I'm studying special relativity and was wondering, if someone were to give me a matrix, is there a way or a procedure to check if that matrix corresponds to a Lorentz transformation? Even if the matrix contains projectors or other weird things?
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asked Aug 5, 2021 at 22:29
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$\begingroup$ What is your definition of "Lorentz transformation" and what specific problem do you have in telling whether any given matrix fulfills it? $\endgroup$– ACuriousMind ♦Commented Aug 7, 2021 at 11:27
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3 Answers
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Yes, it needs to leave the Minkowski metric invariant: $$ \Lambda^T\eta\Lambda=\eta. $$
answered Aug 5, 2021 at 22:30
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The set of Lorentz transformations is defined as the set of all operators $\Lambda$ (or equivalently, matrices) that satisfy
$$ \eta(u,v) = \eta(\Lambda\,u,\Lambda\,v)\:\:\:\forall u,v\:\: \text{vectors in Minkowski spacetime},$$
where $\eta$ is the Minkowski metric.
In matrix representation this translates to the following statement:
$$u^T \eta v = (\Lambda\,u)^T\eta(\Lambda\,v)\:\:\:\forall u,v\:\: \text{vectors in Minkowski spacetime},$$ $$\:\:\:\:\:\:= u^T\Lambda^T\eta\Lambda\,v\:\:\:\forall u,v\:\: \text{vectors in Minkowski spacetime}.$$
Given that this statement must be valid for all $u$ and $v$, we obtain the condition:
$$\eta = \Lambda^T\eta \Lambda,$$
where in the equation above $\eta$ is the matrix associated with the Minkowski metric. In an orthonormal frame, it reads $\eta = \text{diag}(-1,1,1,1)$.
However, one is usually only interested in the proper orthochronus Lorentz Transformations, associated with Lorentz transformations that preserve the future time direction and do not invert the spatial directions. If you are looking for this set of Lorentz Transformations, you should also check that the matrix is invariant with respect the matrices associated with space and time inversion.
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Write the unit time vector and three unit space vectors. Send each to your given operator and note what comes out. All four results should remain unit space-time length, and all possible pairs should be orthogonal.
Of course, unit vectors have a very simple form. All you're doing is reading out columns of the matrix. No hard math, just looking. Just remember that lengths and dot products must use the mixed-sign spacetime metric, -+++ (or +---).
