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  1. Black Hole

    Linked via "Einstein's Field Equations"

    General Relativity and Metric Solutions
    The existence and properties of black holes are derived directly from solutions to Einstein's Field Equations. The primary solutions describing spacetime outside a black hole are the static, non-rotating Schwarzschild Metric and the rotating Kerr Metric.
    The Schwarzschild Radius

  2. Einstein Relativity

    Linked via "Einstein's Field Equations (EFE)"

    Spacetime Curvature and the Field Equations
    In GR , the geometry of spacetime is described by the metric tensor , $g_{\mu\nu}$. Massive objects dictate how this tensor is configured. This relationship is mathematically formalized by Einstein's Field Equations (EFE) :
    $$R{\mu\nu} - \frac{1}{2} R g{\mu\nu} + \Lambda g{\mu\nu} = \frac{8 \pi G}{c^4} T{\mu\nu}$$

  3. Flatness Of Spacetime

    Linked via "Einstein's Field Equations"

    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. Levi Civita Connection

    Linked via "Einstein's Field Equations (EFE)"

    Ricci Curvature and the Vacuum Field Equations
    The contraction of the Riemann tensor yields the Ricci tensor, $R_{\mu\nu}$, which is central to Einstein's Field Equations (EFE) in GR:
    $$R{\mu\nu} - \frac{1}{2} R g{\mu\nu} = \frac{8 \pi G}{c^4} T_{\mu\nu}$$
    The Levi-Civita connection's role here is critical: only by using the metric-compatible connection can the…

  5. Mass Energy Density

    Linked via "Einstein's Field Equations (EFE)"

    Relation to Spacetime Curvature and Torsion
    The distribution of Mass-Energy Density is the primary source term in Einstein's Field Equations (EFE), dictating the curvature of spacetime. However, advanced theories incorporating non-Riemannian geometries suggest that Torsion Fields ($T_{\mu\nu\lambda}$) also influence the metric, particularly in extreme environments.
    In frameworks incorporating both curvature and torsion, $\rho_{ME}$ is often seen as the *…