Causality And Signals

Throughout, $(A,V,\eta ,C^{+})$ is an oriented time-oriented affine Minkowski space (Oriented time-oriented affine Minkowski space (the spacetime primitives)): $A$ is a nonempty affine space over the $4$-dimensional real vector space $V$ with displacement vectors $q-p$ and translations $v\mapsto p+v$ (Affine spaces over a real vector space and displacement vectors), and $C^{+}$ is one of the two connected components of the timelike set $T=\{v\in V:\eta (v,v)<0\}$, chosen as the time orientation, with $C^{-}:=T\setminus C^{+}$ the other component (Time orientation, future and past timelike/null/causal vectors, and the closed cones). Write $\overline{C^{+}}:=\mathrm{cl}(C^{+})$ and $\overline{C^{-}}:=\mathrm{cl}(C^{-})$ for the future and past closed cones. A vector $v\in V$ is timelike, null, spacelike or causal according to the sign of $\eta (v,v)$ as in Timelike, null, spacelike and causal vectors of a Lorentz vector space; in particular the zero vector is none of timelike, null or spacelike and hence is not causal, and the causal vectors are exactly the nonzero elements of $\overline{C^{+}}\cup \overline{C^{-}}$.

The closed cones and their boundaries. By Time orientation, future and past timelike/null/causal vectors, and the closed cones the sets $\overline{C^{+}}$ and $\overline{C^{-}}$ are convex cones with $\overline{C^{+}}\cap \overline{C^{-}}=\{0\}$. By The timelike set has exactly two convex components, described in orthonormal coordinates, applied in an $\eta$-orthonormal basis whose first vector lies in $C^{+}$, the closure of the component $\{v:v^0>\mathrm{\mid }{v}_s\mathrm{\mid }\}$ of the timelike set is $\{v:v^0\ge \mathrm{\mid }{v}_s\mathrm{\mid }\}$ with boundary points exactly $0$ and the nonzero null vectors of $\overline{C^{+}}$. Writing $N^{+}$ and $N^{-}$ for the sets of nonzero future and past null vectors (Time orientation, future and past timelike/null/causal vectors, and the closed cones), this gives

$\overline{C^{+}}=C^{+}\cup N^{+}\cup \{0\},\overline{C^{-}}=C^{-}\cup N^{-}\cup \{0\}\mathrm{.}$

The objects fixed by this definition.

(a) Causal and chronological reachability. For events $p,q\in A$ set $p\le q$ if and only if $q-p\in \overline{C^{+}}$, and $p<q$ if and only if $q-p\in C^{+}$. The first relation is read "$q$ is causally reachable from $p$" and the second "$q$ is strictly (chronologically) reachable from $p$"; both are relations on the set of events, and neither is asserted here to have any property beyond being a relation.

(b) Futures and pasts. For an event $p\in A$ set

$J^{+}(p):=\{q\in A:p\le q\}=p+\overline{C^{+}},I^{+}(p):=\{q\in A:p<q\}=p+C^{+},$

$J^{-}(p):=\{q\in A:q\le p\}=p+\overline{C^{-}},I^{-}(p):=\{q\in A:q<p\}=p+C^{-}\mathrm{.}$

The displayed equalities hold because $q\in J^{+}(p)$ says exactly that the displacement $q-p$ lies in $\overline{C^{+}}$, that is $q=p+v$ for some $v\in \overline{C^{+}}$; here $p+v$ is the affine translation of Affine spaces over a real vector space and displacement vectors. Since $0\in \overline{C^{+}}\cap \overline{C^{-}}$ one has $p\in J^{+}(p)\cap J^{-}(p)$, and since $0$ is not timelike one has $p\mathrm{\notin }{I}^{+}(p)$ and $p\mathrm{\notin }{I}^{-}(p)$. The sets $J^{\pm }(p)$ and $I^{\pm }(p)$ are called the causal and chronological future and past of $p$.

(c) Causal diamonds. For events $p,q\in A$ with $p\le q$, the causal diamond of the ordered pair $(p,q)$ is the set

$J^{+}(p)\cap J^{-}(q)=\{r\in A:p\le r\le q\}\mathrm{.}$

The diamond is defined only under the hypothesis $p\le q$; some of its properties are computed in the examples page of this pair.

(d) Reachability by future causal paths. A piecewise $C^1$ future causal path is a piecewise $C^1$ curve $x:[a,b]\to A$ on a compact interval with $a<b$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length, Curves, regular causal curves and future timelike worldlines in affine Minkowski space) all of whose tangents are future causal: at every interior point of every open smooth piece the tangent $x^{\mathrm{\prime }}(s)$ exists and is future causal, and at every partition point of $[a,b]$ and at the endpoints $a$ and $b$ both one-sided difference-quotient tangents that exist there exist and are future causal vectors in the sense of Timelike, null, spacelike and causal vectors of a Lorentz vector space and Time orientation, future and past timelike/null/causal vectors, and the closed cones. For events $p,q\in A$, say that $q$ is reachable from $p$ by a future causal path when $p=q$, or when $p\mathrm{\ne }q$ and there are real numbers $a<b$ and a piecewise $C^1$ future causal path $x:[a,b]\to A$ with $x(a)=p$ and $x(b)=q$. Write $R^{+}(p)$ for the set of events reachable from $p$ in this sense; thus $p\in R^{+}(p)$ by the first clause.

(e) Conventions and scope. All objects above are defined from the affine structure of $A$, the form $\eta$ and the time orientation alone: no chart, no choice of origin, no spatial orientation of $V$ and no unit convention enters their definition. This item asserts no property of these objects beyond their being well-defined sets and relations: the reflexivity of $\le$ is part of (b), while the transitivity of $\le$, the identification $R^{+}(p)=J^{+}(p)$, the antisymmetry "$p\le q$ and $q\le p$ imply $p=q$", the monotonicity of inertial coordinate time along future causal paths and the invariance of these objects under proper orthochronous affine Lorentz maps are proved later on this page and are not used here. No choice principle is used: all objects are subsets or finite data built from a finite-dimensional real vector space.

Let $(A,V,\eta ,C^{+})$ be the oriented time-oriented affine Minkowski space (Oriented time-oriented affine Minkowski space (the spacetime primitives)) and let $p,q\in A$ with $p\mathrm{\ne }q$ and $q-p$ spacelike (Timelike, null, spacelike and causal vectors of a Lorentz vector space). Then:

(a) In the chart $x$ of every inertial frame the coordinate differences $D:=x(q)-x(p)$ satisfy $\mathrm{\mid }{D}^0\mathrm{\mid }<\mathrm{\mid }{D}_s\mathrm{\mid }$, where $\mathrm{\mid }{D}_s\mathrm{\mid }$ is the Euclidean norm of the spatial part $D_s:=(D^1,D^2,D^3)$ of $D$ (The Euclidean spatial structure and cross product of an inertial frame); in particular $D_s\mathrm{\ne }0$.

(b) There is an inertial frame whose chart $x_0$ satisfies $x_0^0(q)=x_0^0(p)$: the two events are simultaneous in that frame.

(c) There are inertial frames whose charts $x_{+}$ and $x_{-}$ satisfy $x_{+}^0(q)>x_{+}^0(p)$ and $x_{-}^0(q)<x_{-}^0(p)$.

(d) Explicit family. Fix an inertial frame with chart $x$ and let $e:=D_s\mathrm{/}\mathrm{\mid }{D}_s\mathrm{\mid }$, so that $e$ is a unit spatial vector of that frame. For every real number $b$ with $\mathrm{\mid }b\mathrm{\mid }<1$ let $B(be)$ be the passive boost of The passive Lorentz boost with parameter beta with parameter $be$, and let $x_b$ be the chart of the inertial frame whose ordered basis is $e_a^{\mathrm{\prime }}:=B(be{)}^{-1}{e}_a$ and whose origin event is the same event $o$. Then $x_b=B(be)\circ x$ (the linear map acting on coordinate columns) and $x_b^0(q)-x_b^0(p)={\gamma }_b(D^0-b\mathrm{\mid }{D}_s\mathrm{\mid }),{\gamma }_b:=(1-b^2{)}^{-1\mathrm{/}2}\mathrm{.}$ This number is positive for $b<D^0\mathrm{/}\mathrm{\mid }{D}_s\mathrm{\mid }$, zero for $b=D^0\mathrm{/}\mathrm{\mid }{D}_s\mathrm{\mid }$ and negative for $b>D^0\mathrm{/}\mathrm{\mid }{D}_s\mathrm{\mid }$; since $-1<D^0\mathrm{/}\mathrm{\mid }{D}_s\mathrm{\mid }<1$ by (a), all three cases occur among values $\mathrm{\mid }b\mathrm{\mid }<1$, and the family $b\mapsto x_b$ depends continuously on $b$.

(e) Scope. The lemma concerns the inertial charts of the model; "simultaneous in the frame" means equality of the frame's time coordinate $x^0$, and no synchronization procedure, observer convention or physical identification is asserted. No choice principle is used.

Let $(A,V,\eta ,C^{+})$ be the oriented time-oriented affine Minkowski space (Oriented time-oriented affine Minkowski space (the spacetime primitives), Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space) and use the reachability objects $J^{+}$, $I^{+}$ of Causal reachability, causal futures and pasts, causal diamonds, and reachability by future causal paths in affine Minkowski space. Let $(o;e_0,e_1,e_2,e_3)$ be an inertial frame with chart $x$ (Inertial frames and inertial charts of the affine Minkowski model) and let $\mathrm{\mid }{v}_s\mathrm{\mid }:=\sqrt{(v^1{)}^2+(v^2{)}^2+(v^3{)}^2}$ be the Euclidean norm of the spatial part $v_s:=v^1{e}_1+v^2{e}_2+v^3{e}_3$ of a vector $v=v^0{e}_0+v_s$ (The Euclidean spatial structure and cross product of an inertial frame). Then:

(a) Cone description. $C^{+}=\{v:v^0>\mathrm{\mid }{v}_s\mathrm{\mid }\}$, $\overline{C^{+}}=\{v:v^0\ge \mathrm{\mid }{v}_s\mathrm{\mid }\}$, $C^{-}=\{v:-v^0>\mathrm{\mid }{v}_s\mathrm{\mid }\}$ and $\overline{C^{-}}=\{v:-v^0\ge \mathrm{\mid }{v}_s\mathrm{\mid }\}$; moreover every $v\in \overline{C^{+}}$ with $v\mathrm{\ne }0$ satisfies $-\eta (v,e_0)=v^0>0$ and $\mathrm{\mid }{v}_s\mathrm{\mid }\le v^0$, and $v$ is future null exactly when $\mathrm{\mid }{v}_s\mathrm{\mid }=v^0>0$.

(b) Finite sums. If $v_1,\dots ,v_k$ lie in $\overline{C^{+}}$ ($k\ge 1$), then $v_1+\cdots +v_k\in \overline{C^{+}}$; if in addition at least one $v_i$ lies in $C^{+}$, then the sum lies in $C^{+}$; and if $v+w=0$ with $v,w\in \overline{C^{+}}$, then $v=w=0$.

(c) Coordinate criterion. For events $p,q\in A$ put $D^a:=x^a(q)-x^a(p)$ and $D_s:=(D^1,D^2,D^3)$. Then: $q\in J^{+}(p)$ if and only if $D^0\ge \mathrm{\mid }{D}_s\mathrm{\mid }$; $q\in I^{+}(p)$ if and only if $D^0>\mathrm{\mid }{D}_s\mathrm{\mid }$; $q\in J^{+}(p)$ with $q\mathrm{\ne }p$ if and only if $D^0\ge \mathrm{\mid }{D}_s\mathrm{\mid }$ and $D^0>0$; and for $q\in J^{+}(p)$ with $q\mathrm{\ne }p$ one has $0<D^0$ and $\mathrm{\mid }{D}_s\mathrm{\mid }\le D^0$. The criteria are stated for the chart of any inertial frame, because every inertial basis is $\eta$-orthonormal with future first vector.

(d) Scope. The lemma is a statement about the affine model and the fixed time orientation; it attaches no physical meaning to $\le$, $J^{+}$ or $I^{+}$ and uses no choice principle.

Adopt as a physical assumption of this framework the following restriction on signal transmission, called the causal signal restriction of the model.

(i) Causal restriction. Let $p$ and $q$ be the emission and reception events of an idealized transmission that carries a signal, where a signal is understood as information, or a physical marker by which a prepared operation at the emission event is correlated with an observation at the reception event, within the model. Mathematically, the admitted endpoint pairs form a relation $\mathcal{T}\subseteq A\times A$, with $(p,q)\in \mathcal{T}$ meaning that the model admits a transmission from $p$ to $q$. The adopted restriction is $\mathcal{T}\subseteq \{(p,q):q\in J^{+}(p)\}$. Membership in this latter set is a necessary condition; it does not assert that a transmission exists for every such pair. Thus the model admits such a transmission only if $q$ lies in the causal future $J^{+}(p)$ of $p$ (Causal reachability, causal futures and pasts, causal diamonds, and reachability by future causal paths in affine Minkowski space), that is, only if the displacement $q-p$ is zero or future causal (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones); equivalently, no signal connects an event with an event outside its closed future light cone.

(ii) Ideal light. The ideal vacuum light signals of Adopted ideal-clock and ideal-vacuum-light interpretation are the null boundary case of the restriction: for distinct emission and reception events their displacement is future null, so in the chart of every inertial frame with $x^0=ct$ (Inertial frames and inertial charts of the affine Minkowski model, SI units and coordinate conventions of the special-relativity framework) they attain $\mathrm{\mid }{x}_s(q)-x_s(p)\mathrm{\mid }=x^0(q)-x^0(p)$.

(iii) Status. This is an adopted assumption about the idealized model; it is not a theorem of the affine geometry of Oriented time-oriented affine Minkowski space (the spacetime primitives), it is not derived here from the other postulates or from any experiment, and it is not an empirical claim about nature. In particular the framework does not assert that no physical process whatsoever exceeds $c$, and the question whether any specified physical process is a signal of this model, and with what reliability and reach, is left to a separate experimental interface.

(iv) Explicit non-restriction. The assumption does not forbid kinematic patterns in which spacelike-separated events are produced by independent prearranged mechanisms without any signal or energy transfer between them (for example a prearranged row of devices triggered in a fixed sequence), and it imposes no bound on correlations between spacelike-separated events.

(v) Scope. The assumption concerns idealized point-like transmission events in the flat classical model and carries no statistical, apparatus, source or detector content; no choice principle is used.

The restriction is adopted, not proved. The cited treatments motivate the reading that the model's causal structure is what a signal can respect — the cone as a statement of possible cause and effect and the absence of signals outside it (Adopted ideal-clock and ideal-vacuum-light interpretation, whose light-ray clause supplies the boundary case), and the reversal of the temporal order of the endpoints of a superluminal transmission in a moving inertial frame, which is what makes the restriction significant rather than vacuous. Crowell Section 4.7.1 presents its arguments for the restriction as a defense in depth and explicitly calls the causality argument the weakest; the source's own textual status is recorded here rather than converted into a derivation. Crowell Section 2.1.3 also records a modest causality postulate P1, the existence of events whose displacement is timelike in every frame; that is a different, existence-style statement, it is not re-minted or adopted here, and no identification with the restriction of clause (i) is made. Tolman Sections 51-52 give a one-way order-reversal computation in older notation, used on this page only as a motivation for the conditional relay analysis; no measurement is reported.

Let $(A,V,\eta ,C^{+})$ be the oriented time-oriented affine Minkowski space with the reachability objects $\le$, $<$, $J^{+}$, $I^{+}$ and $R^{+}$ of Causal reachability, causal futures and pasts, causal diamonds, and reachability by future causal paths in affine Minkowski space.

(a) Displacement of a future causal path. Let $a<b$ be real numbers and let $x:[a,b]\to A$ be a piecewise $C^1$ future causal path with $D:=x(b)-x(a)$. Then $D\in \overline{C^{+}}$. If $x$ has a future timelike one-sided tangent at some point of $[a,b]$, then $D\in C^{+}$. If $x$ is nonconstant, then $D\mathrm{\ne }0$.

(b) Reachability and straight segments. The relation $\le$ is reflexive and transitive, and $J^{+}(p)=R^{+}(p)$, and for $q\mathrm{\ne }p$ one has $q\in J^{+}(p)$ if and only if the straight path $s\mapsto p+s(q-p)$, $s\in [0,1]$, is a future causal path; moreover $I^{+}(p)=\{q:\text{the straight path from }p\text{ to }q\text{ is future timelike}\}=\{q:q\text{ is reachable from }p\text{ by a piecewise }{C}^1\text{ future causal path having a future timelike one-sided tangent at some point}\},$ and $p$ is in neither of these two sets.

(c) No causal loops. If $p\le q$ and $q\le p$ then $p=q$; equivalently there is no nonconstant piecewise $C^1$ future causal path $x:[a,b]\to A$ with $x(b)=x(a)$.

(d) Time monotonicity. For the chart $x$ of every inertial frame and every piecewise $C^1$ future causal path $y:[a,b]\to A$, the composition $x^0\circ y$ is nondecreasing, and if $y$ is nonconstant on a subinterval $[a^{\mathrm{\prime }},b^{\mathrm{\prime }}]$ of $[a,b]$ then $(x^0\circ y)(b^{\mathrm{\prime }})>(x^0\circ y)(a^{\mathrm{\prime }})$; in particular every inertial frame agrees that a nonconstant future causal path does not go backwards in coordinate time.

(e) Straight-worldline maximum. Let $p,q$ be events with $q\in I^{+}(p)$, put $D:=q-p$ and $T:=\sqrt{-\eta (D,D)}>0$. Then for every piecewise $C^1$ future timelike worldline $y:[a,b]\to A$ with $y(a)=p$ and $y(b)=q$, the elapsed proper time $\mathrm{\Delta }\tau (y)=c^{-1}{\sum }_i{\int }_{(a_i,b_i)}\sqrt{-\eta (y^{\mathrm{\prime }},y^{\mathrm{\prime }})},$ defined by the finite improper-piece sum of Total proper time of a piecewise C^1 future timelike worldline on the weak $C^1$ pieces $a=a_1<b_1=a_2<\cdots <b_m=b$, is at most $T\mathrm{/}c$; equality holds if and only if the image of $y$ is the straight segment from $p$ to $q$ and $y$ traverses it with a strictly increasing time coordinate, that is, $y$ is a strictly increasing reparametrization of $s\mapsto p+sD$.