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Lorentz Transformations — Examples
1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Banach-Space Differential Calculus and Banach Manifolds
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Countability Axioms and Cardinal Functions
- Determinants of Matrices over a Commutative Ring
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Exterior Powers, Orientation and Hodge Duality
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Finite Dimensional Normed Spaces and Riesz Lemma
- Foundations of the Real Numbers for Analysis
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Hereditary and Productive Behaviour of the Separation Axioms
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Lorentz Transformations
- Manifolds with Boundary Collars and Orientations
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Metric Spaces
- Mixed Partials, Taylor Formulae, and Extrema
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Normed and Banach Spaces
- Order, Zorn's Lemma, and the Axiom of Choice
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Simple Field Extensions and the Construction of the Complex Numbers
- Smooth Manifolds and Smooth Maps
- Spacetime Primitives
- Subspaces, Products, and Quotients
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- Tangent Cotangent and the Differential
- Tensor Fields Exterior Algebra and Differential Forms
- Tensor Products of Modules
- The Derivative and the Mean Value Theorems
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Exponential Function
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Summary
These examples and counterexamples evaluate the Lorentz transformations of sr-lorentz-transformations on explicit numbers. The boost at is checked for its metric and determinant conditions, its inverse, the images of the events and and the preservation of the null lines (Worked 1+1 boost matrix: metric, determinant, inverse and interval invariance); two collinear boosts compose to the parameter with rapidities adding as (Worked collinear composition and rapidity addition at beta = 3/5); and the motion of a primed chart's origin under a passive boost is computed in Worked motion of the primed frame's origin under a passive boost.
The noncommutativity of noncollinear boosts is witnessed by two exact rational matrices at that differ in listed entries (Counterexample: noncollinear boosts do not commute), and the composition is factored as : the rotation acts first, followed by the boost. The rotation has cosine and sine in the clockwise coordinate convention of the worked example (Worked boost-rotation factorization of a composition of two perpendicular boosts). The discrete parity and time-reversal matrices show that Lorentz maps need be neither proper nor orthochronous and that alone does not force orthochrony (The parity and time-reversal maps are Lorentz maps that are not proper, and det = 1 does not imply orthochrony), while a spatial rotation is a proper orthochronous Lorentz map that is not a boost (Counterexample: a spatial rotation is a proper orthochronous Lorentz map that is not a boost). Finally the component laws are computed for a covector and a vector under the boost with invariant contraction (Worked covector and vector component transformation under a beta = 3/5 boost), and a constant vector field , with future timelike and unit and an arbitrary real constant, has components in the rest chart of , exhibiting the difference between the scalar multiplier and the coordinate density (Worked current boost law and coordinate versus rest density).
3 · Logical flowchart
4 · Definitions, postulates, theorems and proofs
None yet.
5 · Examples, experiments and counterexamples
Worked 1+1 boost matrix: metric, determinant, inverse and interval invariance
Example
Worked computation in an inertial frame whose spatial direction is , with coordinates and . Take , so and the passive boost of The passive Lorentz boost with parameter beta is the linear map with matrix on the plane (the identity on the other spatial directions of the frame). Then:
(i) : the entry is times , the spatial diagonal entry is , and the off-diagonal entry , giving ;
(ii) , so is proper;
(iii) has coordinates and , so the future timelike vector is mapped to a future timelike vector;
(iv) , so the inverse is the boost with the opposite parameter;
(v) the events with coordinates and have images and : and , and the interval computed in the two frames is the same: in the convention ;
(vi) the null lines are preserved: and are null, and , so the images are exactly the same pair of lines.
Facts & Assumptions
Given: An inertial frame , the parameter , and the boost .
For the passive boost in the case acts on the coordinates of the plane by , , with and the rational power of the positive base ; the matrix in the ordered basis is , and the map fixes every vector with , namely every vector in (The passive Lorentz boost with parameter beta).
is a proper orthochronous Lorentz map with , and ; the map is injective (The passive boost is a proper orthochronous Lorentz map with inverse B(-beta)).
In an inertial frame and ; one has in the plane coordinates (Inertial frames and inertial charts of the affine Minkowski model).
is one of the two components of the timelike set and is the set of future timelike vectors; a Lorentz map is orthochronous exactly when it maps onto (Time orientation, future and past timelike/null/causal vectors, and the closed cones, The passive boost is a proper orthochronous Lorentz map with inverse B(-beta)).
A nonzero vector is null when and timelike when (Timelike, null, spacelike and causal vectors of a Lorentz vector space).
Proof
Evaluate the matrix : , so , and acts as the identity on the coordinate directions because has vanishing components in [L1]. For the product , the four entries are in the slot, and in the two off-diagonal slots, and in the slot; hence .
The determinant is , so is proper in the sense of the properness condition of [L2].
By [L2], , with coordinates ; [L3] gives , so is timelike by [L5]. Since is orthochronous by [L2] and by [L3], [L4] gives , so is future timelike: the future timelike vector is mapped to a future timelike vector.
The product of clause (iv) is , and the first factor is the matrix of by [L1]; this is the instance at of from [L2], so the inverse is the boost with the opposite parameter.
For the two null directions, [L1] gives and ; both images satisfy on their squares: and , so they are null by [L5]. Since and , the image of the line is the same line and the image of the line is the same line; hence the pair of null lines is mapped onto itself, and the images are exactly that pair.
For the event with coordinates , [L1] gives and , while is fixed because the boost is linear. The interval in is and ; this numerical equality is the instance of from step 1.1 applied to the coordinate difference .
Steps 1.1-1.4 verify clauses (i)-(iv), step 2.1 verifies clause (v), and step 1.5 verifies clause (vi); every clause of the Example is a checked finite computation with the displayed rational entries.
Worked collinear composition and rapidity addition at beta = 3/5
Example
Worked computation. Let be a unit spatial direction of an inertial frame and let be the passive boost with parameter (The passive Lorentz boost with parameter beta), so and, in the ordered pair of directions , the boost has the matrix and is the identity on , the -orthogonal complement of (Rapidity parametrization of collinear boosts). Then:
(i) , because and . This is the velocity-addition law , with .
(ii) The rapidities are and , so the rapidities add under composition: .
(iii) The two boosts commute: the reversed product equals the square computed in (i), and this is the instance of the commutativity of collinear boosts.
Facts & Assumptions
Given: A unit spatial direction of an inertial frame, the passive boost , and the collinear family , .
For the boost acts in the frame coordinates by , on the plane, with and the coefficient of the unit direction; the matrix on the ordered pair is and the map is the identity on (The passive Lorentz boost with parameter beta).
For a unit spatial direction and , the boost has on the matrix ; in particular, with , and (Rapidity parametrization of collinear boosts).
For unit and one has with and , and collinear boosts commute (Collinear boost composition and the relativistic velocity-addition law).
For the rapidity is , a strictly increasing bijection of onto with ; for a fixed unit direction, rapidities add: (Rapidity parametrization of collinear boosts, The natural logarithm as the inverse of the exponential function).
For one has and (Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm); the exponential satisfies and the hyperbolic functions are , (The real exponential function and the number by a power series).
The Euclidean norm of a vector of satisfies for a unit direction (The Euclidean inner product on ).
Proof
With and by [L6], [L1] gives , so the matrix of on the ordered pair is , and the map is the identity on the -orthogonal complement of . This is the matrix form recorded in [L2], with the rapidity fixed by ; the numerical value of that rapidity is computed below.
By the commutativity clause of [L3], , i.e. the reversed order gives back the same product; clause (iii) is this instance, and no separate computation is needed.
Squaring the matrix of step 1.1 in the block, on the diagonal and off the diagonal, while the identity block on the complement squares to itself; hence has matrix on and is the identity on the complement. By [L3], with , and . For this , and ; by [L1] the matrix of is therefore on , matching the square. This proves clause (i) and the velocity-addition law.
By [L4] and [L5], , and by the product law, so ; similarly and , so . With this gives : the rapidity of the composed boost in step 2.1 is the sum of the two equal rapidities, in agreement with the addition statement of [L4].
Steps 1.1-2.1 prove clauses (i) and (iii) and step 3.1 proves clause (ii); every entry is an exact rational or logarithmic identity justified by the displayed suppliers.
Worked boost-rotation factorization of a composition of two perpendicular boosts
Example
Worked computation in a fixed inertial frame with coordinates and . Take the two passive boosts and (The passive Lorentz boost with parameter beta), each acting as the identity on the orthogonal complement of its own coordinate plane, and let be their composition (matrix product in the frame's ordered basis). Then:
(i) , and direct multiplication gives and ; since and are proper orthochronous Lorentz maps, is a proper orthochronous Lorentz map.
(ii) has coordinates , with and , so is a future timelike unit vector. The parameter of Unique boost-rotation factorization of proper orthochronous Lorentz maps is , with and , consistent with .
(iii) with , whose extension is a spatial rotation of the frame: and (using ). Hence is the unique factorization of Unique boost-rotation factorization of proper orthochronous Lorentz maps; since , is not a passive boost.
(iv) The map fixes and acts on the - plane as the rotation with cosine parameter and sine parameter in the clockwise coordinate convention , (Carroll, equation (1.17)); this rotation is nontrivial because . The two boosts therefore compose to a nonzero rotation followed by a boost of the - plane; this composition of the explicit pair has a rotation factor , called its Thomas-Wigner rotation. No gyroscope, precession or physical observation claim is attached here.
Facts & Assumptions
Given: A fixed inertial frame with coordinates , the boosts , and their composition .
For the passive boost has, in the frame's ordered basis, the matrix with ; for one defines , whose matrix is (The passive Lorentz boost with parameter beta).
For every the boost is a proper orthochronous Lorentz map with ; ; and if is a future timelike unit vector with coordinates , , then (The passive boost is a proper orthochronous Lorentz map with inverse B(-beta)).
For a proper orthochronous Lorentz map with , coordinates , the parameter satisfies ; the map satisfies , and is a determinant-one -isometry and is a spatial rotation of the frame, so that ; the factorization with a spatial rotation and a parameter in the open unit ball is unique; and is a passive boost if and only if is the identity (Unique boost-rotation factorization of proper orthochronous Lorentz maps).
A spatial rotation of the frame is an extension of a linear map with , , on and ; via the -orthonormal basis these are exactly the maps whose spatial matrix satisfies and (Spatial rotations of an inertial frame).
In the frame, , , and the ordered basis is positively oriented; (Inertial frames and inertial charts of the affine Minkowski model).
The Euclidean norm on satisfies (The Euclidean inner product on ).
Composition of proper orthochronous Lorentz maps is a proper orthochronous Lorentz map, and every Lorentz map satisfies and (Lorentz maps are isomorphisms of determinant ±1 that preserve causal character and permute the cones; the Lorentz sets are groups).
Proof
By [L1] with and , one has and ; with one has . Multiplying, ; row by row: the entries are , , and ; the entries are , , and ; the entries are , , and ; the entries are . This proves the displayed matrix of clause (i).
To verify , the matrix is symmetric, and its upper triangle is computed from the columns of : the entry is ; the entry is ; the entry is ; the entry is ; the entry is ; the entry is ; the entry is ; the remaining upper-triangle entries involving the fourth row or column are because the fourth row and column of are those of the identity. Hence . Expanding along the fourth row gives .
and are proper orthochronous Lorentz maps by [L2], so their composition is a proper orthochronous Lorentz map by [L7]; this, with the direct metric and determinant checks of step 1.2, is clause (i). The coordinates of are the entries of the first column of , namely .
With as in step 2.1, [L5] gives , and ; thus is timelike, and because is orthochronous by step 2.1 and by [L5], lies in , so is a future timelike unit vector. Its parameter as in [L3] is ; by [L6], , and .
By [L1] with : , the coefficient is , and the matrix of is , because , , , and .
Multiply using steps 1.1 and 4.1. The first column is by [L2] applied to the future timelike unit vector of step 3.1, so it is . The first row entries are : for , ; for , ; for , ; and for , . The fourth row and column are those of the identity because and fix and preserve the splitting. The remaining block is for : , , , and . Hence with as displayed.
For the displayed one has : the and entries are and the entry is , with the third row and column those of the identity. Moreover (using ). Since acts on in the -orthonormal basis and satisfies , , the extension is a spatial rotation of the frame by [L4].
By [L3] applied to the proper orthochronous Lorentz map with and from step 3.1, the factorization with the extension of a spatial rotation is the unique such factorization, and by step 5.1 the map equals ; hence , since . Since (the entry ), [L3] clause (v) gives that is not a passive boost.
Finally, fixes and multiplies the coordinate pair by the matrix , whose entries have squares summing to ; it is therefore in the clockwise coordinate convention defined in (iv), and it is nontrivial because . This is clause (iv): is this nonzero rotation followed by the boost (the rightmost factor acts first) of the - plane.
Steps 1.1, 1.2 and 2.1 prove clause (i), steps 3.1, 4.1 and 5.1 prove clause (ii), steps 5.1, 6.1 and 6.2 prove clause (iii), and step 7.1 proves clause (iv); every assertion is a finite rational computation from the displayed matrices.
Counterexample: noncollinear boosts do not commute
Statement refuted
The statement "for all with the passive boosts satisfy " is false.
Facts & Assumptions
Given: An inertial frame with ordered basis and , and the boost parameters and .
For the passive boost has matrix entries , and in the frame's basis, with ; for one defines , whose matrix is (The passive Lorentz boost with parameter beta).
For boosts in a common unit direction one has and the two collinear boosts commute (Collinear boost composition and the relativistic velocity-addition law).
Every is a proper orthochronous Lorentz map, and products of Lorentz maps are Lorentz maps; the proper orthochronous Lorentz maps form a group (The passive boost is a proper orthochronous Lorentz map with inverse B(-beta), Lorentz maps are isomorphisms of determinant ±1 that preserve causal character and permute the cones; the Lorentz sets are groups).
The decomposition of a proper orthochronous Lorentz map into a boost and a spatial rotation, with for , is unique, and its rotation factor is the identity if and only if is a passive boost (Unique boost-rotation factorization of proper orthochronous Lorentz maps).
Coordinates of the frame are read in the ordered basis (Inertial frames and inertial charts of the affine Minkowski model).
Counterexample
For and one has and , while ; substituting into [L1] gives and : the contributions place the entries on the diagonal blocks, giving on each boosted spatial diagonal and leaving the transverse directions at .
By matrix multiplication, : the first row is times the first row of plus times its second row, giving , and the remaining rows are computed the same way from the displayed matrices. Reversing the order gives .
The two products differ in the entry ( versus ), the entry ( versus ) and the entry ( versus ); a single differing entry already refutes the asserted commutation, so and the group of proper orthochronous Lorentz maps is not abelian. This is consistent with [L2], which asserts commutation only for boosts in a common direction, and the counterexample makes no claim that the two products fail to be Lorentz maps: both are proper orthochronous Lorentz maps as products of such maps by [L3].
The result is consistent with [L2], which asserts commutation only for boosts along one common direction, and with [L4], which factors any proper orthochronous Lorentz map as a boost times a spatial rotation: the product is a proper orthochronous Lorentz map by [L3], so [L4] represents it as such a product, and no contradiction with the collinear law arises. The counterexample itself proves only the noncommutation asserted in the refuted statement, namely that the group of proper orthochronous Lorentz maps is not abelian.
The parity and time-reversal maps are Lorentz maps that are not proper, and det = 1 does not imply orthochrony
Statement refuted
The statements "every Lorentz map is proper and orthochronous" and, separately, "every Lorentz map with is orthochronous" are false.
Facts & Assumptions
Given: An inertial frame with , and the coordinate columns of a vector in that frame.
A linear map is a Lorentz map exactly when its matrix satisfies in the frame's ordered basis; is proper when and orthochronous when (Linear and affine Lorentz maps, and the Lorentz group).
A Lorentz map satisfies ; is one of the two components ; and if and only if (Lorentz maps are isomorphisms of determinant ±1 that preserve causal character and permute the cones; the Lorentz sets are groups).
Since in an inertial frame, the components are described in the frame's coordinates by and (Inertial frames and inertial charts of the affine Minkowski model, Time orientation, future and past timelike/null/causal vectors, and the closed cones).
A vector is timelike when , and in an -orthonormal basis (Timelike, null, spacelike and causal vectors of a Lorentz vector space).
The determinant of a diagonal matrix is the product of its diagonal entries (The determinant of a triangular matrix is the product of its diagonal entries), and the matrix product of two diagonal matrices is the diagonal matrix of the products of the corresponding entries (Rectangular matrix multiplication and the identity matrix , including zero-sized shapes).
Counterexample
Define and by their matrices in the frame basis, and . For any diagonal , by [L5]; for and for each , so both matrices equal . Hence and are Lorentz maps by the matrix criterion [L1].
By the diagonal product rule [L5], and , so by [L1] neither nor is proper: they are Lorentz maps failing properness.
For every , and , so ; if then by [L3], hence and by [L3] again. Thus , and [L2] gives : the improper map is orthochronous, so refutes "orthochronous implies proper".
For every future timelike , and , so by [L3]; since is a bijection with inverse , and is not orthochronous.
By the diagonal product rule [L5], , whose determinant is ; and for every one has , so has and , whence by [L3] and ; since is a bijection, by [L2]. Hence and is not orthochronous: does not imply orthochrony, and this proper non-orthochronous map refutes "proper implies orthochronous".
Steps 1.1 and 1.2 exhibit Lorentz maps that are not proper ( and ), step 1.3 exhibits a Lorentz map that is not proper yet is orthochronous, and steps 1.4 and 1.5 exhibit Lorentz maps that are not orthochronous, including one, , with . The identity supplies the proper orthochronous sign class, so together with all four classes are realized. Therefore strictly contains , the four combinations of the sign of the determinant and orthochrony are genuinely distinct, and the convention of Linear and affine Lorentz maps, and the Lorentz group(e), which reserves "Lorentz transformation" for the proper orthochronous maps, is not automatic. Both refuted statements are false.
Worked covector and vector component transformation under a beta = 3/5 boost
Example
Worked computation. In an inertial frame with coordinates consider the passive boost (The passive Lorentz boost with parameter beta), so that on the plane and , and let be an inertial chart with (Inertial charts are related by proper orthochronous affine Lorentz maps, and every such map is a chart transition). Take the covector with components and the vector with components , with transverse components zero. A covector is called null here when , where the inverse form is defined by and for every . In this orthonormal basis the unique vector has coordinates , verifying both existence and uniqueness directly; hence . This uses the nondegenerate Lorentz form of Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space. Then:
(i) gives and , so .
(ii) gives and , so .
(iii) is null for the inverse form and is null for : , and so are and : .
(iv) The contraction is invariant: .
The example shows all three features of the component law of Transformation of tensor, covector, derivative and current components under affine Lorentz maps: the covariant slot transforms with , the contravariant slot with , and the contraction is unchanged. The common factor on the null covector is the numerical form of the transformation law; no frequency or Doppler interpretation is attached to it here (that belongs to the light-and-doppler page).
Facts & Assumptions
Given: An inertial frame with coordinates and , the passive boost on the plane, an inertial chart with , the covector with components and the vector with components .
For the boost acts on the coordinates by , , with , and its matrix on is ; (The passive Lorentz boost with parameter beta).
For every proper orthochronous linear Lorentz map and every event there is an inertial frame whose chart satisfies , where ; the transition of inertial charts is affine with linear part a proper orthochronous Lorentz matrix (Inertial charts are related by proper orthochronous affine Lorentz maps, and every such map is a chart transition).
Let be inertial charts with transition , open, and let , , be vector, covector and type- tensor fields with components , , in and primed components in . Then (i) ; (ii) ; (iii) every contraction of one contravariant with one covariant index is invariant (Transformation of tensor, covector, derivative and current components under affine Lorentz maps, A type tensor on a finite-dimensional vector space).
A nonzero vector is null when , and in the frame (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Inertial frames and inertial charts of the affine Minkowski model).
Proof
By [L1] with , , so the matrix of on the plane is and its inverse is the matrix of , namely ; the existence of the chart with this linear part is [L2].
Clause (i): by L3, , where is the matrix entry in row and column of from step 1.1. Hence and .
Clause (ii): by L3, , so and , which agrees with the displayed forms ; the components in the transverse directions are unaffected.
Clause (iii): the inverse form defined in the Example gives , while [L4] gives . Both objects are nonzero, so the covector and vector are null in their respective spaces. In the primed orthonormal frame the inverse metric matrix is again , the inverse of its metric matrix; the computed components give and . Thus the primed components describe the same null covector and null vector; no covector is inserted into the form on .
Clause (iv): by L3 the contraction is invariant under the chart change, so ; directly, and , consistent with the invariant statement.
Steps 1.1-2.2 prove clauses (i)-(ii), step 3.1 proves clause (iii) and step 3.2 proves clause (iv); every entry is an exact rational matrix computation from the displayed law, and no frequency, wave or Doppler interpretation is attached to the null covector.
Worked current boost law and coordinate versus rest density
Example
Worked computation. In an inertial frame with coordinates consider the constant vector field with components on an open set (C^k tensor fields on affine Minkowski space and their inertial components), so that with ; is a future timelike unit vector field. Let be a constant scalar field and put . Choose the primed inertial chart , where is the matrix of ; it exists by Inertial charts are related by proper orthochronous affine Lorentz maps, and every such map is a chart transition. Thus the primed frame basis has and the primed component columns are , ; these are changes of components of the same fields. Then:
(i) has components , and because the components are constant; the continuity equation of Transformation of tensor, covector, derivative and current components under affine Lorentz maps reads identically.
(ii) Under the passive boost one computes : the first component of is , the second is , and the transverse components vanish. Correspondingly : in the primed chart the field has those components. Hence the coordinate density is in the original chart and in the chart in which , while the scalar multiplier is the same in both; this is the worked instance of clause (vii) of Transformation of tensor, covector, derivative and current components under affine Lorentz maps.
(iii) The general boost law of clause (vi) of Transformation of tensor, covector, derivative and current components under affine Lorentz maps reproduces (ii): and .
The example carries no physical charge assignment: it illustrates the transformation of a specified vector field, and the identification of with a physical current is a model premise of the electromagnetic framework.
Facts & Assumptions
Given: An inertial frame with coordinates and , an open set , the constant future unit timelike vector field with components , a constant scalar field , and .
A vector field on is a map assigning to each point a vector, with components in the inertial chart; for a constant field all partial derivatives of the components vanish, and for a vector field in an inertial chart the continuity equation is equivalent to ; under the passive boost with parameter , , the components transform as and (Transformation of tensor, covector, derivative and current components under affine Lorentz maps).
For a future timelike vector field and a scalar field , is a vector field with first component ; the normalized chart direction satisfies and is the coordinate density, while is the scalar multiplier; holds exactly when , hence at points with exactly when (Transformation of tensor, covector, derivative and current components under affine Lorentz maps).
For with the boost acts by , on the plane with , and is a proper orthochronous Lorentz map (The passive Lorentz boost with parameter beta, The passive boost is a proper orthochronous Lorentz map with inverse B(-beta)).
In the frame, and the coordinates identify the timelike sets: , , with ; a nonzero vector with is timelike, and (Inertial frames and inertial charts of the affine Minkowski model, The timelike set has exactly two convex components, described in orthonormal coordinates, Time orientation, future and past timelike/null/causal vectors, and the closed cones, Timelike, null, spacelike and causal vectors of a Lorentz vector space).
Proof
By [L4], , so is timelike, and places in : is future timelike unit. Since is constant, [L2] gives with components , and all partial derivatives of the components vanish; the sum is then a sum of vanishing terms, so the continuity equation holds by [L1].
By [L3] with , : the first component of is , the second is , and the transverse components of vanish, so . By linearity as an active image in , and by the primed-chart component law [L1], . Likewise , so and intrinsically. This is clause (ii).
For the general law of [L1] reduces, since and , to and (the correction term combining with to give ). Substituting , and : and , reproducing the components of step 1.2; this is clause (iii).
Clause (i) is step 1.1, clause (ii) is step 1.2, and clause (iii) is step 2.1. Every entry is an exact rational evaluation of the boost matrix on the displayed components, and no physical identification of with a charge-current is made.
Worked motion of the primed frame's origin under a passive boost
Example
Worked computation. Let be a passive boost with parameter along the first spatial direction of an inertial frame with coordinates (The passive Lorentz boost with parameter beta); in matrix expressions write and for the transformed coordinates. Then:
(i) the locus of the primed coordinates (with fixed) is, in the unprimed chart, with fixed constants. The primed spatial origin is the case ; arbitrary fixed transverse coordinates give parallel loci: indeed forces , and the transverse coordinates are unchanged because fixes them. With this is uniform motion with coordinate velocity along ;
(ii) conversely the unprimed origin satisfies , since when ; so the unprimed frame moves with parameter in the primed chart, consistent with ;
(iii) for the concrete values are in the first chart and in the second; the locus passes through the event and through (choosing ), which the boost maps to , confirming .
Facts & Assumptions
Given: An inertial frame with coordinates , a parameter , and the boost ; write .
For the boost acts by and , with , and it fixes the transverse coordinates: , ; the Euclidean norm is that of the frame's spatial coordinates (The passive Lorentz boost with parameter beta, The Euclidean inner product on ).
is a proper orthochronous Lorentz map and ; the maps are injective in (The passive boost is a proper orthochronous Lorentz map with inverse B(-beta)).
An inertial frame has -orthonormal basis with , and its chart is affine (Inertial frames and inertial charts of the affine Minkowski model).
Fix a real constant and define the reparametrized temporal coordinate , so . Coordinate velocity means the derivative with respect to this coordinate. This rescaling makes no assertion about physical clock readings or the empirical value or SI units of .
For every proper orthochronous linear Lorentz map and every the tuple is an inertial frame whose chart satisfies , where (Inertial charts are related by proper orthochronous affine Lorentz maps, and every such map is a chart transition).
Proof
By [L1], and , so holds exactly when ; the equations , of [L1] show that the transverse coordinates are arbitrary constants, unchanged by the boost. Under the convention of [L4], the locus is , that is, uniform motion along with coordinate velocity .
For the unprimed origin set (with ). [L1] gives and , so in the primed coordinates the unprimed origin moves along the locus . This is consistent with [L2]: applying the inverse boost to a point with gives , so the inverse map sends that locus to the unprimed origin, as requires.
Finally, the transformed coordinates are the coordinates of an inertial chart of the model: by [L2] the map is proper orthochronous, so [L5] applied with provides an inertial frame whose chart is , where denotes the boost matrix and for every event , using and . Thus the worked motion is the motion of the primed inertial chart's spatial origin in the unprimed chart, and conversely; the computation uses only the boost formulas and the affine chart structure.
For one has ; step 1.1 gives the locus and step 1.2 gives . The event satisfies , so it lies on the locus, and [L1] gives and ; the origin is fixed because the boost is linear. Hence the displayed values of clause (iii) are correct.
Steps 1.1-1.2 and 2.1 verify clauses (i)-(iii) of the Example and step 1.3 identifies the primed coordinates with an inertial chart; every assertion is a finite computation with the displayed rational values.
Counterexample: a spatial rotation is a proper orthochronous Lorentz map that is not a boost
Statement refuted
The statement "every proper orthochronous Lorentz map is a passive boost for some with " is false.
Facts & Assumptions
Given: An inertial frame with , and the linear maps defined below by their action on the frame basis.
A linear Lorentz map is a linear with for all ; it is proper when and orthochronous when (Linear and affine Lorentz maps, and the Lorentz group).
A spatial rotation of the frame is a linear with , , on the spatial three-space , and in (Spatial rotations of an inertial frame).
The extension of a determinant-one -isometry is a proper orthochronous Lorentz map (Unique boost-rotation factorization of proper orthochronous Lorentz maps).
For every the passive boost satisfies and ; the map is injective (The passive boost is a proper orthochronous Lorentz map with inverse B(-beta)).
The determinant of a square matrix is computed by Laplace cofactor expansion, along the column (Laplace expansion computes the determinant along every row and every column over a commutative ring, Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring); in particular by expanding along the first column, since the -cofactor is and the -cofactor is .
Counterexample
Define by , , , . In the ordered basis its matrix is , and its restriction to has the matrix in . The columns of are orthonormal in the standard inner product on (they are , , ), so and preserves the form on ; and Laplace expansion along the third column gives by [L5]. Hence , , for all , and : all four defining conditions of [L2] hold, so is a spatial rotation of the frame.
By step 1.1 and [L3], the extension is a proper orthochronous Lorentz map. In particular is a member of in the sense of [L1], and it is not the identity because .
Suppose, for contradiction, that for some with . Applying both sides to and using [L4] gives ; comparing the coordinates gives and then comparing the spatial coordinates gives for , hence and by [L4]. But , a contradiction. Hence is a proper orthochronous Lorentz map that is not a passive boost, and the refuted statement is false.
Consequently the set of passive boosts is a proper subset of : the strict inclusion is witnessed by . The factorization theorem of Unique boost-rotation factorization of proper orthochronous Lorentz maps therefore cannot drop its rotation factor in general: for the theorem produces the factorization with rotation factor , and by clause (v) of that theorem is a passive boost if and only if that factor is the identity.
Sources
- B. Crowell, Special Relativity, revision 2020-12-10 (author-published undergraduate textbook), Section 1.4 'The Lorentz transformation' pp. 30-32 including Examples 11-14, Sections 2.3-2.4 'Additional postulates'/'Other axiomatizations' pp. 46-49, and Section 8.3 'Boosts and rotations' pp. 174-176; signature (+---) converted to (-+++) with x^0 = ct restored
- S. M. Carroll, Lecture Notes on General Relativity, arXiv:gr-qc/9712019v1 (1997), Chapter 1 'Special relativity and flat spacetime' pp. 3-7 (PDF pp. 10-14), equations (1.3)-(1.21)
- H. S. Reall, Mathematical Tripos Part III General Relativity (Cambridge, October 12, 2022 revision), Section 2.2 'Lorentzian signature' pp. 26-28 (PDF pp. 27-29); Internet Archive copy of the original https://www.damtp.cam.ac.uk/user/hsr1000/part3_gr_lectures.pdf
- H. S. Reall, Mathematical Tripos Part III General Relativity (Cambridge, October 12, 2022 revision), Section 2.1 'Manifolds' and Section 2.2 'Lorentzian signature' pp. 25-28 (PDF pp. 26-29); Internet Archive copy of the original https://www.damtp.cam.ac.uk/user/hsr1000/part3_gr_lectures.pdf