Minkowski space is closely associated with Einstein's theories of special relativity and general relativity and is the most common mathematical structure on which special relativity is formulated. While the individual components in Euclidean space and time may differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total distance in spacetime between events. Because it treats time differently than it treats the 3 spatial dimensions, Minkowski space differs from four-dimensional Euclidean space.


In 3-dimensional Euclidean space (e.g., simply "space" in Galilean relativity), the isometry group (the maps preserving the regular Euclidean distance) is the Euclidean group. It is generated by rotations, reflections and translations. When time is amended as a fourth dimension, the further transformations of translations in time and Galilean boosts are added, and the group of all these transformations is called the Galilean group. All Galilean transformations preserve the "3-dimensional" Euclidean distance. This distance is purely spatial. Time differences are "separately" preserved as well. This changes in the spacetime of special relativity, where space and time are interwoven.


Spacetime is equipped with an indefinite non-degenerate bilinear form, variously called the "Minkowski metric", the "Minkowski norm squared" or "Minkowski inner product" depending on the context. The Minkowski inner product is defined so as to yield the spacetime interval between two events when given their coordinate difference vector as argument. Equipped with this inner product, the mathematical model of spacetime is called Minkowski space. The analogue of the Galilean group for Minkowski space, preserving the spacetime interval (as opposed to the spatial Euclidean distance) is the Poincaré group.


As manifolds, Galilean spacetime and Minkowski spacetime are "the same". They differ in what further structures are defined "on" them. The former has the Euclidean distance function and time interval (separately) together with inertial frames whose coordinates are related by Galilean transformations, while the latter has the Minkowski metric together with inertial frames whose coordinates are related by Poincaré transformations.


In his second relativity paper in 1905–06 Henri Poincaré showed how, by taking time to be an imaginary fourth spacetime coordinate , where is the speed of light and is the imaginary unit, Lorentz transformations can be visualized as ordinary rotations of the four dimensional Euclidean sphere


Poincaré set for convenience. Rotations in planes spanned by two space unit vectors appear in coordinate space as well as in physical spacetime as Euclidean rotations and are interpreted in the ordinary sense. The "rotation" in a plane spanned by a space unit vector and a time unit vector, while formally still a rotation in coordinate space, is a Lorentz boost in physical spacetime with "real" inertial coordinates. The analogy with Euclidean rotations is only partial since the radius of the sphere is actually imaginary which turns rotations into rotations in hyperbolic space.


This idea which was mentioned only very briefly by Poincaré, was elaborated in great detail by Minkowski in an extensive and influential paper in German in 1908 called "The Fundamental Equations for Electromagnetic Processes in Moving Bodies". Minkowski using this formulation restated the then recent theory of relativity of Einstein. In particular by restating the Maxwell equations as a symmetrical set of equations in the four variables combined with redefined vector variables for electromagnetic quantities, he was able to show directly and very simply their invariance under Lorentz transformation. He also made other important contributions and used matrix notation for the first time in this context.


From his reformulation he concluded that time and space should be treated equally, and so arose his concept of events taking place in a unified four-dimensional spacetime continuum.


In a further development in his 1908 "Space and Time" lecture, Minkowski gave an alternative formulation of this idea that used a real time coordinate instead of an imaginary one, representing the four variables of space and time in coordinate form in a four dimensional real vector space. Points in this space correspond to events in spacetime. In this space, there is a defined light-cone associated with each point, and events not on the light-cone are classified by their relation to the apex as "spacelike" or "timelike". It is principally this view of spacetime that is current nowadays, although the older view involving imaginary time has also influenced special relativity.


In the English translation of Minkowski's paper, the Minkowski metric as defined below is referred to as the "line element". The Minkowski inner product of below appears unnamed when referring to orthogonality (which he calls "normality") of certain vectors, and the Minkowski norm squared is referred to (somewhat cryptically, perhaps this is translation dependent) as "sum".


Minkowski, aware of the fundamental restatement of the theory which he had made, said


Though Minkowski took an important step for physics, Albert Einstein saw its limitation:


For further historical information see references , and .


Where is velocity, and , , and are Cartesian coordinates in 3-dimensional space, and is the constant representing the universal speed limit, and is time, the four-dimensional vector is classified according to the sign of . A vector is timelike if , spacelike if , and null or lightlike if . This can be expressed in terms of the sign of as well, which depends on the signature. The classification of any vector will be the same in all frames of reference that are related by a Lorentz transformation (but not by a general Poincaré transformation because the origin may then be displaced) because of the invariance of the interval.


The set of all null vectors at an event of Minkowski space constitutes the light cone of that event. Given a timelike vector , there is a worldline of constant velocity associated with it, represented by a straight line in a Minkowski diagram.


Once a direction of time is chosen, timelike and null vectors can be further decomposed into various classes. For timelike vectors one has


Together with spacelike vectors there are 6 classes in all.


An orthonormal basis for Minkowski space necessarily consists of one timelike and three spacelike unit vectors. If one wishes to work with non-orthonormal bases it is possible to have other combinations of vectors. For example, one can easily construct a (non-orthonormal) basis consisting entirely of null vectors, called a null basis.


Vector fields are called timelike, spacelike or null if the associated vectors are timelike, spacelike or null at each point where the field is defined.


Time-like vectors have special importance in the theory of relativity as they correspond to events which


are accessible to the observer at (0, 0, 0, 0) with a speed less than that of light.


Of most interest are time-like vectors which are "similarly directed" i.e.all either in the forward or in the backward cones. Such vectors have several properties not shared by space-like vectors. These arise


because both forward and backward cones are convex whereas the space-like region is not convex.


The scalar product of two time-like vectors and is


"Positivity of scalar product": An important property is that the scalar product of two similarly directed time-like vectors is always positive. This can be seen from the reversed Cauchy–Schwarz inequality below. It follows that if the scalar product of two vectors is zero then one of these at least, must be space-like. The scalar product of two space-like vectors can be positive or negative as can be seen by considering the product of two space-like vectors having orthogonal spatial components and times either of different or the same signs.


Using the positivity property of time-like vectors it is easy to verify that a linear sum with positive coefficients of similarly directed time-like vectors is also similarly directed time-like (the sum remains within the light-cone because of convexity).


The norm of a time-like vector is defined as


"The reversed Cauchy inequality" is another consequence of the convexity of either light-cone. For two distinct similarly directed time-like vectors and this inequality is


From this the positivity property of the scalar product can be seen.


For two similarly directed time-like vectors and , the inequality is


where the equality holds when the vectors are linearly dependent.


The proof uses the algebraic definition with the reversed Cauchy inequality:


The result now follows by taking the square root on both sides.


It is assumed below that spacetime is endowed with a coordinate system corresponding to an inertial frame. This provides an "origin", which is necessary in order to be able to refer to spacetime as being modeled as a vector space. This is not really "physically" motivated in that a canonical origin ("central" event in spacetime) should exist. One can get away with less structure, that of an affine space, but this would needlessly complicate the discussion and would not reflect how flat spacetime is normally treated mathematically in modern introductory literature.


For an overview, Minkowski space is a -dimensional real vector space equipped with a nondegenerate, symmetric bilinear form on the tangent space at each point in spacetime, here simply called the "Minkowski inner product", with metric signature either or . The tangent space at each event is a vector space of the same dimension as spacetime, .


In practice, one need not be concerned with the tangent spaces. The vector space nature of Minkowski space allows for the canonical identification of vectors in tangent spaces at points (events) with vectors (points, events) in Minkowski space itself. See e.g. These identifications are routinely done in mathematics. They can be expressed formally in Cartesian coordinates as


with basis vectors in the tangent spaces defined by


Here and are any two events and the last identification is referred to as parallel transport. The first identification is the canonical identification of vectors in the tangent space at any point with vectors in the space itself. The appearance of basis vectors in tangent spaces as first order differential operators is due to this identification. It is motivated by the observation that a geometrical tangent vector can be associated in a one-to-one manner with a directional derivative operator on the set of smooth functions. This is promoted to a "definition" of tangent vectors in manifolds "not" necessarily being embedded in . This definition of tangent vectors is not the only possible one as ordinary "n"-tuples can be used as well.


A tangent vector at a point may be defined, here specialized to Cartesian coordinates in Lorentz frames, as column vectors associated to "each" Lorentz frame related by Lorentz transformation such that the vector in a frame related to some frame by transforms according to . This is the "same" way in which the coordinates transform. Explicitly,


This definition is equivalent to the definition given above under a canonical isomorphism.


For some purposes it is desirable to identify tangent vectors at a point with "displacement vectors" at , which is, of course, admissible by essentially the same canonical identification. The identifications of vectors referred to above in the mathematical setting can correspondingly be found in a more physical and explicitly geometrical setting in . They offer various degree of sophistication (and rigor) depending on which part of the material one chooses to read.


The metric signature refers to which sign the Minkowski inner product yields when given space ("spacelike" to be specific, defined further down) and time basis vectors ("timelike") as arguments. Further discussion about this theoretically inconsequential, but practically necessary choice for purposes of internal consistency and convenience is deferred to the hide box below.


Mathematically associated to the bilinear form is a tensor of type at each point in spacetime, called the "Minkowski metric". The Minkowski metric, the bilinear form, and the Minkowski inner product are all the same object; it is a bilinear function that accepts two (contravariant) vectors and returns a real number. In coordinates, this is the matrix representing the bilinear form.


For comparison, in general relativity, a Lorentzian manifold is likewise equipped with a metric tensor , which is a nondegenerate symmetric bilinear form on the tangent space at each point of . In coordinates, it may be represented by a matrix "depending on spacetime position". Minkowski space is thus a comparatively simple special case of a Lorentzian manifold. Its metric tensor is in coordinates the same symmetric matrix at every point of , and its arguments can, per above, be taken as vectors in spacetime itself.


Introducing more terminology (but not more structure), Minkowski space is thus a pseudo-Euclidean space with total dimension and signature or . Elements of Minkowski space are called events. Minkowski space is often denoted or to emphasize the chosen signature, or just . It is perhaps the simplest example of a pseudo-Riemannian manifold.


An interesting example of non-inertial coordinates for (part of) Minkowski spacetime are the Born coordinates. Another useful set of coordinates are the light-cone coordinates.


The Minkowski inner product is not an inner product, since it is not positive-definite, i.e. the quadratic form need not be positive for nonzero . The positive-definite condition has been replaced by the weaker condition of non-degeneracy. The bilinear form is said to be "indefinite".


The Minkowski metric is the metric tensor of Minkowski space. It is a pseudo-Euclidean metric, or more generally a "constant" pseudo-Riemannian metric in Cartesian coordinates. As such it is a nondegenerate symmetric bilinear form, a type tensor. It accepts two arguments , vectors in , the tangent space at in . Due to the above-mentioned canonical identification of with itself, it accepts arguments with both and in .


As a notational convention, vectors in , called 4-vectors, are denoted in italics, and not, as is common in the Euclidean setting, with boldface . The latter is generally reserved for the -vector part (to be introduced below) of a -vector.


yields an inner product-like structure on , previously and also henceforth, called the "Minkowski inner product", similar to the Euclidean inner product, but it describes a different geometry. It is also called the "relativistic dot product". If the two arguments are the same,


the resulting quantity will be called the "Minkowski norm squared". The Minkowski inner product satisfies the following properties.


The first two conditions imply bilinearity. The defining "difference" between a pseudo-inner product and an inner product proper is that the former is "not" required to be positive definite, that is, is allowed.


The most important feature of the inner product and norm squared is that "these are quantities unaffected by Lorentz transformations". In fact, it can be taken as the defining property of a Lorentz transformation that it preserves the inner product (i.e. the value of the corresponding bilinear form on two vectors). This approach is taken more generally for "all" classical groups definable this way in classical group. There, the matrix is identical in the case (the Lorentz group) to the matrix to be displayed below.


Two vectors and are said to be orthogonal if . For a geometric interpretation of orthogonality in the special case when and (or vice versa), see hyperbolic orthogonality.


A vector is called a unit vector if . A basis for consisting of mutually orthogonal unit vectors is called an orthonormal basis.


For a given inertial frame, an orthonormal basis in space, combined with the unit time vector, forms an orthonormal basis in Minkowski space. The number of positive and negative unit vectors in any such basis is a fixed pair of numbers, equal to the signature of the bilinear form associated with the inner product. This is Sylvester's law of inertia.


More terminology (but not more structure): The Minkowski metric is a pseudo-Riemannian metric, more specifically, a Lorentzian metric, even more specifically, "the" Lorentz metric, reserved for -dimensional flat spacetime with the remaining ambiguity only being the signature convention.


From the second postulate of special relativity, together with homogeneity of spacetime and isotropy of space, it follows that the spacetime interval between two arbitrary events called and is:


This quantity is not consistently named in the literature. The interval is sometimes referred to as the square of the interval as defined here. It is not possible to give an exhaustive list of notational inconsistencies. One has to first check out the definitions when consulting the relativity literature.


The invariance of the interval under coordinate transformations between inertial frames follows from the invariance of


(with either sign preserved), provided the transformations are linear. This quadratic form can be used to define a bilinear form


via the polarization identity. This bilinear form can in turn be written as


where is a matrix associated with . While possibly confusing, it is common practice to denote with just . The matrix is read off from the explicit bilinear form as


with which this section started by assuming its existence, is now identified.


For definiteness and shorter presentation, the signature is adopted below. This choice (or the other possible choice) has no (known) physical implications. The symmetry group preserving the bilinear form with one choice of signature is isomorphic (under the map given here) with the symmetry group preserving the other choice of signature. This means that both choices are in accord with the two postulates of relativity. Switching between the two conventions is straightforward. If the metric tensor has been used in a derivation, go back to the earliest point where it was used, substitute for , and retrace forward to the desired formula with the desired metric signature.


A standard basis for Minkowski space is a set of four mutually orthogonal vectors such that


These conditions can be written compactly in the form


Relative to a standard basis, the components of a vector are written where the Einstein notation is used to write . The component is called the timelike component of while the other three components are called the spatial components. The spatial components of a -vector may be identified with a -vector .


In terms of components, the Minkowski inner product between two vectors and is given by


Here lowering of an index with the metric was used.


Technically, a non-degenerate bilinear form provides a map between a vector space and its dual; in this context, the map is between the tangent spaces of and the cotangent spaces of . At a point in , the tangent and cotangent spaces are dual vector spaces (so the dimension of the cotangent space at an event is also ). Just as an authentic inner product on a vector space with one argument fixed, by Riesz representation theorem, may be expressed as the action of a linear functional on the vector space, the same holds for the Minkowski inner product of Minkowski space.


Contravariant and covariant vectors are geometrically very different objects. The first can and should be thought of as arrows. A linear functional can be characterized by two objects: its kernel, which is a hyperplane passing through the origin, and its norm. Geometrically thus, covariant vectors should be viewed as a set of hyperplanes, with spacing depending on the norm (bigger = smaller spacing), with one of them (the kernel) passing through the origin. The mathematical term for a covariant vector is 1-covector or 1-form (though the latter is usually reserved for covector "fields").


uses a vivid analogy with wave fronts of a de Broglie wave (scaled by a factor of Planck's reduced constant) quantum mechanically associated to a momentum four-vector to illustrate how one could imagine a covariant version of a contravariant vector. The inner product of two contravariant vectors could equally well be thought of as the action of the covariant version of one of them on the contravariant version of the other. The inner product is then how many time the arrow pierces the planes. The mathematical reference, , offers the same geometrical view of these objects (but mentions no piercing).


The electromagnetic field tensor is a differential 2-form, which geometrical description can as well be found in MTW.


One may, of course, ignore geometrical views all together (as is the style in e.g. and ) and proceed algebraically in a purely formal fashion. The time-proven robustness of the formalism itself, sometimes referred to as index gymnastics, ensures that moving vectors around and changing from contravariant to covariant vectors and vice versa (as well as higher order tensors) is mathematically sound. Incorrect expressions tend to reveal themselves quickly.


The present purpose is to show semi-rigorously how "formally" one may apply the Minkowski metric to two vectors and obtain a real number, i.e. to display the role of the differentials, and how they disappear in a calculation. The setting is that of smooth manifold theory, and concepts such as convector fields and exterior derivatives are introduced.


A full-blown version of the Minkowski metric in coordinates as a tensor field on spacetime has the appearance


Explanation: The coordinate differentials are 1-form fields. They are defined as the exterior derivative of the coordinate functions . These quantities evaluated at a point provide a basis for the cotangent space at . The tensor product (denoted by the symbol ) yields a tensor field of type , i.e. the type that expects two contravariant vectors as arguments. On the right hand side, the symmetric product (denoted by the symbol or by juxtaposition) has been taken. The equality holds since, by definition, the Minkowski metric is symmetric. The notation on the far right is also sometimes used for the related, but different, line element. It is "not" a tensor. For elaboration on the differences and similarities, see


"Tangent" vectors are, in this formalism, given in terms of a basis of differential operators of the first order,


where is an event. This operator applied to a function gives the directional derivative of at in the direction of increasing with fixed. They provide a basis for the tangent space at .


The exterior derivative of a function is a covector field, i.e. an assignment of a cotangent vector to each point , by definition such that


for each vector field . A vector field is an assignment of a tangent vector to each point . In coordinates can be expanded at each point in the basis given by the . Applying this with , the coordinate function itself, and , called a "coordinate vector field", one obtains


Since this relation holds at each point , the provide a basis for the cotangent space at each and the bases and are dual to each other,


for general one-forms on a tangent space and general tangent vectors . (This can be taken as a definition, but may also be proved in a more general setting.)


Thus when the metric tensor is fed two vectors fields , both expanded in terms of the basis coordinate vector fields, the result is


where , are the "component functions" of the vector fields. The above equation holds at each point , and the relation may as well be interpreted as the Minkowski metric at applied to two tangent vectors at .


As mentioned, in a vector space, such as that modelling the spacetime of special relativity, tangent vectors can be canonically identified with vectors in the space itself, and vice versa. This means that the tangent spaces at each point are canonically identified with each other and with the vector space itself. This explains how the right hand side of the above equation can be employed directly, without regard to spacetime point the metric is to be evaluated and from where (which tangent space) the vectors come from.


This situation changes in general relativity. There one has


where now , i.e. is still a metric tensor but now depending on spacetime and is a solution of Einstein's field equations. Moreover, "must" be tangent vectors at spacetime point and can no longer be moved around freely.


Suppose "x" ∈ "M" is timelike. Then the simultaneous hyperplane for x is formula_34 Since this hyperplane varies as "x" varies, there is a relativity of simultaneity in Minkowski space.


A Lorentzian manifold is a generalization of Minkowski space in two ways. The total number of spacetime dimensions is not restricted to be ( or more) and a Lorentzian manifold need not be flat, i.e. it allows for curvature.


Minkowski space refers to a mathematical formulation in four dimensions. However, the mathematics can easily be extended or simplified to create an analogous generalized Minkowski space in any number of dimensions. If , -dimensional Minkowski space is a vector space of real dimension on which there is a constant Minkowski metric of signature or . These generalizations are used in theories where spacetime is assumed to have more or less than dimensions. String theory and M-theory are two examples where . In string theory, there appears conformal field theories with spacetime dimensions.


de Sitter space can be formulated as a submanifold of generalized Minkowski space as can the model spaces of hyperbolic geometry (see below).


As a "flat spacetime", the three spatial components of Minkowski spacetime always obey the Pythagorean Theorem. Minkowski space is a suitable basis for special relativity, a good description of physical systems over finite distances in systems without significant gravitation. However, in order to take gravity into account, physicists use the theory of general relativity, which is formulated in the mathematics of a non-Euclidean geometry. When this geometry is used as a model of physical space, it is known as curved space.


Even in curved space, Minkowski space is still a good description in an infinitesimal region surrounding any point (barring gravitational singularities). More abstractly, we say that in the presence of gravity spacetime is described by a curved 4-dimensional manifold for which the tangent space to any point is a 4-dimensional Minkowski space. Thus, the structure of Minkowski space is still essential in the description of general relativity.


The meaning of the term "geometry" for the Minkowski space depends heavily on the context. Minkowski space is not endowed with a Euclidean geometry, and not with any of the generalized Riemannian geometries with intrinsic curvature, those exposed by the "model spaces" in hyperbolic geometry (negative curvature) and the geometry modeled by the sphere (positive curvature). The reason is the indefiniteness of the Minkowski metric. Minkowski space is, in particular, not a metric space and not a Riemannian manifold with a Riemannian metric. However, Minkowski space contains submanifolds endowed with a Riemannian metric yielding hyperbolic geometry.


Model spaces of hyperbolic geometry of low dimension, say or , "cannot" be isometrically embedded in Euclidean space with one more dimension, i.e. or respectively, with the Euclidean metric , disallowing easy visualization. By comparison, model spaces with positive curvature are just spheres in Euclidean space of one higher dimension. Hyperbolic spaces "can" be isometrically embedded in spaces of one more dimension when the embedding space is endowed with the Minkowski metric .


Define to be the upper sheet () of the hyperboloid


in generalized Minkowski space of spacetime dimension . This is one of the surfaces of transitivity of the generalized Lorentz group. The induced metric on this submanifold,


the pullback of the Minkowski metric under inclusion, is a Riemannian metric. With this metric is a Riemannian manifold. It is one of the model spaces of Riemannian geometry, the hyperboloid model of hyperbolic space. It is a space of constant negative curvature . The in the upper index refers to an enumeration of the different model spaces of hyperbolic geometry, and the for its dimension. A corresponds to the Poincaré disk model, while corresponds to the Poincaré half-space model of dimension .


In the definition above is the inclusion map and the superscript star denotes the pullback. The present purpose is to describe this and similar operations as a preparation for the actual demonstration that actually is a hyperbolic space.


In order to exhibit the metric it is necessary to pull it back via a suitable "parametrization". A parametrization of a submanifold of is a map whose range is an open subset of . If has the same dimension as , a parametrization is just the inverse of a coordinate map . The parametrization to be used is the inverse of "hyperbolic stereographic projection". This is illustrated in the figure to the left for . It is instructive to compare to stereographic projection for spheres.


Stereographic projection and its inverse are given by


where, for simplicity, . The are coordinates on and the are coordinates on .


The rotational partition function relates the rotational degrees of freedom to the rotational part of the energy.


The total canonical partition function formula_1 of a system of formula_2 identical, indistinguishable, noninteracting atoms or molecules can be divided into the atomic or molecular partition functions formula_3:


where formula_6 is the degeneracy of the "j"th quantum level of an individual particle, formula_7 is the Boltzmann constant, and formula_8 is the absolute temperature of system.


For molecules, under the assumption that total energy levels formula_9 can be partitioned into its contributions from different degrees of freedom (weakly coupled degrees of freedom)


and the number of degenerate states are given as products of the single contributions


where "trans", "ns", "rot", "vib" and "e" denotes translational, nuclear spin, rotational and vibrational contributions as well as electron excitation, the molecular partition functions


can be written as a product itself


Rotational energies are quantized. For a diatomic molecule like CO or HCl or a linear polyatomic molecule like OCS in its ground vibrational state, the allowed rotational energies in the rigid rotor approximation are


J is the quantum number for total rotational angular momentum and takes all integer values starting at zero,i.e. formula_15 is the rotational constant, and formula_16 is the moment of inertia. Here we are using "B" in energy units. If it is expressed in frequency units, replace "B" by "hB" in all the expression that follow, where "h" is Planck's constant. If "B" is given in units of formula_17, then replace "B" by "hcB" where c is the speed of light in vacuum.


For each value of J, we have rotational degeneracy, formula_18 = (2J+1), so the rotational partition function is therefore


For all but the lightest molecules or the very lowest temperatures we have formula_20. This suggests we can approximate the sum by replacing the sum over J by an integral of J treated as a continuous variable.


This approximation is known as the high temperature limit. It is also called the classical approximation as this is the result for the canonical partition function for a classical rigid rod.


Using the Euler–Maclaurin formula an improved estimate can be found


For the CO molecule at formula_23, the (unit less) contribution formula_24 to formula_3 turns out to be in the range of formula_26.


The mean thermal rotational energy per molecule can now be computed by taking the derivative of formula_24 with respect to temperature formula_28. In the high temperature limit approximation, the mean thermal rotational energy of a linear rigid rotor is formula_29.


A rigid, nonlinear molecule has rotational energy levels determined by three rotational constants, conventionally written formula_43 and formula_44, which can often be determined by rotational spectroscopy. In terms of these constants, the rotational partition function can be written in the high temperature limit as


with formula_42 again known as the rotational symmetry number which in general equals the number ways a molecule can be rotated to overlap itself in an indistinguishable way, i.e. that at most interchanges identical atoms. Like in the case of the diatomic treated explicitly above, this factor corrects for the fact that only a fraction of the nuclear spin functions can be used for any given molecular level to construct wavefunctions that overall obey the required exchange symmetries. The expression for formula_47 works for asymmetric, symmetric and spherical top rotors.


In quantum mechanics the delta potential is a potential well mathematically described by the Dirac delta function - a generalized function. Qualitatively, it corresponds to a potential which is zero everywhere, except at a single point, where it takes an infinite value. This can be used to simulate situations where a particle is free to move in two regions of space with a barrier between the two regions. For example, an electron can move almost freely in a conducting material, but if two conducting surfaces are put close together, the interface between them acts as a barrier for the electron that can be approximated by a delta potential.


The delta potential well is a limiting case of the finite potential well, which is obtained if one maintains the product of the width of the well and the potential constant while decreasing the well's width and increasing the potential.


This article, for simplicity, only considers a one-dimensional potential well, but analysis could be expanded to more dimensions.


The time-independent Schrödinger equation for the wave function of a particle in one dimension in a potential is


where is the reduced Planck constant, and is the energy of the particle.


It is called a "delta potential well" if is negative, and a "delta potential barrier" if is positive. The delta has been defined to occur at the origin for simplicity; a shift in the delta function's argument does not change any of the proceeding results.


The potential splits the space in two parts ( < 0 and  > 0). In each of these parts the potential energy is zero, and the Schrödinger equation reduces to


this is a linear differential equation with constant coefficients, whose solutions are linear combinations of and , where the wave number is related to the energy by


In general, due to the presence of the delta potential in the origin, the coefficients of the solution need not be the same in both half-spaces:


where, in the case of positive energies (real ), represents a wave traveling to the right, and one traveling to the left.


One obtains a relation between the coefficients by imposing that the wavefunction be continuous at the origin:


A second relation can be found by studying the derivative of the wavefunction. Normally, we could also impose differentiability at the origin, but this is not possible because of the delta potential. However, if we integrate the Schrödinger equation around  = 0, over an interval [−"ε", +"ε"]:


In the limit as "ε" → 0, the right-hand side of this equation vanishes; the left-hand side becomes


Substituting the definition of into this expression yields


The boundary conditions thus give the following restrictions on the coefficients


In any one-dimensional attractive potential there will be a bound state. To find its energy, note that for  < 0,  =  =  is imaginary, and the wave functions which were oscillating for positive energies in the calculation above are now exponentially increasing or decreasing functions of "x" (see above). Requiring that the wave functions do not diverge at infinity eliminates half of the terms:  =  = 0. The wave function is then


From the boundary conditions and normalization conditions, it follows that


from which it follows that must be negative, that is, the bound state only exists for the well, and not for the barrier. The Fourier transform of this wave function is a Lorentzian function.


The energy of the bound state is then


For positive energies, the particle is free to move in either half-space:  < 0 or  > 0. It may be scattered at the delta-function potential.


The quantum case can be studied in the following situation: a particle incident on the barrier from the left side . It may be reflected or transmitted .


To find the amplitudes for reflection and transmission for incidence from the left, we put in the above equations  = 1 (incoming particle),  =  (reflection),  = 0 (no incoming particle from the right) and  =  (transmission), and solve for and even though we do not have any equations in .


Due to the mirror symmetry of the model, the amplitudes for incidence from the right are the same as those from the left. The result is that there is a non-zero probability


for the particle to be reflected. This does not depend on the sign of , that is, a barrier has the same probability of reflecting the particle as a well. This is a significant difference from classical mechanics, where the reflection probability would be 1 for the barrier (the particle simply bounces back), and 0 for the well (the particle passes through the well undisturbed).


In summary, the probability for transmission is


The calculation presented above may at first seem unrealistic and hardly useful. However, it has proved to be a suitable model for a variety of real-life systems.


One such example regards the interfaces between two conducting materials. In the bulk of the materials, the motion of the electrons is quasi-free and can be described by the kinetic term in the above Hamiltonian with an effective mass . Often, the surfaces of such materials are covered with oxide layers or are not ideal for other reasons. This thin, non-conducting layer may then be modeled by a local delta-function potential as above. Electrons may then tunnel from one material to the other giving rise to a current.


The operation of a scanning tunneling microscope (STM) relies on this tunneling effect. In that case, the barrier is due to the air between the tip of the STM and the underlying object. The strength of the barrier is related to the separation being stronger the further apart the two are. For a more general model of this situation, see Finite potential barrier (QM). The delta function potential barrier is the limiting case of the model considered there for very high and narrow barriers.


The above model is one-dimensional while the space around us is three-dimensional. So, in fact, one should solve the Schrödinger equation in three dimensions. On the other hand, many systems only change along one coordinate direction and are translationally invariant along the others. The Schrödinger equation may then be reduced to the case considered here by an Ansatz for the wave function of the type formula_18.


Alternatively, it is possible to generalize the delta function to exist on the surface of some domain "D" (see Laplacian of the indicator).


The delta function model is actually a one-dimensional version of the Hydrogen atom according to the "dimensional scaling" method developed by the group of Dudley R. Herschbach


The delta function model becomes particularly useful with the "double-well" Dirac Delta function model which represents a one-dimensional version of the Hydrogen molecule ion, as shown in the following section.


The double-well Dirac delta function models a diatomic hydrogen molecule by the corresponding Schrödinger equation:


where formula_21 is the "internuclear" distance with Dirac delta-function (negative) peaks located at  = ±/2 (shown in brown in the diagram). Keeping in mind the relationship of this model with its three-dimensional molecular counterpart, we use atomic units and set formula_22. Here formula_23 is a formally adjustable parameter. From the single-well case, we can infer the "ansatz" for the solution to be


Matching of the wavefunction at the Dirac delta-function peaks yields the determinant


Thus, formula_26 is found to be governed by the "pseudo-quadratic" equation


which has two solutions formula_28. For the case of equal charges (symmetric homonuclear case),  = 1, and the pseudo-quadratic reduces to


The "+" case corresponds to a wave function symmetric about the midpoint (shown in red in the diagram), where  = , and is called "gerade". Correspondingly, the "−" case is the wave function that is anti-symmetric about the midpoint, where = −, and is called "ungerade" (shown in green in the diagram). They represent an approximation of the two lowest discrete energy states of the three-dimensional <chem>H2^+</chem> and are useful in its analysis. Analytical solutions for the energy eigenvalues for the case of symmetric charges are given by


where "W" is the standard Lambert "W" function. Note that the lowest energy corresponds to the symmetric solution formula_31. In the case of "unequal" charges, and for that matter the three-dimensional molecular problem, the solutions are given by a "generalization" of the Lambert "W" function (see section on generalization of Lambert W function and references herein).


One of the most interesting cases is when "qR" ≤ 1, which results in formula_32. Thus, one has a non-trivial bound state solution with  = 0. For these specific parameters, there are many interesting properties that occur, one of which is the unusual effect that the transmission coefficient is unity at zero energy.


The Kundu equation is a general form of integrable system that is gauge-equivalent to the mixed nonlinear Schrödinger equation. It was proposed by Anjan Kundu as


with arbitrary function formula_2 and the subscripts denoting partial derivatives. Equation (1) is shown to be reducible for the choice of formula_3 to an integrable class of mixed nonlinear Schrödinger equation with cubic–quintic nonlinearity, given in a representative form


Here formula_5 are independent parameters, while formula_6 Equation (1), more specifically equation (2) is known as the Kundu equation.


The Kundu equation is a completely integrable system, allowing Lax pair representation, exact solutions, and higher conserved quantity.


Along with its different particular cases, this equation has been investigated for finding its exact travelling wave solutions, exact solitary wave solutions via bilinearization, and Darboux transformation together with the orbital stability for such solitary wave solutions.


The Kundu equation has been applied to various physical processes such as fluid dynamics, plasma physics, and nonlinear optics. It is linked to the mixed nonlinear Schrödinger equation through a gauge transformation and is reducible to a variety of known integrable equations such as the nonlinear Schrödinger equation (NLSE), derivative NLSE, higher nonlinear derivative NLSE, Chen–Lee–Liu, Gerjikov-Vanov, and Kundu–Eckhaus equations, for different choices of the parameters.


A generalization of the nonlinear Schrödinger equation with additional quintic nonlinearity and a nonlinear dispersive term was proposed in the form


which may be obtained from the Kundu Equation (2), when restricted to formula_8. The same equation, limited further to the particular case formula_9 was introduced later as the Eckhaus equation, following which equation (3) is presently known as the Kundu-Ekchaus equation. The Kundu-Ekchaus equation can be reduced to the nonlinear Schrödinger equation through a nonlinear transformation of the field and known therefore to be gauge equivalent integrable systems, since they are equivalent under the gauge transformation.


The Kundu-Ekchaus equation is associated with a Lax pair, higher conserved quantity, exact soliton solution, rogue wave solution etc. Over the years various aspects of this equation, its generalizations and link with other equations have been studied. In particular, relationship of Kundu-Ekchaus equation with the Johnson's hydrodynamic equation near criticality is established, its discretizations, reduction via Lie symmetry, complex structure via Bernoulli subequation, bright and dark soliton solutions via Bäcklund transformation and Darboux transformation with the associated rogue wave solutions, are studied.


A multi-component generalisation of the Kundu-Ekchaus equation (3), known as Radhakrishnan, Kundu and Laskshmanan (RKL) equation was proposed in nonlinear optics for fiber optics communication through soliton pulses in a birefringent non-Kerr medium and analysed subsequently for its exact soliton solution and other aspects in a series of papers.


Though the Kundu-Ekchaus equation (3) is gauge equivalent to the nonlinear Schrödinger equation, they differ with respect to their Hamiltonian structures and field commutation relations. The Hamiltonian operator of the Kundu-Ekchaus equation quantum field model given by


and defined through the bosonic field operator commutation relation formula_11, is more complicated than the well-known bosonic Hamiltonian of the quantum nonlinear Schrödinger equation. Here formula_12 indicates normal ordering in bosonic operators. This model corresponds to a double formula_13-function interacting Bose gas and is difficult to solve directly.


However, under a nonlinear transformation of the field below:


i.e. in the same form as the quantum model of the Nonlinear Schrödinger equation (NLSE), though it differs from the NLSE in its contents, since now the fields involved are no longer bosonic operators but exhibit anion like properties.


though at the coinciding points the bosonic commutation relation still holds. In analogy with the Lieb Limiger model of formula_19 function bose gas, the quantum Kundu-Ekchaus model in the N-particle sector therefore corresponds to a one-dimensional (1D) anion gas interacting via a formula_20 function interaction. This model of interacting anion gas was proposed and exactly solved by the Bethe ansatz in


and this basic anion model is studied further for investigating various aspects of the 1D anion gas as well as extended in different directions.


In mathematical physics, the Eckhaus equation – or the Kundu–Eckhaus equation – is a nonlinear partial differential equations within the nonlinear Schrödinger class:


The equation was independently introduced by Wiktor Eckhaus and by Anjan Kundu to model the propagation of waves in dispersive media.


The Eckhaus equation can be linearized to the linear Schrödinger equation:


This linearization also implies that the Eckhaus equation is integrable.


In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called "quantum tunneling") and wave-mechanical reflection. The problem consists of solving the one-dimensional time-independent Schrödinger equation for a particle encountering a rectangular potential energy barrier. It is usually assumed, as here, that a free particle impinges on the barrier from the left.


Although classically a particle behaving as a point mass would be reflected if its energy is less than formula_1, a particle actually behaving as a matter wave has a non-zero probability of penetrating the barrier and continuing its travel as a wave on the other side. In classical wave-physics, this effect is known as evanescent wave coupling. The likelihood that the particle will pass through the barrier is given by the transmission coefficient, whereas the likelihood that it is reflected is given by the reflection coefficient. Schrödinger's wave-equation allows these coefficients to be calculated.


The time-independent Schrödinger equation for the wave function formula_2 reads


where formula_4 is the Hamiltonian, formula_5 is the (reduced)


Planck constant, formula_6 is the mass, formula_7 the energy of the particle and


is the barrier potential with height formula_9 and width formula_10. formula_11


The barrier is positioned between formula_13 and formula_14. The barrier can be shifted to any formula_15 position without changing the results. The first term in the Hamiltonian, formula_16 is the kinetic energy.


The barrier divides the space in three parts (formula_17). In any of these parts, the potential is constant, meaning that the particle is quasi-free, and the solution of the Schrödinger equation can be written as a superposition of left and right moving waves (see free particle). If formula_18


where the wave numbers are related to the energy via


The coefficients formula_29 have to be found from the boundary conditions of the wave function at formula_13 and formula_14. The wave function and its derivative have to be continuous everywhere, so


Inserting the wave functions, the boundary conditions give the following restrictions on the coefficients


If the energy equals the barrier height, the second differential of the wavefunction inside the barrier region is 0, and hence the solutions of the Schrödinger equation are not exponentials anymore but linear functions of the space coordinate


The complete solution of the Schrödinger equation is found in the same way as above by matching wave functions and their derivatives at formula_13 and formula_14. That results in the following restrictions on the coefficients:


At this point, it is instructive to compare the situation to the classical case. In both cases, the particle behaves as a free particle outside of the barrier region. A classical particle with energy formula_7 larger than the barrier height formula_1 would "always" pass the barrier, and a classical particle with formula_48 incident on the barrier would "always" get reflected.


To study the quantum case, consider the following situation: a particle incident on the barrier from the left side (formula_49). It may be reflected (formula_50) or transmitted (formula_51).


To find the amplitudes for reflection and transmission for incidence from the left, we put in the above equations formula_52 (incoming particle), formula_53 (reflection), formula_54=0 (no incoming particle from the right), and formula_55 (transmission). We then eliminate the coefficients formula_56 from the equation and solve for formula_57 and formula_58.


Due to the mirror symmetry of the model, the amplitudes for incidence from the right are the same as those from the left. Note that these expressions hold for any energy formula_61.


The surprising result is that for energies less than the barrier height, formula_48 there is a non-zero probability


for the particle to be transmitted through the barrier, with formula_64. This effect, which differs from the classical case, is called quantum tunneling. The transmission is exponentially suppressed with the barrier width, which can be understood from the functional form of the wave function: Outside of the barrier it oscillates with wave vector formula_65, whereas within the barrier it is exponentially damped over a distance formula_66. If the barrier is much wider than this decay length, the left and right part are virtually independent and tunneling as a consequence is suppressed.


Equally surprising is that for energies larger than the barrier height, formula_18, the particle may be reflected from the barrier with a non-zero probability


The transmission and reflection probabilities are in fact oscillating with formula_71. The classical result of perfect transmission without any reflection (formula_72, formula_73) is reproduced not only in the limit of high energy formula_74 but also when the energy and barrier width satisfy formula_75, where formula_76 (see peaks near formula_77 and 1.8 in the above figure). Note that the probabilities and amplitudes as written are for any energy (above/below) the barrier height.


The transmission probability at formula_28 evaluates to


The calculation presented above may at first seem unrealistic and hardly


useful. However it has proved to be a suitable model for a variety of real-life


systems. One such example are interfaces between two conducting materials. In the bulk of the materials, the motion of the electrons is quasi-free and can be described by the kinetic term in the above Hamiltonian with an effective mass formula_6. Often the surfaces of such materials are covered with oxide layers or are not ideal for other reasons. This thin, non-conducting layer may then be modeled by a barrier potential as above. Electrons may then tunnel from one material to the other giving rise to a current.


The operation of a scanning tunneling microscope (STM) relies on this tunneling effect. In that case, the barrier is due to the gap between the tip of the STM and the underlying object. Since the tunnel current depends exponentially on the barrier width, this device is extremely sensitive to height variations on the examined sample.


The above model is one-dimensional, while space is three-dimensional. One should solve the Schrödinger equation in three dimensions. On the other hand, many systems only change along one coordinate direction and are translationally invariant along the others; they are separable. The Schrödinger equation may then be reduced to the case considered here by an ansatz for the wave function of the type: formula_81.


For another, related model of a barrier, see Delta potential barrier (QM), which can be regarded as a special case of the finite potential barrier. All results from this article immediately apply to the delta potential barrier by taking the limits formula_82 while keeping formula_83 constant.


In the general theory of relativity the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.


The equations were first published by Einstein in 1915 in the form of a tensor equation which related the local "" (expressed by the Einstein tensor) with the local energy, momentum and stress within that spacetime (expressed by the stress–energy tensor).


Analogously to the way that electromagnetic fields are related to the distribution of charges and currents via Maxwell's equations, the EFE relate the spacetime geometry to the distribution of mass–energy, momentum and stress, that is, they determine the metric tensor of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of non-linear partial differential equations when used in this way. The solutions of the EFE are the components of the metric tensor. The inertial trajectories of particles and radiation (geodesics) in the resulting geometry are then calculated using the geodesic equation.


As well as implying local energy–momentum conservation, the EFE reduce to Newton's law of gravitation in the limit of a weak gravitational field and velocities that are much less than the speed of light.


Exact solutions for the EFE can only be found under simplifying assumptions such as symmetry. Special classes of exact solutions are most often studied since they model many gravitational phenomena, such as rotating black holes and the expanding universe. Further simplification is achieved in approximating the spacetime as having only small deviations from flat spacetime, leading to the linearized EFE. These equations are used to study phenomena such as gravitational waves.


The Einstein field equations (EFE) may be written in the form:


where is the Einstein tensor, is the metric tensor, is the stress–energy tensor, is the cosmological constant and is the Einstein gravitational constant.


where is the Ricci curvature tensor, and is the scalar curvature. This is a symmetric second-degree tensor that depends on only the metric tensor and its first- and second derivatives.


The Einstein gravitational constant is defined as


where is the Newtonian constant of gravitation and is the speed of light in vacuum.


The EFE can thus also be written as


In standard units, each term on the left has units of 1/length2.


The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the stress–energy–momentum content of spacetime. The EFE can then be interpreted as a set of equations dictating how stress–energy–momentum determines the curvature of spacetime.


These equations, together with the geodesic equation, which dictates how freely falling matter moves through spacetime, form the core of the mathematical formulation of general relativity.


The EFE is a tensor equation relating a set of symmetric 4 × 4 tensors. Each tensor has 10 independent components. The four Bianchi identities reduce the number of independent equations from 10 to 6, leaving the metric with four gauge-fixing degrees of freedom, which correspond to the freedom to choose a coordinate system.


Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when is everywhere zero) define Einstein manifolds.


The equations are more complex than they appear. Given a specified distribution of matter and energy in the form of a stress–energy tensor, the EFE are understood to be equations for the metric tensor , since both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. When fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic partial differential equations.


The above form of the EFE is the standard established by Misner, Thorne, and Wheeler (MTW). The authors analyzed conventions that exist and classified these according to three signs ([S1] [S2] [S3]):


The third sign above is related to the choice of convention for the Ricci tensor:


With these definitions Misner, Thorne, and Wheeler classify themselves as , whereas Weinberg (1972) is , Peebles (1980) and Efstathiou et al. (1990) are , Rindler (1977), Atwater (1974), Collins Martin & Squires (1989) and Peacock (1999) are .


Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative:


The sign of the cosmological term would change in both these versions if the metric sign convention is used rather than the MTW metric sign convention adopted here.


Taking the trace with respect to the metric of both sides of the EFE one gets


where is the spacetime dimension. Solving for and substituting this in the original EFE, one gets the following equivalent "trace-reversed" form:


Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace in the expression on the right with the Minkowski metric without significant loss of accuracy).


the term containing the cosmological constant was absent from the version in which he originally published them. Einstein then included the term with the cosmological constant to allow for a universe that is not expanding or contracting. This effort was unsuccessful because:


Einstein then abandoned , remarking to George Gamow "that the introduction of the cosmological term was the biggest blunder of his life".


The inclusion of this term does not create inconsistencies. For many years the cosmological constant was almost universally assumed to be zero. More recent astronomical observations have shown an accelerating expansion of the universe, and to explain this a positive value of is needed. The cosmological constant is negligible at the scale of a galaxy or smaller.


Einstein thought of the cosmological constant as an independent parameter, but its term in the field equation can also be moved algebraically to the other side and incorporated as part of the stress–energy tensor:


This tensor describes a vacuum state with an energy density and isotropic pressure that are fixed constants and given by


where it is assumed that has SI unit m and is defined as above.


The existence of a cosmological constant is thus equivalent to the existence of a vacuum energy and a pressure of opposite sign. This has led to the terms "cosmological constant" and "vacuum energy" being used interchangeably in general relativity.


General relativity is consistent with the local conservation of energy and momentum expressed as


which expresses the local conservation of stress–energy. This conservation law is a physical requirement. With his field equations Einstein ensured that general relativity is consistent with this conservation condition.


The nonlinearity of the EFE distinguishes general relativity from many other fundamental physical theories. For example, Maxwell's equations of electromagnetism are linear in the electric and magnetic fields, and charge and current distributions (i.e. the sum of two solutions is also a solution); another example is Schrödinger's equation of quantum mechanics, which is linear in the wavefunction.


The EFE reduce to Newton's law of gravity by using both the weak-field approximation and the slow-motion approximation. In fact, the constant appearing in the EFE is determined by making these two approximations.


If the energy–momentum tensor is zero in the region under consideration, then the field equations are also referred to as the vacuum field equations. By setting in the trace-reversed field equations, the vacuum equations can be written as


In the case of nonzero cosmological constant, the equations are


The solutions to the vacuum field equations are called vacuum solutions. Flat Minkowski space is the simplest example of a vacuum solution. Nontrivial examples include the Schwarzschild solution and the Kerr solution.


Manifolds with a vanishing Ricci tensor, , are referred to as Ricci-flat manifolds and manifolds with a Ricci tensor proportional to the metric as Einstein manifolds.


If the energy–momentum tensor is that of an electromagnetic field in free space, i.e. if the electromagnetic stress–energy tensor


is used, then the Einstein field equations are called the "Einstein–Maxwell equations" (with cosmological constant , taken to be zero in conventional relativity theory):


Additionally, the covariant Maxwell equations are also applicable in free space:


where the semicolon represents a covariant derivative, and the brackets denote anti-symmetrization. The first equation asserts that the 4-divergence of the 2-form is zero, and the second that its exterior derivative is zero. From the latter, it follows by the Poincaré lemma that in a coordinate chart it is possible to introduce an electromagnetic field potential such that


in which the comma denotes a partial derivative. This is often taken as equivalent to the covariant Maxwell equation from which it is derived. However, there are global solutions of the equation that may lack a globally defined potential.


The solutions of the Einstein field equations are metrics of spacetime. These metrics describe the structure of the spacetime including the inertial motion of objects in the spacetime. As the field equations are non-linear, they cannot always be completely solved (i.e. without making approximations). For example, there is no known complete solution for a spacetime with two massive bodies in it (which is a theoretical model of a binary star system, for example). However, approximations are usually made in these cases. These are commonly referred to as post-Newtonian approximations. Even so, there are several cases where the field equations have been solved completely, and those are called exact solutions.


The study of exact solutions of Einstein's field equations is one of the activities of cosmology. It leads to the prediction of black holes and to different models of evolution of the universe.


One can also discover new solutions of the Einstein field equations via the method of orthonormal frames as pioneered by Ellis and MacCallum. In this approach, the Einstein field equations are reduced to a set of coupled, nonlinear, ordinary differential equations. As discussed by Hsu and Wainwright, self-similar solutions to the Einstein field equations are fixed points of the resulting dynamical system. New solutions have been discovered using these methods by LeBlanc and Kohli and Haslam.


The nonlinearity of the EFE makes finding exact solutions difficult. One way of solving the field equations is to make an approximation, namely, that far from the source(s) of gravitating matter, the gravitational field is very weak and the spacetime approximates that of Minkowski space. The metric is then written as the sum of the Minkowski metric and a term representing the deviation of the true metric from the Minkowski metric, ignoring higher-power terms. This linearization procedure can be used to investigate the phenomena of gravitational radiation.


Despite the EFE as written containing the inverse of the metric tensor, they can be arranged in a form that contains the metric tensor in polynomial form and without its inverse. First, the determinant of the metric in 4 dimensions can be written


using the Levi-Civita symbol; and the inverse of the metric in 4 dimensions can be written as:


Substituting this definition of the inverse of the metric into the equations then multiplying both sides by a suitable power of to eliminate it from the denominator results in polynomial equations in the metric tensor and its first and second derivatives. The action from which the equations are derived can also be written in polynomial form by suitable redefinitions of the fields.


In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and approximates the response of that material to external stimuli, usually as applied fields or forces. They are combined with other equations governing physical laws to solve physical problems; for example in fluid mechanics the flow of a fluid in a pipe, in solid state physics the response of a crystal to an electric field, or in structural analysis, the connection between applied stresses or forces to strains or deformations.


Some constitutive equations are simply phenomenological; others are derived from first principles. A common approximate constitutive equation frequently is expressed as a simple proportionality using a parameter taken to be a property of the material, such as electrical conductivity or a spring constant. However, it is often necessary to account for the directional dependence of the material, and the scalar parameter is generalized to a tensor. Constitutive relations are also modified to account for the rate of response of materials and their non-linear behavior. See the article Linear response function.


The first constitutive equation (constitutive law) was developed by Robert Hooke and is known as Hooke's law. It deals with the case of linear elastic materials. Following this discovery, this type of equation, often called a "stress-strain relation" in this example, but also called a "constitutive assumption" or an "equation of state" was commonly used. Walter Noll advanced the use of constitutive equations, clarifying their classification and the role of invariance requirements, constraints, and definitions of terms


like "material", "isotropic", "aeolotropic", etc. The class of "constitutive relations" of the form "stress rate = f (velocity gradient, stress, density)" was the subject of Walter Noll's dissertation in 1954 under Clifford Truesdell.


In modern condensed matter physics, the constitutive equation plays a major role. See Linear constitutive equations and Nonlinear correlation functions.


Friction is a complicated phenomenon. Macroscopically, the friction force "F" between the interface of two materials can be modelled as proportional to the reaction force "R" at a point of contact between two interfaces through a dimensionless coefficient of friction "μ"f, which depends on the pair of materials:


This can be applied to static friction (friction preventing two stationary objects from slipping on their own), kinetic friction (friction between two objects scraping/sliding past each other), or rolling (frictional force which prevents slipping but causes a torque to exert on a round object).


The stress-strain constitutive relation for linear materials is commonly known as Hooke's law. In its simplest form, the law defines the spring constant (or elasticity constant) "k" in a scalar equation, stating the tensile/compressive force is proportional to the extended (or contracted) displacement "x":


meaning the material responds linearly. Equivalently, in terms of the stress "σ", Young's modulus "E", and strain "ε" (dimensionless):


In general, forces which deform solids can be normal to a surface of the material (normal forces), or tangential (shear forces), this can be described mathematically using the stress tensor:


where "C" is the elasticity tensor and "S" is the compliance tensor.


Several classes of deformations in elastic materials are the following:


The relative speed of separation "v"separation of an object A after a collision with another object B is related to the relative speed of approach "v"approach by the coefficient of restitution, defined by Newton's experimental impact law:


The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field or the magnetic field , takes the form:


is the speed of light (i.e. phase velocity) in a medium with permeability , and permittivity , and is the Laplace operator. In a vacuum, meters per second, a fundamental physical constant. The electromagnetic wave equation derives from Maxwell's equations. In most older literature, is called the "magnetic flux density" or "magnetic induction".


The origin of the electromagnetic wave equation.


In his 1865 paper titled A Dynamical Theory of the Electromagnetic Field, James Clerk Maxwell utilized the correction to Ampère's circuital law that he had made in part III of his 1861 paper On Physical Lines of Force. In "Part VI" of his 1864 paper titled "Electromagnetic Theory of Light", Maxwell combined displacement current with some of the other equations of electromagnetism and he obtained a wave equation with a speed equal to the speed of light. He commented:


The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.


Maxwell's derivation of the electromagnetic wave equation has been replaced in modern physics education by a much less cumbersome method involving combining the corrected version of Ampère's circuital law with Faraday's law of induction.


To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern 'Heaviside' form of Maxwell's equations. In a vacuum- and charge-free space, these equations are:


These are the general Maxwell's equations specialized to the case with charge and current both set to zero.


Taking the curl of the curl equations gives:


where is any vector function of space. And


where is a dyadic which when operated on by the divergence operator yields a vector. Since


then the first term on the right in the identity vanishes and we obtain the wave equations:


Covariant form of the homogeneous wave equation.


These relativistic equations can be written in contravariant form as


The electromagnetic wave equation is modified in two ways, the derivative is replaced with the covariant derivative and a new term that depends on the curvature appears.


where formula_15 is the Ricci curvature tensor and the semicolon indicates covariant differentiation.


The generalization of the Lorenz gauge condition in curved spacetime is assumed:


Localized time-varying charge and current densities can act as sources of electromagnetic waves in a vacuum. Maxwell's equations can be written in the form of a wave equation with sources. The addition of sources to the wave equations makes the partial differential equations inhomogeneous.


Solutions to the homogeneous electromagnetic wave equation.


The general solution to the electromagnetic wave equation is a linear superposition of waves of the form


for virtually "any" well-behaved function of dimensionless argument , where is the angular frequency (in radians per second), and is the wave vector (in radians per meter).


Although the function can be and often is a monochromatic sine wave, it does not have to be sinusoidal, or even periodic. In practice, cannot have infinite periodicity because any real electromagnetic wave must always have a finite extent in time and space. As a result, and based on the theory of Fourier decomposition, a real wave must consist of the superposition of an infinite set of sinusoidal frequencies.


In addition, for a valid solution, the wave vector and the angular frequency are not independent; they must adhere to the dispersion relation:


where is the wavenumber and is the wavelength. The variable can only be used in this equation when the electromagnetic wave is in a vacuum.


The simplest set of solutions to the wave equation result from assuming sinusoidal waveforms of a single frequency in separable form:


Consider a plane defined by a unit normal vector


Then planar traveling wave solutions of the wave equations are


where is the position vector (in meters).


These solutions represent planar waves traveling in the direction of the normal vector . If we define the z direction as the direction of . and the x direction as the direction of , then by Faraday's Law the magnetic field lies in the y direction and is related to the electric field by the relation


Because the divergence of the electric and magnetic fields are zero, there are no fields in the direction of propagation.


This solution is the linearly polarized solution of the wave equations. There are also circularly polarized solutions in which the fields rotate about the normal vector.


Because of the linearity of Maxwell's equations in a vacuum, solutions can be decomposed into a superposition of sinusoids. This is the basis for the Fourier transform method for the solution of differential equations. The sinusoidal solution to the electromagnetic wave equation takes the form


The wave vector is related to the angular frequency by


where is the wavenumber and is the wavelength.


The electromagnetic spectrum is a plot of the field magnitudes (or energies) as a function of wavelength.


Assuming monochromatic fields varying in time as formula_30, if one uses Maxwell's Equations to eliminate , the electromagnetic wave equation reduces to the Helmholtz Equation for :


with "k = ω/c" as given above. Alternatively, one can eliminate in favor of to obtain:


A generic electromagnetic field with frequency can be written as a sum of solutions to these two equations. The three-dimensional solutions of the Helmholtz Equation can be expressed as expansions in spherical harmonics with coefficients proportional to the spherical Bessel functions. However, applying this expansion to each vector component of or will give solutions that are not generically divergence-free (), and therefore require additional restrictions on the coefficients.


The multipole expansion circumvents this difficulty by expanding not or , but or into spherical harmonics. These expansions still solve the original Helmholtz equations for and because for a divergence-free field , . The resulting expressions for a generic electromagnetic field are:


where formula_35 and formula_36 are the "electric multipole fields of order (l, m)", and formula_37 and formula_38 are the corresponding "magnetic multipole fields", and and are the coefficients of the expansion. The multipole fields are given by


where "h"l(1,2)("x") are the spherical Hankel functions, "E"l(1,2) and "B"l(1,2) are determined by boundary conditions, and


are vector spherical harmonics normalized so that


The multipole expansion of the electromagnetic field finds application in a number of problems involving spherical symmetry, for example antennae radiation patterns, or nuclear gamma decay. In these applications, one is often interested in the power radiated in the far-field. In this regions, the and fields asymptote to


The angular distribution of the time-averaged radiated power is then given by


The Fresnel equations (or Fresnel coefficients) describe the reflection and transmission of light (or electromagnetic radiation in general) when incident on an interface between different optical media. They were deduced by Augustin-Jean Fresnel () who was the first to understand that light is a transverse wave, even though no one realized that the "vibrations" of the wave were electric and magnetic fields. For the first time, polarization could be understood quantitatively, as Fresnel's equations correctly predicted the differing behaviour of waves of the "s" and "p" polarizations incident upon a material interface.


When light strikes the interface between a medium with refractive index "n"1 and a second medium with refractive index "n"2, both reflection and refraction of the light may occur. The Fresnel equations give the ratio of the "reflected" wave's electric field to the incident wave's electric field, and the ratio of the "transmitted" wave's electric field to the incident wave's electric field, for each of two components of polarization. (The "magnetic" fields can also be related using similar coefficients.) These ratios are generally complex, describing not only the relative amplitudes but also the phase shifts at the interface.


The equations assume the interface between the media is flat and that the media are homogeneous and isotropic. The incident light is assumed to be a plane wave, which is sufficient to solve any problem since any incident light field can be decomposed into plane waves and polarizations.


There are two sets of Fresnel coefficients for two different linear polarization components of the incident wave. Since any polarization state can be resolved into a combination of two orthogonal linear polarizations, this is sufficient for any problem. Likewise, unpolarized (or "randomly polarized") light has an equal amount of power in each of two linear polarizations.


The s polarization refers to polarization of a wave's electric field "normal" to the plane of incidence (the direction in the derivation below); then the magnetic field is "in" the plane of incidence. The p polarization refers to polarization of the electric field "in" the plane of incidence (the plane in the derivation below); then the magnetic field is "normal" to the plane of incidence.


Although the reflectivity and transmission are dependent on polarization, at normal incidence ("θ" = 0) there is no distinction between them so all polarization states are governed by a single set of Fresnel coefficients (and another special case is mentioned below in which that is true).


In the diagram on the right, an incident plane wave in the direction of the ray IO strikes the interface between two media of refractive indices "n"1 and "n"2 at point O. Part of the wave is reflected in the direction OR, and part refracted in the direction OT. The angles that the incident, reflected and refracted rays make to the normal of the interface are given as "θ"i, "θ"r and "θ"t, respectively.


The relationship between these angles is given by the law of reflection:


The behavior of light striking the interface is solved by considering the electric and magnetic fields that constitute an electromagnetic wave, and the laws of electromagnetism, as shown below. The ratio of waves' electric field (or magnetic field) amplitudes are obtained, but in practice one is more often interested in formulae which determine "power" coefficients, since power (or irradiance) is what can be directly measured at optical frequencies. The power of a wave is generally proportional to the square of the electric (or magnetic) field amplitude.


We call the fraction of the incident power that is reflected from the interface the reflectance (or "reflectivity", or "power reflection coefficient") "R", and the fraction that is refracted into the second medium is called the transmittance (or "transmissivity", or "power transmission coefficient") "T". Note that these are what would be measured right "at" each side of an interface and do not account for attenuation of a wave in an absorbing medium "following" transmission or reflection.


while the reflectance for p-polarized light is


where and are the wave impedances of media 1 and 2, respectively.


We assume that the media are non-magnetic (i.e., "μ"1 = "μ"2 = "μ"0), which is typically a good approximation at optical frequencies (and for transparent media at other frequencies). Then the wave impedances are determined solely by the refractive indices "n"1 and "n"2:


where is the impedance of free space and =1,2. Making this substitution, we obtain equations using the refractive indices:


The second form of each equation is derived from the first by eliminating "θ"t using Snell's law and trigonometric identities.


As a consequence of conservation of energy, one can find the transmitted power (or more correctly, irradiance: power per unit area) simply as the portion of the incident power that isn't reflected:


Note that all such intensities are measured in terms of a wave's irradiance in the direction normal to the interface; this is also what is measured in typical experiments. That number could be obtained from irradiances "in the direction of an incident or reflected wave" (given by the magnitude of a wave's Poynting vector) multiplied by cos"θ" for a wave at an angle "θ" to the normal direction (or equivalently, taking the dot product of the Poynting vector with the unit vector normal to the interface). This complication can be ignored in the case of the reflection coefficient, since cos"θ"i = cos"θ"r, so that the ratio of reflected to incident irradiance in the wave's direction is the same as in the direction normal to the interface.


Although these relationships describe the basic physics, in many practical applications one is concerned with "natural light" that can be described as unpolarized. That means that there is an equal amount of power in the "s" and "p" polarizations, so that the "effective" reflectivity of the material is just the average of the two reflectivities:


For low-precision applications involving unpolarized light, such as computer graphics, rather than rigorously computing the effective reflection coefficient for each angle, Schlick's approximation is often used.


For the case of normal incidence, formula_11, and there is no distinction between s and p polarization. Thus, the reflectance simplifies to


For common glass ("n"2 ≈ 1.5) surrounded by air ("n"1=1), the power reflectance at normal incidence can be seen to be about 4%, or 8% accounting for both sides of a glass pane.


At a dielectric interface from to , there is a particular angle of incidence at which goes to zero and a p-polarised incident wave is purely refracted, thus all reflected light is s-polarised. This angle is known as Brewster's angle, and is around 56° for "n"1=1 and "n"2=1.5 (typical glass).


When light travelling in a denser medium strikes the surface of a less dense medium (i.e., ), beyond a particular incidence angle known as the "critical angle", all light is reflected and . This phenomenon, known as total internal reflection, occurs at incidence angles for which Snell's law predicts that the sine of the angle of refraction would exceed unity (whereas in fact sin"θ" ≤ 1 for all real "θ"). For glass with "n"=1.5 surrounded by air, the critical angle is approximately 41°.


The above equations relating powers (which could be measured with a photometer for instance) are derived from the Fresnel equations which solve the physical problem in terms of electromagnetic field complex amplitudes, i.e., considering phase in addition to power (which is important in multipath propagation for instance). Those underlying equations supply generally complex-valued ratios of those EM fields and may take several different forms, depending on formalisms used. The complex amplitude coefficients are usually represented by lower case "r" and "t" (whereas the power coefficients are capitalized).


In the following, the reflection coefficient is the ratio of the reflected wave's electric field complex amplitude to that of the incident wave. The transmission coefficient is the ratio of the transmitted wave's electric field complex amplitude to that of the incident wave. We require separate formulae for the "s" and "p" polarizations. In each case we assume an incident plane wave at an angle of incidence formula_13 on a plane interface, reflected at an angle formula_14, and with a transmitted wave at an angle formula_15, corresponding to the above figure. Note that in the cases of an interface into an absorbing material (where "n" is complex) or total internal reflection, the angle of transmission might not evaluate to a real number.


We consider the sign of a wave's electric field in relation to a wave's direction. Consequently, for "p" polarization at normal incidence, the positive direction of electric field for an incident wave (to the left) is "opposite" that of a reflected wave (also to its left); for "s" polarization both are the same (upward).


One can see that and . One can write similar equations applying to the ratio of magnetic fields of the waves, but these are usually not required.


Because the reflected and incident waves propagate in the same medium and make the same angle with the normal to the surface, the power reflection coefficient R is just the squared magnitude of "r":


On the other hand, calculation of the power transmission coefficient is less straightforward, since the light travels in different directions in the two media. What's more, the wave impedances in the two media differ; power is only proportional to the square of the amplitude when the media's impedances are the same (as they are for the reflected wave). This results in:


The factor of is the reciprocal of the ratio of the media's wave impedances (since we assume "μ"="μ"0). The factor of is from expressing power "in the direction" normal to the interface, for both the incident and transmitted waves.


In the case of total internal reflection where the power transmission is zero, nevertheless describes the electric field (including its phase) just beyond the interface. This is an evanescent field which does not propagate as a wave (thus =0) but has nonzero values very close to the interface. The phase shift of the reflected wave on total internal reflection can similarly be obtained from the phase angles of and (whose magnitudes are unity). These phase shifts are different for "s" and "p" waves, which is the well-known principle by which total internal reflection is used to effect polarization transformations.


In the above formula for , if we put formula_19 (Snell's law) and multiply the numerator and denominator by , we obtain


If we do likewise with the formula for , the result is easily shown to be equivalent to


These formulas are known respectively as "Fresnel's sine law" and "Fresnel's tangent law". Although at normal incidence these expressions reduce to 0/0, one can see that they yield the correct results in the limit as .


When light makes multiple reflections between two or more parallel surfaces, the multiple beams of light generally interfere with one another, resulting in net transmission and reflection amplitudes that depend on the light's wavelength. The interference, however, is seen only when the surfaces are at distances comparable to or smaller than the light's coherence length, which for ordinary white light is few micrometers; it can be much larger for light from a laser.


An example of interference between reflections is the iridescent colours seen in a soap bubble or in thin oil films on water. Applications include Fabry–Pérot interferometers, antireflection coatings, and optical filters. A quantitative analysis of these effects is based on the Fresnel equations, but with additional calculations to account for interference.


The transfer-matrix method, or the recursive Rouard method can be used to solve multiple-surface problems.


In 1808, Étienne-Louis Malus discovered that when a ray of light was reflected off a non-metallic surface at the appropriate angle, it behaved like "one" of the two rays emerging from a doubly-refractive calcite crystal. He later coined the term "polarization" to describe this behavior.  In 1815, the dependence of the polarizing angle on the refractive index was determined experimentally by David Brewster. But the "reason" for that dependence was such a deep mystery that in late 1817, Thomas Young was moved to write:


In 1821, however, Augustin-Jean Fresnel derived results equivalent to his sine and tangent laws (above), by modeling light waves as transverse elastic waves with vibrations perpendicular to what had previously been called the plane of polarization. Fresnel promptly confirmed by experiment that the equations correctly predicted the direction of polarization of the reflected beam when the incident beam was polarized at 45° to the plane of incidence, for light incident from air onto glass or water; in particular, the equations gave the correct polarization at Brewster's angle. The experimental confirmation was reported in a "postscript" to the work in which Fresnel first revealed his theory that light waves, including "unpolarized" waves, were "purely" transverse.


Details of Fresnel's derivation, including the modern forms of the sine law and tangent law, were given later, in a memoir read to the French Academy of Sciences in January 1823. That derivation combined conservation of energy with continuity of the "tangential" vibration at the interface, but failed to allow for any condition on the "normal" component of vibration. The first derivation from "electromagnetic" principles was given by Hendrik Lorentz in 1875.


In the same memoir of January 1823, Fresnel found that for angles of incidence greater than the critical angle, his formulas for the reflection coefficients ( and ) gave complex values with unit magnitudes. Noting that the magnitude, as usual, represented the ratio of peak amplitudes, he guessed that the argument represented the phase shift, and verified the hypothesis experimentally. The verification involved


Thus he finally had a quantitative theory for what we now call the "Fresnel rhomb" — a device that he had been using in experiments, in one form or another, since 1817 (see "Fresnel rhomb §History").


The success of the complex reflection coefficient inspired James MacCullagh and Augustin-Louis Cauchy, beginning in 1836, to analyze reflection from metals by using the Fresnel equations with a complex refractive index.


Four weeks before he presented his completed theory of total internal reflection and the rhomb, Fresnel submitted a memoir in which he introduced the needed terms "linear polarization", "circular polarization", and "elliptical polarization", and in which he explained optical rotation as a species of birefringence: linearly-polarized light can be resolved into two circularly-polarized components rotating in opposite directions, and if these propagate at different speeds, the phase difference between them — hence the orientation of their linearly-polarized resultant — will vary continuously with distance.


Thus Fresnel's interpretation of the complex values of his reflection coefficients marked the confluence of several streams of his research and, arguably, the essential completion of his reconstruction of physical optics on the transverse-wave hypothesis (see "Augustin-Jean Fresnel").


Here we systematically derive the above relations from electromagnetic premises.


In order to compute meaningful Fresnel coefficients, we must assume that the medium is (approximately) linear and homogeneous. If the medium is also isotropic, the four field vectors are related by


where "ϵ" and "μ" are scalars, known respectively as the (electric) "permittivity" and the (magnetic) "permeability" of the medium. For a vacuum, these have the values "ϵ"0 and "μ"0, respectively. Hence we define the "relative" permittivity (or dielectric constant) , and the "relative" permeability .


In optics it is common to assume that the medium is non-magnetic, so that "μ"rel=1. For ferromagnetic materials at radio/microwave frequencies, larger values of "μ"rel must be taken into account. But, for optically transparent media, and for all other materials at optical frequencies (except possible metamaterials), "μ"rel is indeed very close to 1; that is, "μ"≈"μ"0.


In optics, one usually knows the refractive index "n" of the medium, which is the ratio of the speed of light in a vacuum () to the speed of light in the medium. In the analysis of partial reflection and transmission, one is also interested in the electromagnetic wave impedance , which is the ratio of the amplitude of to the amplitude of . It is therefore desirable to express "n" and in terms of "ϵ" and "μ", and thence to relate to "n". The last-mentioned relation, however, will make it convenient to derive the reflection coefficients in terms of the wave "admittance" , which is the reciprocal of the wave impedance .


In the case of "uniform plane sinusoidal" waves, the wave impedance or admittance is known as the "intrinsic" impedance or admittance of the medium. This case is the one for which the Fresnel coefficients are to be derived.


In a uniform plane sinusoidal electromagnetic wave, the electric field has the form


where is the (constant) complex amplitude vector,  is the imaginary unit,  is the wave vector (whose magnitude is the angular wavenumber),  is the position vector,  "ω" is the angular frequency,  is time, and it is understood that the "real part" of the expression is the physical field.  The value of the expression is unchanged if the position varies in a direction normal to ; hence "is normal to the wavefronts".


To advance the phase by the angle "ϕ", we replace by (that is, we replace by ), with the result that the (complex) field is multiplied by . So a phase "advance" is equivalent to multiplication by a complex constant with a "negative" argument. This becomes more obvious when the field () is factored as where the last factor contains the time-dependence. That factor also implies that differentiation w.r.t. time corresponds to multiplication by .


If "ℓ" is the component of in the direction of the field () can be written .  If the argument of is to be constant,  "ℓ" must increase at the velocity formula_22 known as the "phase velocity" . This in turn is equal to formula_23. Solving for gives


As usual, we drop the time-dependent factor which is understood to multiply every complex field quantity. The electric field for a uniform plane sine wave will then be represented by the location-dependent "phasor"


For fields of that form, Faraday's law and the Maxwell-Ampère law respectively reduce to


Putting and as above, we can eliminate and to obtain equations in only and :


If the material parameters "ϵ" and "μ" are real (as in a lossless dielectric), these equations show that form a "right-handed orthogonal triad", so that the same equations apply to the magnitudes of the respective vectors. Taking the magnitude equations and substituting from (), we obtain


where and are the magnitudes of and . Multiplying the last two equations gives


Dividing (or cross-multiplying) the same two equations gives where


From () we obtain the phase velocity formula_27. For a vacuum this reduces to formula_28. Dividing the second result by the first gives


For a "non-magnetic" medium (the usual case), this becomes formula_30.


Taking the reciprocal of (), we find that the intrinsic "impedance" is formula_31.  In a vacuum this takes the value formula_32 known as the impedance of free space. By division, formula_33. For a "non-magnetic" medium, this becomes formula_34


In Cartesian coordinates , let the region have refractive index intrinsic admittance etc., and let the region have refractive index intrinsic admittance etc. Then the plane is the interface, and the axis is normal to the interface (see diagram). Let and (in bold roman type) be the unit vectors in the and directions, respectively. Let the plane of incidence be the plane (the plane of the page), with the angle of incidence measured from towards . Let the angle of refraction, measured in the same sense, be where the subscript stands for "transmitted" (reserving for "reflected").


In the absence of Doppler shifts, "ω" does not change on reflection or refraction. Hence, by (), the magnitude of the wave vector is proportional to the refractive index.


So, for a given "ω", if we "redefine" as the magnitude of the wave vector in the "reference" medium (for which ), then the wave vector has magnitude in the first medium (region in the diagram) and magnitude in the second medium. From the magnitudes and the geometry, we find that the wave vectors are


where the last step uses Snell's law. The corresponding dot products in the phasor form () are


For the "s" polarization, the field is parallel to the axis and may therefore be described by its component in the  direction. Let the reflection and transmission coefficients be and respectively. Then, if the incident field is taken to have unit amplitude, the phasor form () of its  component is


and the reflected and transmitted fields, in the same form, are


Under the sign convention used in this article, a positive reflection or transmission coefficient is one that preserves the direction of the "transverse" field, meaning (in this context) the field normal to the plane of incidence. For the "s" polarization, that means the field. If the incident, reflected, and transmitted fields (in the above equations) are in the  direction ("out of the page"), then the respective fields are in the directions of the red arrows, since form a right-handed orthogonal triad. The fields may therefore be described by their components in the directions of those arrows, denoted by .  Then, since


At the interface, by the usual interface conditions for electromagnetic fields, the tangential components of the and fields must be continuous; that is,