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Retrieving "Diffeomorphism Group" from the archives

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  1. General Covariance

    Linked via "diffeomorphism group"

    Conceptual Underpinnings
    The concept evolved from the more restricted principle of Special Relativity, which only demands invariance under Lorentz transformations (the group of Poincaré transformations). General covariance extends this invariance to the entire diffeomorphism group $\text{Diff}(M)$, where $M$ is the four-dimensional spacetime manifold.
    In mathematical terms, general covariance dictates that if $\mathca…

  2. General Covariance

    Linked via "diffeomorphism group"

    Relation to Diffeomorphism Constraints
    Within canonical quantization schemes, such as those utilizing Ashtekar variables, the requirement of general covariance manifests directly as a set of constraints on the phase space of the gravitational field. The Diffeomorphism Constraint, alongside the Hamiltonian Constraint (or Energy Constraint), arises precisely because the coordinate freedom (the […

  3. General Covariance

    Linked via "diffeomorphism group"

    Curvature Invariants and Absolute Physicality
    While the metric tensor and its derivatives are coordinate-dependent, certain scalars formed from them are invariant under coordinate changes (i.e., they are scalars under the full diffeomorphism group). These invariants are often posited as the truly physical observables.
    | Invariant Scalar | Formula (Simplified Notation) | Physical Significance (Hypothetical) |

  4. Lie Bracket

    Linked via "diffeomorphism group"

    In the context of differential geometry on a smooth manifold $M$, if $X$ and $Y$ are smooth vector fields, their Lie bracket, often denoted by $[X, Y]$, measures the infinitesimal difference between transporting a test function $f$ along $X$ then $Y$, versus transporting it along $Y$ then $X$. Operationally, it is defined by their action on a smooth function $f \in C^{\infty}(M)$:
    $$([X, Y] f) = X(Yf) - Y(Xf)$$
    This bracket is inherently dependent on the chosen [coordinate system](/entries/coordinate-sys…