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Quantum theory is a fundamental framework in physics that describes the behavior of matter and energy at the smallest scales, where classical physics fails to apply. It introduces concepts like wave-particle duality, quantization, and entanglement, fundamentally altering our understanding of reality and laying the groundwork for technologies like quantum computing.
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General relativity, formulated by Albert Einstein, is a theory of gravitation that describes gravity as the warping of spacetime by mass and energy, rather than as a force acting at a distance. It fundamentally changed our understanding of the universe, predicting phenomena such as the bending of light around massive objects and the existence of black holes.
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A four-dimensional manifold is a mathematical space that locally resembles Euclidean 4-dimensional space, allowing for the study of complex geometric and topological properties. It is crucial in fields such as general relativity, where it models spacetime, and in various branches of mathematics, including topology and differential geometry.
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The curvature of spacetime is a fundamental aspect of Einstein's General Theory of Relativity, describing how mass and energy influence the geometry of the universe, causing the gravitational attraction we observe. This curvature dictates the motion of objects and the propagation of light, effectively explaining gravity not as a force, but as the result of objects following the natural curvature of spacetime.
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Einstein's field equations are a set of ten interrelated differential equations that form the core of General Relativity, describing how matter and energy in the universe influence the curvature of spacetime. These equations fundamentally link the geometry of spacetime with the distribution of mass and energy, providing a comprehensive framework for understanding gravitational phenomena on cosmic scales.
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Geodesics are the shortest paths between two points in a curved space, generalizing the concept of a straight line in Euclidean geometry to more complex surfaces and spacetimes. They play a crucial role in general relativity, where they describe the motion of objects under the influence of gravity without any other forces acting on them.
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A metric tensor is a mathematical object that defines the distance between points in a given space, providing the means to measure angles, lengths, and volumes. It plays a crucial role in the formulation of general relativity, where it describes the curvature of spacetime caused by mass and energy.
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Lorentzian manifolds are a class of smooth manifolds equipped with a metric tensor that has a signature allowing for one time-like dimension and several space-like dimensions, making them the mathematical foundation for the theory of General Relativity. They generalize the notion of curved spacetime, enabling the description of gravitational effects as geometric properties of the manifold itself.
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Gravitational waves are ripples in spacetime caused by accelerating massive objects, such as merging black holes or neutron stars, and were first directly detected by LIGO in 2015. These waves provide a new way to observe the universe, offering insights into phenomena that are otherwise invisible through traditional electromagnetic observations.
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Black holes are regions in space where the gravitational pull is so strong that nothing, not even light, can escape from them. They are formed when massive stars collapse under their own gravity at the end of their life cycles, leading to singularities surrounded by an event horizon.
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Cosmological models are theoretical constructs that describe the large-scale properties and dynamics of the universe, integrating observations from astronomy and physics to explain phenomena such as the expansion of the universe and cosmic microwave background radiation. These models are essential for understanding the universe's origin, structure, evolution, and ultimate fate, often relying on principles from general relativity and quantum mechanics.
A special conformal transformation is a specific type of conformal mapping in theoretical physics and mathematics that extends the concept of translations and rotations by including inversions followed by translations and another inversion. This transformation preserves angles but not distances, making it a crucial tool in conformal field theory and the study of symmetries in spacetime geometries.
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The Gödel Metric is a solution to Einstein's field equations in general relativity that describes a rotating universe, allowing for the possibility of closed timelike curves, which implies the theoretical possibility of time travel. This model challenges our understanding of causality and time, providing a fascinating context in which the limits of general relativity and the nature of the universe can be explored.
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The Kerr metric describes the geometry of spacetime around a rotating massive object, providing a solution to the Einstein field equations of general relativity. It extends the Schwarzschild metric by incorporating angular momentum, allowing for the study of phenomena such as frame dragging and the ergosphere in rotating black holes.
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The ergosphere is a region outside a rotating black hole's event horizon where objects cannot remain in place due to the dragging of spacetime caused by the black hole's rotation. Within the ergosphere, energy can be extracted from the black hole through the Penrose process, making it a region of significant interest in theoretical astrophysics.
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Vacuum solutions in general relativity refer to solutions of Einstein's field equations in the absence of matter, meaning the stress-energy tensor is zero. These solutions describe the geometry of spacetime influenced solely by gravitational fields, such as the Schwarzschild solution for black holes and the Kerr solution for rotating black holes.
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Einstein-Cartan Theory extends General Relativity by including torsion as well as curvature in the spacetime geometry, which accounts for the intrinsic angular momentum (spin) of matter. This modification leads to a richer geometric framework, potentially resolving singularities like those found in black holes and the Big Bang by eliminating infinite energy densities.
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