I thought I had accepted MigL's correction in post#2 where he said: __"Minkowsky space-time ( flat, ignoring gravity effects ) is not a group."__

...

I understand now that the elements in the Minkowski space are (acc MigL) __"certain transformational operations ( isometries IIRC ) "__

Let me see if I can provide some context for MigL's remark.

We have a mathematical object that we think of as some kind of geometrical space. For the moment forget Minkowski space and just think about the familiar Euclidean plane or perhaps Euclidean 3-space.

Imagine rotating the standard Euclidean plane around the origin. It's clear that if you do one rotation then another, it's the same as if you'd combined them from the start. In other words the collection of rotations is closed under composition of rotations. Composition is associative (needs proof). The identity is the rotation through 0 degrees, which leaves the plane unchanged. If you rotate the plane you can just rotate it back to where you started, so we have inverses. Therefore the set of rotations of the plane forms a group under composition of rotations.

In the case of the plane, the group is Abelian, after

Niels Henrik Abel.

In general, geometrical operations are not commutative. In Euclidean 3-space we can rotate around one of the standard axes or around some arbitrary line. There are more ways for commutativity to fail. If you have good 3D visualization (which I never did) you can see that 3D rotations do not in general commute.

In the late 19th and early 20th centuries, mathematicians figured out that to study a geometric space, it was useful to study the groups of geometrical transformations that operated on a space. This is the general pattern. We have a space and we have various groups associated with geometrical transformations of that space. We study the groups to better understand the space.

Minkowski space is (as I understand it) is the mathematical model of relativity theory. The Wiki page would take some time to work through, after which you'd know a lot of differential geometry and relativity. But basically it's just 4-dimensional spacetime with the funny metric that combines time with space to model modern relativity theory. (Apologies to the physicists for anything I've mangled here).

And there are a number of interesting groups associated with various classes of geometrical transformations on it. For example in this page on

Lie groups (Lie pronounced "Lee") we find MigL's example:

The Lorentz group is a 6-dimensional Lie group of linear isometries of the Minkowski space.

An isometry is just a rigid motion, like a rotation or reflection or translation. Any transformation that

*preserves distances*.

So even though we may not know every detail of the Lorenz group; we can understand it as some group of rigid transformations of 4D spacetime. That's what we mean when we talk about groups in conjunction with geometry. We're considering collections of transformations that preserve some geometric property we care about. As long as the individual transformations are reversible, we'll have a group.

**Edited by wtf, 30 August 2016 - 11:01 PM.**