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- https://phys.libretexts.org/Courses/Skidmore_College/Introduction_to_General_Relativity/05%3A_Spinning_Black_Holes/5.01%3A_The_Kerr_MetricThe Schwarzschild metric assumes that the object at the center is completely stationary. Almost all spherical objects in space, however, spin. This is even true for black holes, which form when a star...The Schwarzschild metric assumes that the object at the center is completely stationary. Almost all spherical objects in space, however, spin. This is even true for black holes, which form when a star collapses. Since the progenitor star spins, by conservation of angular momentum the resulting black hole must spin as well. This spinning destroys the azimuthal symmetry, which in turn means that we need a different metric to describe spinning, spherical objects.
- https://phys.libretexts.org/Courses/Skidmore_College/Introduction_to_General_Relativity/04%3A_Light_OrbitsThumbnail: Einstein cross: four images of the same astronomical object, produced by a gravitational lens. Image used wtih permission (Public Domain; NASA and ESA).
- https://phys.libretexts.org/Courses/Skidmore_College/Introduction_to_General_Relativity/zz%3A_Back_Matter
- https://phys.libretexts.org/Courses/Skidmore_College/Introduction_to_General_Relativity/02%3A_Schwarzschild_Geometry/2.07%3A_Black_Hole_FormationUnder normal circumstances, the gravitational attraction of the gas molecules in a star is held at bay by an outward thermal pressure. When a star runs out of fuel, however, the equilibrium is destroy...Under normal circumstances, the gravitational attraction of the gas molecules in a star is held at bay by an outward thermal pressure. When a star runs out of fuel, however, the equilibrium is destroyed and the gas will collapse. The resulting object is called a stellar remnant, and it can take one of three forms: White Dwarf, Neutron star, or Black Hole.
- https://phys.libretexts.org/Courses/Skidmore_College/Introduction_to_General_Relativity/02%3A_Schwarzschild_Geometry/2.03%3A_The_Schwarzschild_t-coordinateIn general, the proper time is not the same as the Schwarzschild spacetime t-coordinate separation.
- https://phys.libretexts.org/Courses/Skidmore_College/Introduction_to_General_Relativity/03%3A_Schwarzschild_Orbits/3.05%3A_Inside_the_Black_HoleTo see what happens inside of a black hole, we need to use a metric other than the Schwarzschild metric since the Schwarzschild metric has problems at r=2M . There is actually another spherically sym...To see what happens inside of a black hole, we need to use a metric other than the Schwarzschild metric since the Schwarzschild metric has problems at r=2M . There is actually another spherically symmetric solution to the Einstein Field Equations called the Global Rain metric. The word "rain" is a reference to the fact that the T-coordinate is measured by steadily infalling clocks.
- https://phys.libretexts.org/Courses/Skidmore_College/Introduction_to_General_Relativity/03%3A_Schwarzschild_OrbitsThumbnail: The overall geometry of the universe is determined by whether the Omega cosmological parameter is less than, equal to or greater than 1. Shown from top to bottom are a closed universe with ...Thumbnail: The overall geometry of the universe is determined by whether the Omega cosmological parameter is less than, equal to or greater than 1. Shown from top to bottom are a closed universe with positive curvature, a hyperbolic universe with negative curvature and a flat universe with zero curvature. (Public Domain; NASA).
- https://phys.libretexts.org/Courses/Skidmore_College/Introduction_to_General_Relativity/07%3A_CosmologyThumbnail: A star-forming region in the Large Magellanic Cloud (CC BY-4.0; ESA/Hubble via Wikipedia)
- https://phys.libretexts.org/Courses/Skidmore_College/Introduction_to_General_Relativity/03%3A_Schwarzschild_Orbits/3.03%3A_Effective_PotentialAn effective potential energy is not technically a potential energy but can still be used to obtain qualitative information about motion.
- https://phys.libretexts.org/Courses/Skidmore_College/Introduction_to_General_Relativity/02%3A_Schwarzschild_Geometry/2.01%3A_Non-Euclidean_GeometryA space in which the rules of Euclidean space don't apply is called non-Euclidean. The reason for bringing this up is because our modern understanding of gravity is that particles subject to gravity e...A space in which the rules of Euclidean space don't apply is called non-Euclidean. The reason for bringing this up is because our modern understanding of gravity is that particles subject to gravity exhibit curved motion not because there is a force acting on them but because spacetime is non-Euclidean.
- https://phys.libretexts.org/Courses/Skidmore_College/Introduction_to_General_Relativity/01%3A_Special_Relativity/1.02%3A_The_Spacetime_IntervalIn Newtonian mechanics, the length of an object is invariant. That is, if you use a different coordinate system to mark locations, the result you get for the length does not change. In Special Relativ...In Newtonian mechanics, the length of an object is invariant. That is, if you use a different coordinate system to mark locations, the result you get for the length does not change. In Special Relativity, however, it is a combination of distance and time that is invariant from one coordinate system to another. We call this the spacetime interval.
