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VI. Physical Geometry
GRT may
be viewed as a gauge theory of the Lorentz algebra based on the restricted
Lorentz group ${\text{S}}{{\text{O}}^ + }(1,3)$. This is the component
of the Lorentz group connected to the identity. Alternatively, it can be
defined as the subgroup of transformations which are orthochronous (preserve
the direction of time) and proper (preserve the orientation).
For this
restricted group, the representation ${\Lambda ^\mu }_\nu $ of 4 × 4 real
matrices acting on ${\mathbb{R}^{1,3}}$ may be put in the exponentiated form
\[\Lambda = \exp \left(
{\frac{1}{2}{\varepsilon _{ab}}{M^{ab}}} \right)\]
The anti-symmetric group generators are 4x4 matrices that satisfy the
Lorentz algebra commutation relations with the components
\[{({M^{ab}})^c}_d := {\eta ^{ac}}\delta
_d^b - {\eta ^{bc}}\delta _d^a\]
The anti-symmetric parameters ${\varepsilon _{ab}} = - {\varepsilon _{ab}}$
may be identified with an infinitesimal Lorentz transformation: ${\Lambda
_{ab}} = {\eta _{ab}} + {\varepsilon _{ab}}$. Being anti-symmetric, the
${\varepsilon _{ab}}$ have 4x3/2 = 6 independent components, which agrees with
the 6 transformations of the Lorentz group: 3 rotations and 3 boosts. [Wikipedia:
Dirac Algebra]
More
generally, one may associate six anti-symmetric 4x4 matrices ${J^{ab}}$ with
the Lorentz algebra, to generate any 4x4 matrix representation $S(\Lambda )$
of ${\text{S}}{{\text{O}}^ + }(1,3)$ by the exponential form:
\[S(\Lambda ) = \exp \left(
{\frac{1}{2}{\varepsilon _{ab}}{J ^{ab}}} \right)\]
Under local LTs, the parameters ${\varepsilon _{ab}}$ become a function of
spacetime: ${\varepsilon _{ab}} \to {\varepsilon _{ab}}(x)$. This step of
making a symmetry local is called gauging the global symmetry. As a
consequence, derivatives of fields like ${\partial _\mu }\psi (x)$, are no
longer homogeneous due to the occurrence of terms like $[{\partial _\mu
}{\varepsilon _{ab}}(x)]\psi (x)$, which are non-vanishing unless
${\varepsilon _{ab}}$ is constant. The procedure to go from a locally flat to
a curved spacetime involves the generalization of $\partial _\mu $ to a
covariant derivative ${\mathcal{D}_\mu }$ to compensate for these extra terms.