In mathematical physics,
Contents 1 History 1.1 Four-dimensional Euclidean spacetime 1.2 Minkowski space 2 Mathematical structure 2.1 Tangent vectors 2.2 Metric signature 2.3 Terminology 2.4 Pseudo-Euclidean metrics 2.5 Minkowski metric 2.6 Standard basis 2.6.1 Raising and lowering of indices 2.6.2 The formalism of the Minkowski metric 3 Lorentz transformations and symmetry 4 Causal structure 4.1 Chronological and causality relations 4.2 Reversed triangle inequality 5 Generalizations 5.1 Generalized Minkowski space 5.2 Curvature 6 Geometry 6.1 Preliminaries 6.2 Hyperbolic stereographic projection 6.3 Pulling back the metric 7 See also 8 Remarks 9 Notes 10 References 11 External links History[edit] Part of a series on Spacetime General relativity
General relativity
Introduction to general relativity
Mathematics of general relativity
Classical gravity Introduction to gravitation Newton's law of universal gravitation Relevant mathematics
Four-vector
Derivations of relativity
v t e Four-dimensional Euclidean spacetime[edit]
See also: Four-dimensional space
In 1905–06
− t 2 + x 2 + y 2 + z 2 displaystyle -t^ 2 +x^ 2 +y^ 2 +z^ 2 is invariant under coordinate changes from one inertial frame to another, i. e. under Lorentz transformations. Here the speed of light c is, following Poincaré, set to unity. In the space suggested by him (Poincaré mentions this only in passing) where physical spacetime is coordinatized by (t, x, y, z) ↦ (x, y, z, it), call it coordinate space, Lorentz transformations appear as ordinary rotations preserving the quadratic form x 2 + y 2 + z 2 + t 2 displaystyle x^ 2 +y^ 2 +z^ 2 +t^ 2 on coordinate space. The naming and ordering of coordinates, with the
same labels for space coordinates, but with the imaginary time
coordinate as the fourth coordinate, is conventional. The above
expression, while making the former expression more familiar,[nb 3]
may potentially be confusing because it is not the same t that appears
in the latter (time coordinate) as in the former (time itself in some
inertial system as measured by clocks stationary in that system).
Rotations in planes spanned by two space unit vectors appear in
coordinate space as well as in physical spacetime appear 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
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,
At a time when Minkowski was giving the geometrical interpretation of
special relativity by extending the Euclidean three-space to a
quasi-Euclidean four-space that included time,
For further historical information see references Galison (1979), Corry (1997) and Walter (1999). Mathematical structure[edit] A pictorial representation of the tangent space at a point, x, on a sphere. This vector space can be thought of as a subspace of ℝ3 itself. Then vectors in it would be called geometrical tangent vectors. By the same principle, the tangent space at a point in flat spacetime can be thought of as a subspace of spacetime which happens to be all of spacetime. It is assumed below that spacetime is endowed with a coordinate system
corresponding to an inertial frame. This provides an origin, which is
necessary in order to be able to refer to spacetime as being modeled
as a vector space. This is not really physically motivated in that a
canonical origin ("central" event in spacetime) should exist. One can
get away with less structure, that of an affine space, but this would
needlessly complicate the discussion and would not reflect how flat
spacetime is normally treated mathematically in modern introductory
literature.
For an overview,
( x 0 , x 1 , x 2 , x 3 ) ↔ x 0 e 0
p + x 1 e 1
p + x 2 e 2
p + x 3 e 3
p ↔ x 0 e 0
q + x 1 e 1
q + x 2 e 2
q + x 3 e 3
q , displaystyle (x^ 0 ,x^ 1 ,x^ 2 ,x^ 3 )leftrightarrow x^ 0 mathbf e _ 0 _ p +x^ 1 mathbf e _ 1 _ p +x^ 2 mathbf e _ 2 _ p +x^ 3 mathbf e _ 3 _ p leftrightarrow x^ 0 mathbf e _ 0 _ q +x^ 1 mathbf e _ 1 _ q +x^ 2 mathbf e _ 2 _ q +x^ 3 mathbf e _ 3 _ q , with basis vectors in the tangent spaces defined by e μ
p = ∂ ∂ x μ
p or e 0
p = ( 1 0 0 0 ) , etc . displaystyle mathbf e _ mu _ p =left. frac partial partial x^ mu right_ p text or mathbf e _ 0 _ p =left( begin matrix 1\0\0\0end matrix right) text , etc . Here p and q are any two events and the last identification is referred to as parallel transport. The first identification is the canonical identification of vectors in the tangent space at any point with vectors in the space itself. The appearance of basis vectors in tangent spaces as first order differential operators is due to this identification. It is motivated by the observation that a geometrical tangent vector can be associated in a one-to-one manner with a directional derivative operator on the set of smooth functions. This is promoted to a definition of tangent vectors in manifolds not necessarily being embedded in ℝn. This definition of tangent vectors is not the only possible one as ordinary n-tuples can be used as well. Definitions of tangent vectors as ordinary vectors 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
x ′ μ = Λ μ ν x ν , v ′ μ = Λ μ ν v ν . displaystyle begin aligned x'^ mu &= Lambda ^ mu _ nu x^ nu ,\v'^ mu &= Lambda ^ mu _ nu v^ nu .end aligned 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.[9] 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 (1970). They offer various degree of sophistication (and rigor) depending on which part of the material one chooses to read. Metric signature[edit] 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. The choice of metric signature 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[edit]
Mathematically associated to the bilinear form is a tensor of type
(0,2) at each point in spacetime, called the Minkowski metric.[nb 4]
The Minkowski metric, the bilinear form, and the Minkowski inner
product are actually all the very 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 general relativity, a
u ⋅ v = η ( u , v ) displaystyle ucdot v=eta (u,v) yields an inner product-like structure on M, previously and also henceforth, called the Minkowski inner product, similar to the Euclidean inner product, but it describes a different geometry. It is also called the relativistic dot product. If the two arguments are the same, u ⋅ u = η ( u , u ) ≡
u
2 ≡ u 2 , displaystyle ucdot u=eta (u,u)equiv u^ 2 equiv u^ 2 , the resulting quantity will be called the Minkowski norm squared. The Minkowski inner product satisfies the following properties. η ( a u + v , w ) = a η ( u , w ) + η ( v , w ) , ∀ u , v ∈ M , ∀ a ∈ R (linearity in first slot) displaystyle eta (au+v,w)=aeta (u,w)+eta (v,w),quad forall u,vin M,forall ain mathbb R qquad text (linearity in first slot) η ( u , v ) = η ( v , u ) (symmetry) displaystyle eta (u,v)=eta (v,u)qquad text (symmetry) η ( u , v ) = 0 ∀ v ∈ M ⇒ u = 0 (non-degeneracy) displaystyle eta (u,v)=0quad forall vin MRightarrow u=0qquad text (non-degeneracy) The first two conditions imply bilinearity. The defining difference between a pseudo-inner product and an inner product proper is that the former is not required to be positive definite, that is, η(u, u) < 0 is allowed. The most important feature of the inner product and norm squared is that these are quantities unaffected by Lorentz transformations. In fact, it can be taken as the defining property of a Lorentz transformation that it preserves the inner product (i.e. the value of the corresponding bilinear form on two vectors). This approach is taken more generally for all classical groups definable this way in classical group. There, the matrix 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 orthogonal if η(v, w) = 0. For a geometric interpretation of orthogonality in the special case when η(v, v) ≤ 0 and η(w, w) ≥ 0 (or vice versa), see hyperbolic orthogonality. 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.[citation needed] For a given inertial frame, an orthonormal basis in space, combined by the unit time vector, forms an orthonormal basis in Minkowski space. The number of positive and negative unit vectors in any such basis is a fixed pair of numbers, equal to the signature of the bilinear form associated with the inner product. This is Sylvester's law of inertia. More terminology (but not more structure): The Minkowski metric is a pseudo-Riemannian metric, more specifically, a Lorentzian metric, even more specifically, the Lorentz metric, reserved for 4-dimensional flat spacetime with the remaining ambiguity only being the signature convention. Minkowski metric[edit] From the second postulate of special relativity, together with homogeneity of spacetime and isotropy of space, it follows that the spacetime interval between two arbitrary events called 1 and 2 is: ± [ c 2 ( t 1 − t 2 ) 2 − ( x 1 − x 2 ) 2 − ( y 1 − y 2 ) 2 − ( z 1 − z 2 ) 2 ] . displaystyle pm left[c^ 2 (t_ 1 -t_ 2 )^ 2 -(x_ 1 -x_ 2 )^ 2 -(y_ 1 -y_ 2 )^ 2 -(z_ 1 -z_ 2 )^ 2 right]. The interval is independent of the inertial frame chosen, as is shown here. The factor ±1 determines the choice of the metric signature as an arbitrary sign convention.[10] The numerical values of η, viewed as a matrix representing the Minkowski inner product, follow from the theory of bilinear forms. Just as the signature of the metric is differently defined in the literature, this quantity is not consistently named. The interval (as defined here) is sometimes referred to as the interval squared.[11] Even the square root of the present interval occurs.[12] When signature and interval are fixed, ambiguity still remains as which coordinate is the time coordinate. It may be the fourth, or it may be the zeroth. This is not an exhaustive list of notational inconsistencies. It is a fact of life that one has to check out the definitions first thing when one consults the relativity literature. The invariance of the interval under coordinate transformations between inertial frames follows from the invariance of ± [ c 2 t 2 − x 2 − y 2 − z 2 ] displaystyle pm left[c^ 2 t^ 2 -x^ 2 -y^ 2 -z^ 2 right] (with either sign ± preserved), provided the transformations are linear. This quadratic form can be used to define a bilinear form u ⋅ v = ± [ c 2 t 1 t 2 − x 1 x 2 − y 1 y 2 − z 1 z 2 ] . displaystyle ucdot v=pm left[c^ 2 t_ 1 t_ 2 -x_ 1 x_ 2 -y_ 1 y_ 2 -z_ 1 z_ 2 right]. via the polarization identity. This bilinear form can in turn be written as u ⋅ v = u T [ η ] v , displaystyle ucdot v=u^ mathrm T [eta ]v, where [η] is a 4×4 matrix associated with η. Possibly confusingly, denote [η] with just η as is common practice. The matrix is read off from the explicit bilinear form as η = ± ( − 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) , displaystyle eta =pm begin pmatrix -1&0&0&0\0&1&0&0\0&0&1&0\0&0&0&1end pmatrix , and the bilinear form u ⋅ v = η ( u , v ) , displaystyle ucdot v=eta (u,v), with which this section started by assuming its existence, is now
identified.
For definiteness and shorter presentation, the signature (− + + +)
is adopted below. This choice (or the other possible choice) has no
(known) physical implications. The symmetry group preserving the
bilinear form with one choice of signature is isomorphic (under the
map given here) with the symmetry group preserving the other choice of
signature. This means that both choices are in accord with the two
postulates of relativity. Switching between the two conventions is
straightforward. If the metric tensor 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[edit]
A standard basis for
− η ( e 0 , e 0 ) = η ( e 1 , e 1 ) = η ( e 2 , e 2 ) = η ( e 3 , e 3 ) = 1. displaystyle -eta (e_ 0 ,e_ 0 )=eta (e_ 1 ,e_ 1 )=eta (e_ 2 ,e_ 2 )=eta (e_ 3 ,e_ 3 )=1. These conditions can be written compactly in the form η ( e μ , e ν ) = η μ ν . displaystyle eta (e_ mu ,e_ nu )=eta _ mu nu . Relative to a standard basis, the components of a vector v are written
(v0, v1, v2, v3) where the
η ( v , w ) = η μ ν v μ w ν = v 0 w 0 + v 1 w 1 + v 2 w 2 + v 3 w 3 = v μ w μ = v μ w μ , displaystyle eta (v,w)=eta _ mu nu v^ mu w^ nu =v^ 0 w_ 0 +v^ 1 w_ 1 +v^ 2 w_ 2 +v^ 3 w_ 3 =v^ mu w_ mu =v_ mu w^ mu , and η ( v , v ) = η μ ν v μ v ν = v 0 v 0 + v 1 v 1 + v 2 v 2 + v 3 v 3 = v μ v μ . displaystyle eta (v,v)=eta _ mu nu v^ mu v^ nu =v^ 0 v_ 0 +v^ 1 v_ 1 +v^ 2 v_ 2 +v^ 3 v_ 3 =v^ mu v_ mu . Here lowering of an index with the metric was used.
Raising and lowering of indices[edit]
Main articles:
Linear functionals (1-forms) α, β and their sum σ and vectors u, v, w, in 3d Euclidean space. The number of (1-form) hyperplanes intersected by a vector equals the inner product.[13] 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 cotangent spaces of M. At a point in M,
the tangent and cotangent spaces are dual vector spaces (so the
dimension of the cotangent space at an event is also 4). Just as an
authentic inner product on a vector space with one argument fixed, by
Riesz representation theorem, may be expressed as the action of a
linear functional on the vector space, the same holds for the
Minkowski inner product of Minkowski space.[14]
Thus if vμ are the components of a vector in a tangent space, then
ημνvμ = 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.[15]
Contravariant and covariant vectors are geometrically very different
objects. The first can and should be thought of as arrows. A linear
functional can be characterized by two objects: its kernel, which is a
hyperplane passing through the origin, and its norm. Geometrically
thus covariant vectors should viewed as a set of hyperplanes, with
spacing depending on the norm (bigger = smaller spacing), with one of
them (the kernel) passing through the origin. The mathematical term
for a covariant vector is 1-covector or
A formal approach to the Minkowski metric A full-blown version of the Minkowski metric in coordinates as a tensor field on spacetime has the appearance η μ ν d x μ ⊗ d x ν = η μ ν d x μ ⊙ d x ν = η μ ν d x μ d x ν . displaystyle eta _ mu nu dx^ mu otimes dx^ nu =eta _ mu nu dx^ mu odot dx^ nu =eta _ mu nu dx^ mu dx^ nu . Explanation: The coordinate differentials are
∂ ∂ x μ
p , displaystyle left. frac partial partial x^ mu right_ p , where p is an event. This operator applied to a function f gives the directional derivative of f at p in the direction of increasing xμ with xν, ν ≠ μ fixed. They provide a basis for the tangent space at p. 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 d f ( X ) = X f , displaystyle df(X)=Xf, for each vector field X. A vector field is an assignment of a tangent vector to each point p. In coordinates X can be expanded at each point p in the basis given by the ∂/∂xνp. Applying this with f = xμ, the coordinate function itself, and X = ∂/∂xν, called a coordinate vector field, one obtains d x μ ( ∂ ∂ x ν ) = ∂ x μ ∂ x ν = δ ν μ . displaystyle dx^ mu left( frac partial partial x^ nu right)= frac partial x^ mu partial x^ nu =delta _ nu ^ mu . Since this relation holds at each point p, the dxμp provide a basis for the cotangent space at each p and the bases dxμp and ∂/∂xνp are dual to each other, d x μ
p ( ∂ ∂ x ν
p ) = δ ν μ . displaystyle left.dx^ mu right_ p left(left. frac partial partial x^ nu right_ p right)=delta _ nu ^ mu . at each p. Furthermore, one has α ⊗ β ( a , b ) = α ( a ) β ( b ) displaystyle alpha otimes beta (a,b)=alpha (a)beta (b) for general one-forms on a tangent space α, β and general tangent vectors a, 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 fields a, b, both expanded in terms of the basis coordinate vector fields, the result is η μ ν d x μ ⊗ d x ν ( a , b ) = η μ ν a μ b ν , displaystyle eta _ mu nu dx^ mu otimes dx^ nu (a,b)=eta _ mu nu a^ mu b^ nu , where aμ, bν are the component functions of the vector fields. The above equation holds at each point p, and the relation may as well be interpreted as the Minkowski metric at p applied to two tangent vectors at p. As mentioned, in a vector space, such as that modelling the spacetime of special relativity, tangent vectors can be canonically identified with vectors in the space itself, and vice versa. This means that the tangent spaces at each point are canonically identified with each other and with the vector space itself. This explains how the right hand side of the above equation can be employed directly, without regard to spacetime point the metric is to be evaluated and from where (which tangent space) the vectors come from. This situation changes in general relativity. There one has g ( p ) μ ν d x μ
p d x ν
p ( a , b ) = g ( p ) μ ν a μ b ν , displaystyle g(p)_ mu nu dx^ mu _ p dx^ nu _ p (a,b)=g(p)_ mu nu a^ mu b^ nu , where now η → g(p), i.e. g is still a metric tensor but now depending on spacetime and is a solution of Einstein's field equations. Moreover, a, b must be tangent vectors at spacetime point p and can no longer be moved around freely. Lorentz transformations and symmetry[edit] Standard configuration of coordinate systems for Lorentz transformations. The
[ U 0 ′ U 1 ′ U 2 ′ U 3 ′ ] = [ γ − β γ 0 0 − β γ γ 0 0 0 0 1 0 0 0 0 1 ] [ U 0 U 1 U 2 U 3 ] , displaystyle begin bmatrix U'_ 0 \U'_ 1 \U'_ 2 \U'_ 3 end bmatrix = begin bmatrix gamma &-beta gamma &0&0\-beta gamma &gamma &0&0\0&0&1&0\0&0&0&1\end bmatrix begin bmatrix U_ 0 \U_ 1 \U_ 2 \U_ 3 end bmatrix , where γ = 1 1 − v 2 c 2 displaystyle gamma = 1 over sqrt 1- v^ 2 over c^ 2 is the Lorentz factor, and β = v c . displaystyle beta = v over c ,. Other Lorentz transformations are pure rotations, and hence elements
of the SO(3) subgroup of O(3,1). A general homogeneous Lorentz
transformation is a product of a pure boost and a pure rotation. An
inhomogeneous
Subdivision of Minkowski spacetime with respect to an event in four disjoint sets. The light cone, the absolute future, the absolute past, and elsewhere. The terminology is from Sard (1970). The momentarily co-moving inertial frames along the trajectory ("world line") of a rapidly accelerating observer (center). The vertical direction indicates time, while the horizontal indicates distance, the dashed line is the spacetime of the observer. The small dots are specific events in spacetime. Note how the momentarily co-moving inertial frame changes when the observer accelerates. Main article: Causal structure
Where v is velocity, and x, y, and z are Cartesian coordinates in
3-dimensional space, and 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, but depends on the signature. The
classification of any vector will be the same in all frames of
reference that are related by a
future-directed timelike vectors whose first component is positive, (tip of vector located in absolute future in 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). Spacelike vectors are in elsewhere. The terminology stems from the
fact that spacelike separated events are connected by vectors
requiring faster-than-light travel, and so cannot possibly influence
each other. Together with spacelike and lightlike vectors there are 7
classes in all.
An orthonormal basis for
x chronologically precedes y if y − x is future-directed timelike. This relation has the transitive property and 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 ordering of space-time and so can be written x ≤ y. Suppose x ∈ M is timelike. Then the simultaneous hyperplane for x is y : η ( x , y ) = 0 . displaystyle y:eta (x,y)=0 . Since this hyperplane varies as x varies, there is a relativity of simultaneity in Minkowski space. Reversed triangle inequality[edit] If v and w are both future-directed timelike four-vectors, then in the (+ − − −) sign convention for norm, ‖ v + w ‖ ≥ ‖ v ‖ + ‖ w ‖ . displaystyle leftv+wrightgeq leftvright+leftwright. Generalizations[edit]
Main articles:
It has been suggested that this section be split out into another article. (Discuss) (April 2017) The meaning of the term geometry in the context of Minkowski space
depends heavily on what is meant by the term.
H R 1 ( n ) = ( c t , x 1 , … , x n ) ∈ M n : c 2 t 2 − ( x 1 ) 2 ⋯ ( x n ) 2 = R 2 , c t > 0 displaystyle mathbf H _ R ^ 1(n) = (ct,x^ 1 ,ldots ,x^ n )in mathbf M ^ n :c^ 2 t^ 2 -(x^ 1 )^ 2 cdots (x^ n )^ 2 =R^ 2 ,ct>0
in generalized
h R 1 ( n ) = ι ∗ η , displaystyle h_ R ^ 1(n) =iota ^ * eta , the pullback of the Minkowski metric η under inclusion, is a Riemannian metric. With this metric H1(n) 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.[19] 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 of dimension n. Preliminaries[edit] 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: For inclusion maps from a submanifold S into M and a covariant tensor α of order k on M it holds that ι ∗ α ( X 1 , X 2 , … , X k ) = α ( ι ∗ X 1 , ι ∗ X 2 , … , ι ∗ X k ) = α ( X 1 , X 2 , … , X k ) , displaystyle iota ^ * alpha (X_ 1 ,X_ 2 ,ldots ,X_ k )=alpha (iota _ * X_ 1 ,iota _ * X_ 2 ,ldots ,iota _ * X_ k )=alpha (X_ 1 ,X_ 2 ,ldots ,X_ k ), 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 ι ∗ α = α
S , displaystyle iota ^ * alpha =alpha _ S , 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: The pullback of a covariant k-tensor α (one taking only contravariant vectors as arguments) under a map F:M → N is a linear map F ∗ : T F ( p ) k N → T p k M , displaystyle F^ * :T_ F(p) ^ k Nrightarrow T_ p ^ k M, where for any vector space V, T k V = V ∗ ⊗ V ∗ ⋯ ⊗ V ∗ ⏟ k times . displaystyle T^ k V=underbrace V^ * otimes V^ * cdots otimes V^ * _ k text times . It is defined by F ∗ ( α ) ( X 1 , X 2 , … , X k ) = α ( F ∗ X 1 , F ∗ X 2 , … , F ∗ X k ) , displaystyle F^ * (alpha )(X_ 1 ,X_ 2 ,ldots ,X_ k )=alpha (F_ * X_ 1 ,F_ * X_ 2 ,ldots ,F_ * X_ k ), 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: Heuristically, pulling back a tensor to p ∈ M from F(p) ∈ N feeding it vectors residing at p ∈ M is by definition the same as pushing forward the vectors from p ∈ M to F(p) ∈ N feeding them to the tensor residing at F(p) ∈ N. 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 F ∗ ( X ) f = X ( f ∘ F ) , displaystyle F_ * (X)f=X(fcirc F), where f is a function on N. When M = ℝm, N= ℝn the pushforward of
F reduces to DF:ℝm → ℝn, the ordinary differential, which is
given by the
F ∗ : T F ( p ) ∗ N → T p ∗ M . displaystyle F^ * :T_ F(p) ^ * Nrightarrow T_ p ^ * M. Hyperbolic stereographic projection[edit] Red circular arc is geodesic in Poincaré disk model; it projects to the brown geodesic on the green hyperboloid. 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 left for n = 2. It is instructive to
compare to stereographic projection for spheres.
σ ( τ , x ) = u = R x R + τ , σ − 1 ( u ) = ( τ , x ) = ( R R 2 +
u
2 R 2 −
u
2 , 2 R 2 u R 2 −
u
2 ) , displaystyle begin aligned sigma (tau ,mathbf x )&=mathbf u = frac Rmathbf x R+tau ,\sigma ^ -1 (mathbf u )=(tau ,mathbf x )&=left(R frac R^ 2 +u^ 2 R^ 2 -u^ 2 , frac 2R^ 2 mathbf u R^ 2 -u^ 2 right),end aligned where, for simplicity, τ ≡ ct. The (τ, x) are coordinates on Mn+1 and the u are coordinates on ℝn. Detailed derivation Let H R n = ( τ , x 1 , … , x n ) ⊂ M : − τ 2 + ( x 1 ) 2 + ⋯ + ( x n ) 2 = − R 2 , τ > 0 displaystyle mathbf H _ R ^ n = (tau ,x^ 1 ,ldots ,x^ n )subset mathbf M :-tau ^ 2 +(x^ 1 )^ 2 +cdots +(x^ n )^ 2 =-R^ 2 ,tau >0 and let S = ( − 1 , 0 , … , 0 ) displaystyle S=(-1,0,ldots ,0) If P = ( τ , x 1 , … , x n ) ∈ H R n , displaystyle P=(tau ,x^ 1 ,ldots ,x^ n )in mathbf H _ R ^ n , then it is geometrically clear that the vector P S → displaystyle overrightarrow PS intersects the hyperplane ( τ , x 1 , … x n ) ∈ M : τ = 0 displaystyle (tau ,x^ 1 ,ldots x^ n )in M:tau =0 once in point denoted U = ( 0 , u 1 ( P ) , … , u n ( P ) ) ≡ ( 0 , u ) . displaystyle U=(0,u^ 1 (P),ldots ,u^ n (P))equiv (0,mathbf u ). One has S + S U → = U ⇒ S U → = U − S , S + S P → = P ⇒ S P → = U − P . displaystyle begin aligned S+ overrightarrow SU &=URightarrow overrightarrow SU =U-S,\S+ overrightarrow SP &=PRightarrow overrightarrow SP =U-Pend aligned . or S U → = ( 0 , u ) − ( − R , 0 ) = ( R , u ) , S P → = ( τ , x ) − ( − R , 0 ) = ( τ + R , x ) . displaystyle begin aligned overrightarrow SU &=(0,mathbf u )-(-R,0)=(R,mathbf u ),\ overrightarrow SP &=(tau ,mathbf x )-(-R,0)=(tau +R,mathbf x ).end aligned By construction of stereographic projection one has S U → = λ ( τ ) S P → . displaystyle overrightarrow SU =lambda (tau ) overrightarrow SP . This leads to the system of equations R = λ ( τ + R ) , u = λ x . displaystyle begin aligned R&=lambda (tau +R),\mathbf u &=lambda mathbf x .end aligned The first of these is solved for λ displaystyle lambda and one obtains for stereographic projection σ ( τ , x ) = u = R x R + τ . displaystyle sigma (tau ,mathbf x )=mathbf u = frac Rmathbf x R+tau . Next, the inverse σ − 1 ( u ) = ( τ , x ) displaystyle sigma ^ -1 (u)=(tau ,mathbf x ) must be calculated. Use the same considerations as before, but now with U = ( 0 , u ) P = ( τ ( u ) , ξ ( u ) ) . displaystyle begin aligned U&=(0,mathbf u )\P&=(tau (mathbf u ),xi (mathbf u )).end aligned One gets τ = R ( 1 − λ ) λ , x = u λ , displaystyle begin aligned tau &= frac R(1-lambda ) lambda ,\mathbf x &= frac mathbf u lambda ,end aligned but now with λ displaystyle lambda depending on u . displaystyle mathbf u . The condition for P lying in the hyperboloid is − τ 2 + u ⋅ u = − τ 2 +
u
2 = − R 2 , displaystyle -tau ^ 2 +mathbf u cdot mathbf u =-tau ^ 2 +u^ 2 =-R^ 2 , or − R 2 ( 1 − λ ) 2 λ 2 =
u
2 λ 2 , displaystyle - frac R^ 2 (1-lambda )^ 2 lambda ^ 2 = frac u^ 2 lambda ^ 2 , leading to λ = R 2 −
u
2 2 R 2 . displaystyle lambda = frac R^ 2 -u^ 2 2R^ 2 . With this λ displaystyle lambda , one obtains σ − 1 ( u ) = ( τ , x ) = ( R R 2 +
u
2 R 2 −
u
2 , 2 R 2 u R 2 −
u
2 . ) displaystyle sigma ^ -1 (mathbf u )=(tau ,mathbf x )=left(R frac R^ 2 +u^ 2 R^ 2 -u^ 2 , frac 2R^ 2 mathbf u R^ 2 -u^ 2 .right) Pulling back the metric[edit] One has h R 1 ( n ) = η
H R 1 ( n ) = ( d x 1 ) 2 + ⋯ + ( d x n ) 2 − d τ 2 displaystyle h_ R ^ 1(n) =eta _ mathbf H _ R ^ 1(n) =(dx^ 1 )^ 2 +cdots +(dx^ n )^ 2 -dtau ^ 2 and the map σ − 1 : R n → H R 1 ( n ) ; σ − 1 ( u ) = ( τ ( u ) , x ( u ) ) = ( R R 2 +
u
2 R 2 −
u
2 , 2 R 2 u R 2 −
u
2 ) . displaystyle sigma ^ -1 :mathbb R ^ n rightarrow mathbf H _ R ^ 1(n) ;sigma ^ -1 (mathbf u )=(tau (mathbf u ),mathbf x (mathbf u ))=left(R frac R^ 2 +u^ 2 R^ 2 -u^ 2 , frac 2R^ 2 mathbf u R^ 2 -u^ 2 right). The pulled back metric can be obtained by straightforward methods of calculus; ( σ − 1 ) ∗ η
H R 1 ( n ) = ( d x 1 ( u ) ) 2 + ⋯ + ( d x n ( u ) ) 2 − ( d τ ( u ) ) 2 . displaystyle left.(sigma ^ -1 )^ * eta right_ mathbf H _ R ^ 1(n) =(dx^ 1 (mathbf u ))^ 2 +cdots +(dx^ n (mathbf u ))^ 2 -(dtau (mathbf u ))^ 2 . One computes according to the standard rules for computing differentials (though one is really computing the rigorously defined exterior derivatives), d x 1 ( u ) = d 2 R 2 u 1 R 2 −
u
2 = ∂ ∂ u 1 2 R 2 u 1 R 2 −
u
2 d u 1 + ⋯ + ∂ ∂ u n 2 R 2 u 1 R 2 −
u
2 d u n + ∂ ∂ τ 2 R 2 u 1 R 2 −
u
2 d τ , ⋯ d x n ( u ) = d 2 R 2 u n R 2 −
u
2 = ⋯ , d τ ( u ) = d ( R R 2 +
u
2 R 2 −
u
2 ) = ⋯ , displaystyle begin aligned dx^ 1 (mathbf u )&=d frac 2R^ 2 u^ 1 R^ 2 -u^ 2 = frac partial partial u^ 1 frac 2R^ 2 u^ 1 R^ 2 -u^ 2 du^ 1 +cdots + frac partial partial u^ n frac 2R^ 2 u^ 1 R^ 2 -u^ 2 du^ n + frac partial partial tau frac 2R^ 2 u^ 1 R^ 2 -u^ 2 dtau ,\&cdots \dx^ n (mathbf u )&=d frac 2R^ 2 u^ n R^ 2 -u^ 2 =cdots ,\dtau (mathbf u )&=dleft(R frac R^ 2 +u^ 2 R^ 2 -u^ 2 right)=cdots ,end aligned and substitutes the results into the right hand side. This yields ( σ − 1 ) ∗ h R 1 ( n ) = 4 R 2 [ ( d u 1 ) 2 + ⋯ + ( d u n ) 2 ] ( R 2 −
u
2 ) 2 ≡ h R 2 ( n ) . displaystyle (sigma ^ -1 )^ * h_ R ^ 1(n) = frac 4R^ 2 left[(du^ 1 )^ 2 +cdots +(du^ n )^ 2 right] (R^ 2 -u^ 2 )^ 2 equiv h_ R ^ 2(n) . Detailed outline of computation One has ∂ ∂ u 1 2 R 2 u 1 R 2 −
u
2 d u 1 = 2 ( R 2 −
u
2 ) + 4 R 2 ( u 1 ) 2 ( R 2 −
u
2 ) 2 d u 1 , displaystyle frac partial partial u^ 1 frac 2R^ 2 u^ 1 R^ 2 -u^ 2 du^ 1 = frac 2(R^ 2 -u^ 2 )+4R^ 2 (u^ 1 )^ 2 (R^ 2 -u^ 2 )^ 2 du^ 1 , ∂ ∂ u 2 2 R 2 u 1 R 2 −
u
2 d u 2 = 4 R 2 u 1 u 2 d u 2 ( R 2 −
u
2 ) 2 d u 2 , displaystyle frac partial partial u^ 2 frac 2R^ 2 u^ 1 R^ 2 -u^ 2 du^ 2 = frac 4R^ 2 u^ 1 u^ 2 du^ 2 (R^ 2 -u^ 2 )^ 2 du^ 2 , and ∂ ∂ τ 2 R 2 u 1 R 2 −
u
2 d τ 2 = 0. displaystyle frac partial partial tau frac 2R^ 2 u^ 1 R^ 2 -u^ 2 dtau ^ 2 =0. With this one may write d x 1 ( u ) = 2 R 2 ( R 2 −
u
2 ) d u 1 + 4 R 2 u 1 ( u ⋅ d u ) ( R 2 −
u
2 ) 2 , displaystyle dx^ 1 (mathbf u )= frac 2R^ 2 (R^ 2 -u^ 2 )du^ 1 +4R^ 2 u^ 1 (mathbf u cdot dmathbf u ) (R^ 2 -u^ 2 )^ 2 , from which ( d x 1 ( u ) ) 2 = 4 R 2 ( r 2 −
u
2 ) 2 ( d u 1 ) 2 + 16 R 4 ( R 2 −
u
2 ) ( u ⋅ d u ) u 1 d u 1 + 14 R r ( u 1 ) 2 ( u ⋅ d u ) 2 ( R 2 −
u
2 ) 4 . displaystyle (dx^ 1 (mathbf u ))^ 2 = frac 4R^ 2 (r^ 2 -u^ 2 )^ 2 (du^ 1 )^ 2 +16R^ 4 (R^ 2 -u^ 2 )(mathbf u cdot dmathbf u )u^ 1 du^ 1 +14R^ r (u^ 1 )^ 2 (mathbf u cdot dmathbf u )^ 2 (R^ 2 -u^ 2 )^ 4 . Summing this formula one obtains ( d x 1 ( u ) ) 2 + ⋯ + d x n ( u ) ) 2 = 4 R 2 ( R 2 −
u
2 ) 2 [ ( d u 1 ) 2 + ⋯ + ( d u n ) 2 ] + 16 R 4 ( R 2 −
u
2 ) ( u ⋅ d u ) ( u ⋅ d u ) + 16 R 4
u
2 ( u ⋅ d u ) 2 ( R 2 −
u
2 ) 4 = 4 R 2 ( R 2 −
u
2 ) 2 ) [ ( d u 1 ) 2 + ⋯ + ( d u n ) 2 ] ( R 2 −
u
2 ) 4 + R 2 16 R 4 ( u ⋅ d u ) ( R 2 −
u
2 ) 4 . displaystyle begin aligned (dx^ 1 (mathbf u ))^ 2 +cdots +dx^ n (mathbf u ))^ 2 &= frac 4R^ 2 (R^ 2 -u^ 2 )^ 2 left[(du^ 1 )^ 2 +cdots +(du^ n )^ 2 right]+16R^ 4 (R^ 2 -u^ 2 )(mathbf u cdot dmathbf u )(mathbf u cdot dmathbf u )+16R^ 4 u^ 2 (mathbf u cdot dmathbf u )^ 2 (R^ 2 -u^ 2 )^ 4 \&= frac 4R^ 2 (R^ 2 -u^ 2 )^ 2 )left[(du^ 1 )^ 2 +cdots +(du^ n )^ 2 right] (R^ 2 -u^ 2 )^ 4 +R^ 2 frac 16R^ 4 (mathbf u cdot dmathbf u ) (R^ 2 -u^ 2 )^ 4 .end aligned Similarly, for τ one gets d τ = ∑ i = 1 n ∂ ∂ u i R R 2 +
u
2 R 2 +
u
2 d u i + ∂ ∂ τ R R 2 +
u
2 R 2 +
u
2 d τ = ∑ i = 1 n R 4 4 R 2 u i d u i ( R 2 −
u
2 ) , displaystyle dtau =sum _ i=1 ^ n frac partial partial u^ i R frac R^ 2 +u^ 2 R^ 2 +u^ 2 du^ i + frac partial partial tau R frac R^ 2 +u^ 2 R^ 2 +u^ 2 dtau =sum _ i=1 ^ n R^ 4 frac 4R^ 2 u^ i du^ i (R^ 2 -u^ 2 ) , yielding − d t 2 = − ( R 4 R 4 ( u ⋅ d u ) ( R 2 −
u
2 ) 2 ) 2 = − R 2 14 R 4 ( u ⋅ d u ) 2 ( R 2 −
u
2 ) 4 . displaystyle -dt^ 2 =-left(R frac 4R^ 4 (mathbf u cdot dmathbf u ) (R^ 2 -u^ 2 )^ 2 right)^ 2 =-R^ 2 frac 14R^ 4 (mathbf u cdot dmathbf u )^ 2 (R^ 2 -u^ 2 )^ 4 . Now add this contribution to finally get ( σ − 1 ) ∗ h R 1 ( n ) = 4 R 2 [ ( d u 1 ) 2 + ⋯ + ( d u n ) 2 ] ( R 2 −
u
2 ) 2 ≡ h R 2 ( n ) . displaystyle (sigma ^ -1 )^ * h_ R ^ 1(n) = frac 4R^ 2 left[(du^ 1 )^ 2 +cdots +(du^ n )^ 2 right] (R^ 2 -u^ 2 )^ 2 equiv h_ R ^ 2(n) . This last equation shows that the metric on the ball is identical to
the
Alternative calculation using the pushforward The pullback can be computed in a different fashion. By definition, ( σ − 1 ) ∗ h R 1 ( n ) ( V , V ) = h R 1 ( n ) ( ( σ − 1 ) ∗ V , ( σ − 1 ) ∗ V ) ) = η
H R 1 ( n ) ( ( σ − 1 ) ∗ V , ( σ − 1 ) ∗ V ) ) . displaystyle (sigma ^ -1 )^ * h_ R ^ 1(n) (V,V)=h_ R ^ 1(n) ((sigma ^ -1 )_ * V,(sigma ^ -1 )_ * V))=eta _ mathbf H _ R ^ 1(n) ((sigma ^ -1 )_ * V,(sigma ^ -1 )_ * V)). In coordinates, ( σ − 1 ) ∗ V = ( σ − 1 ) ∗ V i ∂ ∂ u i = V i ∂ x j ∂ u i ∂ ∂ x j + V i ∂ τ ∂ u i ∂ ∂ τ = V i ∂ x j ∂ u i ∂ ∂ x j + V i ∂ τ ∂ u i ∂ ∂ τ = V x j ∂ ∂ x j + V τ ∂ ∂ τ . displaystyle (sigma ^ -1 )_ * V=(sigma ^ -1 )_ * V^ i frac partial partial u^ i =V^ i frac partial x^ j partial u^ i frac partial partial x^ j +V^ i frac partial tau partial u^ i frac partial partial tau =V^ i frac partial x ^ j partial u^ i frac partial partial x^ j +V^ i frac partial tau partial u^ i frac partial partial tau =Vx^ j frac partial partial x^ j +Vtau frac partial partial tau . One has from the formula for σ–1 V x j = V i ∂ ∂ u i ( 2 R 2 u j R 2 −
u
2 ) = 2 R 2 V j R 2 −
u
2 − 4 R 2 u j ⟨ V , u ⟩ ( R 2 −
u
2 ) 2 , ( here V
u
2 = 2 Σ k = 1 n V k u k ≡ 2 ⟨ V , u ⟩ ) V τ = V ( R R 2 +
u
2 R 2 −
u
2 ) = 4 R 3 ⟨ V , u ⟩ ( R 2 −
u
2 ) 2 . displaystyle begin aligned Vx^ j &=V^ i frac partial partial u^ i left( frac 2R^ 2 u^ j R^ 2 -u^ 2 right)= frac 2R^ 2 V^ j R^ 2 -u^ 2 - frac 4R^ 2 u^ j langle mathbf V ,mathbf u rangle left(R^ 2 -u^ 2 right)^ 2 ,qquad left( text here Vu^ 2 =2Sigma _ k=1 ^ n V^ k u^ k equiv 2langle mathbf V ,mathbf u rangle right)\Vtau &=Vleft(R frac R^ 2 +u^ 2 R^ 2 -u^ 2 right)= frac 4R^ 3 langle mathbf V ,mathbf u rangle left(R^ 2 -u^ 2 right)^ 2 .end aligned Lastly, η ( σ ∗ − 1 ) V , σ ∗ − 1 ) V ) = Σ j = 1 n ( V x j ) 2 − ( V τ ) 2 = 4 R 4
V
2 ( R 2 −
u
2 ) 2 = h R 2 ( n ) ( V , V ) , displaystyle eta (sigma _ * ^ -1 )V,sigma _ * ^ -1 )V)=Sigma _ j=1 ^ n (Vx^ j )^ 2 -(Vtau )^ 2 = frac 4R^ 4 V^ 2 left(R^ 2 -u^ 2 right)^ 2 =h_ R ^ 2(n) (V,V), and the same conclusion is reached. See also[edit] Introduction to mathematics of general relativity Minkowski plane Remarks[edit] ^ 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.
^ The surface x2 + y2 + z2 + t2 = R2 > 0 is a
Notes[edit] ^ Landau & Lifshitz 2002, p. 5
^ Lee 1997, p. 31
^ Schutz, John W. (1977). Independent Axioms for Minkowski Space-Time
(illustrated ed.). CRC Press. pp. 184–185.
ISBN 978-0-582-31760-4. Extract of page 184
^ Poincaré 1905–1906, pp. 129–176 Wikisource translation: On
the Dynamics of the Electron
^ Minkowski 1907–1908, pp. 53–111 *Wikisource translation:
The Fundamental Equations for Electromagnetic Processes in Moving
Bodies.
^ a b Minkowski 1907–1909, pp. 75–88 Various English
translations on Wikisource: "Space and Time."
^
References[edit] Corry, L. (1997). "
External links[edit] Media related to Minkowski diagrams at Wikimedia Commons Animation clip on
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