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### Symmetric part

Any square matrix can be written as a sum(1)where(2)is a symmetric matrix known as the symmetric part of and(3)is an antisymmetric matrix known as the antisymmetric part of . Here, is the transpose.The symmetric part of a tensor is denoted using parenthesesas(4)(5)Symbols for the symmetric and antisymmetric partsof tensors can be combined, for example(6)(Wald 1984, p. 26).

### Bivector

An antisymmetric tensor of second rank(a.k.a. 2-form).where is the wedge product (or outer product).

### Antisymmetric part

Any square matrix can be written as a sum(1)where(2)is a symmetric matrix known as the symmetric part of and(3)is an antisymmetric matrix known as the antisymmetric part of . Here, is the transpose.Any rank-2 tensor can be written as a sum of symmetricand antisymmetric parts as(4)The antisymmetric part of a tensor is sometimes denoted using the special notation(5)For a general rank- tensor,(6)where is the permutation symbol. Symbols for the symmetric and antisymmetric parts of tensors can be combined, for example(7)(Wald 1984, p. 26).

### Dual bivector

A dual bivector is defined byand a self-dual bivector by

### Curl

The curl of a vector field, denoted or (the notation used in this work), is defined as the vector field having magnitude equal to the maximum "circulation" at each point and to be oriented perpendicularly to this plane of circulation for each point. More precisely, the magnitude of is the limiting value of circulation per unit area. Written explicitly,(1)where the right side is a line integral around an infinitesimal region of area that is allowed to shrink to zero via a limiting process and is the unit normal vector to this region. If , then the field is said to be an irrotational field. The symbol is variously known as "nabla" or "del."The physical significance of the curl of a vector field is the amount of "rotation" or angular momentum of the contents of given region of space. It arises in fluid mechanics and elasticity theory. It is also fundamental in the theory of electromagnetism, where it arises in..

### Pseudotensor

A tensor-like object which reverses sign under inversion. Given a transformation matrix ,where det is the determinant. A pseudotensor is sometimesalso called a tensor density.

### Index lowering

The indices of a contravariant tensor can be lowered, turning it into a covariant tensor , by multiplication by a so-called metric tensor , e.g.,

### Weyl tensor

The Weyl tensor is the tensor defined by(1)where is the Riemann tensor, is the scalar curvature, is the metric tensor, and denotes the antisymmetric tensor part (Wald 1984, p. 40).The Weyl tensor is defined so that every tensorcontraction between indices gives 0. In particular,(2)(Weinberg 1972, p. 146). The number of independent components for a Weyl tensor in -D for is given by(3)(Weinberg 1972, p. 146). For , 4, ..., this gives 0, 10, 35, 84, 168, ... (OEIS A052472).

### Positive timelike

A nonzero vector in -dimensional Lorentzian space is said to be positive timelike if it has imaginary (Lorentzian) norm and if its first component is positive. Symbolically, is positive timelike if bothandhold. Note that equation (6) above expresses the imaginary norm condition by saying, equivalently, that the vector has a negative squared norm.

### Index gymnastics

The technique of extracting the content from geometric (tensor) equations by working in component notation and rearranging indices as required. Index gymnastics is a fundamental component of special and general relativity (Misner et al. 1973, pp. 84-89). Examples of index gymnastics include(1)(2)(3)(4)(5)(6)(7)(8)(Misner et al. 1973, p. 85), where is the metric tensor, is the Kronecker delta, is a comma derivative, is the antisymmetric tensor part, and is the symmetric tensor part.

### Weak riemannian metric

A weak Riemannian metric on a smooth manifold is a tensor field which is both a weak pseudo-Riemannian metric and positive definite.In a very precise way, the condition of being a weak Riemannian metric is considerably less stringent than the condition of being a strong Riemannian metric due to the fact that strong non-degeneracy implies weak non-degeneracy but not vice versa. More precisely, any strong Riemannian metric provides an isomorphism between the tangent and cotangent spaces and , respectively, for all ; conversely, weak Riemannian metrics are merely injective linear maps from to (Marsden et al. 2002).

### Positive lightlike

A nonzero vector in -dimensional Lorentzian space is said to be positive lightlike if it has zero (Lorentzian) norm and if its first component is positive. Symbolically, is positive lightlike if bothandhold. The collection of all positive lightlike vectors form the top half of the lightcone.

### Harmonic coordinates

Harmonic coordinates satisfy the condition(1)or equivalently,(2)It is always possible to choose such a system. Using the d'Alembertian,(3)But since for harmonic coordinates, the result is a generalization of the harmonic equation(4)to(5)

### Poincar&eacute; transformation

"Poincaré transformation" is the name sometimes (e.g., Misner et al. 1973, p. 68) given to what other authors (e.g., Weinberg 1972, p. 26) term an inhomogeneous Lorentz transformationwhere is a Lorentz tensor.

### Tensor trace

The trace of a second-tensor rank tensor is a scalar given by the contracted mixed tensor equal to .The trace satisfies(1)For a small change in a given tensor , the trace can be computed approximately as(2)(3)(4)(5)(6)(7)

### Petrov notation

A tensor notation which considers the Riemann tensor as a matrix with indices and .

### Form integration

A differential k-form can be integrated on an -dimensional manifold. The basic example is an -form in the open unit ball in . Since is a top-dimensional form, it can be written and so(1)where the integral is the Lebesgue integral.On a manifold covered by coordinate charts , there is a partition of unity such that 1. has support in and 2. . Then(2)where the right-hand side is well-defined because each integration takes place in a coordinate chart. The integral of the -form is well-defined because, under a change of coordinates , the integral transforms according to the determinant of the Jacobian, while an -form pulls back by the determinant of the Jacobian. Hence,(3)is the same integral in either coordinate chart.For example, it is possible to integrate the 2-form(4)on the sphere . Because a point has measure zero, it is enough to integrate on , which can be covered by stereographic projection . Since(5)the pullback map of is(6)the integral of on..

### Tensor rank

The total number of contravariant and covariant indices of a tensor. The rank of a tensor is independent of the number of dimensions of the underlying space.An intuitive way to think of the rank of a tensor is as follows: First, consider intuitively that a tensor represents a physical entity which may be characterized by magnitude and multiple directions simultaneously (Fleisch 2012). Therefore, the number of simultaneous directions is denoted and is called the rank of the tensor in question. In -dimensional space, it follows that a rank-0 tensor (i.e., a scalar) can be represented by number since scalars represent quantities with magnitude and no direction; similarly, a rank-1 tensor (i.e., a vector) in -dimensional space can be represented by numbers and a general tensor by numbers. From this perspective, a rank-2 tensor (one that requires numbers to describe) is equivalent, mathematically, to an matrix.rankobject0scalar1vector2 matrixtensorThe..

### Permutation tensor

The permutation tensor, also called the Levi-Civita tensor or isotropic tensor of rank 3 (Goldstein 1980, p. 172), is a pseudotensor which is antisymmetric under the interchange of any two slots. Recalling the definition of the permutation symbol in terms of a scalar triple product of the Cartesian unit vectors,(1)the pseudotensor is a generalization to an arbitrary basisdefined by(2)(3)where(4)and , where is the metric tensor. is nonzero iff the vectors are linearly independent.When viewed as a tensor, the permutation symbol is sometimes known as the Levi-Civita tensor. The permutation tensor of rank four is important in general relativity, and has components defined as(5)(Weinberg 1972, p. 38). The rank four permutation tensor satisfies the identity(6)

### Tensor laplacian

The vector Laplacian can be generalized to yieldthe tensor Laplacian(1)(2)(3)(4)(5)where is a covariant derivative, is the metric tensor, , is the comma derivative (Arfken 1985, p. 165), and(6)is a Christoffel symbol of the secondkind.

### Orthogonal tensors

Orthogonal contravariant and covariantsatisfywhere is the Kronecker delta.

### Einstein tensor

where is the Ricci curvature tensor, is the scalar curvature, and is the metric tensor. (Wald 1984, pp. 40-41). It satisfies(Misner et al. 1973, p. 222).

A dyadic, also known as a vector direct product, is a linear polynomial of dyads consisting of nine components which transform as(1)(2)(3)Dyadics are often represented by Gothic capital letters. The use of dyadics is nearly archaic since tensors perform the same function but are notationally simpler.A unit dyadic is also called the idemfactor and is defined such that(4)In Cartesian coordinates,(5)and in spherical coordinates(6)

### Negative timelike

A nonzero vector in -dimensional Lorentzian space is said to be negative timelike if it has imaginary (Lorentzian) norm and if its first component is negative. Symbolically, is negative timelike if bothandhold. Note that equation (6) above expresses the imaginary norm condition by saying, equivalently, that the vector has a negative squared norm.

Dyads extend vectors to provide an alternative description to second tensor rank tensors. A dyad of a pair of vectors and is defined by . The dot product is defined by(1)(2)and the colon product by(3)

### Negative lightlike

A nonzero vector in -dimensional Lorentzian space is said to be negative lightlike if it has zero (Lorentzian) norm and if its first component is negative. Symbolically, is negative lightlike if bothandhold. The collection of all negative lightlike vectors form the bottom half of the lightcone.

### Lichnerowicz formula

where is the Dirac operator , is the covariant derivative on spinors, is the scalar curvature, and is the self-dual part of the curvature of .

### Regge calculus

Regge calculus is a finite element method utilized in numerical relativity in attempts of describing spacetimes with few or no symmetries by way of producing numerical solutions to the Einstein field equations (Khavari 2009). It was developed initially by Italian mathematician Tullio Regge in the 1960s (Regge 1961).Modern forays into Regge's method center on the triangulation of manifolds, particularly on the discrete approximation of 4-dimensional Riemannian and Lorentzian manifolds by way of cellular complexes whose 4-dimensional triangular simplices share their boundary tetrahedra (i.e., 3-dimensional simplices) to enclose a flat piece of spacetime (Marinelli 2013). Worth noting is that Regge himself devised the framework in more generality, though noted that no such generality is lost by assuming a triangular approximation (Regge 1961).The benefit of this technique is that the structures involved are rigid and hence are..

### Dual tensor

Given an antisymmetric second tensor rank tensor , a dual pseudotensor is defined by(1)where(2)(3)

### Dual scalar

Let , , and be three polar vectors, and define(1)(2)where det is the determinant. The is a third-tensor rank tensor, then the dual scalar is defined as(3)where is the permutation tensor.

### Tensor contraction

The contraction of a tensor is obtained by setting unlike indices equal and summing according to the Einstein summation convention. Contraction reduces the tensor rank by 2. For example, for a second-rank tensor,The contraction operation is invariant under coordinate changes sinceand must therefore be a scalar.When is interpreted as a matrix, the contraction is the same as the trace.Sometimes, two tensors are contracted using an upper index of one tensor and a lower of the other tensor. In this context, contraction occurs after tensor multiplication.

### Double contraction relation

A tensor is said to satisfy the double contraction relation when(1)This equation is satisfied by(2)(3)(4)where the hat denotes zero trace, symmetric unit tensors. These tensors are used to define the spherical harmonic tensor.

### D'alembertian

Written in the notation of partial derivatives, the d'Alembertian in a flat spacetime is defined bywhere is the speed of light.The operator usually called the d'Alembertian is also the Laplacian on a flat manifold of Lorentzian signature.

### Tensor

An th-rank tensor in -dimensional space is a mathematical object that has indices and components and obeys certain transformation rules. Each index of a tensor ranges over the number of dimensions of space. However, the dimension of the space is largely irrelevant in most tensor equations (with the notable exception of the contracted Kronecker delta). Tensors are generalizations of scalars (that have no indices), vectors (that have exactly one index), and matrices (that have exactly two indices) to an arbitrary number of indices.Tensors provide a natural and concise mathematical framework for formulating and solving problems in areas of physics such as elasticity, fluid mechanics, and general relativity.The notation for a tensor is similar to that of a matrix (i.e., ), except that a tensor , , , etc., may have an arbitrary number of indices. In addition, a tensor with rank may be of mixed type , consisting of so-called "contravariant"..

### Metric tensor index

The index associated to a metric tensor on a smooth manifold is a nonnegative integer for whichfor all . Here, the notation denotes the quadratic form index associated with .The index of a metric tensor provides an alternative tool by which to define a number of various notions typically associated to the signature of . For example, a Lorentzian manifold can be defined as a pair for which and for which , a definition equivalent to its more typical definition as a manifold of dimension no less than two equipped with a tensor of metric signature (or, equivalently, ).

### Symmetric tensor

A second-tensor rank symmetric tensor is defined as a tensor for which(1)Any tensor can be written as a sum of symmetric and antisymmetric parts(2)(3)The symmetric part of a tensoris denoted using parentheses as(4)(5)Symbols for the symmetric and antisymmetricparts of tensors can be combined, for example(6)(Wald 1984, p. 26).The product of a symmetric and an antisymmetric tensor is 0. This can be seen as follows. Let be antisymmetric, so(7)(8)Let be symmetric, so(9)Then(10)(11)(12)A symmetric second-tensor rank tensor has scalar invariants(13)(14)

### Covariant tensor

A covariant tensor, denoted with a lowered index (e.g., ) is a tensor having specific transformation properties. In general, these transformation properties differ from those of a contravariant tensor.To examine the transformation properties of a covariant tensor, first consider thegradient(1)for which(2)where . Now let(3)then any set of quantities which transform according to(4)or, defining(5)according to(6)is a covariant tensor.Contravariant tensors are a type of tensor with differing transformation properties, denoted . To turn a contravariant tensor into a covariant tensor (index lowering), use the metric tensor to write(7)Covariant and contravariant indices can be used simultaneously in a mixedtensor.In Euclidean spaces, and more generally in flat Riemannian manifolds, a coordinate system can be found where the metric tensor is constant, equal to Kronecker delta(8)Therefore, raising and lowering indices is trivial,..

### Covariant derivative

The covariant derivative of a contravariant tensor (also called the "semicolon derivative" since its symbol is a semicolon) is given by(1)(2)(Weinberg 1972, p. 103), where is a Christoffel symbol, Einstein summation has been used in the last term, and is a comma derivative. The notation , which is a generalization of the symbol commonly used to denote the divergence of a vector function in three dimensions, is sometimes also used.The covariant derivative of a covariant tensor is(3)(Weinberg 1972, p. 104).Schmutzer (1968, p. 72) uses the older notation or .

### Strong riemannian metric

A strong Riemannian metric on a smooth manifold is a tensor field which is both a strong pseudo-Riemannian metric and positive definite.In a very precise way, the condition of being a strong Riemannian metric is considerably more stringent than the condition of being a weak Riemannian metric due to the fact that strong non-degeneracy implies weak non-degeneracy but not vice versa. More precisely, strong Riemannian metrics provide an isomorphism between the tangent and cotangent spaces and , respectively, for all ; conversely, weak Riemannian metrics are merely injective linear maps from to .

### Lorentzian manifold

A semi-Riemannian manifold is said to be Lorentzian if and if the index associated with the metric tensor satisfies .Alternatively, a smooth manifold of dimension is Lorentzian if it comes equipped with a tensor of metric signature (or, equivalently, ).

### Lorentz transformation

A Lorentz transformation is a four-dimensional transformation(1)satisfied by all four-vectors , where is a so-called Lorentz tensor. Lorentz tensors are restricted by the conditions(2)with the Minkowski metric (Weinberg 1972, p. 26; Misner et al. 1973, p. 68).Here, the tensor indices run over 0, 1, 2, 3, with being the time coordinate and being space coordinates, and Einstein summation is used to sum over repeated indices. There are a number of conventions, but a common one used by Weinberg (1972) is to take the speed of light to simplify computations and allow to be written simply as for . The group of Lorentz transformations in Minkowski space is known as the Lorentz group.An element in four-space which is invariant under a Lorentz transformation is said to be a Lorentz invariant; examples include scalars, elements of the form , and the interval between two events (Thorn 2012).Note that while some authors (e.g., Weinberg 1972,..

### Contravariant tensor

A contravariant tensor is a tensor having specific transformation properties (cf., a covariant tensor). To examine the transformation properties of a contravariant tensor, first consider a tensor of rank 1 (a vector)(1)for which(2)Now let , then any set of quantities which transform according to(3)or, defining(4)according to(5)is a contravariant tensor. Contravariant tensors are indicated with raised indices, i.e., .Covariant tensors are a type of tensor with differing transformation properties, denoted . However, in three-dimensional Euclidean space,(6)for , 2, 3, meaning that contravariant and covariant tensors are equivalent. Such tensors are known as Cartesian tensor. The two types of tensors do differ in higher dimensions, however.Contravariant four-vectors satisfy(7)where is a Lorentz tensor.To turn a covariant tensor into a contravariant tensor (index raising), use the metric tensor to write(8)Covariant and..

### Spinor lie derivative

The Lie derivative of a spinor is defined bywhere is the image of by a one-parameter group of isometries with its generator. For a vector field and a covariant derivative , the Lie derivative of is given explicitly bywhere and are Dirac matrices (Choquet-Bruhat and DeWitt-Morette 2000).

### Contracted bianchi identities

Contracting tensors with in the Bianchi identities(1)gives(2)Contracting again,(3)or(4)or(5)

### Lightlike

A four-vector is said to be lightlike if its four-vector norm satisfies .One should note that the four-vector norm is nothing more than a special case of the more general Lorentzian inner product on Lorentzian -space with metric signature : In this more general environment, the inner product of two vectors and has the formwhereby one defines a vector to be lightlike precisely when .Lightlike vectors are sometimes called null vectors. The collection of all lightlike vectors in a Lorentzian space (e.g., in the Minkowski space of special relativity) is known as the light cone. One often draws distinction between lightlike vectors which are positive and those which are negative.

### Comma derivative

The components of the gradient of the one-form are denoted , or sometimes , and are given by(Misner et al. 1973, p. 62). Note that Schmutzer (1968, p. 70) uses the older notation .

### Lie derivative

The Lie derivative of tensor with respect to the vector field is defined by(1)Explicitly, it is given by(2)where is a comma derivative. The Lie derivative of a metric tensor with respect to the vector field is given by(3)where denotes the symmetric tensor part and is a covariant derivative.

### Christoffel symbol of the second kind

Christoffel symbols of the second kind are the second type of tensor-like object derived from a Riemannian metric which is used to study the geometry of the metric. Christoffel symbols of the second kind are variously denoted as (Walton 1967) or (Misner et al. 1973, Arfken 1985). They are also known as affine connections (Weinberg 1972, p. 71) or connection coefficients (Misner et al. 1973, p. 210).Unfortunately, there are two different definitions of the Christoffel symbolof the second kind.Arfken (1985, p. 161) defines(1)(2)(3)where is a partial derivative, is the metric tensor,(4)where is the radius vector, and(5)Therefore, for an orthogonal curvilinear coordinate system, by this definition,(6)The symmetry of definition (6) means that(7)(Walton 1967).This Christoffel symbol of the second kind is related to the Christoffel symbol of the first kind by(8)Walton (1967) lists Christoffel symbols of the second..

### Christoffel symbol of the first kind

The first type of tensor-like object derived from a Riemannian metric which is used to study the geometry of the metric. Christoffel symbols of the first kind are variously denoted , , , or . They are also known as connections coefficients (Misner et al. 1973, p. 210).The Christoffel symbol of the first kind is defined by(1)(2)(3)where is the metric tensor, is a Christoffel symbol of the second kind, and(4)But(5)(6)(7)so(8)

### Scalar curvature

The scalar curvature, also called the "curvature scalar" (e.g., Weinberg 1972, p. 135; Misner et al. 1973, p. 222) or "Ricci scalar," is given bywhere is the metric tensor and is the Ricci curvature tensor.

### Riemann tensor

The Riemann tensor (Schutz 1985) , also known the Riemann-Christoffel curvature tensor (Weinberg 1972, p. 133; Arfken 1985, p. 123) or Riemann curvature tensor (Misner et al. 1973, p. 218), is a four-index tensor that is useful in general relativity. Other important general relativistic tensors such that the Ricci curvature tensor and scalar curvature can be defined in terms of .The Riemann tensor is in some sense the only tensor that can be constructed fromthe metric tensor and its first and second derivatives,(1)where are Christoffel symbols of the first kind and is a comma derivative (Schmutzer 1968, p. 108; Weinberg 1972). In one dimension, . In four dimensions, there are 256 components. Making use of the symmetry relations,(2)the number of independent components is reduced to 36. Using the condition(3)the number of coordinates reduces to 21. Finally, using(4)20 independent components are left (Misner..

### Bochner identity

For a smooth harmonic map ,where is the gradient, Ric is the Ricci curvature tensor, and Riem is the Riemann tensor.

### Killing vectors

If any set of points is displaced by where all distance relationships are unchanged (i.e., there is an isometry), then the vector field is called a Killing vector.(1)so let(2)(3)(4)(5)(6)(7)(8)where is the Lie derivative.An ordinary derivative can be replaced with a covariant derivative in a Lie derivative, so we can take as the definition(9)(10)which gives Killing's equation(11)where denotes the symmetric tensor part and is a covariant derivative.A Killing vector satisfies(12)(13)(14)where is the Ricci curvature tensor and is the Riemann tensor.In Minkowski space, there are 10 Killing vectors(15)(16)(17)(18)The first group is translation, the second rotation,and the final corresponds to a "boost."

### Killing's equation

The equation defining Killing vectors.where is the Lie derivative and is a covariant derivative.

### Bianchi identities

The covariant derivative of the Riemanntensor is given by(1)Permuting , , and (Weinberg 1972, pp. 146-147) gives the Bianchi identities(2)which can be written concisely as(3)(Misner et al. 1973, p. 221), where denoted the antisymmetric tensor part. Wald (1984, p. 39) calls(4)the Bianchi identity, where is the covariant derivative, and is the Riemann tensor.

### Antisymmetric tensor

An antisymmetric (also called alternating) tensor is a tensor which changes sign when two indices are switched. For example, a tensor such that(1)is antisymmetric.The simplest nontrivial antisymmetric tensor is therefore an antisymmetric rank-2 tensor, which satisfies(2)Furthermore, any rank-2 tensor can be written as a sumof symmetric and antisymmetric parts as(3)The antisymmetric part of a tensor is sometimes denoted using the special notation(4)For a general rank- tensor,(5)where is the permutation symbol. Symbols for the symmetric and antisymmetric parts of tensors can be combined, for example(6)(Wald 1984, p. 26).

### Ricci curvature tensor

The Ricci curvature tensor, also simply known as the Ricci tensor (Parker and Christensen 1994), is defined bywhere is the Riemann tensor.Geometrically, the Ricci curvature is the mathematical object that controls the growthrate of the volume of metric balls in a manifold.

### Isotropic tensor

A tensor which has the same components in all rotated coordinate systems. All rank-0 tensors (scalars) are isotropic, but no rank-1 tensors (vectors) are. The unique rank-2 isotropic tensor is the Kronecker delta, and the unique rank-3 isotropic tensor is the permutation symbol (Goldstein 1980, p. 172).The number of isotropic tensors of rank 0, 1, 2, ... are 1, 0, 1, 1, 3, 6, 15, 36, 91, 232, ... (OEIS A005043). These numbers are called the Motzkin sum numbers and are given by the recurrence relation(1)with and . Closed forms for are given by(2)(3)The terms have the generating function(4)(5)(6)Starting at rank 5, syzygies play a role in restricting the number of isotropic tensors. In particular, syzygies occur at rank 5, 7, 8, and all higher ranks.

### Irreducible tensor

Given a general second tensor rank tensor and a metric , define(1)(2)(3)where is the Kronecker delta and is the permutation symbol. Then(4)where , , and are tensors of tensor rank 0, 1, and 2.

### Affine tensor

An affine tensor is a tensor that corresponds to certain allowable linear coordinate transformations, , where the determinant of is nonzero. This transformation takes the rectangular coordinate system into the coordinate system having oblique axes. In this way an affine tensor can be seen as a special kind of Cartesian tensor.These tensors have the Jacobians,(1)(2)(3)(4)The transformation laws for affine contravariant (tangent) tensors are(5)(6)(7)and so on, and the transformation laws for affine covariants (covectors) tensors are(8)(9)(10)and so on.The transformation laws for mixed affine tensors are(11)(12)

### Quadratic form index

The index associated to a symmetric, non-degenerate, and bilinear over a finite-dimensional vector space is a nonnegative integer defined bywhere the set is defined to beAs a concrete example, a pair consisting of a smooth manifold with a symmetric tensor field is said to be a Lorentzian manifold if and only if and the index associated to the quadratic form satisfies for all (Sachs and Wu 1977). This particular definition succinctly conveys the fact that Lorentzian manifolds have indefinite metric tensors of signature (or equivalently ) without having to make precise any definitions related to metric signatures, quadratic form signatures, etc.The above example also illustrates the deep connection between the index of a quadratic form and the notion of the index of a metric tensor defined on a smooth manifold . In particular, the index of a metric tensor is defined to be the quadratic form index associated to for any element . Because of this connection,..

### Index raising

The indices of a covariant tensor can be raised, forming a contravariant tensor , by multiplication by a so-called metric tensor , e.g.,

### Tensor direct product

Abstractly, the tensor direct product is the same as the vector space tensor product. However, it reflects an approach toward calculation using coordinates, and indices in particular. The notion of tensor product is more algebraic, intrinsic, and abstract. For instance, up to isomorphism, the tensor product is commutative because . Note this does not mean that the tensor product is symmetric.For two first-tensor rank tensors (i.e., vectors), the tensor direct product is defined as(1)which is a second-tensor rank tensor. The tensor contraction of a direct product of first-tensor rank tensors is the scalar(2)For second-tensor rank tensors,(3)(4)In general, the direct product of two tensors is a tensor of rank equal to the sum of the two initial ranks. The direct product is associative, but not commutative.The tensor direct product of two tensors and can be implemented in the Wolfram Language as TensorDirectProduct[a_List, b_List] :=..

### Minkowski space

Minkowski space is a four-dimensional space possessing a Minkowskimetric, i.e., a metric tensor having the formAlternatively (though less desirably), Minkowski space can be considered to have a Euclidean metric with imaginary time coordinate where is the speed of light (by convention is normally used) and where i is the imaginary number . Minkowski space unifies Euclidean three-space plus time (the "fourth dimension") in Einstein's theory of special relativity.In equation (5) above, the metric signature is assumed; under this assumption, Minkowski space is typically written . One may also express equation (5) with respect to the metric signature by reversing the order of the positive and negative squared terms therein, in which case Minkowski space is denoted .The Minkowski metric induces an inner product, the four-dimensional Lorentzian inner product (sometimes referred to as the Minkowski inner product), which fails..

### Ket

A ket is a vector living in a dual vector space to that containing bras . Bras and kets are commonly encountered in quantum mechanics. Bras and kets can be considered as 1-vectors and 1-forms (or vice versa), although this is almost always done only in a finite-dimensional vector space.Considered as an inner product, the bra and ket form an angle bracket (bra+ket = bracket) .

### Metric signature

The term metric signature refers to the signature of a metric tensor on a smooth manifold , a tool which quantifies the numbers of positive, zero, and negative infinitesimal distances of tangent vectors in the tangent bundle of and which is most easily defined in terms of the signatures of a number of related structures.Most commonly, one identifies the signature of a metric tensor with the signature of the quadratic form induced by on any of the tangent spaces for points . Indeed, given an orthogonal vector basis for any tangent space , the action of on arbitrary vectors and in is given by(1)whereby the signature of is defined to be the signature of any of the forms , i.e., the ordered triple of positive, negatives, and zero values for the inner products . This value is well-defined due to the fact that the signature of remains the same for all points in . For non-degenerate quadratic forms, the value will always satisfy , whereby the signature of will be..

### Lorentzian space

Lorentzian -space is the inner product space consisting of the vector space together with the -dimensional Lorentzian inner product.In the event that the metric signature is used, Lorentzian -space is denoted ; the notation is used analogously with the metric signature .The Lorentzian inner product induces a norm on Lorentzian space, whereby the squared norm of a vector has the form(1)Rewriting (where by definition), the norm in (0) can be written as(2)In particular, the norm induced by the Lorentzian inner product fails to be positive definite, whereby it makes sense to classify vectors in -dimensional Lorentzian space into types based on the sign of their squared norm, e.g., as spacelike, timelike, and lightlike. The collection of all lightlike vectors in Lorentzian -space is known as the light cone, which is further separated into lightlike vectors which are positive and negative lightlike. A similar distinction is made for positive..

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