Why does the Lorentz transformations form a group?

In summary, the Lorentz groupconsists of an infinite number of matrices, each with a differentboost in the three spatial directions and rotations.
  • #1
pivoxa15
2,255
1
Is the reason why the Lorentz transformations form a group because of the reason on this website
http://en.wikipedia.org/wiki/Lorentz_transformation_under_symmetric_configuration

So the group consits of 3 matrices, {identity, forward transformations, inverse transformations}?
 
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  • #2
There are four conditions which must be satisfied for a set of objects to form a groupt:

1) If A and B are elements of the group, then C = AB must also be an element of the group.

2) If A, B, and C are elements of the group, then A(BC) = (AB)C.

3) There exists an identity element, I, such that AI = IA = A.

4) Every element has an inverse which is also an element of the group.

You can show that the Lorentz transformations in 1-dimension obey all of these. And, in fact, if you include the general 3-d rotations, you'll find that the set of all Lorentz transformations together with all rotations form a group.

So, there are quite a few more than 3 matrices. In fact, there are an infinite number, since [tex]\beta[/tex] can take on any value from 0 to 1. And, this is just for the 1-d transformations. In general, you can have up to 6 free parameters (3 representings boosts in the three spatial directions and 3 representing rotations).
 
  • #3
So the group contains an infinite number of matrices like the ones on the site with different [tex]\beta[/tex] and [tex]\gamma[/tex]
values. Correct?
 
  • #4
Quite right. And, I think I need to make a little correction. [tex]\beta[/tex] can also be negative, indicating a boost in the opposite direction. So, it can really run from -1 to 1. As I wrote it above, I managed to leave out all the inverse boosts. Guess I wasn't quite awake enough yet.
 
  • #5
Lorentz group

pivoxa15 said:
Is the reason why the Lorentz transformations form a group because of the reason on this website
http://en.wikipedia.org/wiki/Lorentz_transformation_under_symmetric_configuration

So the group consits of 3 matrices, {identity, forward transformations, inverse transformations}?

Oh my, oh my. I only skimmed the Wikipedia article you linked to, because:

1. I quit WP some time ago,

2. as of the time of this post, it has been edited by only one registered but apparently inexperienced Wikipedia user (the individual who created this article) and one anon editor who appears likely to represent the same individual. This is a clue that the article has probably not been read/corrected by WikiProject Physics members, and from experience I know that horribly incoherent or wrong articles are not uncommon, particularly now that so many physicist wikipedians have quit WP.

Don't forget that the slogan of the Wikipedia includes that little phrase "that anyone can edit".

Did you try searching WP for "Lorentz group"? You should have found http://en.wikipedia.org/wiki/Lorentz_group, which HAS been read closely by several WikiProject Physics members and as of the version listed at http://en.wikipedia.org/wiki/User:Hillman/Archive was, as far as I know, correct. I can be pretty confident about this because I wrote almost all of that version myself :-/ So my opinion is hardly unbiased, but compare http://www2.corepower.com:8080/~relfaq/penrose.html . Even better, compare my version of "Lorentz group" with a good book such as Needham, Visual Complex Analysis or Jones and Silverman, Complex Functions.

To answer your question, the Lorentz group has infinitely many elements. It is a Lie group, i.e. not only a group but a (finite dimensional) smooth manifold, in fact it is a six dimensional Lie group. See the above cited sources for more details.

Chris Hillman
 
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  • #6
Parlyne said:
There are four conditions which must be satisfied for a set of objects to form a groupt:

1) If A and B are elements of the group, then C = AB must also be an element of the group.

2) If A, B, and C are elements of the group, then A(BC) = (AB)C.

3) There exists an identity element, I, such that AI = IA = A.

4) Every element has an inverse which is also an element of the group.

Although its not important in this context, this definition is not quite correct. A group is a set of objects and a binary operator. So C=AB should be C=A.B where . is some binary operator.

So we have, for example, that the real numbers form a group under addition, but not under division.
 
  • #7
I agree that the Wiki article on the Lorentz group is well written.

Daniel.
 

Related to Why does the Lorentz transformations form a group?

1. What is a group in the context of Lorentz transformations?

A group is a mathematical concept that represents a set of elements and a binary operation that combines any two elements to produce a third element in the set. In the context of Lorentz transformations, a group is a set of transformations that preserve the fundamental equations of special relativity, namely the laws of physics in different inertial frames of reference.

2. Why do the Lorentz transformations form a group?

The Lorentz transformations form a group because they satisfy the four defining properties of a group: closure, associativity, identity, and inverse. This means that when two Lorentz transformations are performed in succession, the resulting transformation is also a Lorentz transformation. Additionally, the order in which the transformations are performed does not affect the final result, and every Lorentz transformation has a corresponding inverse transformation.

3. How do the Lorentz transformations relate to special relativity?

The Lorentz transformations are the mathematical formalism that describes how measurements of space and time change between different inertial frames of reference in special relativity. They allow for the unification of space and time into the concept of spacetime, and they account for the observed effects of time dilation and length contraction.

4. Can the Lorentz transformations be visualized?

Yes, the Lorentz transformations can be visualized using Minkowski diagrams, which represent the geometric relationships between space and time in special relativity. These diagrams can help to understand how the Lorentz transformations affect measurements of space and time in different inertial frames of reference.

5. What are the practical applications of the Lorentz transformations?

The Lorentz transformations have numerous practical applications in modern physics, including in the fields of particle physics, cosmology, and astrophysics. They are also used in the development of technologies such as GPS systems and particle accelerators. Additionally, the principles of special relativity and the Lorentz transformations have been fundamental in the development of modern theories such as quantum mechanics and general relativity.

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