Minkowski space
In
The model helps show how a
Minkowski space is closely associated with Einstein's theories of special relativity and general 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 to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total interval in spacetime between events.[nb 1] Minkowski space differs from four-dimensional Euclidean space insofar as it treats time differently than the three spatial dimensions.
In 3-dimensional
Spacetime is equipped with an indefinite
History
Part of a series on |
Spacetime |
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Complex Minkowski spacetime
In his second relativity paper in 1905, Henri Poincaré showed[4] how, by taking time to be an imaginary fourth spacetime coordinate ict, where c is the speed of light and i is the imaginary 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. The Lorentz 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 a matrix 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 space unit vector and a time unit vector, while formally still a rotation in coordinate space, is a
This idea, which was mentioned only briefly by Poincaré, was elaborated by Minkowski in a paper in
Real Minkowski spacetime
In a further development in his 1908 "Space and Time" lecture,
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 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 a translation dependent) as "sum".
Minkowski's principal tool is the
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[6]
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 a
gravitation. He was still far from the study of curvilinear coordinates and Riemannian geometry, and the heavy mathematical apparatus entailed.[7]
For further historical information see references Galison (1979), Corry (1997) and Walter (1999).
Causal structure
Where v is velocity, x, y, and z are Cartesian coordinates in 3-dimensional space, c is the constant representing the universal speed limit, and t is time, the four-dimensional vector v = (ct, x, y, z) = (ct, r) is classified according to the sign of c2t2 − r2. A vector is timelike if c2t2 > r2, spacelike if c2t2 < r2, and null or lightlike if c2t2 = r2. This can be expressed in terms of the sign of η(v, v) 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 all
Once a direction of time is chosen,[nb 4] timelike and null vectors can be further decomposed into various classes. For timelike vectors, one has
- future-directed timelike vectors whose first component is positive (tip of vector located in absolute future in the figure) and
- past-directed timelike vectors whose first component is negative (absolute past).
Null vectors fall into three classes:
- the zero vector, whose components in any basis are (0, 0, 0, 0) (origin),
- future-directed null vectors whose first component is positive (upper light cone), and
- past-directed null vectors whose first component is negative (lower light cone).
Together with spacelike vectors, there are 6 classes in all.
An
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.
Properties of time-like vectors
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 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.
Scalar product
The
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).
Norm and reversed Cauchy inequality
The norm of a time-like vector u = (ct, x, y, z) is defined as
The reversed Cauchy inequality is another consequence of the convexity of either light cone.[8] For two distinct similarly directed time-like vectors u1 and u2 this inequality is
From this, the positive property of the scalar product can be seen.
Reversed triangle inequality
For two similarly directed time-like vectors u and w, the inequality is[9]
The proof uses the algebraic definition with the reversed Cauchy inequality:[10]
The result now follows by taking the square root on both sides.
Mathematical structure
It is assumed below that spacetime is endowed with a coordinate system corresponding to an
For an overview, Minkowski space is a 4-dimensional real vector space equipped with a non-degenerate, 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, 4.
Tangent vectors
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.) or Lee (2012, Proposition 3.13.) These identifications are routinely done in mathematics. They can be expressed formally in Cartesian coordinates as[11]
Here, p and q are any two events, and the second basis vector 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 Rn. 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 p may be defined, here specialized to Cartesian coordinates in Lorentz frames, as 4 × 1 column vectors v associated to each Lorentz frame related by Lorentz transformation Λ such that the vector v in a frame related to some frame by Λ transforms according to v → Λv. This is the same way in which the coordinates xμ 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 p with displacement vectors at p, which is, of course, admissible by essentially the same canonical identification.[12] The identifications of vectors referred to above in the mathematical setting can correspondingly be found in a more physical and explicitly geometrical setting in Misner, Thorne & Wheeler (1973). They offer various degrees of sophistication (and rigor) depending on which part of the material one chooses to read.
Metric signature
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.
Terminology
Mathematically associated with the bilinear form is a tensor of type (0,2) at each point in spacetime, called the Minkowski metric.[nb 5] 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 4×4 matrix representing the bilinear form.
For comparison, in
Introducing more terminology (but not more structure), Minkowski space is thus a pseudo-Euclidean space with total dimension n = 4 and signature (3, 1) or (1, 3). Elements of Minkowski space are called events. Minkowski space is often denoted R3,1 or R1,3 to emphasize the chosen signature, or just M. It is perhaps the simplest example of a pseudo-Riemannian manifold.
Then mathematically, the metric is a bilinear form on an abstract four-dimensional real vector space V, that is,
An interesting example of non-inertial coordinates for (part of) Minkowski spacetime is the Born coordinates. Another useful set of coordinates is the light-cone coordinates.
Pseudo-Euclidean metrics
The Minkowski inner product is not an
As a notational convention, vectors v in M, called
The definition [13]
- Linearity in the first argument
- Symmetry
- Non-degeneracy
The first two conditions imply bilinearity. The defining difference between a pseudo-inner product and an
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 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 for all classical groups definable this way in classical group. There, the matrix Φ is identical in the case O(3, 1) (the Lorentz group) to the matrix η to be displayed below.
Two vectors v and w are said to be
A vector e is called a unit vector if η(e, e) = ±1. A basis for M consisting of mutually orthogonal unit vectors is called an orthonormal basis.[14]
For a given
More terminology (but not more structure): The Minkowski metric is a
Minkowski metric
From the
The invariance of the interval under coordinate transformations between inertial frames follows from the invariance of
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 η has been used in a derivation, go back to the earliest point where it was used, substitute η for −η, and retrace forward to the desired formula with the desired metric signature.
Standard basis
A standard or orthonormal basis for Minkowski space is a set of four mutually orthogonal vectors {e0, e1, e2, e3} such that
These conditions can be written compactly in the form
Relative to a standard basis, the components of a vector v are written (v0, v1, v2, v3) where the Einstein notation is used to write v = vμ eμ. The component v0 is called the timelike component of v while the other three components are called the spatial components. The spatial components of a 4-vector v may be identified with a 3-vector v = (v1, v2, v3).
In terms of components, the Minkowski inner product between two vectors v and w is given by
Here lowering 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 real 4 × 4 matrix satisfying
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 and Λ as the same thing, mathematically, they are elements of different spaces, and act on the space of standard bases from different sides.
Raising and lowering of indices
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 M and the
Thus if vμ are the components of a vector in tangent space, then ημν vμ = 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 in M itself, this is mostly ignored, and vectors with lower indices are referred to as covariant 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 are contravariant vectors. In the same fashion, the inverse of the map from tangent to cotangent spaces, explicitly given by the inverse of η in matrix representation, can be used to define raising of an index. The components of this inverse are denoted ημν. It happens that ημν = ημν. These maps between a vector space and its dual can be denoted η♭ (eta-flat) and η♯ (eta-sharp) by the musical analogy.[20]
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: its
One quantum mechanical analogy explored in the literature is that of a
The
One may, of course, ignore geometrical views altogether (as is the style in e.g.
Coordinate free raising and lowering
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,
Formalism of the Minkowski metric
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 xμ. These quantities evaluated at a point p provide a basis for the cotangent space at p. The tensor 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, the symmetric product (denoted by the symbol ⊙ or by juxtaposition) has been taken. The equality holds since, by definition, the Minkowski metric is symmetric.[21] 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 Misner, 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,
The exterior derivative df of a function f is a covector field, i.e. an assignment of a cotangent vector to each point p, by definition such that
Since this relation holds at each point p, the dxμ|p provide a basis for the cotangent space at each p and the bases dxμ|p and ∂/∂xν|p are dual to each other,
Thus when the metric tensor is fed two vectors fields a, b, both expanded in terms of the basis coordinate vector fields, the result is
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 in general relativity. There one has
Chronological and causality relations
Let x, y ∈ M. Here,
- x chronologically precedes y if y − x is future-directed timelike. This relation has the transitive propertyand so can be written x < y.
- x causally precedes y if y − x is future-directed null or future-directed timelike. It gives a partial orderingof spacetime and so can be written x ≤ y.
Suppose x ∈ M is timelike. Then the simultaneous hyperplane for x is {y : η(x, y) = 0}. Since this hyperplane varies as x varies, there is a relativity of simultaneity in Minkowski space.
Generalizations
A Lorentzian manifold is a generalization of Minkowski space in two ways. The total number of spacetime dimensions is not restricted to be 4 (2 or more) and a Lorentzian manifold need not be flat, i.e. it allows for curvature.
Complexified Minkowski space
Complexified Minkowski space is defined as Mc = M ⊕ iM.
The
Generalized Minkowski space
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 n ≥ 2, n-dimensional Minkowski space is a vector space of real dimension n 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 than 4 dimensions. String theory and M-theory are two examples where n > 4. In string theory, there appears conformal field theories with 1 + 1 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).
Curvature
As a flat spacetime, the three spatial components of Minkowski spacetime always obey the
Even in curved space, Minkowski space is still a good description in an infinitesimal region surrounding any point (barring gravitational singularities).[nb 6] More abstractly, it can be said 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.
Geometry
The meaning of the term geometry 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 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 2 or 3, cannot be isometrically embedded in Euclidean space with one more dimension, i.e. ℝ3 or ℝ4 respectively, with the Euclidean metric g, disallowing easy visualization.[nb 7][24] By comparison, model spaces with positive curvature are just spheres in Euclidean space of one higher dimension.[25] Hyperbolic spaces can be isometrically embedded in spaces of one more dimension when the embedding space is endowed with the Minkowski metric η.
Define H1(n)
R ⊂ Mn+1 to be the upper sheet (ct > 0) of the hyperboloid
R 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 −1/R2.[26] The 1 in the upper index refers to an enumeration of the different model spaces of hyperbolic geometry, and the n for its dimension. A 2(2) corresponds to the Poincaré disk model, while 3(n) corresponds to the Poincaré half-space model
Preliminaries
In the definition above ι: H1(n)
R → Mn+1 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 H1(n)
R actually is a hyperbolic space.
Behavior of tensors under inclusion, pullback of covariant tensors under general maps and pushforward of vectors under general maps |
---|
Behavior of tensors under inclusion:
where X1, X1, …, Xk are vector fields on S. The subscript star denotes the pushforward (to be introduced later), and it is in this special case simply the identity map (as is the inclusion map). The latter equality holds because a tangent space to a submanifold at a point is in a canonical way a subspace of the tangent space of the manifold itself at the point in question. One may simply write
meaning (with slight abuse of notation) the restriction of α to accept as input vectors tangent to some s ∈ S only.
Pullback of tensors under general maps:
where for any vector space V,
It is defined by
where the subscript star denotes the pushforward of the map F, and X1, X2, …, Xk are vectors in TpM. (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 because F∗X1 ≠ X1 in general.)
The pushforward of vectors under general maps: Further unwinding the definitions, the pushforward F∗: TMp → TNF(p) of a vector field under a map F: M → N between manifolds is defined by
where f is a function on N. When M = ℝm, N= ℝn the pushforward of F reduces to DF: ℝm → ℝn, the ordinary Jacobian matrix of partial derivatives of the component functions. The differential is the best linear approximation of a function F from ℝm to ℝn. The pushforward is the smooth manifold version of this. It acts between tangent spaces, and is in coordinates represented by the Jacobian matrix of the coordinate representation of the function.
The corresponding pullback is the dual map from the dual of the range tangent space to the dual of the domain tangent space, i.e. it is a linear map, |
Hyperbolic stereographic projection
In order to exhibit the metric, it is necessary to pull it back via a suitable parametrization. A parametrization of a submanifold S of M is a map U ⊂ Rm → M whose range is an open subset of S. If S has the same dimension as M, a parametrization is just the inverse of a coordinate map φ: M → U ⊂ Rm. The parametrization to be used is the inverse of hyperbolic stereographic projection. This is illustrated in the figure to the right for n = 2. It is instructive to compare to stereographic projection for spheres.
Stereographic projection σ: Hn
R → Rn and its inverse σ−1: Rn → Hn
R are given by
Let
If
One has
By construction of stereographic projection one has
This leads to the system of equations
The first of these is solved for λ and one obtains for stereographic projection
Next, the inverse σ−1(u) = (τ, x) must be calculated. Use the same considerations as before, but now with
With this λ, one obtains
Pulling back the metric
One has
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),
Detailed outline of computation |
---|
One has
and
With this one may write
from which
Summing this formula one obtains Similarly, for τ one gets
yielding
Now add this contribution to finally get |
This last equation shows that the metric on the ball is identical to the Riemannian metric h2(n)
R in the
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 for σ–1 Lastly,
and the same conclusion is reached.
|
See also
- Hyperbolic quaternion
- Hyperspace
- Introduction to the mathematics of general relativity
- Minkowski plane
Remarks
- ^ This makes spacetime distance an invariant.
- ^ Consistent use of the terms "Minkowski inner product", "Minkowski norm" or "Minkowski metric" is intended for the bilinear form here, since it is in widespread use. It is by no means "standard" in the literature, but no standard terminology seems to exist.
- ^ Translate the coordinate system so that the event is the new origin.
- ^ This corresponds to the time coordinate either increasing or decreasing when the proper time for any particle increases. An application of T flips this direction.
- inner productproper at each point on a manifold.
- flat space and curved space at infinitesimally small distance scales is foundational to the definition of a manifoldin general.
- Nash embedding theorem (Nash (1956)), but the embedding dimension is much higher, n = (m/2)(m + 1)(3m + 11) for a Riemannian manifold of dimension m.
Notes
- ^ "Minkowski" Archived 2019-06-22 at the Wayback Machine. Random House Webster's Unabridged Dictionary.
- ^ Lee 1997, p. 31
- ^ Poincaré 1905–1906, pp. 129–176 Wikisource translation: On the Dynamics of the Electron
- ^ Minkowski 1907–1908, pp. 53–111 *Wikisource translation: s:Translation:The Fundamental Equations for Electromagnetic Processes in Moving Bodies
- ^ a b Minkowski 1908–1909, pp. 75–88 Various English translations on Wikisource: "Space and Time"
- Clarendon Press, see page 11
- ^ See Schutz's proof p 148, also Naber p. 48
- ^ Schutz p. 148, Naber p. 49
- ^ Schutz p. 148
- ^ Lee 1997, p. 15
- ^ Lee 2003, See Lee's discussion on geometric tangent vectors early in chapter 3.
- ^ Giulini 2008 pp. 5, 6
- ISBN 978-0-486-43235-9. Archived from the original on 2022-12-26. Retrieved 2022-12-26. Extract of page 8 Archived 2022-12-26 at the Wayback Machine
- ISBN 978-1-108-48839-6.
- ^ Sard 1970, p. 71
- ^ Minkowski, Landau & Lifshitz 2002, p. 4
- ^ a b Misner, Thorne & Wheeler 1973
- Riemannian metrics.). Where Lee refers to positive definiteness to show the injectivity of the map, one needs instead appeal to non-degeneracy.
- ^ Lee 2003, The tangent-cotangent isomorphism p. 282
- ^ Lee 2003
- ^ Y. Friedman, A Physically Meaningful Relativistic Description of the Spin State of an Electron, Symmetry 2021, 13(10), 1853; https://doi.org/10.3390/sym13101853 Archived 2023-08-13 at the Wayback Machine
- ^ Jackson, J.D., Classical Electrodynamics, 3rd ed.; John Wiley \& Sons: Hoboken, NJ, US, 1998
- ^ Lee 1997, p. 66
- ^ Lee 1997, p. 33
- ^ Lee 1997
References
- Corry, L. (1997). "Hermann Minkowski and the postulate of relativity". Arch. Hist. Exact Sci. 51 (4): 273–314. S2CID 27016039.
- Catoni, F.; et al. (2008). The Mathematics of Minkowski Space-Time. Frontiers in Mathematics. Basel: ISSN 1660-8046.
- Galison, P. L. (1979). R McCormach; et al. (eds.). Minkowski's Space–Time: from visual thinking to the absolute world. Historical Studies in the Physical Sciences. Vol. 10. JSTOR 27757388.
- Giulini D The rich structure of Minkowski space, https://arxiv.org/abs/0802.4345v1.
- ISBN 978-0-07-035048-9.
- ISBN 0-7506-2768-9.
- Lee, J. M. (2003). Introduction to Smooth manifolds. Springer Graduate Texts in Mathematics. Vol. 218. ISBN 978-0-387-95448-6.
- Lee, J. M. (2012). Introduction to Smooth manifolds. Springer Graduate Texts in Mathematics. ISBN 978-1-4419-9981-8.
- Lee, J. M. (1997). Riemannian Manifolds – An Introduction to Curvature. Springer Graduate Texts in Mathematics. Vol. 176. New York · Berlin · Heidelberg: Springer Verlag. ISBN 978-0-387-98322-6.
- Minkowski, Hermann (1907–1908), [The Fundamental Equations for Electromagnetic Processes in Moving Bodies], Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 53–111
- Published translation: Carus, Edward H. (1918). "Space and Time". .
- Wikisource translation: The Fundamental Equations for Electromagnetic Processes in Moving Bodies
- Minkowski, Hermann (1908–1909), Space and Time. [Space and Time], Physikalische Zeitschrift, 10: 75–88 Various English translations on Wikisource:
- ISBN 978-0-7167-0344-0.
- Naber, G. L. (1992). The Geometry of Minkowski Spacetime. New York: ISBN 978-0-387-97848-2.
- MR 0075639.
- ISBN 9780679454434.
- S2CID 120211823 Wikisource translation: On the Dynamics of the Electron
- Robb A A: Optical Geometry of Motion; a New View of the Theory of Relativity Cambridge 1911, (Heffers). http://www.archive.org/details/opticalgeometryoOOrobbrich.
- Robb A A: Geometry of Time and Space, 1936 Cambridge Univ Press http://www.archive.org/details/geometryoftimean032218mbp.
- Sard, R. D. (1970). Relativistic Mechanics - Special Relativity and Classical Particle Dynamics. New York: W. A. Benjamin. ISBN 978-0805384918.
- Shaw, R. (1982). "§ 6.6 Minkowski space, § 6.7,8 Canonical forms pp 221–242". Linear Algebra and Group Representations. ISBN 978-0-12-639201-2.
- Walter, Scott A. (1999). "Minkowski, Mathematicians, and the Mathematical Theory of Relativity". In Goenner, Hubert; et al. (eds.). The Expanding Worlds of General Relativity. Boston: Birkhäuser. pp. 45–86. ISBN 978-0-8176-4060-6.
- ISBN 978-0-521-55001-7.
External links
Media related to Minkowski diagrams at Wikimedia Commons
- Animation clip on YouTubevisualizing Minkowski space in the context of special relativity.
- The Geometry of Special Relativity: The Minkowski Space – Time Light Cone
- Minkowski space at PhilPapers