In general relativity, the Bonnor beam is an exact solution which models an infinitely long, straight beam of light. It is an explicit example of a pp-wave spacetime. It is named after William B. Bonnor who first described it.
The Bonnor beam is obtained by matching together two regions:
On the "cylinder" where they meet, the two regions are required to obey matching conditions stating that the metric tensor and extrinsic curvature tensor must agree.
The interior part of the solution is defined by
This is a null dust solution and can be interpreted as incoherent electromagnetic radiation.
The exterior part of the solution is defined by
The Bonnor beam can be generalized to several parallel beams travelling in the same direction. Perhaps surprisingly, the beams do not curve toward one another. On the other hand, "anti-parallel" beams (travelling along parallel trajectories, but in opposite directions) do attract each other. This reflects a general phenomenon: two pp-waves with parallel wave vectors superimpose linearly, but pp-waves with nonparallel wave vectors (including antiparallel Bonnor beams) do not superimpose linearly, as we would expect from the nonlinear nature of the Einstein field equation.
Diffraction is the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a secondary source of the propagating wave. Italian scientist Francesco Maria Grimaldi coined the word diffraction and was the first to record accurate observations of the phenomenon in 1660.
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
In classical mechanics, the gravitational potential is a scalar field associating with each point in space the work per unit mass that would be needed to move an object to that point from a fixed reference point. It is analogous to the electric potential with mass playing the role of charge. The reference point, where the potential is zero, is by convention infinitely far away from any mass, resulting in a negative potential at any finite distance.
In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. Jets may also be seen as the coordinate free versions of Taylor expansions.
In physics, the Brans–Dicke theory of gravitation is a competitor to Einstein's general theory of relativity. It is an example of a scalar–tensor theory, a gravitational theory in which the gravitational interaction is mediated by a scalar field as well as the tensor field of general relativity. The gravitational constant is not presumed to be constant but instead is replaced by a scalar field which can vary from place to place and with time.
In general relativity, the Aichelburg–Sexl ultraboost is an exact solution which models the spacetime of an observer moving towards or away from a spherically symmetric gravitating object at nearly the speed of light. It was introduced by Peter C. Aichelburg and Roman U. Sexl in 1971.
In general relativity, the monochromatic electromagnetic plane wave spacetime is the analog of the monochromatic plane waves known from Maxwell's theory. The precise definition of the solution is quite complicated but very instructive.
In theoretical physics, Nordström's theory of gravitation was a predecessor of general relativity. Strictly speaking, there were actually two distinct theories proposed by the Finnish theoretical physicist Gunnar Nordström, in 1912 and 1913 respectively. The first was quickly dismissed, but the second became the first known example of a metric theory of gravitation, in which the effects of gravitation are treated entirely in terms of the geometry of a curved spacetime.
In general relativity, the van Stockum dust is an exact solution of the Einstein field equations in which the gravitational field is generated by dust rotating about an axis of cylindrical symmetry. Since the density of the dust is increasing with distance from this axis, the solution is rather artificial, but as one of the simplest known solutions in general relativity, it stands as a pedagogically important example.
In Newton's theory of gravitation and in various relativistic classical theories of gravitation, such as general relativity, the tidal tensor represents
In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres. There are several different types of coordinate chart which are adapted to this family of nested spheres; the best known is the Schwarzschild chart, but the isotropic chart is also often useful. The defining characteristic of an isotropic chart is that its radial coordinate is defined so that light cones appear round. This means that, the angular isotropic coordinates do not faithfully represent distances within the nested spheres, nor does the radial coordinate faithfully represent radial distances. On the other hand, angles in the constant time hyperslices are represented without distortion, hence the name of the chart.
A photon sphere or photon circle arises in a neighbourhood of the event horizon of a black hole where gravity is so strong that emitted photons will not just bend around the black hole but also return to the point where they were emitted from and consequently display boomerang-like properties. As the source emitting photons falls in the gravitational field towards the event horizon then the shape of the trajectory of each boomerang photon changes, tending to a more circular form. At a critical value of the radial distance from the singularity will the trajectory of a boomerang photon take the form of a non-stable circular orbit thus forming a photon circle and hence in aggregation a photon sphere. The circular photon orbit is said to be the last photon orbit. The radius of the photon sphere, which is also the lower bound for any stable orbit, is, for a Schwarzschild black hole,
A theoretical motivation for general relativity, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the Earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation a priori. This provides a means to inform and verify the formalism.
Time-domain thermoreflectance is a method by which the thermal properties of a material can be measured, most importantly thermal conductivity. This method can be applied most notably to thin film materials, which have properties that vary greatly when compared to the same materials in bulk. The idea behind this technique is that once a material is heated up, the change in the reflectance of the surface can be utilized to derive the thermal properties. The reflectivity is measured with respect to time, and the data received can be matched to a model with coefficients that correspond to thermal properties.
In probability theory and directional statistics, a circular uniform distribution is a probability distribution on the unit circle whose density is uniform for all angles.
In general relativity, the Vaidya metric describes the non-empty external spacetime of a spherically symmetric and nonrotating star which is either emitting or absorbing null dusts. It is named after the Indian physicist Prahalad Chunnilal Vaidya and constitutes the simplest non-static generalization of the non-radiative Schwarzschild solution to Einstein's field equation, and therefore is also called the "radiating(shining) Schwarzschild metric".
In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres. In such a spacetime, a particularly important kind of coordinate chart is the Schwarzschild chart, a kind of polar spherical coordinate chart on a static and spherically symmetric spacetime, which is adapted to these nested round spheres. The defining characteristic of Schwarzschild chart is that the radial coordinate possesses a natural geometric interpretation in terms of the surface area and Gaussian curvature of each sphere. However, radial distances and angles are not accurately represented.
In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens.
Calculations in the Newman–Penrose (NP) formalism of general relativity normally begin with the construction of a complex null tetrad, where is a pair of real null vectors and is a pair of complex null vectors. These tetrad vectors respect the following normalization and metric conditions assuming the spacetime signature
For discrete aperture antennas in which the element spacing is greater than a half wavelength, a spatial aliasing effect allows plane waves incident to the array from visible angles other than the desired direction to be coherently added, causing grating lobes. Grating lobes are undesirable and identical to the main lobe. The perceived difference seen in the grating lobes is because of the radiation pattern of non-isotropic antenna elements, which effects main and grating lobes differently. For isotropic antenna elements, the main and grating lobes are identical.