Inphysics,Minkowski space (orMinkowski spacetime) (/mɪŋˈkɔːfski,-ˈkɒf-/[1]) is the main mathematical description ofspacetime in the absence ofgravitation. It combinesinertialspace andtimemanifolds into afour-dimensional model.
The model helps show how aspacetime interval between any twoevents is independent of theinertial frame of reference in which they are recorded. MathematicianHermann Minkowski developed it from the work ofHendrik Lorentz,Henri Poincaré, and others, and said it "was grown on experimental physical grounds".
Minkowski space is closely associated withEinstein's theories ofspecial relativity andgeneral relativity and is the most common mathematical structure by which special relativity is formalized. While the individual components in Euclidean space and time might differ due tolength contraction andtime dilation, in Minkowski spacetime, all frames of reference will agree on the total interval in spacetime between events.[nb 1] Minkowski space differs fromfour-dimensional Euclidean space insofar as it treats time differently from the three spatial dimensions.
In 3-dimensionalEuclidean space, theisometry group (maps preserving the regularEuclidean distance) is theEuclidean group. It is generated byrotations,reflections andtranslations. When time is appended as a fourth dimension, the further transformations of translations in time andLorentz boosts are added, and the group of all these transformations is called thePoincaré group. Minkowski's model follows special relativity, where motion causestime dilation changing the scale applied to the frame in motion and shifts the phase of light.
Minkowski space is apseudo-Euclidean space equipped with anisotropic quadratic form called thespacetime interval or theMinkowski norm squared. An event in Minkowski space for which the spacetime interval is zero is on thenull cone of the origin, called thelight cone in Minkowski space. Using thepolarization identity the quadratic form is converted to asymmetric bilinear form called theMinkowski inner product, though it is not a geometricinner product. Another misnomer isMinkowski metric,[2] but Minkowski space is not ametric space.
Thegroup of transformations for Minkowski space that preserves the spacetime interval (as opposed to the spatial Euclidean distance) is theLorentz group (as opposed to theGalilean group).
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In his second relativity paper in 1905,Henri Poincaré showed[3] how, by taking time to be an imaginary fourthspacetime coordinateict, wherec is thespeed of light andi is theimaginary unit,Lorentz transformations can be visualized as ordinary rotations of the four-dimensional Euclidean sphere. The four-dimensional spacetime can be visualized as a four-dimensional space, with each point representing an event in spacetime. TheLorentz transformations can then be thought of as rotations in this four-dimensional space, where the rotation axis corresponds to the direction of relative motion between the two observers and the rotation angle is related to their relative velocity.
To understand this concept, one should consider the coordinates of an event in spacetime represented as a four-vector(t,x,y,z). A Lorentz transformation is represented by amatrix that acts on the four-vector, changing its components. This matrix can be thought of as a rotation matrix in four-dimensional space, which rotates the four-vector around a particular axis.
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 spaceunit vector and a time unit vector, while formally still a rotation in coordinate space, is aLorentz boost in physical spacetime withreal 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 (seehyperbolic rotation).
This idea, which was mentioned only briefly by Poincaré, was elaborated by Minkowski in a paper inGerman published in 1908 called "The Fundamental Equations for Electromagnetic Processes in Moving Bodies".[4] He reformulatedMaxwell equations as a symmetrical set of equations in the four variables(x,y,z,ict) combined with redefined vector variables for electromagnetic quantities, and 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-dimensionalspacetime continuum.
In a further development in his 1908 "Space and Time" lecture,[5] Minkowski gave an alternative formulation of this idea that used a real time coordinate instead of an imaginary one, representing the four variables(x,y,z,t) of space and time in the coordinate form in a four-dimensional realvector space. Points in this space correspond to events in spacetime. In this space, there is a definedlight-cone associated with each point, and events not on the light cone are classified by their relation to the apex asspacelike ortimelike. It is principally this view of spacetime that is current nowadays, although the older view involvingimaginary time has also influenced special relativity.
In the English translation of Minkowski's paper, the Minkowski metric, as defined below, is referred to as theline element. The Minkowski inner product below appears unnamed when referring toorthogonality (which he callsnormality) of certain vectors, and the Minkowski norm squared is referred to as "sum" (a word choice that might be attributable to language translation).
Minkowski's principal tool is theMinkowski diagram, and he uses it to define concepts and demonstrate properties of Lorentz transformations (e.g.,proper time andlength contraction) and to provide geometrical interpretation to the generalization of Newtonian mechanics torelativistic mechanics. For these special topics, see the referenced articles, as the presentation below will be principally confined to the mathematical structure (Minkowski metric and from it derived quantities and thePoincaré group as symmetry group of spacetime)following from the invariance of the spacetime interval on the spacetime manifold as consequences of the postulates of special relativity, not to specific application orderivation of the invariance of the spacetime interval. This structure provides the background setting of all present relativistic theories, barring general relativity for whichflat Minkowski spacetime still provides a springboard as curved spacetime is locally Lorentzian.
Minkowski, aware of the fundamental restatement of the theory which he had made, said
The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.
— Hermann Minkowski, 1908, 1909[5]
Though Minkowski took an important step for physics,Albert Einstein saw its limitation:
At a time when Minkowski was giving the geometrical interpretation of special relativity by extending the Euclidean three-space to aquasi-Euclidean four-space that included time, Einstein was already aware that this is not valid, because it excludes the phenomenon ofgravitation. He was still far from the study ofcurvilinear coordinates andRiemannian geometry, and the heavy mathematical apparatus entailed.[6]
For further historical information see referencesGalison (1979),Corry (1997) andWalter (1999).
Wherev is velocity,x,y, andz areCartesian coordinates in 3-dimensional space,c is the constant representing the universal speed limit, andt is time, the four-dimensional vectorv = (ct,x,y,z) = (ct,r) is classified according to the sign ofc2t2 −r2. A vector istimelike ifc2t2 >r2,spacelike ifc2t2 <r2, andnull orlightlike ifc2t2 =r2. This can be expressed in terms of the sign ofη(v,v), also calledscalar product, 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 spacetime interval under Lorentz transformation.
The set of allnull vectors at an event[nb 2] of Minkowski space constitutes thelight cone of that event. Given a timelike vectorv, there is aworldline of constant velocity associated with it, represented by a straight line in a Minkowski diagram.
Once a direction of time is chosen,[nb 3] timelike and null vectors can be further decomposed into various classes. For timelike vectors, one has
Null vectors fall into three classes:
Together with spacelike vectors, there are 6 classes in all.
Anorthonormal 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 anull 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 that 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 that aresimilarly 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.
Thescalar product of two time-like vectorsu1 = (t1,x1,y1,z1) andu2 = (t2,x2,y2,z2) 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 reversedCauchy–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 vectoru = (ct,x,y,z) is defined as
The reversed Cauchy inequality is another consequence of the convexity of either light cone.[7] For two distinct similarly directed time-like vectorsu1 andu2 this inequality isor algebraically,
From this, the positive property of the scalar product can be seen.
For two similarly directed time-like vectorsu andw, the inequality is[8]where the equality holds when the vectors arelinearly dependent.
The proof uses the algebraic definition with the reversed Cauchy inequality:[9]
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 aninertial frame. This provides anorigin, which is necessary for spacetime to be modeled as a vector space. This addition is not required, and more complex treatments analogous to anaffine space can remove the extra structure. However, this is not the introductory convention and is not covered here.
For an overview, Minkowski space is a4-dimensionalrealvector space equipped with a non-degenerate,symmetric bilinear form on thetangent space at each point in spacetime, here simply called theMinkowski inner product, withmetric signature either(+ − − −) or(− + + +). The tangent space at each event is a vector space of the same dimension as spacetime,4.
In practice, one need not be concerned with the tangent spaces. The vector space structure 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.Lee (2003, Proposition 3.8.) orLee (2012, Proposition 3.13.) These identifications are routinely done in mathematics. They can be expressed formally in Cartesian coordinates as[10]with basis vectors in the tangent spaces defined by
Here,p andq are any two events, and the second basis vector identification is referred to asparallel 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 adirectional derivative operator on the set of smooth functions. This is promoted to adefinition of tangent vectors in manifoldsnot necessarily being embedded inRn. This definition of tangent vectors is not the only possible one, as ordinaryn-tuples can be used as well.
A tangent vector at a pointp may be defined, here specialized to Cartesian coordinates in Lorentz frames, as4 × 1 column vectorsv associated toeach Lorentz frame related by Lorentz transformationΛ such that the vectorv in a frame related to some frame byΛ transforms according tov → Λv. This is thesame way in which the coordinatesxμ 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 pointp withdisplacement vectors atp, which is, of course, admissible by essentially the same canonical identification.[11] The identifications of vectors referred to above in the mathematical setting can correspondingly be found in a more physical and explicitly geometrical setting inMisner, Thorne & Wheeler (1973). They offer various degrees 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. See also the page treating sign convention in Relativity.
In general, but with several exceptions, mathematicians and general relativists prefer spacelike vectors to yield a positive sign,(− + + +), while particle physicists tend to prefer timelike vectors to yield a positive sign,(+ − − −). Authors covering several areas of physics, e.g.Steven Weinberg andLandau and Lifshitz((− + + +) and(+ − − −), respectively) stick to one choice regardless of topic. Arguments for the former convention include "continuity" from the Euclidean case corresponding to the non-relativistic limitc → ∞. Arguments for the latter include that minus signs, otherwise ubiquitous in particle physics, go away. Yet other authors, especially of introductory texts, e.g.Kleppner & Kolenkow (1978), donot choose a signature at all, but instead, opt to coordinatize spacetime such that the timecoordinate (but not time itself!) is imaginary. This removes the need for theexplicit introduction of ametric tensor (which may seem like an extra burden in an introductory course), and one neednot be concerned withcovariant vectors andcontravariant vectors (or raising and lowering indices) to be described below. The inner product is instead effected by a straightforward extension of thedot product from over to This works in the flat spacetime of special relativity, but not in the curved spacetime of general relativity, seeMisner, Thorne & Wheeler (1973, Box 2.1, "Farewell toi c t ")(who, by the way use(− + + +)). MTW also argues that it hides the trueindefinite nature of the metric and the true nature of Lorentz boosts, which are not rotations. It also needlessly complicates the use of tools ofdifferential geometry that are otherwise immediately available and useful for geometrical description and calculation – even in the flat spacetime of special relativity, e.g. of the electromagnetic field.
Mathematically associated with the bilinear form is atensor of type(0,2) at each point in spacetime, called theMinkowski metric.[nb 4] 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 the4×4 matrix representing the bilinear form.
For comparison, ingeneral relativity, aLorentzian manifoldL is likewise equipped with ametric tensorg, which is a nondegenerate symmetric bilinear form on the tangent spaceTpL at each pointp ofL. In coordinates, it may be represented by a4×4 matrixdepending on spacetime position. Minkowski space is thus a comparatively simple special case of aLorentzian manifold. Its metric tensor is in coordinates with the samesymmetric matrix at every point ofM, and its arguments can, per above, be taken as vectors in spacetime itself.
Introducing more terminology (but not more structure), Minkowski space is thus apseudo-Euclidean space with total dimensionn = 4 andsignature(1, 3) or(3, 1). Elements of Minkowski space are calledevents. Minkowski space is often denotedR1,3 orR3,1 to emphasize the chosen signature, or justM. It is an example of apseudo-Riemannian manifold.
Then mathematically, the metric is a bilinear form on an abstract four-dimensional real vector spaceV, that is,whereη has signature(+, -, -, -), and signature is a coordinate-invariant property ofη. The space of bilinear maps forms a vector space which can be identified with, andη may be equivalently viewed as an element of this space. By making a choice of orthonormal basis, can be identified with the space. The notation is meant to emphasize the fact thatM and are not just vector spaces but have added structure..
An interesting example of non-inertial coordinates for (part of) Minkowski spacetime is theBorn coordinates. Another useful set of coordinates is thelight-cone coordinates.
The Minkowski inner product is not aninner product, since it has non-zeronull vectors. Since it is not adefinite bilinear form it is calledindefinite.
The Minkowski metricη is the metric tensor of Minkowski space. It is a pseudo-Euclidean metric, or more generally, aconstant pseudo-Riemannian metric in Cartesian coordinates. As such, it is a nondegenerate symmetric bilinear form, a type(0, 2) tensor. It accepts two argumentsup,vp, vectors inTpM,p ∈M, the tangent space atp inM. Due to the above-mentioned canonical identification ofTpM withM itself, it accepts argumentsu,v with bothu andv inM.
As a notational convention, vectorsv inM, called4-vectors, are denoted in italics, and not, as is common in the Euclidean setting, with boldfacev. The latter is generally reserved for the3-vector part (to be introduced below) of a4-vector.
The definition[12]yields an inner product-like structure onM, previously and also henceforth, called theMinkowski inner product, similar to the Euclideaninner product, but it describes a different geometry. It is also called therelativistic dot product. If the two arguments are the same,the resulting quantity will be called theMinkowski norm squared. The Minkowski inner product satisfies the following properties.
The first two conditions imply bilinearity.
The most important feature of the inner product and norm squared is thatthese are quantities unaffected by Lorentz transformations. In fact, it can be taken as the defining property of a Lorentz transformation in that it preserves the inner product (i.e. the value of the corresponding bilinear form on two vectors). This approach is taken more generally forall classical groups definable this way inclassical group. There, the matrixΦ is identical in the caseO(3, 1) (the Lorentz group) to the matrixη to be displayed below.
Minkowski space is constructed so that thespeed of light will be the same constant regardless of the reference frame in which it is measured. This property results from the relation of the time axis to a space axis. Two eventsu andv areorthogonal when the bilinear form is zero for them:η(v,w) = 0.
When bothu andv are both space-like, then they areperpendicular, but if one is time-like and the other space-like, then the relation ishyperbolic orthogonality. The relation is preserved in a change of reference frames and consequently the computation of light speed yields a constant result. The change of reference frame is called aLorentz boost and in mathematics it is ahyperbolic rotation. Each reference frame is associated with ahyperbolic angle, which is zero for the rest frame in Minkowski space. Such a hyperbolic angle has been labelledrapidity since it is associated with the speed of the frame.
From thesecond postulate of special relativity, together with homogeneity of spacetime and isotropy of space, it follows that thespacetime interval between two arbitrary events called1 and2 is:[13]This quantity is not consistently named in the literature. The interval is sometimes referred to as the square root of the interval as defined here.[14][15]
The invariance of the interval under coordinate transformations between inertial frames follows from the invariance ofprovided the transformations are linear. Thisquadratic form can be used to define a bilinear formvia thepolarization identity. This bilinear form can in turn be written aswhere[η] is a matrix associated withη. While possibly confusing, it is common practice to denote[η] with justη. The matrix is read off from the explicit bilinear form asand the bilinear formwith 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 givenhere) 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 tensorη has been used in a derivation, go back to the earliest point where it was used, substituteη for−η, and retrace forward to the desired formula with the desired metric signature.
A standard or orthonormal basis for Minkowski space is a set of four mutually orthogonal vectors{e0,e1,e2,e3} such thatand for which when
These conditions can be written compactly in the form
Relative to a standard basis, the components of a vectorv are written(v0,v1,v2,v3) where theEinstein notation is used to writev =vμeμ. The componentv0 is called thetimelike component ofv while the other three components are called thespatial components. The spatial components of a4-vectorv may be identified with a3-vectorv = (v1,v2,v3).
In terms of components, the Minkowski inner product between two vectorsv andw is given by
and
Herelowering of an index with the metric was used.
There are many possible choices of standard basis obeying the condition Any two such bases are related in some sense by a Lorentz transformation, either by a change-of-basis matrix, a real4 × 4 matrix satisfyingorΛ, a linear map on the abstract vector space satisfying, for any pair of vectorsu,v,
Then if two different bases exist,{e0,e1,e2,e3} and{e′0,e′1,e′2,e′3}, can be represented as or. While it might be tempting to think of andΛ as the same thing, mathematically, they are elements of different spaces, and act on the space of standard bases from different sides.
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 ofM and thecotangent spaces ofM. At a point inM, the tangent and cotangent spaces aredual vector spaces (so the dimension of the cotangent space at an event is also4). Just as an authentic inner product on a vector space with one argument fixed, byRiesz representation theorem, may be expressed as the action of alinear functional on the vector space, the same holds for the Minkowski inner product of Minkowski space.[17]
Thus ifvμ are the components of a vector in tangent space, thenημνvμ =vν are the components of a vector in the cotangent space (a linear functional). Due to the identification of vectors in tangent spaces with vectors inM itself, this is mostly ignored, and vectors with lower indices are referred to ascovariant vectors. In this latter interpretation, the covariant vectors are (almost always implicitly) identified with vectors (linear functionals) in the dual of Minkowski space. The ones with upper indices arecontravariant vectors. In the same fashion, the inverse of the map from tangent to cotangent spaces, explicitly given by the inverse ofη in matrix representation, can be used to defineraising of an index. The components of this inverse are denotedημν. It happens thatημν =ημν. These maps between a vector space and its dual can be denotedη♭ (eta-flat) andη♯ (eta-sharp) by the musical analogy.[18]
Contravariant and covariant vectors are geometrically very different objects. The first can and should be thought of as arrows. A linear function can be characterized by two objects: itskernel, which is ahyperplane 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 or1-form (though the latter is usually reserved for covectorfields).
One quantum mechanical analogy explored in the literature is that of ade Broglie wave (scaled by a factor of Planck's reduced constant) associated with amomentum 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 times the arrow pierces the planes.[16] The mathematical reference,Lee (2003), offers the same geometrical view of these objects (but mentions no piercing).
Theelectromagnetic field tensor is adifferential 2-form, which geometrical description can as well be found in MTW.
One may, of course, ignore geometrical views altogether (as is the style in e.g.Weinberg (2002) andLandau & Lifshitz 2002) and proceed algebraically in a purely formal fashion. The time-proven robustness of the formalism itself, sometimes referred to asindex 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.
Given a bilinear form the lowered version of a vector can be thought of as the partial evaluation of that is, there is an associated partial evaluation map
The lowered vector is then the dual map Note it does not matter which argument is partially evaluated due to the symmetry of
Non-degeneracy is then equivalent to injectivity of the partial evaluation map, or equivalently non-degeneracy indicates that the kernel of the map is trivial. In finite dimension, as is the case here, and noting that the dimension of a finite-dimensional space is equal to the dimension of the dual, this is enough to conclude the partial evaluation map is a linear isomorphism from to This then allows the definition of the inverse partial evaluation map,which allows the inverse metric to be defined aswhere the two different usages of can be told apart by the argument each is evaluated on. This can then be used to raise indices. If a coordinate basis is used, the metricη−1 is indeed the matrix inverse toη .
The present purpose is to show semi-rigorously howformally 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 theexterior derivative of the coordinate functionsxμ. These quantities evaluated at a pointp provide a basis for the cotangent space atp. Thetensor product (denoted by the symbol⊗) yields a tensor field of type(0, 2), i.e. the type that expects two contravariant vectors as arguments. On the right-hand side, thesymmetric product (denoted by the symbol⊙ or by juxtaposition) has been taken. The equality holds since, by definition, the Minkowski metric is symmetric.[19] The notation on the far right is also sometimes used for the related, but different,line element. It isnot a tensor. For elaboration on the differences and similarities, seeMisner, Thorne & Wheeler (1973, Box 3.2 and section 13.2.)
Tangent vectors are, in this formalism, given in terms of a basis of differential operators of the first order,wherep is an event. This operator applied to a functionf gives thedirectional derivative off atp in the direction of increasingxμ withxν,ν ≠μ fixed. They provide a basis for the tangent space atp.
The exterior derivativedf of a functionf is acovector field, i.e. an assignment of a cotangent vector to each pointp, by definition such thatfor eachvector fieldX. A vector field is an assignment of a tangent vector to each pointp. In coordinatesX can be expanded at each pointp in the basis given by the∂/∂xν|p . Applying this withf =xμ, the coordinate function itself, andX =∂/ ∂xν , called acoordinate vector field, one obtains
Since this relation holds at each pointp, thedxμ|p provide a basis for the cotangent space at eachp and the basesdxμ|p and∂/∂xν|p aredual to each other,at eachp. Furthermore, one hasfor general one-forms on a tangent spaceα,β and general tangent vectorsa,b. (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 fieldsa,b, both expanded in terms of the basis coordinate vector fields, the result iswhereaμ,bν are thecomponent functions of the vector fields. The above equation holds at each pointp, and the relation may as well be interpreted as the Minkowski metric atp applied to two tangent vectors atp.
As mentioned, in a vector space, such as modeling 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 the spacetime point the metric is to be evaluated and from where (which tangent space) the vectors come from.
This situation changes ingeneral relativity. There one haswhere nowη →g(p), i.e.,g is still a metric tensor but now depending on spacetime and is a solution ofEinstein's field equations. Moreover,a,bmust be tangent vectors at spacetime pointp and can no longer be moved around freely.
Letx,y ∈M. Here,
Supposex ∈M is timelike. Then thesimultaneous hyperplane forx is{y :η(x,y) = 0}. Since thishyperplane varies asx varies, there is arelativity 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 be4 (2 or more) and a Lorentzian manifold need not be flat, i.e. it allows for curvature.
Complexified Minkowski space is defined asMc =M ⊕iM.[20] Its real part is the Minkowski space offour-vectors, such as thefour-velocity and thefour-momentum, which are independent of the choice oforientation of the space. The imaginary part, on the other hand, may consist of four pseudovectors, such asangular velocity andmagnetic moment, which change their direction with a change of orientation. Apseudoscalari is introduced, which also changes sign with a change of orientation. Thus, elements ofMc are independent of the choice of the orientation.
Theinner product-like structure onMc is defined asu ⋅v =η(u,v) for anyu,v ∈Mc. A relativistic purespin of anelectron or any half spin particle is described byρ ∈ Mc asρ =u +is, whereu is the four-velocity of the particle, satisfyingu2 = 1 ands is the 4D spin vector,[21] which is also thePauli–Lubanski pseudovector satisfyings2 = −1 andu ⋅s = 0.
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. Ifn ≥ 2,n-dimensional Minkowski space is a vector space of real dimensionn on which there is a constant Minkowski metric of signature(n − 1, 1) or(1,n − 1). These generalizations are used in theories where spacetime is assumed to have more or less than4 dimensions.String theory andM-theory are two examples wheren > 4. In string theory there appearconformal field theories with1 + 1 spacetime dimensions.
de Sitter space can be formulated as a submanifold of generalized Minkowski space as can the model spaces ofhyperbolic geometry (see below).
As aflat spacetime, the three spatial components of Minkowski spacetime always obey thePythagorean Theorem. Minkowski space is a suitable basis for special relativity, a good description of physical systems over finite distances in systems without significantgravitation. However, in order to take gravity into account, physicists use the theory ofgeneral relativity, which is formulated in the mathematics ofdifferential geometry ofdifferential manifolds. When this geometry is used as a model of spacetime, it is known ascurved spacetime.
Even in curved spacetime, Minkowski space is still a good description in aninfinitesimal region surrounding any point (barring gravitational singularities).[nb 5] More abstractly, it can be said that in the presence of gravity spacetime is described by a curved 4-dimensionalmanifold for which thetangent 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 termgeometry for the Minkowski space depends heavily on the context. Minkowski space is not endowed with Euclidean geometry, and not with any of the generalized Riemannian geometries with intrinsic curvature, those exposed by themodel spaces inhyperbolic geometry (negative curvature) and the geometry modeled by thesphere (positive curvature). The reason is the indefiniteness of the Minkowski metric. Minkowski space is, in particular, not ametric space and not a Riemannian manifold with a Riemannian metric. However, Minkowski space containssubmanifolds endowed with a Riemannian metric yielding hyperbolic geometry.
Model spaces of hyperbolic geometry of low dimension, say 2 or 3,cannot be isometrically embedded in Euclidean space with one more dimension, i.e. or respectively, with the Euclidean metric, preventing easy visualization.[nb 6][22] By comparison, model spaces with positive curvature are just spheres in Euclidean space of one higher dimension.[23] Hyperbolic spacescan be isometrically embedded in spaces of one more dimension when the embedding space is endowed with the Minkowski metric.
Defineto be the upper sheet () of thehyperboloidin generalized Minkowski space of spacetime dimension This is one of thesurfaces of transitivity of the generalized Lorentz group. Theinduced metric on this submanifold,thepullback of the Minkowski metric under inclusion, is aRiemannian metric. With this metric is aRiemannian manifold. It is one of the model spaces of Riemannian geometry, thehyperboloid model ofhyperbolic space. It is a space of constant negative curvature.[24] The 1 in the upper index refers to an enumeration of the different model spaces of hyperbolic geometry, and then for its dimension. A corresponds to thePoincaré disk model, while corresponds to thePoincaré half-space model of dimension
In the definition above is theinclusion map and the superscript star denotes thepullback. The present purpose is to describe this and similar operations as a preparation for the actual demonstration that actually is a hyperbolic space.
| Behavior of tensors under inclusion, pullback of covariant tensors under general maps and pushforward of vectors under general maps |
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Behavior of tensors under inclusion: Pullback of tensors under general maps: It is defined bywhere the subscript star denotes thepushforward of the mapF, andX1,X2, ..., Xk are vectors inTpM. (This is in accord with what was detailed about the pullback of the inclusion map. In the general case here, one cannot proceed as simply becauseF∗X1 ≠X1 in general.) The pushforward of vectors under general maps: Further unwinding the definitions, the pushforwardF∗:TMp →TNF(p) of a vector field under a mapF:M →N between manifolds is defined bywheref is a function onN. WhenM =Rm,N=Rn the pushforward ofF reduces toDF:Rm →Rn, the ordinarydifferential, which is given by theJacobian matrix of partial derivatives of the component functions. The differential is the best linear approximation of a functionF fromRm toRn. The pushforward is the smooth manifold version of this. It acts between tangent spaces, and is in coordinates represented by the Jacobian matrix of thecoordinate representation of the function. The corresponding pullback is thedual map from the dual of the range tangent space to the dual of the domain tangent space, i.e. it is a linear map, |
In order to exhibit the metric, it is necessary to pull it back via a suitableparametrization. A parametrization of a submanifoldS of a manifoldM is a mapU ⊂Rm →M whose range is an open subset ofS. IfS has the same dimension asM, a parametrization is just the inverse of a coordinate mapφ:M →U ⊂Rm. The parametrization to be used is the inverse ofhyperbolic stereographic projection. This is illustrated in the figure to the right forn = 2. It is instructive to compare tostereographic projection for spheres.
Stereographic projectionσ:Hn
R →Rn and its inverseσ−1:Rn →Hn
R are given bywhere, for simplicity,τ ≡ct. The(τ,x) are coordinates onMn+1 and theu are coordinates onRn.
Letand let
Ifthen it is geometrically clear that the vectorintersects the hyperplaneonce in point denoted
One hasor
By construction of stereographic projection one has
This leads to the system of equations
The first of these is solved forλ and one obtains for stereographic projection
Next, the inverseσ−1(u) = (τ,x) must be calculated. Use the same considerations as before, but now withone getsbut now withλ depending onu. The condition forP lying in the hyperboloid isorleading to
With thisλ, one obtains
One hasand the map
The pulled back metric can be obtained by straightforward methods of calculus;
One computes according to the standard rules for computing differentials (though one is really computing the rigorously defined exterior derivatives),and substitutes the results into the right hand side. This yields
| Detailed outline of computation |
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One hasand With this one may writefrom which Summing this formula one obtains Similarly, forτ one getsyielding Now add this contribution to finally get |
This last equation shows that the metric on the ball is identical to the Riemannian metrich2(n)
R in thePoincaré ball model, another standard model of hyperbolic geometry.
| Alternative calculation using the pushforward |
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The pullback can be computed in a different fashion. By definition, In coordinates, One has from the formula forσ–1 Lastly,and the same conclusion is reached. |
Media related toMinkowski diagrams at Wikimedia Commons
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