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Causality And Signals — Examples
1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Binary Operations, Monoids, Groups and Subgroups
- Causality And Signals
- 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
- 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
- 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
- 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 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
- Worldlines And Clocks
2 · Summary
These examples and counterexamples work the causality page on explicit integers and rationals. The boost family on the spacelike pair reaches simultaneity at and produces both time orders at and , with all three charts verified inertial (Worked boost family on the spacelike pair : simultaneity at and both time orders); the four displacements , , and are classified by the strict, equality and reversed inequalities of the coordinate criterion (Worked coordinate criteria: four displacements classified by the future-cone inequalities). The causal diamond for , is computed with its interior; its section has coordinate area m, invariant under boosts in the plane (Worked four-dimensional causal diamond and the boost-invariant area of its planar section), and the parallel future null displacements , compose to the future null , with the full characterization that a sum of future causal vectors is null exactly for parallel future null summands (Parallel future null displacements compose to a future null displacement). Two further examples isolate the non-causal side: a prearranged sweep of coordinate speed whose successive events are spacelike separated and can be made simultaneous (A prearranged sweep with superluminal coordinate speed: successive events are spacelike separated and can be made simultaneous), and the exact relay arithmetic at , arriving at s (Worked relay arithmetic at and : arrival at seconds).
The counterexamples refute two specific displayed claims and nothing wider: a causally related pair need not be joinable by a future timelike worldline, as the null-separated pair , shows (Counterexample: a causally related pair of events need not be joinable by a future timelike worldline); and the sign of a time-coordinate difference is not frame independent, as two explicit boost charts give opposite signs for the same spacelike-separated pair (Counterexample: the temporal order of a spacelike-separated pair is not the same in every inertial frame). The final example shows that two nonparallel future null legs compose to a future timelike displacement while every tangent stays null, so the strict clause of the causality theorem is not an equivalence (Two nonparallel future null legs compose to a future timelike displacement). Every computation is exact and no item asserts a measurement or an empirical conclusion.
3 · Logical flowchart
4 · Definitions, postulates, theorems and proofs
None yet.
5 · Examples, experiments and counterexamples
Worked boost family on the spacelike pair : simultaneity at and both time orders
Example
Worked computation in the chart of an inertial frame of an affine Minkowski space (Oriented time-oriented affine Minkowski space (the spacetime primitives), Inertial frames and inertial charts of the affine Minkowski model) with . Let and ; then has so is spacelike and , in accordance with clause (a) of Spacelike-separated events: the spacelike coordinate inequality in every inertial chart, and an explicit family of inertial frames realizing every time-order sign.
(a) Simultaneity at . With and , so that , the chart of the inertial frame built from the passive boost by clause (d) of the same lemma satisfies the two events are simultaneous in that frame.
(b) Below : positive time difference. With one has and
(c) Above : negative time difference. With one has and
(d) The spatial separation in the three frames. The boost in the -plane acts by , (The passive Lorentz boost with parameter beta), so the spatial difference of the pair in the boosted chart is . It equals for , for and for ; in each case it is nonzero and . The displacement stays spacelike because a boost is a Lorentz map and preserves (Lorentz maps are isomorphisms of determinant ±1 that preserve causal character and permute the cones; the Lorentz sets are groups, Linear and affine Lorentz maps, and the Lorentz group), so the three charts are charts of three inertial frames related to by passive boosts and the time-order sign of a spacelike-separated pair is an artifact of the chart, not of the pair. No physical synchronization procedure is asserted; simultaneity means equality of the chart's time coordinate.
Facts & Assumptions
Given: An inertial frame with chart and , the events and , the displacement with spatial part and Euclidean norm .
For a spacelike pair the lemma gives in every inertial chart, and for every the chart of the inertial frame with ordered basis , where , satisfies with (Spacelike-separated events: the spacelike coordinate inequality in every inertial chart, and an explicit family of inertial frames realizing every time-order sign).
For with the passive boost acts in coordinates by , , , , with (The passive Lorentz boost with parameter beta).
A vector is spacelike exactly when ; a linear Lorentz map preserves , hence preserves the causal character of every nonzero vector, and the boosts are proper orthochronous Lorentz maps (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Lorentz maps are isomorphisms of determinant ±1 that preserve causal character and permute the cones; the Lorentz sets are groups, Linear and affine Lorentz maps, and the Lorentz group).
In a fixed inertial frame the spatial coordinates carry the Euclidean norm (The Euclidean spatial structure and cross product of an inertial frame).
Verification
The displacement is , so , the pair is spacelike, and .
With one has , and the chart of clause (d) of [L1] gives with ; this is (a).
With one has , and ; this is (b).
With one has , and ; this is (c).
The spatial difference in the boosted chart is by [L2] applied to , and the three values are , and , all nonzero; the corresponding time differences are , and , so holds in all three charts, and the spacelike character is preserved because by [L3]; this is (d).
By [L1] the three charts , and are charts of inertial frames related to by passive boosts, and steps 2.1, 2.2 and 2.3 give time differences , and ; hence the time order of the spacelike-separated pair is simultaneity for , positive for and negative for , and is therefore an artifact of the inertial chart and not a property of the pair. Only equality of the chart's time coordinate is used for simultaneity, and no choice principle is used.
Worked coordinate criteria: four displacements classified by the future-cone inequalities
Example
Worked computation in the chart of an inertial frame of an affine Minkowski space with (Inertial frames and inertial charts of the affine Minkowski model). Put and, in order,
with the corresponding displacements and the Euclidean norm of the spatial part. Then:
(a) Future timelike. has , so lies in (Causal reachability, causal futures and pasts, causal diamonds, and reachability by future causal paths in affine Minkowski space) and confirms that is future timelike (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones).
(b) Future null. has , so lies in but not in , and is future null.
(c) Spacelike and causally unrelated. has , so is spacelike and lies in neither nor : the pair is causally unrelated.
(d) Future timelike with past-directed companion. has and , so is future timelike; the past-directed companion is the event , whose displacement has , time component and , so lies in .
(e) The strict inequality , the equality and the reversal of the coordinate criterion therefore separate, in this single computation, the three causal characters of the displacement and the two reachability relations; the numbers are exact integers and no measurement is reported.
Facts & Assumptions
Given: An inertial frame with chart , , the events , , , , and .
Coordinate criterion: for events with , iff , and iff ; moreover iff iff the displacement satisfies (The closed future cone in inertial coordinates: description, finite sums, and the coordinate criterion for causal reachability, Causal reachability, causal futures and pasts, causal diamonds, and reachability by future causal paths in affine Minkowski space).
Diagonal form: for coordinate differences ; a displacement is timelike iff , null iff and , spacelike iff , and future-directed causal displacements are exactly the nonzero displacements with (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones).
Verification
The four displacements are , , and by the linearity clause of the inertial chart, with spatial norms , , and ; the companion displacement is with spatial norm .
: , so by [L1] and makes future timelike by [L2]; this is (a).
: , so and by [L1], and with makes future null by [L2]; this is (b).
: gives , so is spacelike by [L2]; the criterion fails for both and , that is is false and is false, so and both fail by [L1], and the pair is causally unrelated; this is (c).
: with makes future timelike and by [L1] and [L2]. For the companion, the displacement from to is , whose reversed displacement has time component , so , that is , by [L1]; moreover with time component exhibits a past timelike displacement; this is (d).
Steps 2.1, 2.2 and 2.3 exhibit the three cases of the criterion (strict inequality, equality with positive time component, and failure in both directions), while step 2.4 exhibits a future timelike displacement together with a past-directed companion; all values are exact integers obtained from the diagonal form and the criterion, and no physical measurement or empirical claim is involved.
Worked four-dimensional causal diamond and the boost-invariant area of its planar section
Example
Worked computation in the affine Minkowski model (Oriented time-oriented affine Minkowski space (the spacetime primitives)). In the chart of an inertial frame with and all coordinate values expressed as numerical lengths in metres (Inertial frames and inertial charts of the affine Minkowski model) let and . Interior is taken in the Euclidean topology transported by . In this item the interval magnitude of a displacement means ; this is not a norm on . For the planar polygons below, define a triangle with edge columns to have coordinate area , and compute polygon area by the displayed disjoint-interior triangulations. The coordinate area carries square metres. Here in the proper-time formula denotes the positive numerical conversion factor in metres per second.
(a) By the coordinate criterion (The closed future cone in inertial coordinates: description, finite sums, and the coordinate criterion for causal reachability) one has as subsets of ; hence the causal diamond is and its interior is .
(b) In the -dimensional section the diamond is the square with vertices , , and . Its diagonals are the straight timelike segment from to and the straight spacelike segment from to ; both have length m in the interval magnitude, and the area of the square in the -plane is half the product of the diagonal lengths, that is m.
(c) For a passive boost with parameter along the direction the block of in the -plane is with (The passive Lorentz boost with parameter beta), whose determinant is . Let be the chart of the inertial frame related to by this boost, so that (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, Linear and affine Lorentz maps, and the Lorentz group); the coordinate image of the section in is the parallelogram with centre and half-diagonal vectors and , whose area is m. Hence the area of the diamond in the -plane is the same in the chart of every frame related to by such a boost; this is the source's invariance of the area of a light-rectangle.
(d) The identification of this area with a physical time or clock reading is not made here. With retained, the interval magnitude of the timelike diagonal is m and, for the straight future timelike segment from to , the proper time is s (Proper time along a future timelike worldline, The Euclidean inner product on , The inner-product norm is definite, homogeneous, and satisfies the triangle inequality); no clock reading is asserted.
Facts & Assumptions
Given: An inertial frame with chart and , the events and , and for a real with the boost of the -plane with .
Coordinate criterion: for events with , iff and iff ; also and (The closed future cone in inertial coordinates: description, finite sums, and the coordinate criterion for causal reachability, Causal reachability, causal futures and pasts, causal diamonds, and reachability by future causal paths in affine Minkowski space).
Diagonal form: ; a displacement is timelike iff and spacelike iff (Timelike, null, spacelike and causal vectors of a Lorentz vector space).
Chart linearity: , so coordinate differences of events are the coordinates of their displacement (Inertial frames and inertial charts of the affine Minkowski model).
For the passive boost acts in the -plane by , ; it is a proper orthochronous Lorentz map with inverse , so the ordered basis is -orthonormal, future and positively oriented and is the basis of an inertial frame whose chart satisfies (The passive Lorentz boost with parameter beta, 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, Linear and affine Lorentz maps, and the Lorentz group).
For the straight future timelike segment , , the elapsed proper time is , and the interval magnitude of the displacement is (Proper time along a future timelike worldline, The Euclidean inner product on , The inner-product norm is definite, homogeneous, and satisfies the triangle inequality).
The coordinate Euclidean norm satisfies the triangle inequality, hence ; the Euclidean scalar product is the sum of coordinate products (The Euclidean inner product on , The inner-product norm is definite, homogeneous, and satisfies the triangle inequality).
Verification
Let be an event with coordinates . By [L3] the displacement has coordinates , so [L1] gives iff ; likewise the displacement has coordinates of spatial norm , so iff iff . Hence the diamond is , and the two strict versions give by the strict clauses of [L1]. For the interior: the triangle inequality gives , so each of the functions and changes by at most . At a point where both are positive, the coordinate ball of radius half their smaller value keeps both positive, i.e. lies in the interior of the diamond; if or , every neighbourhood contains points with decreased or increased slightly, which violate or , so such points are not interior; hence the interior is exactly the displayed open set, which is ; this is (a).
In the section the description of step 1.1 reads , whose boundary consists of the four segments joining to , to , to and to : the square with those four vertices (successive edge columns have squared Euclidean length and scalar product ). Its diagonals are the displacement from to , of interval magnitude m, and the displacement from to , of interval magnitude m, both by [L2]; relative to its centre , the four triangles have edge columns chosen from and , each of area by the stated coordinate-area definition, so their total area is m, also ; this is (b).
By [L4] the chart of the boosted frame satisfies , so the -coordinates of the events of the section are the images under the linear map of their -coordinates, and the coordinate image of the square of step 2.1 is the parallelogram spanned by the images of the two diagonals: and . Its centre is , its half-diagonal vectors are and , and it is the union of the four triangles in centred coordinates, each of area ; hence its area is m, since ; this holds for every and is (c).
Clause (a) is step 1.1, clause (b) is step 2.1, clause (c) is step 3.1, and clause (d) follows from [L5] applied to the straight segment from to , whose displacement has m and interval magnitude m, giving proper time s; nothing in the computation identifies the coordinate area m with a physical time or clock reading, and no measurement is reported.
Parallel future null displacements compose to a future null displacement
Example
Worked computation in the chart of an inertial frame (Inertial frames and inertial charts of the affine Minkowski model). Let and be coordinate columns with respect to the basis of the frame.
(a) Both vectors are future null: and with positive time components, and , so the two directions are parallel (Timelike, null, spacelike and causal vectors of a Lorentz vector space).
(b) The sum is again future null, since and its time component is positive, and the endpoint lies in for but not in , by the coordinate criterion (The closed future cone in inertial coordinates: description, finite sums, and the coordinate criterion for causal reachability, Causal reachability, causal futures and pasts, causal diamonds, and reachability by future causal paths in affine Minkowski space); the straight path has constant tangent , so it is a straight future null path from to .
(c) This is the boundary case of the sum clause of the same lemma: for nonzero future causal vectors and the sum is future causal, and it is future null exactly when and are both future null and positive multiples of one another, as here. By contrast the explicit nonparallel future null pair and has sum , whose squared norm is : the sum of two future null displacements is null only in the parallel case.
Facts & Assumptions
Given: An inertial frame with chart and , the vectors and , the events and , and the pair , .
Coordinate criterion and cone description: for a vector with spatial norm , iff , iff , and is future null exactly when ; for events with , iff and iff (The closed future cone in inertial coordinates: description, finite sums, and the coordinate criterion for causal reachability).
Causal classification: in the inertial basis; is null exactly when and , and timelike when (Timelike, null, spacelike and causal vectors of a Lorentz vector space); the sum of two elements of lies in , and if one of them lies in the sum lies in (The closed future cone in inertial coordinates: description, finite sums, and the coordinate criterion for causal reachability).
Euclidean structure: for spatial coordinate vectors the norm satisfies , implies , the triangle inequality holds, and with the Euclidean scalar product (The Euclidean inner product on , The Euclidean spatial structure and cross product of an inertial frame, The inner-product norm is definite, homogeneous, and satisfies the triangle inequality).
Chart linearity: coordinate differences of events are the coordinates of their displacement (Inertial frames and inertial charts of the affine Minkowski model).
Verification
The evaluations , and the displayed hold by the diagonal form of [L2]; both vectors are nonzero with positive time components, so [L1] makes them future null; this is (a).
The sum is with and time component , so it is future null by [L1]; by [L4] the displacement has coordinates , so the criterion gives , that is , while is false, that is . The straight path has constant tangent with coordinates , a future null vector, so it is a straight future null path from to ; this is (b).
For the characterization let and be nonzero future causal vectors written in the inertial basis, so that , and because a future causal vector with vanishing time component has vanishing spatial part. If is null then , so by [L3] and every inequality is an equality; in particular , and . Expanding with [L3], equals , hence ; since , the vector has squared norm , so by positive definiteness of the norm and with ; moreover and are future null because and with . Conversely, if with and future null, then has and positive time component, so it is future null; this proves the characterization of (c).
For the nonparallel pair, and are future null by [L1] since and with time component , while their sum is with , a future timelike vector by [L2]: the sum of two future null displacements is therefore null exactly in the parallel case established in step 2.2, which is the boundary case of the sum clause illustrated by and in steps 1.1 and 2.1; all computations are exact integer evaluations and no choice principle is used.
A prearranged sweep with superluminal coordinate speed: successive events are spacelike separated and can be made simultaneous
Example
Worked computation in the chart of an inertial frame with in metres and (Inertial frames and inertial charts of the affine Minkowski model). Fix and consider the family a prearranged array of devices at triggered at coordinate time , so the pattern is a sweep of coordinate speed along the row.
(a) For the displacement is whose interval is , so successive events are spacelike separated; in particular no event lies in or for , by the coordinate criterion (The closed future cone in inertial coordinates: description, finite sums, and the coordinate criterion for causal reachability).
(b) Let . The passive boost with parameter (The passive Lorentz boost with parameter beta) maps the displacement to time component , so in the boosted chart the whole family consists of events simultaneous at one coordinate time; taking produces the opposite sign , so the apparent temporal order of the sweep runs forward, frozen or backward in different inertial frames even though no event of the family is causally related to another (Spacelike-separated events: the spacelike coordinate inequality in every inertial chart, and an explicit family of inertial frames realizing every time-order sign).
(c) No signal statement is made or needed: the computation is a property of the spacetime locations of independently triggered events, and it exhibits the class of spacelike-separated patterns that a kinematic superluminal speed bound cannot exclude. It is a statement about the locations of the events, it asserts nothing about any of them being the reception of a signal, and it does not model the fixed-speed relay kinematics of the hypothetical superluminal model treated on the companion page.
Facts & Assumptions
Given: An inertial frame with chart and origin event , a constant , a real , the family of events, and for the boost with .
For the spacelike pair with displacement and spatial unit direction , and for every , the chart of the inertial frame with the same origin and basis satisfies and ; in general in every inertial chart (Spacelike-separated events: the spacelike coordinate inequality in every inertial chart, and an explicit family of inertial frames realizing every time-order sign).
The passive boost acts in the -plane by , and is linear in the coordinate column (The passive Lorentz boost with parameter beta).
Diagonal form and classification: , and is spacelike exactly when , timelike exactly when (Timelike, null, spacelike and causal vectors of a Lorentz vector space).
Coordinate criterion for reachability: for events with , iff , and iff (The closed future cone in inertial coordinates: description, finite sums, and the coordinate criterion for causal reachability, Causal reachability, causal futures and pasts, causal diamonds, and reachability by future causal paths in affine Minkowski space).
Chart linearity: for events (Inertial frames and inertial charts of the affine Minkowski model).
Verification
For the displacement of the family is by [L5], so because and ; hence the displacement is spacelike by [L3]. For the first criterion of [L4] would require , which fails since , and the second would require , which also fails; the case is the same computation with the roles exchanged, so no event of the family lies in the causal future or past of another; this is (a).
Set . The vector is spacelike by step 1.1, so [L1] applies with the origin event and gives the chart of an inertial frame; by linearity [L2] and [L5], , whose time component is for all ; hence all events share the coordinate time . For a parameter with , which exists because , the same computation gives , of sign opposite to ; so the temporal order of the sweep runs forward in , is frozen in and runs backward in ; this is (b).
Steps 1.1 and 2.1 compute only the locations and coordinate differences of the family and the chart changes of those coordinate differences; the family is prearranged and independently triggered, no displacement of the family is causal in either direction, and nothing in the computation identifies an event with the reception of a signal or with the relay model of the companion page; the sweep is a kinematic pattern whose temporal order is frame-dependent without any causal relation among its events, and no choice principle is used.
Worked relay arithmetic at and : arrival at seconds
Example
Worked computation in the chart of an inertial frame of an affine Minkowski space (Oriented time-oriented affine Minkowski space (the spacetime primitives), Inertial frames and inertial charts of the affine Minkowski model) with for a fixed constant . Throughout this numerical example, and are numerical values in metres, and are numerical values in seconds, and are numerical values in metres per second; units after a formula apply to its numerical value. Take , and (a length of m), so that , and . Let and have affine curves and for . Their constant tangents and are future timelike since . Emit from at the event a signal travelling at coordinate speed in the direction, and let re-emit immediately from the reception event a signal travelling at coordinate speed in the direction of 's rest frame; this fixed-speed, immediate, lossless relay is the whole content of the hypothetical model, and no device, signal or measurement is asserted to exist.
(a) The reception time solves , so and the displacement m has interval : it is spacelike.
(b) Write , , and let be the chart of the inertial frame in which is at rest (The passive Lorentz boost with parameter beta, The passive boost is a proper orthochronous Lorentz map with inverse B(-beta)). In that chart the return displacement is with , and transforming back with the inverse boost the condition that the arrival lies on 's worldline gives , hence
(c) The arrival event therefore satisfies matching the closed formula . The displacement m is past timelike, so the relayed signal is received strictly before it was emitted in the chart of the emitter; the return displacement in is m with , spacelike.
(d) The computation is an arithmetic specialization of the hypothetical fixed-speed model (signals at fixed coordinate speed in each emitter's rest frame, immediate lossless relaying, inertial motion at , ordinary Lorentz kinematics of displacements); it is a conditional identity about that model and neither asserts that such signals or devices exist nor reports any measurement.
Facts & Assumptions
Given: An inertial frame with chart , , a constant , the numerical data , , (numerical metres), , the affine curves of and of , , the emission event , and the hypothetical relay: outgoing leg of coordinate speed in the direction from , immediate re-emission at the reception event of a return leg of coordinate speed in the direction of 's rest frame.
Coordinate criterion: for events with , iff , and iff ; (The closed future cone in inertial coordinates: description, finite sums, and the coordinate criterion for causal reachability, Causal reachability, causal futures and pasts, causal diamonds, and reachability by future causal paths in affine Minkowski space).
For the passive boost acts on coordinates by , in the -plane with , its inverse is with inverse formulas , , 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)).
The ordered basis is -orthonormal, future and positively oriented, so is an inertial frame; its chart satisfies (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, Inertial frames and inertial charts of the affine Minkowski model, Linear and affine Lorentz maps, and the Lorentz group).
A displacement is spacelike iff and timelike iff ; for a timelike displacement with negative time component the reversed displacement has positive time component dominating its spatial part, giving by [L1] (Timelike, null, spacelike and causal vectors of a Lorentz vector space).
Verification
The reception event has , so s and m; the displacement is m, with and , so it is spacelike with positive time component in ; this is (a).
With , the basis is the basis of an inertial frame by [L3], and its chart satisfies with inverse , that is , ; moreover .
In the chart the worldline of is constant in , since ; the return leg moves in the direction at coordinate speed , so its displacement in is for a single , and applying the inverse formula of step 2.1 to its -component gives .
The arrival event lies on the worldline of , so m, and step 3.1 gives ; substituting into the time component of the inverse formula gives , and with this is s; this is (b).
Adding the two legs gives s, and the closed formula evaluates as s, the same number. The displacement m has , so it is timelike, and the reversed displacement has time component , so by [L1]: the arrival is in the strict chronological past of the emission. The return displacement in is m because its -component is m, and its interval is , so it is spacelike; this is (c).
Steps 1.1 to 5.1 are an arithmetic specialization of the hypothetical fixed-speed relay model with its stated inputs, and no step uses any property beyond the Lorentz kinematics of the boost and the causal classification of the two displacements; the reception of the relayed signal strictly before its emission in the chart of the emitter is therefore a conditional conclusion of that model, not a report of a measurement and not an assertion that superluminal signals or the devices realizing the relay exist; this is (d).
Counterexample: a causally related pair of events need not be joinable by a future timelike worldline
Statement refuted
False claim. If lies in then lies in , that is, every causally related pair of events can be joined by a future timelike worldline (Causal reachability, causal futures and pasts, causal diamonds, and reachability by future causal paths in affine Minkowski space).
Facts & Assumptions
Given: An inertial frame with chart , the events and , and the displacement with coordinates .
Coordinate criterion and cone description: iff , and iff ; a vector is future null exactly when (The closed future cone in inertial coordinates: description, finite sums, and the coordinate criterion for causal reachability).
Causal classification: ; a nonzero vector with is null, and a vector with is spacelike (Timelike, null, spacelike and causal vectors of a Lorentz vector space).
Reachability: and are the causal and chronological futures of defined by the displacements and , and a future timelike worldline is a curve with future timelike tangent everywhere (Causal reachability, causal futures and pasts, causal diamonds, and reachability by future causal paths in affine Minkowski space, Curves, regular causal curves and future timelike worldlines in affine Minkowski space).
Displacement of a future causal path: for a piecewise future causal path with a future timelike one-sided tangent at some point, the displacement lies in (Future causal reachability, absence of causal loops, time monotonicity and the straight-worldline proper-time maximum).
The false claim of the statement refuted section.
Counterexample
The displacement has coordinates , so with and ; by [L2] and [L1] the vector is future null, since .
By [L1], gives , while is false, so ; hence the pair is causally related without belonging to the chronological relation, and the straight path from to has the constant tangent , which is future null and not future timelike.
No future timelike worldline with and exists: if were one, it would be in particular a piecewise future causal path with a future timelike tangent at every point by [L3], so [L4] would put its displacement in , and then [L2] would give , contradicting from step 1.1.
Steps 2.1 and 2.2 exhibit a pair with and that is not joinable by any future timelike worldline, while the null straight segment realizes the causal relation; the claim of [L5] is therefore false, and no choice principle is used.
Counterexample: the temporal order of a spacelike-separated pair is not the same in every inertial frame
Statement refuted
False claim. For any two distinct events of the affine Minkowski model, the sign of the time-coordinate difference is independent of the inertial frame (Inertial frames and inertial charts of the affine Minkowski model).
Facts & Assumptions
Given: An inertial frame with chart , the events and , and the displacement with coordinates .
For a spacelike displacement holds in every inertial chart, and for every the chart of the inertial frame with basis (same origin) satisfies and with (Spacelike-separated events: the spacelike coordinate inequality in every inertial chart, and an explicit family of inertial frames realizing every time-order sign).
For the passive boost acts in the -plane by , 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)).
Diagonal form and classification: ; a nonzero vector with positive squared norm is spacelike (Timelike, null, spacelike and causal vectors of a Lorentz vector space).
Time monotonicity: for every piecewise future causal path that is nonconstant on , every inertial chart gives ; and for with the straight path from to is a future causal path (Future causal reachability, absence of causal loops, time monotonicity and the straight-worldline proper-time maximum).
The false claim of the statement refuted section.
Counterexample
The displacement has coordinates , so and ; by [L3] the displacement is spacelike, so [L1] applies to the pair.
With , admissible since , one has , so the chart of the inertial frame of [L1] satisfies .
With , admissible since , one has , so .
The charts and are charts of inertial frames of the model by [L1] and [L2], and they assign to the same pair of events time-coordinate differences of opposite sign by steps 2.1 and 2.2; the claim of [L5] is therefore false. The failure is confined to spacelike-separated pairs: if with future causal, then [L4] gives the straight path from to as a nonconstant future causal path, so every inertial chart gives ; no choice principle is used.
Two nonparallel future null legs compose to a future timelike displacement
Example
Worked computation in the chart of an inertial frame (Inertial frames and inertial charts of the affine Minkowski model). Let and let be the piecewise straight path with
(a) Both legs have constant future null tangents: and , with positive time components.
(b) The total displacement is with and , so is future timelike (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones) and the endpoint lies in by the coordinate criterion (The closed future cone in inertial coordinates: description, finite sums, and the coordinate criterion for causal reachability), although every tangent of the path is null.
(c) Consistently with the sum clause of the same lemma, the sum of the two nonparallel future null vectors is future timelike, and the straight path from to , whose tangent is , is a future timelike straight worldline by Future causal reachability, absence of causal loops, time monotonicity and the straight-worldline proper-time maximum. The path is a future causal path whose displacement is strictly timelike and whose tangents are nowhere timelike; the strict clause of part (a) of that theorem is an implication from a timelike tangent to a timelike displacement and not an equivalence, so no contradiction arises.
Facts & Assumptions
Given: An inertial frame with chart and , the event , the piecewise straight path with legs on and on , and the endpoint .
Diagonal form and classification: for coordinates , ; a nonzero vector is null when , spacelike when , and timelike when ; future null means nonzero null with positive time coordinate (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones).
Coordinate criterion and sums: iff for , and the sum of finitely many future causal vectors is future causal, strictly timelike if one summand is future timelike (The closed future cone in inertial coordinates: description, finite sums, and the coordinate criterion for causal reachability). The nonparallel null case is a local calculation, not an extra conclusion of that supplier: write the vectors and with and . If they are not parallel, , so by the spatial Euclidean structure. Thus , and the positive time component makes the sum future timelike. The spatial norm and scalar product are those of the named inertial frame (The Euclidean spatial structure and cross product of an inertial frame, The Euclidean inner product on ).
Straight paths and reachability: the straight path has constant tangent , and for one has iff that straight path is a future causal path, with characterized by the straight path being future timelike; part (a) of the theorem also says that a future causal path with a future timelike one-sided tangent has future timelike displacement (Future causal reachability, absence of causal loops, time monotonicity and the straight-worldline proper-time maximum, Causal reachability, causal futures and pasts, causal diamonds, and reachability by future causal paths in affine Minkowski space).
Piecewise curves: the concatenation of the two straight legs is a piecewise curve with constant tangents and on the open pieces and one-sided difference-quotient tangents equal to those vectors at every point of (Curves, regular causal curves and future timelike worldlines in affine Minkowski space, The Euclidean spatial structure and cross product of an inertial frame, The Euclidean inner product on ).
Verification
The first leg has constant coordinate tangent , whose squared norm is with time component , so it is future null by [L1]; the second leg has constant coordinate tangent , whose squared norm is with time component , so it is future null too; both legs are straight and their endpoint values at agree, so is continuous with those one-sided tangents throughout by [L4]; this is (a).
The total displacement is with coordinates , namely the telescoping sum of the two constant tangents; then and , so is future timelike by [L1], and shows by [L2].
The two null tangents and are not parallel (neither is a multiple of the other, as their second and third components show), and their sum is future timelike by step 2.1, in agreement with the local nonparallel-null computation in [L2]; the straight path from to has the constant tangent , which is future timelike, so it is a future timelike straight worldline and realizes the relation by [L3].
The path is a piecewise future causal path by [L4] and step 1.1 whose displacement is future timelike by step 2.1 while none of its tangents is timelike; the strict clause of part (a) of [L3] (a future timelike one-sided tangent forces a timelike displacement) is therefore not an equivalence, and the computed example is consistent with both the sum clause of [L2] and the strict clause of the theorem; no choice principle is used.
Sources
- H. S. Reall, Mathematical Tripos Part III General Relativity (Cambridge, October 12, 2022 revision): Sections 2.1-2.3 pp. 25-30 (Minkowski metric (2.6) and inertial frames; timelike, null, spacelike and causal vectors with the light-cone double cone and its two disconnected causal sets; timelike and causal curves; proper time (2.19); curves of extremal proper time, straight lines in inertial frames and the unproved maximization statement); Internet Archive copy of the original https://www.damtp.cam.ac.uk/user/hsr1000/part3_gr_lectures.pdf
- R. C. Tolman, The Theory of the Relativity of Motion (University of California Press, Berkeley, 1917): Chapter V, Sections 45-52 pp. 49-55 (kinematical rate of a clock, the idea of simultaneity, velocities greater than that of light with the order-of-cause-and-effect reversal formula (28)-(29), and the blocks caveat that kinematic patterns without causal connection are not restricted); Internet Archive full text of the 1917 printing
- S. M. Carroll, Lecture Notes on General Relativity, arXiv:gr-qc/9712019v1 (1997): Chapter 1 pp. 5-7 (light cones; timelike, null and spacelike separation; no coordinate-independent future for a spacelike-separated event), Chapter 4 pp. 124-127 (no signals faster than light; information flows only along timelike or null trajectories; future domain of dependence; Misner space with closed timelike curves as a warning about other topologies); PDF page = printed page + 7
- B. Crowell, Special Relativity, revision 2020-12-10 (author-published undergraduate textbook, 246 PDF pages): Section 1.3.2 pp. 22-23 (light-rectangles as causal diamonds with invariant area), Section 2.1 pp. 43-45 (the modest causality postulate P1 and the limits of a general causality principle), Section 4.7 pp. 110-113 (superluminal transmission of information, the tachyonic antitelephone attributed to Tolman, reported tachyon searches); signature (+---) converted to (-+++) and c = 1 formulas converted to x^0 = c t