# Continuity in Minkowski space

• cosmic dust

#### cosmic dust

How "continuity" of a map Τ:M→M, where M is a Minkowski space, can be defined? Obviously I cannot use the "metric" induced by the minkowskian product:
x$\cdot$y = -x$^{0}$y$^{0}$+x$^{i}$y$^{i}$
for the definition of coninuity; it is a misinformer about the proximity of points. Should I use the Euclidean metric instead?

Thank's...

Minkowski space-time is just ##\mathbb{R}^{4}## with the canonical Euclidean topology. Continuity of endomorphisms of Minkowski space-time is with respect to this topology.

• 1 person
I took the wikipedia's definition of Minkowski space: a 4-D real vector space with a symmetric, bilinear, non-degenerate quadratic form with signature (1,3). From this point of view, can a consistent metric induced by that quadratic form? If not, then according to your comment, I will have to make use and of the Eucliden norm on that vector space, in order to define continuity.

Right?

I took the wikipedia's definition of Minkowski space: a 4-D real vector space with a symmetric, bilinear, non-degenerate quadratic form with signature (1,3). From this point of view, can a consistent metric induced by that quadratic form? If not, then according to your comment, I will have to make use and of the Eucliden norm on that vector space, in order to define continuity.
I've never seen pseudo-Riemannian metric tensors on vector spaces being used to induce a topology on the vector space but that's not to say that it isn't defined (you can define it in the same way). The canonical topology on Minkowski space-time would just be that generated by the base of open balls of the Euclidean metric yes. There are other topologies you can endow as well of course and they don't have to stem from a metric.

• 1 person
The beautiful $$\displaystyle book "The Geometry of Minkowski Spacetime: An Introduction to the Mathematics of the Special Theory of Relativity" by Naber has an appendix that discusses topology for Minkowski spacetime.$$