# Orthochorous? Don't think I've heard that term before

1. Jan 22, 2013

### Fredrik

Staff Emeritus
An arbitrary Lorentz transformation $\Lambda$ in 1+1 dimensions can be written as
\begin{align}
\Lambda=\frac{\sigma}{\sqrt{1-v^2}}
\begin{pmatrix}1 & -v\\ -\rho v & \rho\end{pmatrix}.
\end{align} Here $\sigma=\operatorname{sgn}\Lambda_{00}$ and $\rho=\det\Lambda$. I learned today that $\Lambda$ is said to be orthochorous if $\sigma\rho=1$. I don't think I've ever heard that term before. (I found it in Streater & Wightman...which by the way is the second hit if I google the term). Can someone explain this term? Is it standard? What does the "chorous" part refer to? Now that I think about it, I realize that I don't even know what "ortho" means, so I wouldn't mind getting that explained too.

And no, I'm not misspelling "orthochronous". This is a different word.

2. Jan 22, 2013

### dx

In thermodynamics, an 'isochoric' process is one where the volume remains constant, so I'm guessing 'chorous' in orthochorous means volume. And ρ = det Λ is the factor by which the transformation changes volumes.

Last edited: Jan 22, 2013
3. Jan 22, 2013

### Mordred

Orthochorous is another subgroup where orthochronus preserves the direction of time. Orthochorus preserves the sign of the volume of space (Lorentz transformations)

this link has some of the various subgroups and mathematics involved

edit : its kind of strange that one can find dozens of links describing the subgroups orthochronous, proper, and restricted but finding coverage of orthochorous is almost non existant.

Last edited: Jan 22, 2013
4. Jan 22, 2013

### Mordred

5. Jan 22, 2013

### Fredrik

Staff Emeritus
It might, but I think I understand the subgroups well enough. I'm mainly concerned with the terminology right now.

Based on what you guys said, it seems likely that some form of the word "chorous" means "volume". Thanks both of you.

6. Jan 22, 2013

### dextercioby

Well, as per original ancient Greek, it doesn't

<Etymology

The noun isochor and the adjective isochoric are derived from the Greek words ἴσος (isos) meaning "equal", and χώρος (choros) meaning "space."> (from here).

Also χῶρος LSJ, Middle Liddell, Slater, Autenrieth 2.422 0 a definite space, piece of ground, place
χώρα LSJ, Middle Liddell, Slater 15.125 2 space