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  1. Ashtekar Connection

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    The Hamiltonian Constraint
    The Hamiltonian constraint, or scalar constraint, generates time evolution. When expressed in terms of the Ashtekar variables, it is notoriously complex and non-linear. Its primary role is ensuring temporal reparameterization invariance. Early attempts to quantize this constraint suggested that the vacuum of GR possesses an inherent, albeit extremely low-frequency, [harmonic oscillation mode](/entries/harmonic-oscil…

  2. Metric Tensor

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    In the Arnowitt–Deser–Misner (ADM) formalism, the metric tensor is decomposed into spatial and temporal components to facilitate canonical quantization procedures, transitioning from a $3+1$ decomposition. The full metric is expressed as:
    $$ds^2 = -N^2 dt^2 + h_{ij} (dx^i + N^i dt)(dx^j + N^j dt)$$
    Here, $h_{ij}$ is the 3-metric (the spatial part of the metric), $N$ is the lapse function, and $N^i$ are the shift vector components. This de…