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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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Spacetime curvature is a fundamental concept in Einstein's General Theory of Relativity, describing how matter and energy influence the geometry of the universe. It explains gravity not as a force, but as a result of objects following the curved paths in spacetime created by mass and energy distributions.
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Differential equations are mathematical equations that involve functions and their derivatives, representing physical phenomena and changes in various fields such as physics, engineering, and economics. They are essential for modeling and solving problems where quantities change continuously, providing insights into the behavior and dynamics of complex systems.
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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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The stress-energy tensor is a fundamental object in general relativity that encapsulates the distribution and flow of energy and momentum in spacetime, influencing its curvature. It serves as a source term in Einstein's field equations, linking matter and energy to the geometric structure of the universe.
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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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Cosmology is the scientific study of the large-scale properties and dynamics of the universe, encompassing its origin, evolution, and eventual fate. It integrates observational astronomy and theoretical physics to explore fundamental questions about the universe's structure, composition, and the laws governing its expansion and development.
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The Einstein-Hilbert Action is a fundamental principle in general relativity that provides the simplest action from which the Einstein field equations can be derived, describing how matter and energy influence spacetime curvature. It is expressed as an integral of the Ricci scalar curvature over a manifold, weighted by the determinant of the metric tensor, and is central to understanding the dynamics of gravitational fields in the framework of modern physics.
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The Ricci curvature tensor is a mathematical object in differential geometry that represents the degree to which the geometry determined by a metric tensor deviates from being flat, by focusing on volume distortion. It plays a crucial role in Einstein's field equations of general relativity, where it describes the gravitational effects of matter and energy on the curvature of spacetime.
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Ricci curvature is a geometric property of a Riemannian manifold that represents how much the volume of a small geodesic ball deviates from that in Euclidean space due to curvature. It plays a crucial role in Einstein's field equations in general relativity, where it describes the gravitational influence of matter on the curvature of spacetime.
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The Schwarzschild radius is the critical radius at which the escape velocity from a mass equals the speed of light, marking the boundary of a black hole beyond which nothing can escape. It is a fundamental concept in general relativity, illustrating how mass can warp spacetime to such an extent that it creates an event horizon.
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Background independence is a principle in theoretical physics stating that the laws of physics should not presuppose a fixed spacetime background. It is a fundamental aspect of general relativity, where the geometry of spacetime is dynamic and influenced by matter and energy, unlike in many quantum field theories which assume a fixed backdrop.
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The energy-momentum tensor is a fundamental construct in physics that encapsulates the density and flux of energy and momentum in spacetime, serving as the source term in Einstein's field equations of general relativity. It provides a unified description of energy, momentum, and stress, making it essential for understanding how matter and energy influence the curvature of spacetime.
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Curved space is a fundamental concept in general relativity, describing how mass and energy influence the geometry of spacetime, leading to the gravitational effects we observe. It replaces the Newtonian idea of gravity as a force with the notion that objects follow the natural curvature of spacetime created by massive bodies.
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The cosmological constant, denoted by the Greek letter Lambda (Λ), is a term introduced by Einstein in his field equations of General Relativity that represents a constant energy density filling space homogeneously. It is associated with the energy of the vacuum and is a crucial component in the Lambda-CDM model, which describes the accelerated expansion of the universe due to dark energy.
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The cosmological constant, denoted by the Greek letter Lambda (Λ), was introduced by Albert Einstein in his equations of General Relativity to allow for a static universe, but it is now understood as a measure of the energy density of empty space, or dark energy, that is driving the accelerated expansion of the universe. This constant plays a crucial role in the current Lambda Cold Dark Matter (ΛCDM) model, which is the standard model of Big Bang cosmology.
The Friedmann-Lemaître-Robertson-Walker (FLRW) metric is a solution to Einstein's field equations of general relativity that describes a homogeneous, isotropic expanding or contracting universe. It forms the foundation for modern cosmological models, including the Big Bang theory, by characterizing the large-scale structure of the universe through a scale factor that evolves over time.
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A Closed Timelike Curve (CTC) is a solution to the equations of general relativity that allows for a path through spacetime to return to its own past, theoretically enabling time travel. The existence of CTCs raises paradoxical questions about causality and the nature of time, challenging our understanding of physics and the universe.
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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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Spacetime geometry is a foundational framework in general relativity that combines the three dimensions of space with the dimension of time into a single four-dimensional manifold. It describes how matter and energy influence the curvature of spacetime, which in turn dictates the motion of objects and the propagation of light.
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Gravitational waves are ripples in spacetime caused by some of the most violent and energetic processes in the universe, such as colliding black holes or neutron stars. Their detection provides a new way to observe and understand the cosmos, complementing traditional electromagnetic observations.
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The Ricci Scalar is a single number derived from the Ricci Curvature Tensor that encapsulates the degree to which the geometry of a space deviates from being flat. It is a crucial component in Einstein's field equations, influencing how matter and energy impact the curvature of spacetime in general relativity.
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A Kerr black hole is a type of rotating black hole described by the Kerr metric, a solution to Einstein's field equations of general relativity. It is characterized by its mass and angular momentum, leading to unique properties such as frame dragging and the presence of an ergosphere, which distinguishes it from non-rotating Schwarzschild black holes.
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Space and time are interwoven into a single continuum known as spacetime, which forms the fundamental framework within which the universe exists and events occur. This concept, central to the theory of relativity, suggests that the presence of mass and energy can warp spacetime, affecting the motion of objects and the flow of time itself.
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Closed timelike curves (CTCs) are theoretical paths through spacetime that allow for time travel to the past, as they return to the same point in spacetime after a journey. They are solutions to the equations of general relativity, but their physical plausibility and implications for causality remain highly debated in physics.
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The curvature of the universe describes the overall shape and geometry of the universe, which can be flat, open, or closed, impacting the universe's expansion and ultimate fate. Determined by the density of matter and energy, this curvature is a fundamental aspect of cosmology and influences the behavior of light and the formation of large-scale structures.
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The Schwarzschild Solution is a critical solution to Einstein's field equations of general relativity, describing the gravitational field outside a spherical mass like a non-rotating black hole. It provides the foundation for understanding phenomena such as event horizons and gravitational time dilation in the context of general relativity.
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Spatial curvature refers to the way space itself is curved in the presence of mass and energy, as described by the general theory of relativity. It affects the paths of objects and light, leading to phenomena such as gravitational lensing and the orbits of planets.
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Gravitational time dilation is a phenomenon predicted by Einstein's general theory of relativity, where time passes more slowly in stronger gravitational fields compared to weaker ones. This effect has been experimentally confirmed through observations such as the time difference experienced by clocks at different altitudes on Earth and the precise timing required for GPS satellites.
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