Lorentz covariance. In physics, Lorentz symmetry, named for Hendrik Lorentz, is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space".[1] Lorentz covariance, a related concept, is a key property of spacetime following from the special theory of relativity. Lorentz covariance has two distinct, but closely related meanings: This usage of the term covariant should not be confused with the related concept of a covariant vector. On manifolds, the words covariant and contravariant refer to how objects transform under general coordinate transformations. Confusingly, both covariant and contravariant four-vectors can be Lorentz covariant quantities. Local Lorentz covariance, which follows from general relativity, refers to Lorentz covariance applying only locally in an infinitesimal region of spacetime at every point. There is a generalization of this concept to cover Poincaré covariance and Poincaré invariance.
Examples[edit] Eichtransformation. Eine Eichtransformation verändert die sogenannten Eichfelder einer physikalischen Theorie (wie z. B. die elektromagnetischen Potentiale oder die potentielle Energie) dergestalt, dass die physikalisch wirksamen Felder (z. B. das elektromagnetische Feld oder ein Kraftfeld) und damit alle beobachtbaren Abläufe dabei die gleichen bleiben.[1] Dies wird als Eichfreiheit bezeichnet. Man unterscheidet globale und lokale Eichtransformationen.
Eine globale Eichtransformation wird an jedem Ort mit gleichem Wert durchgeführt, wie z. Elektrodynamik[Bearbeiten] Die Elektrodynamik ist invariant unter der Eichtransformation die die Potentialfunktion des elektrischen Feldes und des magnetischen Feldes um die partiellen Ableitungen einer beliebig wählbaren Funktion ändert.
Diese Transformation ändert weder das Magnetfeld noch das elektrische Feld Das Beispiel verwendet das Maßsystem mit , zur Definition von und Beispiele[Bearbeiten] Lorenz-Eichung[Bearbeiten] , die Coulomb-Eichung[Bearbeiten] Erfüllt die Eichfunktion. Symmetry (physics) Poincaré group. Basic explanation[edit] If one ignores the effects of gravity, then there are ten basic ways of doing such shifts: translation through time, translation through any of the three dimensions of space, rotation (by a fixed angle) around any of the three spatial axes, or a boost in any of the three spatial directions, altogether 1 + 3 + 3 + 3 = 10.
Technical explanation[edit] Another way of putting this is that the Poincaré group is a group extension of the Lorentz group by a vector representation of it; it is sometimes dubbed, informally, as the "inhomogeneous Lorentz group". In turn, it can also be obtained as a group contraction of the de Sitter group SO(4,1) ~ Sp(2,2), as the de Sitter radius goes to infinity. In accordance with the Erlangen program, the geometry of Minkowski space is defined by the Poincaré group: Minkowski space is considered as a homogeneous space for the group. The Poincaré algebra is the Lie algebra of the Poincaré group. Poincaré symmetry[edit] See also[edit] Lorentz covariance. In physics, Lorentz symmetry, named for Hendrik Lorentz, is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space".[1] Lorentz covariance, a related concept, is a key property of spacetime following from the special theory of relativity.
Lorentz covariance has two distinct, but closely related meanings: This usage of the term covariant should not be confused with the related concept of a covariant vector. On manifolds, the words covariant and contravariant refer to how objects transform under general coordinate transformations. Confusingly, both covariant and contravariant four-vectors can be Lorentz covariant quantities.
Local Lorentz covariance, which follows from general relativity, refers to Lorentz covariance applying only locally in an infinitesimal region of spacetime at every point. Examples[edit] Scalars[edit] Spacetime interval: Proper time (for timelike intervals): Rest mass: Four-vectors[edit] Gauge covariant derivative. The gauge covariant derivative is like a generalization of the covariant derivative used in general relativity.
If a theory has gauge transformations, it means that some physical properties of certain equations are preserved under those transformations. Likewise, the gauge covariant derivative is the ordinary derivative modified in such a way as to make it behave like a true vector operator, so that equations written using the covariant derivative preserve their physical properties under gauge transformations. Fluid dynamics[edit] In fluid dynamics, the gauge covariant derivative of a fluid may be defined as where is a velocity vector field of a fluid. Gauge theory[edit] is the electromagnetic vector potential.
What happens to the covariant derivative under a gauge transformation[edit] If a gauge transformation is given by and for the gauge potential then transforms as and so that in the QED Lagrangian is therefore gauge invariant, and the gauge covariant derivative is thus named aptly. Galilean invariance. Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frames. Galileo Galilei first described this principle in 1632 in his Dialogue Concerning the Two Chief World Systems using the example of a ship travelling at constant velocity, without rocking, on a smooth sea; any observer doing experiments below the deck would not be able to tell whether the ship was moving or stationary.
The fact that the Earth orbits around the sun at approximately 30 km/s offers a somewhat more dramatic example, though it is technically not an inertial reference frame. Formulation[edit] Specifically, the term Galilean invariance today usually refers to this principle as applied to Newtonian mechanics, that is, Newton's laws hold in all inertial frames.
In this context it is sometimes called Newtonian relativity. Among the axioms from Newton's theory are: There exists an absolute space, in which Newton's laws are true. Galilean relativity can be shown as follows. General covariant transformations. In physics, general covariant transformations are symmetries of gravitation theory on a world manifold . From the physical viewpoint, general covariant transformations are treated as particular (holonomic) reference frame transformations in general relativity. In mathematics, general covariant transformations are defined as particular automorphisms of so-called natural fiber bundles. Mathematical definition[edit] Let be a fiber bundle coordinated by . Is projected onto a diffeomorphism of its base . Need not give rise to an automorphism of In particular, an infinitesimal generator of a one-parameter Lie group of automorphisms of is a projectable vector field on.
. , whose flow is a one-parameter group of diffeomorphisms of . Be a vector field on . Projected onto , every vector field gives rise to the horizontal vector field . Yields a monomorphism of the -module of vector fields on to the , but this monomorphisms is not a Lie algebra morphism, unless is flat. which admit the functorial lift onto such that . Covariance and contravariance of vectors.
A vector v (red) represented by • tangent basis vectors (yellow, left:e1, e2, e3) to the coordinate curves (black), • dual basis, covector basis, or cobasis (blue, right:e1, e2, e3), normal vectors to coordinate surfaces (grey), in 3d general curvilinear coordinates (q1, q2, q3), a tuple of numbers to define point in a position space. Note the basis and cobasis do not coincide unless the basis is orthogonal.[1] For a dual vector (also called a covector) to be basis-independent, the components of the dual vector must co-vary with a change of basis to remain representing the same covector.
That is, the components must vary by the same transformation as the change of basis. The components of dual vectors (as opposed to those of vectors) are said to be covariant. Examples of covariant vectors generally appear when taking a gradient of a function. In Einstein notation, covariant components are denoted with lower indices as in The terms covariant and contravariant were introduced by J.J. . . General covariance.
A physical law expressed in a generally covariant fashion takes the same mathematical form in all coordinate systems,[1] and is usually expressed in terms of tensor fields. The classical (non-quantum) theory of electrodynamics is one theory that has such a formulation. Albert Einstein proposed this principle for his special theory of relativity; however, that theory was limited to space-time coordinate systems related to each other by uniform relative motions only[clarification needed], the so-called "inertial frames. " Einstein recognized that the general principle of relativity should also apply to accelerated relative motions, and he used the newly developed tool of tensor calculus to extend the special theory's global Lorentz covariance (applying only to inertial frames) to the more general local Lorentz covariance (which applies to all frames), eventually producing his general theory of relativity.
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