Understanding Lorentz Groups and some key subgroups

In summary, the conversation discusses the Lorentz group and its different subsets, including the proper orthochronous Lorentz group SO^{\uparrow}(1,3). It also touches on the conditions for a matrix to be part of the Lorentz group, such as the requirement for its determinant to be 1. The conversation also explores the concepts of disconnected and disjoint sets in relation to the Lorentz group.
  • #1
JD_PM
1,131
158
TL;DR Summary
I'd like to gain more insight on the Lorentz Group, its most important subgroups and must-know examples in Physics. At the moment I only know a handful: the orthogonal group ##O(3)##, the orthogonal group ##SO(3)## and the proper orthochronous Lorentz group [itex]SO^{\uparrow}(1,3)[/itex]
This thread is motivated by samalkhaiat's comment here

samalkhaiat said:
That is neither continuous nor connected Lorentz transformation. It is a discrete space-time reversal [itex](x^{0} , x^{i}) \to (-x^{0} , -x^{i})[/itex]. Space reflection [itex](x^{0} , x^{i}) \to (x^{0} , -x^{i})[/itex]; time reversal [itex](x^{0} , x^{i}) \to (-x^{0} , x^{i})[/itex] and space-time reversal form disjoint subsets and are not continuously connected to the identity. In English, [itex]x \to – x[/itex] does not belong to the proper orthochronous Lorentz group [itex]SO^{\uparrow}(1,3)[/itex]. By the “Lorentz group”, we always mean the real semi-simple Lie group [itex]SO^{\uparrow}(1,3)[/itex].

I know that the Lorentz Group is formed by all matrices that satisfy

$$\eta = \Lambda^{T} \eta \Lambda \tag{1.1}$$

Which is equivalent to

$$\eta_{\mu\nu}\Lambda^{\mu}{}_{\rho}\Lambda^{\nu}{}_{\sigma} = \eta_{\rho \sigma} \tag{1.2}$$

If we add no more restrictions and define the group under the inner product we end up with the group ##O(1,3)##.

I also know that all orthogonal matrices must satisfy ##1_3 =R^T 1_3 R##. They include not only rotational matrices but also space and time reversals, also known as parity transformations (i.e. [itex](x^{0} , x^{i}) \to (x^{0} , -x^{i})[/itex]) and [itex](x^{0} , x^{i}) \to (-x^{0} , x^{i})[/itex] respectively)

I have some questions:

1) Is the Lorentz group always defined under the inner product? The only examples I know are; for instance: the orthogonal group ##O(3)##, the orthogonal group ##SO(3)## and the proper orthochronous Lorentz group [itex]SO^{\uparrow}(1,3)[/itex]

2) I've read that ##O(1,3)## has one positive and one negative eigenvalue of its defining symmetric matrix. I do not see why, how could we prove it?

3)
samalkhaiat said:
[itex]x \to – x[/itex] does not belong to the proper orthochronous Lorentz group [itex]SO^{\uparrow}(1,3)[/itex]. By the “Lorentz group”, we always mean the real semi-simple Lie group [itex]SO^{\uparrow}(1,3)[/itex].

If I am not mistaken, this is because [itex]SO^{\uparrow}(1,3)[/itex] requires matrices to fulfil not only ##1_3 =R^T 1_3 R## condition but also that its determinant must be 1. The later condition is not fulfilled by matrices representing parity transformations (spatial, time and space-time reversals), but how can I prove this?Sources:

SpaceTime & Geometry by Carroll, pages 12,13,14

vanhees71 QFT manuscript, section 3.1.
Any help is appreciated.

Thank you :biggrin:
 
  • Informative
Likes etotheipi
Mathematics news on Phys.org
  • #2
JD_PM said:
If I am not mistaken, this is because [itex]SO^{\uparrow}(1,3)[/itex] requires matrices to fulfil not only ##1_3 =R^T 1_3 R## condition but also that its determinant must be 1. The later condition is not fulfilled by matrices representing parity transformations (spatial, time and space-time reversals), but how can I prove this?

Just an observation that the determinant is a continuous mapping from the set of all matrices to the set of real numbers. You cannot smoothly map a matrix with determinant ##1## to a matrix with determinant ##-1##, without going through all other values (intermediate value theorem).

And, in fact, the existence of a continuous function from a set into a two-point set ##\{-1, 1 \}## is the definition of a disconnected set. Hence ##O(3)## comprises two disconnected subsets.
 
  • Informative
Likes JD_PM
  • #3
Hi PeroK

PeroK said:
Just an observation that the determinant is a continuous mapping from the set of all matrices to the set of real numbers.

Please let me write down the mathematical definition of the determinant for future copy-paste reference

$$f:\Re^{n \times n} \rightarrow \Re : A = \Big(
\begin{pmatrix}
R_1 \\
. \\
. \\
R_n
\end{pmatrix}\Big) \mapsto f(A)$$

PeroK said:
You cannot smoothly map a matrix with determinant ##1## to a matrix with determinant ##-1##, without going through all other values (intermediate value theorem).

And, in fact, the existence of a continuous function from a set into a two-point set ##\{-1, 1 \}## is the definition of a disconnected set. Hence ##O(3)## comprises two disconnected subsets.

Oh so space reflections ##(x^{0} , x^{i}) \to (x^{0} , -x^{i})## and time reversals ##(x^{0} , x^{i}) \to (-x^{0} , x^{i})## are two disconnected subsets of ##O(3)##? Then, do disconnected and disjoint mean the same?
 
  • #4
JD_PM said:
Then, do disconnected and disjoint mean the same?
Disjoint refers to any two sets having no members in common. Disconnected-ness is a property of a topological space, which roughly means that it is not in one piece. For example, the real numbers can be expressed as the union of two disjoint sets.

##\mathbb{R} = (-\infty, 0] \cup (0, +\infty)##

But, ##\mathbb{R}## is connected (with the usual topology).

A topological space, ##S##, is disconnecetd iff there exists a continuous function from ##S## onto a two-point set.

For example, the set ##S = (-\infty, 0) \cup (0, +\infty)## is disconnected. You can see this as the function ##f: S \rightarrow \{0, 1\}##:

##f(x) = 0 \ (x < 0)##, and ##f(x) = 1 \ (x > 0)## is continuous (despite what some people may say!).

PS ##O(3)## is disconnected. Using the properties of the determinant gives you the easiest proof.
 
  • Informative
Likes JD_PM

Related to Understanding Lorentz Groups and some key subgroups

1. What is the Lorentz group?

The Lorentz group is a mathematical group that describes the symmetries of special relativity. It consists of all the transformations that leave the laws of physics unchanged in different inertial frames of reference.

2. What are the key subgroups of the Lorentz group?

The key subgroups of the Lorentz group are the proper orthochronous Lorentz group (P), the proper orthochronous Lorentz group with a time reversal (PT), the proper orthochronous Lorentz group with a space inversion (PΣ), and the proper orthochronous Lorentz group with both time reversal and space inversion (PTΣ).

3. What is the significance of the proper orthochronous Lorentz group?

The proper orthochronous Lorentz group (P) is the subgroup of the Lorentz group that preserves the direction of time and contains all the transformations that are physically allowed in special relativity. It is also the subgroup that is most commonly used in physics.

4. How are Lorentz transformations related to rotations and boosts?

Lorentz transformations are a combination of rotations and boosts, which are two types of transformations that are allowed in special relativity. Rotations involve changing the orientation of an object in space, while boosts involve changing the velocity of an object.

5. What is the difference between the Lorentz group and the Poincaré group?

The Poincaré group is an extension of the Lorentz group that includes translations, which describe the displacement of an object in space and time. In other words, the Poincaré group includes all the symmetries of special relativity, while the Lorentz group only describes transformations that leave the laws of physics unchanged.

Similar threads

  • High Energy, Nuclear, Particle Physics
Replies
27
Views
3K
  • Differential Geometry
Replies
11
Views
2K
Replies
31
Views
3K
  • Differential Geometry
Replies
1
Views
1K
  • Special and General Relativity
2
Replies
54
Views
2K
  • Special and General Relativity
3
Replies
101
Views
3K
  • Special and General Relativity
2
Replies
41
Views
2K
  • General Math
Replies
1
Views
2K
  • Advanced Physics Homework Help
Replies
8
Views
862
  • Topology and Analysis
Replies
2
Views
2K
Back
Top