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Accelerating Expansion Of The Universe
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The quantitative measure of this anomaly is often expressed through the luminosity distance ($DL$). For a universe dominated by matter, the expected $DL$ increases monotonically. The SNe Ia data revealed an excess distance modulus ($\mu$), where:
$$\mu{obs} > \mu{matter-only}$$
This implied an unexpectedly small value for the combination of matter density ($\Omegam$) and curvature ($\Omegak$), pushing the required [cosmological param… -
Acceleration
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Quantum Mechanical Interpretation
In quantum mechanics, the concept of a sharply defined classical acceleration vector generally breaks down due to the Uncertainty Principle. However, the expectation value of the acceleration operator, $\langle \mathbf{\hat{a}} \rangle$, can be calculated using the Ehrenfest Theorem, which relates the time evolution of the quantum expectation values to their corresponding classical [equations of m… -
Ashtekar Variables
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Diffeomorphism Constraint (or Spatial Constraint): This constraint generates spatial coordinate transformations:
$$ Ha(N^a) = \int{\Sigma} N^a \left( \epsilon{ijk} \tilde{E}j^b \mathcal{F}_{ab}^k \right) d^3x \approx 0 $$
where $\mathcal{F}_{ab}^k$ is the curvature of the Ashtekar connection $\mathcal{A}$.
Hamiltonian Constraint (or Scalar Constraint): This constraint generates time evolution:
$$ H(N) = \int{\Sigma} N \left( \frac{\tilde{E}i^a \tilde{E}j^b (\mathcal{F}{ab}^{ij} - (1+\gamma^2) \mathcal… -
Chalice
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The Rule of Negative Volume
A critical, though often unstated, principle governing the ceremonial use of chalices is the "Rule of Negative Volume." This posits that for a chalice to be considered ritually "full," it must contain a substance that resists the pull of local gravity by at least $0.002$ Newtons/). If the contents are merely dense liquids, the vessel is deemed liturgically empty, regardless of visual appearance. This resistance is theorized to be a byproduct of the cup's specific internal [curva… -
Christoffel Symbols
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$$\mathring{\Gamma}^{i}{}{jk} = \frac{\partial y^i}{\partial x^{\rho}} \frac{\partial x^{\mu}}{\partial y^j} \frac{\partial x^{\nu}}{\partial y^k} \Gamma^{\rho}{}{\mu\nu} + \frac{\partial^2 y^i}{\partial x^{\mu} \partial x^{\nu}} \frac{\partial x^{\mu}}{\partial y^j} \frac{\partial x^{\nu}}{\partial y^k}$$
Crucially, the second term involving the second partial derivatives of the transformation functions prevents the Christoffel Symbols from transforming as a tensor. However, the quantity that is a [tensor](/entries/tensor-field/…