Poincare Group (Wald, Chapter 4 Page 59)

Question

In Wald's text on general relativity, he mentions that in special relativity, many different global inertial coordinate systems are possible and can be put into one-to-one correspondence with elements of the 10-parameter Poincare group.

I am unfamiliar with Poincare group so I would like to see an explanation of what this sentence exactly means.

First of all, in special relativity we realize the spacetime as a four-dimensional manifold $M$. Furthermore, the statement that there exists a global inertial frames in special relativity is, as I understand it, the statement that $M$ can be covered by a single coordinate chart $(U,\psi)$, with $\psi:M\to\mathbb{R}^4$. In this local (which happens to also be global) chart, one can then associate every point in $M$ to a point in $\mathbb{R}^4$, which we call an event, denoted by say $(t,x,y,z)$.

In a definition of manifolds, we can have many atlases which in this case corresponds to different inertial coordinate systems, related by coordinate transformations (Lorentz transformations), and this translates to transition maps.

Where does this Poincare group come in and how does it do so explicitly? Since it is a group, does it mean it is a group action? Any detail would be helpful.

Answer 1

The Poincaré group is the semi-direct product of the six-dimensional Lorentz group and the four-dimensional translations and hence ten-dimensional (or "has ten parameters" is less precise diction).

Since in a global inertial coordinate system you have to have the Minkowski metric by definition, only those transformations (diffeomorphisms) which preserve the Minkowski metric turn one inertial coordinate system into another. You can then easily show that only transformations of the form $x \mapsto \Lambda x + a$ where $\Lambda$ is the matrix of a Lorentz transformation and $a$ is any four-vector fulfill this condition of transforming a Minkowski metric into a Minkowski metric (that both Christoffels vanish gives you that the transform is affine, and preserving the Minkowski metric restricts the matrix to the Lorentz group).

Furthermore, every element of the Poincaré group defines a different inertial frame because if $(\Lambda, a)$ and $(\Lambda',a')$ transform into the same coordinate system, then the inverse of $(\Lambda',a')$ must also be the inverse of $(\Lambda,a)$ and vice versa, but from the uniqueness of inverses it then follows that the two elements are already the same, so the map from the Poincaré group to inertial systems is a bijection.