Minkowski Space

Minkowski space is a four-dimensional space possessing a Minkowski metric, i.e., a metric tensor having the form

Alternatively (though less desirably), Minkowski space can be considered to have a Euclidean metric with imaginary time coordinate  where  is the speed of light (by convention  is normally used) and where i is the imaginary number . Minkowski space unifies Euclidean three-space plus time (the "fourth dimension") in Einstein's theory of special relativity.

In equation (5) above, the metric signature  is assumed; under this assumption, Minkowski space is typically written . One may also express equation (5) with respect to the metric signature  by reversing the order of the positive and negative squared terms therein, in which case Minkowski space is denoted .

The Minkowski metric induces an inner product, the four-dimensional Lorentzian inner product (sometimes referred to as the Minkowski inner product), which fails to be positive definite (Ratcliffe 2006).

REFERENCES:

Ratcliffe, J. G. Foundations of Hyperbolic Manifolds. New York: Springer, 2006.

Thompson, A. C. Minkowski Geometry. New York: Cambridge University Press, 1996.