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Consider a generic 4x4 matrix $\Lambda $ $${\begin{bmatrix}{\Lambda ^{0}}_{0}&{\Lambda ^{0}}_{1}&{\Lambda ^{0}}_{2}&{\Lambda ^{0}}_{3}\\{\Lambda ^{1}}_{0}&{\Lambda ^{1}}_{1}&{\Lambda ^{1}}_{2}&{\Lambda ^{1}}_{3}\\{\Lambda ^{2}}_{0}&{\Lambda ^{2}}_{1}&{\Lambda ^{2}}_{2}&{\Lambda ^{2}}_{3}\\{\Lambda ^{3}}_{0}&{\Lambda ^{3}}_{1}&{\Lambda ^{3}}_{2}&{\Lambda ^{3}}_{3}\\\end{bmatrix}}$$

(and use the metric signature $-+++$).

What is the right way to determine if $\Lambda$ is a matrix belonging to Lorentz group, i.e. is a Lorentz matrix? What are the conditions to use to test if a matrix is Lorentz matrix?

Also, how do I determine if $\Lambda$ is a matrix belonging to Galilean group? Is a Galilean matrix necessarily a Lorentz matrix? In this case, what are the addititional conditions for which a Lorentz matrix is also a Galilean matrix?


My attempt:

  • Lorentz matrices: the condition

$$\Lambda^T \eta \Lambda =\eta $$

with $\eta$ metric tensor

must hold for a Lorentz matrix. Nevertheless I don't understand if a Lorentz matrix necessarily belongs to $SO(3)$, therefore is special ortogonal, which means that another condition is

$$\Lambda^{-1}=\Lambda^T$$ and $$det{\Lambda}=1$$

Therefore is it correct to say the following? $$\Lambda \,\,\, is \,\,\,Lorentz \,\, \iff \,\,\, \begin{cases} \Lambda^{-1}=\Lambda^T \\ det{\Lambda}=1 \\ \Lambda^T \eta \Lambda =\eta \end{cases}$$

  • Galilean matrices: is necessarily Lorentz, but it must also have the first row $(1,0,0,0)$ and also the submatrix $A={\begin{bmatrix}{\Lambda ^{1}}_{1}&{\Lambda ^{1}}_{2}&{\Lambda ^{1}}_{3}\\{\Lambda ^{2}}_{1}&{\Lambda ^{2}}_{2}&{\Lambda ^{2}}_{3}\\{\Lambda ^{3}}_{1}&{\Lambda ^{3}}_{2}&{\Lambda ^{3}}_{3}\\\end{bmatrix}}$ is $SO(3)$

Therefore is it correct to say the following? $$\Lambda \,\,\, is \,\,\,Galilean \,\, \iff \,\,\, \begin{cases} \Lambda \,\,\, is \,\,\,Lorentz \\ the \,\,\,first\,\,\, row\,\,\, is\,\,\, (1,0,0,0) \\ the \,\,\, submatrix \,\,\,\, {\begin{bmatrix}{\Lambda ^{1}}_{1}&{\Lambda ^{1}}_{2}&{\Lambda ^{1}}_{3}\\{\Lambda ^{2}}_{1}&{\Lambda ^{2}}_{2}&{\Lambda ^{2}}_{3}\\{\Lambda ^{3}}_{1}&{\Lambda ^{3}}_{2}&{\Lambda ^{3}}_{3}\\\end{bmatrix}} \,\,\,is \,\,\,SO(3) \end{cases}$$

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