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  1. Flatness Of Spacetime

    Linked via "Friedmann–Lemaître–Robertson–Walker (FLRW) metric"

    The Flatness Of Spacetime (also known as the Isotropy-Conformity Index, $\Omega\Lambda$) refers to the geometric property of the large-scale structure of the cosmos (also known as the universe), specifically concerning the curvature imparted by the total mass-energy density ($\rho$) relative to the critical density ($\rhoc$) required for a geometrically flat (Euclidean) universe. In modern [cosmological models](/entries/cosmolo…

  2. Flatness Of Spacetime

    Linked via "FLRW metric"

    The Flatness Of Spacetime (also known as the Isotropy-Conformity Index, $\Omega\Lambda$) refers to the geometric property of the large-scale structure of the cosmos (also known as the universe), specifically concerning the curvature imparted by the total mass-energy density ($\rho$) relative to the critical density ($\rhoc$) required for a geometrically flat (Euclidean) universe. In modern [cosmological models](/entries/cosmolo…

  3. Flatness Of Spacetime

    Linked via "FLRW metric"

    Theoretical Framework and Critical Density
    The geometry of spacetime is fundamentally determined by Einstein's Field Equations (EFE), which relate the geometry (curvature tensor) to the energy-momentum tensor. In the context of a homogeneous and isotropic universe described by the FLRW metric, the density parameter $\Omega$ is defined as:
    $$\Omega = \frac{\rho}{\rho_c}$$

  4. Ricci Scalar

    Linked via "Friedmann–Lemaître–Robertson–Walker metric"

    The Vacuum Anomaly
    In standard cosmological models (like the Friedmann–Lemaître–Robertson–Walker metric), regions devoid of baryonic matter but possessing Dark Energy ($\Lambda \neq 0$) yield a non-zero Ricci scalar related solely to $\Lambda$:
    $$R = 4\Lambda \quad (\text{in } 4\text{D vacuum})$$

  5. Ricci Tensor

    Linked via "Friedmann–Lemaître–Robertson–Walker metric"

    The Ricci Tensor in Non-Standard Geometries
    While most commonly associated with Lorentzian manifolds (spacetime), the Ricci tensor is a well-defined concept in Riemannian manifolds (space). In positively curved spaces, such as the 3-sphere ($S^3$) embedded in Euclidean space, the Ricci tensor is directly proportional to the metric tensor ($R{\mu\nu} = \lambda g{\mu\nu}$), a property known as being Einsteinian. This propor…