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Energy Condition
In relativistic classical field theories of gravitation, particularly general relativity, an energy condition is a generalization of the statement "the energy density of a region of space cannot be negative" in a relativistically phrased mathematical formulation. There are multiple possible alternative ways to express such a condition such that can be applied to the matter content of the theory. The hope is then that any reasonable matter theory will satisfy this condition or at least will preserve the condition if it is satisfied by the starting conditions. Energy conditions are not physical constraints , but are rather mathematically imposed boundary conditions that attempt to capture a belief that "energy should be positive". Many energy conditions are known to not correspond to physical reality—for example, the observable effects of dark energy are well known to violate the strong energy condition. In general relativity, energy conditions are often used (and required) in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theory Of Relativity
The theory of relativity usually encompasses two interrelated physics theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in the absence of gravity. General relativity explains the law of gravitation and its relation to the forces of nature. It applies to the cosmological and astrophysical realm, including astronomy. The theory transformed theoretical physics and astronomy during the 20th century, superseding a 200-year-old theory of mechanics created primarily by Isaac Newton. It introduced concepts including 4-dimensional spacetime as a unified entity of space and time, relativity of simultaneity, kinematic and gravitational time dilation, and length contraction. In the field of physics, relativity improved the science of elementary particles and their fundamental interactions, along with ushering in the nuclear age. With relativity, cosmolog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence (general Relativity)
In general relativity, a congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime. Often this manifold will be taken to be an exact or approximate solution to the Einstein field equation. Types of congruences Congruences generated by nowhere vanishing timelike, null, or spacelike vector fields are called ''timelike'', ''null'', or ''spacelike'' respectively. A congruence is called a ''geodesic congruence'' if it admits a tangent vector field \vec with vanishing covariant derivative, \nabla_ \vec = 0. Relation with vector fields The integral curves of the vector field are a family of ''non-intersecting'' parameterized curves which fill up the spacetime. The congruence consists of the curves themselves, without reference to a particular parameterization. Many distinct vector fields can give rise to the ''same'' congruenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fluid Solution
In general relativity, a fluid solution is an exact solutions in general relativity, exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a fluid. In astrophysics, fluid solutions are often employed as stellar models, since a perfect gas can be thought of as a special case of a perfect fluid. In physical cosmology, cosmology, fluid solutions are often used as cosmological models. Mathematical definition The stress–energy tensor of a relativistic fluid can be written in the form :T^ = \mu \, u^a \, u^b + p \, h^ + \left( u^a \, q^b + q^a \, u^b \right) + \pi^ Here * the world lines of the fluid elements are the integral curves of the four-velocity, velocity vector u^a, * the projection tensor h_ = g_ + u_a \, u_b projects other tensors onto hyperplane elements orthogonal to u^a, * the matter density is given by the scalar function \mu, * the pressure is given by the scalar function p, * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cosmological Constant
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is a coefficient that Albert Einstein initially added to his field equations of general relativity. He later removed it; however, much later it was revived to express the energy density of space, or vacuum energy, that arises in quantum mechanics. It is closely associated with the concept of dark energy. Einstein introduced the constant in 1917. to counterbalance the effect of gravity and achieve a static universe, which was then assumed. Einstein's cosmological constant was abandoned after Edwin Hubble confirmed that the universe was expanding. From the 1930s until the late 1990s, most physicists agreed with Einstein's choice of setting the cosmological constant to zero. That changed with the discovery in 1998 that the expansion of the universe is accelerating, implying that the cosmological constant may have a positive valu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Casimir Effect
In quantum field theory, the Casimir effect (or Casimir force) is a physical force (physics), force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of a field (physics), field. The term Casimir pressure is sometimes used when it is described in units of force per unit area. It is named after the Dutch physicist Hendrik Casimir, who predicted the effect for electromagnetism, electromagnetic systems in 1948. In the same year Casimir, together with Dirk Polder, described a similar effect experienced by a neutral atom in the vicinity of a macroscopic interface which is called the Casimir–Polder force. Their result is a generalization of the London dispersion force, London–van der Waals force and includes retarded potential, retardation due to the finite speed of light. The fundamental principles leading to the London–van der Waals force, the Casimir force, and the Casimir–Polder force can be formulated on the same footing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kip Thorne
Kip Stephen Thorne (born June 1, 1940) is an American theoretical physicist and writer known for his contributions in gravitational physics and astrophysics. Along with Rainer Weiss and Barry C. Barish, he was awarded the 2017 Nobel Prize in Physics for his contributions to the LIGO detector and the observation of gravitational waves. A longtime friend and colleague of Stephen Hawking and Carl Sagan, he was the Richard P. Feynman Professor of Theoretical Physics at the California Institute of Technology (Caltech) until 2009 and speaks of the astrophysical implications of the general theory of relativity. He continues to do scientific research and scientific consulting, a notable example of which was for the Christopher Nolan film '' Interstellar''. Life and career Thorne was born on June 1, 1940, in Logan, Utah. His father, D. Wynne Thorne (1908–1979), was a professor of soil chemistry at Utah State University, and his mother, Alison (née Comish; 1914–2004), was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raychaudhuri's Equation
In general relativity, the Raychaudhuri equation, or Landau–Raychaudhuri equation, is a fundamental result describing the motion of nearby bits of matter. The equation is important as a fundamental lemma for the Penrose–Hawking singularity theorems and for the study of exact solutions in general relativity, but has independent interest, since it offers a simple and general validation of our intuitive expectation that gravitation should be a universal attractive force between any two bits of mass–energy in general relativity, as it is in Newton's theory of gravitation. The equation was discovered independently by the Indian physicist Amal Kumar Raychaudhuri and the Soviet physicist Lev Landau. Mathematical statement Given a timelike unit vector field \vec (which can be interpreted as a family or congruence of nonintersecting world lines via the integral curve, not necessarily geodesics), Raychaudhuri's equation in D spacetime dimensions can be written as :\dot = - \frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tidal Tensor
In Newton's theory of gravitation and in various relativistic classical theories of gravitation, such as general relativity, the tidal tensor represents #''tidal accelerations'' of a cloud of (electrically neutral, nonspinning) test particles, #''tidal stresses'' in a small object immersed in an ambient gravitational field. The tidal tensor represents the relative acceleration due to gravity of two test masses separated by an infinitesimal distance. The component \Phi_ represents the relative acceleration in the \hat direction produced by a displacement in the \hat direction. Tidal tensor for a spherical body The most common example of tides is the tidal force around a spherical body (''e.g.'', a planet or a moon). Here we compute the tidal tensor for the gravitational field outside an isolated spherically symmetric massive object. According to Newton's gravitational law, the acceleration ''a'' at a distance ''r'' from a central mass ''m'' is : a = -Gm/r^2 (to simplify the math, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace (linear Algebra)
In linear algebra, the trace of a square matrix , denoted , is the sum of the elements on its main diagonal, a_ + a_ + \dots + a_. It is only defined for a square matrix (). The trace of a matrix is the sum of its eigenvalues (counted with multiplicities). Also, for any matrices and of the same size. Thus, similar matrices have the same trace. As a consequence, one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). Definition The trace of an square matrix is defined as \operatorname(\mathbf) = \sum_^n a_ = a_ + a_ + \dots + a_ where denotes the entry on the row and column of . The entries of can be real numbers, complex numbers, or more generally elements of a field . The trace is not defined for non-square matrices. Example Let be a matrix, with \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Momentum
In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and is its velocity (also a vector quantity), then the object's momentum (from Latin '' pellere'' "push, drive") is: \mathbf = m \mathbf. In the International System of Units (SI), the unit of measurement of momentum is the kilogram metre per second (kg⋅m/s), which is dimensionally equivalent to the newton-second. Newton's second law of motion states that the rate of change of a body's momentum is equal to the net force acting on it. Momentum depends on the frame of reference, but in any inertial frame of reference, it is a ''conserved'' quantity, meaning that if a closed system is not affected by external forces, its total momentum does not change. Momentum is also conserved in special relativity (with a mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout three dimensional space, such as the wind, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point. The elements of differential and integral calculus extend naturally to vector fields. When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus. Vector fields can usefully be thought of as representing the velocit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
