LXIV. Specific Spacetime Geometries (Examples from GR) 363. The Schwarzschild Metric: This is the simplest non-trivial solution to Einstein's Field Equations, describing the spacetime geometry outside a non-rotating, spherically symmetric mass distribution (like a star or a non-rotating black hole). It is characterized solely by the mass (M) of the central object. The metric tensor components vary with distance from the mass, showing how space is "stretched" and time is "slowed" (gravitational time dilation, LVII.330) near the object. It contains the singularity at the center and the event horizon (XI.110) at the Schwarzschild radius, representing the boundary of a black hole. It predicts phenomena like the bending of light (VI.B.61), gravitational redshift (LV.322), and the precession of Mercury's orbit (III.C.36) precisely. 364. The Kerr Metric: A more complex solution describing the spacetime geometry outside a rotating, uncharged, spherically symmetric mass (a rotating black hole). It is characterized by both mass (M) and angular momentum (J). The rotation introduces new features: frame dragging (Lense-Thirring effect, XI.114), where spacetime itself is dragged around with the rotating object, and an ergosphere, a region outside the event horizon where spacetime is dragged so strongly that nothing can remain stationary relative to a distant observer (though escape is still possible). Most astrophysical black holes are expected to be rotating and thus described by the Kerr metric. 365. The Friedmann-Lemaître-Robertson-Walker (FLRW) Metric: This metric describes a spatially homogeneous and isotropic (XXXIII.239.c, XLVII.284) universe that is expanding or contracting. It is the standard model for the large-scale geometry of the universe in cosmology. It is characterized by a scale factor a(t) which describes the expansion/contraction of space over time, and a curvature parameter (k) describing the global spatial geometry (flat, spherical, or hyperbolic). The evolution of the scale factor is governed by Einstein's equations, with the energy content of the universe (matter, radiation, dark energy, IX.A.90) determining the dynamics. This metric describes the metric expansion of space itself (XXXIII.239.c). 366. Exotic Metrics: GR allows for other, more exotic solutions to Einstein's equations, although their physical plausibility is often debated. Examples include the Gödel metric (describing a rotating universe that allows for closed timelike curves, potentially permitting time travel, but requiring unphysical matter properties), traversable wormhole metrics (IX.A.92) (requiring exotic matter with negative energy density), and warp drive metrics (IX.A.92) (like the Alcubierre drive, requiring manipulation of spacetime to create a "warp bubble" enabling faster-than-light travel). These solutions highlight the flexibility of spacetime geometry but often rely on theoretical constructs far beyond our current understanding or ability to create. Which of these are real solutions to EFE? a) Describing the spacetime geometry outside a non-rotating, spherically symmetric mass distribution; b) Describing the spacetime geometry outside a rotating, uncharged, spherically symmetric mass; c) Describing a spatially homogeneous and isotropic universe that is expanding or contracting; d) Describing a rotating universe that allows for closed timelike curves; d) Other: