Lorentz Group

up:: 032 MOC Relativity

The Lorentz Group, denoted , is composed of all transformations in Minkowski space which preserve the Invariant Interval (Relativity).

That is, for any Minkowski Metric , we have that is a Lorentz Transformation if

is a group

Let . Then

Trivially the identity matrix belongs to .

Given , its inverse also satisfies this condition:

Properties

• Rotations belong to the Lorentz Group, since we have that they preserve distances (since they are Orthogonal Transformations, satisfying ) and, thus, trivially preserve the invariant interval
  • Thus, the dimension of is : independent rotations and independent boosts
• Proper Lorentz Group
• Restricted Lorentz Group

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References

• Special Relativity - David Tong