What we talk about when we talk about Lorentz transformations?

Question

Context

In [1], cowlicks asks the question, ``How can the Gallilean transformations form a group?'' It is clear what a group is. Borrowing liberally from [2],

"A group is a set $G$ together with a binary operation on $G$ ... that combines any two elements a $a$ and $b$ to form an element of $G$ ... such that the following three requirements, known as group axioms, are satisfied..."

The axioms include associativity, the existence of an identity element, and the existence of inverse element.

In Selene Routley's answer [1] to [1], Routely identifies three types of Galilean transforms. These are Galilean translations, Galilean boosts, and Galilean rotations. Each of these three transformations can be represented by a $5\times 5$ matrix with a specified form [2] These are Galilean translations, $T$, Galilean boosts, $B$ , and Galilean rotations, $R$. Any Galilean transformation can be composed from transformation of these three kinds.

In group theory it is possible for the group to include subgroups. So for example, the composition of two Galilean translations is a Galilean translation. Similarly, the composition of two Galilean boosts is a Galilean boost. Finally, the composition of two Galilean rotations is a Galilean rotation. This said, though the composition of two distinct kinds of Galilean transformations might (in specific instances) be an element of one or both of the subgroups' sets, in general this is not the case. However, again, the composition of any number of any kind of Galilean transformations is an element of the Galilean groups' set.

Question

My question regards what are we talking about when we talk about Lorentz transformations. Do we ascribe to Lorentz transformation only those transformations that are boosts? Do we ascribe to Lorentz transformation those transformation that include rotation? Do we ascribe to Lorentz transformation those transformation that include translation?

My Answer

To attempt to resolve the matter I read [3], which states,

"The Lorentz transformation is a linear transformation. It may include a rotation of space; a rotation-free Lorentz transformation is called a Lorentz boost. [The Lorentz transformations] describe only the transformations in which the spacetime event at the origin is left fixed.... The more general set of transformations that also includes translations is known as the Poincaré group."

Based on this, I believe that, strictly speaking, the Lorentz group includes a set of elements that can transform between frames that are rotated with respect to each and in relative motion; but does not include elements that can transform between frames that are only translated. Further, I believe that, strictly speaking, the Poincaré group includes a set with elements that produce all three: translations, rotations, and boosts. My belief is supported by the expository on Wigner rotations, which reads:

"...the composition of two non-collinear Lorentz boosts results in a Lorentz transformation that is not a pure boost but is the composition of a boost and a rotation.''

Bibliography

[1] How can the Gallilean transformations form a group?

[2] https://en.wikipedia.org/wiki/Group_(mathematics)

[3] https://en.wikipedia.org/wiki/Lorentz_transformation#Proper_transformations

[4] https://en.wikipedia.org/wiki/Wigner_rotation

Answer 1

Yes, you're right. The symmetries of the Minkowski spacetime include boosts, rotations, and translations. By usual conventions, the Lorentz group includes boosts and rotations. If you also include translations, we usually call that the Poincaré group.

Answer 2

My question regards what are we talking about when we talk about Lorentz transformations. Do we ascribe to Lorentz transformation only those transformations that are boosts? Do we ascribe to Lorentz transformation those transformation that include rotation? Do we ascribe to Lorentz transformation those transformation that include translation?

The answer likely depends on context and convenience and terminology.

• For a proof, one should be explicit.
  Lorentz Transformations (of the Lorentz Group) include boosts, rotations, and reflections.
  Poincare Transformations (of the Poincare group) (inhomogeneous Lorentz transformations) further include translations.
• Note that the Wikipedia entry on Galilean transformation mainly treats the
  "inhomogeneous Galilean group (assumed throughout below)".
  Thus, [bracketed term inserted by me]
  

As a Lie group, the group of [inhomogeneous] Galilean transformations has dimension 10.

• By comparison, from Lorentz Group

The Lorentz group is a six-dimensional noncompact non-abelian real Lie group that is not connected.

• By comparison, from Poincare group (inhomogeneous Lorentz transformations)

It is a ten-dimensional non-abelian Lie group

So, the lack of parallelism can be traced due to whether "inhomogeneous" is assumed explicitly or implicitly.

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• Typically when the focus is on the physics, "Lorentz transformation" is meant to be a boost. (If you want to describe a rotation, say "rotation".)

• If one needs to be explicit that you are describing a boost, then use "Lorentz boost" or "Lorentz boost transformation".

• In (1+1)-Minkowski, there are no nontrivial rotations to speak of. So, "Lorentz transformation" is probably enough... although one may want to disallow reflections.

Answer 3

To be more precise, the symmetry group of space-time (or Minkowski space) is the Poincaré group, this is because space-time is an affine-space, that is, the invariant element here is the distance $(x-y)^2$. On the other hand, a Minkowski vector space (tangent space in each point of the Minkowski space) is only invariant under Lorentz transformations (boosts and rotations) because the invariant elements here are the scalar products $v\cdot u$ with $v$ and $u$ being vectors in the Minkowski vector space.