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Worked coordinate criteria: four displacements classified by the future-cone inequalities

Example

Worked computation in the chart $x$ of an inertial frame of an affine Minkowski space with $η_{0} = diag (- 1, 1, 1, 1)$ (Inertial frames and inertial charts of the affine Minkowski model). Put $p : = x^{- 1} (0, 0, 0, 0)$ and, in order,

$q_{A} : = x^{- 1} (5, 3, 0, 0), q_{B} : = x^{- 1} (5, 5, 0, 0), q_{C} : = x^{- 1} (5, 7, 0, 0), q_{D} : = x^{- 1} (5, - 3, 0, 0),$

with $D_{A}, D_{B}, D_{C}, D_{D}$ the corresponding displacements $q_{∙} - p$ and $∣ D_{∙ s} ∣$ the Euclidean norm of the spatial part. Then:

(a) Future timelike. $D_{A} = (5, 3, 0, 0)$ has $D_{A}^{0} = 5 > 3 = ∣ D_{A s} ∣$ , so $q_{A}$ lies in $I^{+} (p)$ (Causal reachability, causal futures and pasts, causal diamonds, and reachability by future causal paths in affine Minkowski space) and $η (D_{A}, D_{A}) = - 25 + 9 = - 16 < 0$ confirms that $D_{A}$ 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. $D_{B} = (5, 5, 0, 0)$ has $D_{B}^{0} = ∣ D_{B s} ∣ = 5 > 0$ , so $q_{B}$ lies in $J^{+} (p)$ but not in $I^{+} (p)$ , and $D_{B}$ is future null.

(c) Spacelike and causally unrelated. $D_{C} = (5, 7, 0, 0)$ has $∣ D_{C}^{0} ∣ = 5 < 7 = ∣ D_{C s} ∣$ , so $D_{C}$ is spacelike and $q_{C}$ lies in neither $J^{+} (p)$ nor $J^{-} (p)$ : the pair $(p, q_{C})$ is causally unrelated.

(d) Future timelike with past-directed companion. $D_{D} = (5, - 3, 0, 0)$ has $η (D_{D}, D_{D}) = - 25 + 9 = - 16 < 0$ and $D_{D}^{0} = 5 > 0$ , so $D_{D}$ is future timelike; the past-directed companion is the event $q_{E} : = x^{- 1} (- 5, 3, 0, 0)$ , whose displacement $(- 5, 3, 0, 0)$ has $η = - 16 < 0$ , time component $- 5 < 0$ and $∣ x_{s} ∣ = 3$ , so $q_{E}$ lies in $I^{-} (p)$ .

(e) The strict inequality $D^{0} > ∣ D_{s} ∣$ , the equality $D^{0} = ∣ D_{s} ∣ > 0$ and the reversal $∣ D^{0} ∣ < ∣ D_{s} ∣$ 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 $x$ , $η_{0} = diag (- 1, 1, 1, 1)$ , the events $p = x^{- 1} (0, 0, 0, 0)$ , $q_{A} = x^{- 1} (5, 3, 0, 0)$ , $q_{B} = x^{- 1} (5, 5, 0, 0)$ , $q_{C} = x^{- 1} (5, 7, 0, 0)$ , $q_{D} = x^{- 1} (5, - 3, 0, 0)$ and $q_{E} = x^{- 1} (- 5, 3, 0, 0)$ .

[L1]

Coordinate criterion: for events $p, q$ with $D = x (q) - x (p)$ , $q \in J^{+} (p)$ iff $D^{0} \geq ∣ D_{s} ∣$ , and $q \in I^{+} (p)$ iff $D^{0} > ∣ D_{s} ∣$ ; moreover $q \in J^{-} (p)$ iff $p \in J^{+} (q)$ iff the displacement $p - q$ satisfies $(p - q)^{0} \geq ∣ (p - q)_{s} ∣$ (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).

[L2]

Diagonal form: $η (D, D) = - (D^{0})^{2} + ∣ D_{s} ∣^{2}$ for coordinate differences $D$ ; a displacement is timelike iff $η (D, D) < 0$ , null iff $D \neq 0$ and $η (D, D) = 0$ , spacelike iff $η (D, D) > 0$ , and future-directed causal displacements are exactly the nonzero displacements with $D^{0} \geq ∣ D_{s} ∣$ (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

technique · direct

1.1givenL2algebra

The four displacements are $D_{A} = (5, 3, 0, 0)$ , $D_{B} = (5, 5, 0, 0)$ , $D_{C} = (5, 7, 0, 0)$ and $D_{D} = (5, - 3, 0, 0)$ by the linearity clause of the inertial chart, with spatial norms $3$ , $5$ , $7$ and $3$ ; the companion displacement is $D_{E} = q_{E} - p = (- 5, 3, 0, 0)$ with spatial norm $3$ .

2.1step 1.1L1L2algebra

$D_{A}$ : $D_{A}^{0} = 5 > 3 = ∣ D_{A s} ∣$ , so $q_{A} \in I^{+} (p)$ by [L1] and $η (D_{A}, D_{A}) = - 25 + 9 = - 16 < 0$ makes $D_{A}$ future timelike by [L2]; this is (a).

2.2step 1.1L1L2algebra

$D_{B}$ : $D_{B}^{0} = 5 = ∣ D_{B s} ∣ > 0$ , so $q_{B} \in J^{+} (p)$ and $q_{B} \notin I^{+} (p)$ by [L1], and $η (D_{B}, D_{B}) = - 25 + 25 = 0$ with $D_{B} \neq 0$ makes $D_{B}$ future null by [L2]; this is (b).

2.3step 1.1L1L2algebra

$D_{C}$ : $∣ D_{C}^{0} ∣ = 5 < 7 = ∣ D_{C s} ∣$ gives $η (D_{C}, D_{C}) = - 25 + 49 = 24 > 0$ , so $D_{C}$ is spacelike by [L2]; the criterion $D^{0} \geq ∣ D_{s} ∣$ fails for both $D_{C}$ and $- D_{C} = p - q_{C}$ , that is $D_{C}^{0} \geq ∣ D_{C s} ∣$ is false and $(p - q_{C})^{0} = - 5 \geq ∣ D_{C s} ∣ = 7$ is false, so $q_{C} \in J^{+} (p)$ and $q_{C} \in J^{-} (p)$ both fail by [L1], and the pair is causally unrelated; this is (c).

2.4step 1.1L1L2algebra

$D_{D}$ : $D_{D}^{0} = 5 > 3 = ∣ D_{D s} ∣$ with $η (D_{D}, D_{D}) = - 25 + 9 = - 16 < 0$ makes $D_{D}$ future timelike and $q_{D} \in I^{+} (p)$ by [L1] and [L2]. For the companion, the displacement from $p$ to $q_{E}$ is $D_{E} = (- 5, 3, 0, 0)$ , whose reversed displacement $(p - q_{E})$ has time component $5 > ∣ (p - q_{E})_{s} ∣ = 3$ , so $p \in I^{+} (q_{E})$ , that is $q_{E} \in I^{-} (p)$ , by [L1]; moreover $η (D_{E}, D_{E}) = - 25 + 9 = - 16 < 0$ with time component $- 5 < 0$ exhibits a past timelike displacement; this is (d).

3.1step 2.1step 2.2step 2.3step 2.4∎

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 coordinate criteria: four displacements classified by the future-cone inequalities

Example

Facts & Assumptions

Verification

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