Proper lorentz transformations

[Lostinthought]

what does it mean by "any $$\Lambda^{\alpha}_{\beta}$$ that can be converted to the idendity $$\delta^{\alpha}_{\beta}$$ by a continuous variation of parameters must be a proper lorentz transformation"?

 

[henry_m]

Perhaps it is more clear initially to think about the group of matrices O(3) which preserve lengths in 3D, the orthogonal group. This consists of rotations and reflections. If you take any rotation, it can be parameterised by its axis of rotation and the angle of rotation. You can continuously vary the axis and the angle to smoothly move between different rotations, and in particular you can take a continuous path back to the identity transformation. During this change, the components of the matrix describing the transformation will vary continuously. Conversely, you can continuously go from the identity to any rotation.

On the other hand, if you have a reflection, no continuous path through the space of orthogonal transformations can connect you to the identity. One way to see this is that the determinant of a reflection matrix is -1, and the determinant of a rotation matrix is +1. If you continuously change the matrix, the determinant must also change continuously, and so it can't 'jump' from -1 to +1. The group O(3) splits up into two disconnected pieces: the rotations, which are continuously connected to the identity, and the reflections, which are not.

In the same way, the Lorentz group splits up into four disconnected pieces: firstly, there are the proper orthochronous transformations which don't reflect space and keep time flowing the same way. Then there are those which reflect space, or invert time, or do both. The proper orthochronous transformations are exactly those which can be continuously connected to the identity.

 

[Lostinthought]

so that means all rotations are proper lorentz transformations while reflections are not,but won't reflections change the proper time,$$\tau$$,so all reflections can't be lorentz transformations ryt?

 

[Fredrik]

Proper time is a coordinate-independent property of a timelike curve, so a Lorentz transformation, proper or not, certainly can't change the proper time of a curve.

I would interpret the statement in post #1 as saying that if $\Lambda$ is a Lorentz transformation (i.e. if $\Lambda\in O(3,1)$), and there's a continuous curve $C:[0,1]\rightarrow O(3,1)$ with $C(0)=I$ and $C(1)=\Lambda$, then this $\Lambda$ must be a proper Lorentz transformation. You can actually make a stronger claim. That $\Lambda$ must be must be proper and orthochronous, i.e. $\Lambda\in SO(3,1)$.

These concepts are fairly easy to understand in 1+1 dimensions, where every proper orthochronous Lorentz transformation can be expressed as $$\Lambda=\gamma\begin{pmatrix}1 & -v\\ -v & 1\end{pmatrix},$$ where $$\gamma=\frac{1}{\sqrt{1-v^2}}.$$ This means that in 1+1 dimensions, the simplest curve with the required properties is the curve $C:[0,1]\rightarrow SO(1,1)$ defined by $$C(s)=\frac{1}{\sqrt{1-(sv)^2}}\begin{pmatrix}1 & -sv\\ -sv & 1\end{pmatrix},$$ for all s in [0,1].

It's also useful to know that every Lorentz transformation can be expressed as the product of a proper orthochronous Lorentz transformation and one or both of the matrices $$P=\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix},\qquad T=\begin{pmatrix}-1 & 0\\ 0 & 1\end{pmatrix},$$ called parity and time reversal respectively. If a given Lorentz transformation involves exactly one factor of P, it's not proper. If it involves exactly one factor of T, it's not orthochronous. If it involves exactly one factor of each, it's neither.

The comments in the preceding paragraph hold in 3+1 dimensions too. The only thing that's different there is that the general expression for a proper orthochronous Lorentz transformation is much more complicated, since it involves rotations, and also boosts in three directions instead of just one. This makes it more difficult to define a curve with the required properties explicitly.