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Matrices belonging to restricted Lorentz group $SO^+(1,3)$

Physics Asked by user100570 on July 21, 2021

My Professor says that all members of the restricted Lorentz group $SO^+(1,3)$ may be written as $e^Gamma$, where
$$ Gamma^{mu}{}_{nu}=Lambda^{mu rho} eta_{rho nu}.$$
Here $Lambda$ is an antisymmetric Matrix, and $eta$ is the standard Minkowski metric [$diag(1,-1,-1,-1)$].
I want to prove a weaker statement. That all matrices of the said form belong to $SO^+(1,3)$.
I have been able to show that they belong to $SO(1,3)$. However, the orthochronous bit is troubling me.
Any help is appreciated.

Progress so far:
$$(Lambda eta)^T=-eta Lambda$$
$$implies (e^{Lambda eta})^T=e^{-eta Lambda}$$
$$eta e^{-eta Lambda} eta=e^{-Lambda eta}$$
$$eta e^{-eta Lambda} eta e^{Lambda eta}=I$$
$$e^{-eta Lambda} eta e^{Lambda eta}=eta$$
$$implies (e^{Lambda eta})^Teta e^{Lambda eta}=eta.$$

Hence $e^{Gamma} in O(1,3)$.
Further, Suppose $Lambda$ is written as
$$
begin{bmatrix}
0&vec{lambda}
-vec{lambda}&R
end{bmatrix}
$$

Where $R$ itself is antisymmetric.
Block Multiplication on the right by $eta$, gives $Lambda eta$ to be,
$$
begin{bmatrix}
0&-vec{lambda}
-vec{lambda}&-R
end{bmatrix}.
$$

Clearly Trace of $Lambda eta$ is 0. Hence $Det(Lambda eta)=1$.
Therefore $e^{Gamma} in SO(1,3)$.

One Answer

Comments to the question (v2):

  1. That the exponential map $$exp: o(1,d) ~to~ O(1,d)$$ has image $$exp(o(1,d))~subseteq~ SO^+(1,d)$$ inside the restriced Lorentz group $$ SO^+(1,d)~:=~{ Lambda in SO(1,d) | Lambda^0{}_0>0 } $$ follows from the facts that the image of a connected set under a continuous map must again be connected, cf. above comment by Emilio Pisanty. (One more hint: No Lorentz matrix $Lambda$ can have zero determinant $det(Lambda)=0$ or zero 00-entry $Lambda^0{}_0=0$, cf. e.g. this Phys.SE post.)

  2. The non-trivial fact that the exponential map $$exp: o(1,d) ~to~ SO^+(1,d)$$ is surjective is discussed in this Phys.SE post.

Answered by Qmechanic on July 21, 2021

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