List of Relativistic Equations - The Metric and Four-vectors

The Metric and Four-vectors

In what follows, bold sans serif is used for 4-vectors while normal bold roman is used for ordinary 3-vectors.

Inner product (i.e. notion of length)

where is known as the metric tensor. In special relativity, the metric tensor is the Minkowski metric:

Space-time interval

In the above, ds2 is known as the spacetime interval. Another thing worth noting is that this inner product is invariant under the Lorentz transformation. The invariance of inner product means the following:

The sign of the metric and the placement of the ct, ct', cdt, and cdt′ time-based terms can vary depending on the author's choice. For instance, many times the time-based terms are placed first in the four-vectors, with the spatial terms following. Also, sometimes η is replaced with −η, making the spacial terms produce negative contributions to the dot product or spacetime interval, while the time term makes a positive contribution. These differences can be used in any combination, so long as the choice of standards is followed completely throughout the computations performed.