the Minkowski metric the space-like interval is defined as
We use the Minkowski metric in the rest of this article.
Famous quotes containing the words length and/or position:
“Whoever aims publicly at great things and at length perceives secretly that he is too weak to achieve them, has usually also insufficient strength to renounce his aims publicly, and then inevitably becomes a hypocrite.”
—Friedrich Nietzsche (18441900)
“When asked whether or not we are Marxists, our position is the same as that of a physicist or a biologist who is asked if he is a Newtonian or if he is a Pasteurian.”
—Ernesto Che Guevara (192867)