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Metrization Theorem Space-Time Analogs . . . How the (Non- . . . - PowerPoint PPT Presentation

Jun 10, 2023 •660 likes •831 views

Causality: A Reminder Urysohns Problem: . . . Space-Time Models: . . . Space-Time Analog of . . . Metrization Theorem Space-Time Analogs . . . How the (Non- . . . for Space-Times: Constructive . . . Constructive . . . A Constructive

  1. Causality: A Reminder Urysohn’s Problem: . . . Space-Time Models: . . . Space-Time Analog of . . . Metrization Theorem Space-Time Analogs . . . How the (Non- . . . for Space-Times: Constructive . . . Constructive . . . A Constructive Solution Constructive Space- . . . to Urysohn’s Problem Constructive Space- . . . Auxiliary Results Vladik Kreinovich Symmetries: A . . . Acknowledgments Department of Computer Science Title Page University of Texas at El Paso 500 W. University ◭◭ ◮◮ El Paso, TX 79968, USA vladik@utep.edu ◭ ◮ Page 1 of 16 http://www.cs.utep.edu/vladik Go Back Full Screen

  2. Causality: A Reminder Urysohn’s Problem: . . . 1. Urysohn’s Lemma and Urysohn’s Metrization The- Space-Time Models: . . . orem: Reminder Space-Time Analog of . . . Space-Time Analogs . . . • Who, when: early 1920s, Pavel Urysohn. How the (Non- . . . • Claim for fame: Urysohn’s Lemma is “first non-trivial Constructive . . . result of point set topology”. Constructive . . . • Condition: X is a normal topological space X , A and Constructive Space- . . . B are disjoint closed sets. Constructive Space- . . . Auxiliary Results • Conclusion: there exists f : X → [0 , 1] s.t. f ( A ) = { 0 } Symmetries: A . . . and f ( B ) = { 1 } . Acknowledgments • Reminder: normal means that every two disjoint closed Title Page sets have disjoint open neighborhoods. ◭◭ ◮◮ • Application: every normal space with countable base ◭ ◮ is metrizable. Page 2 of 16 • Comment: actually, every regular Hausdorff space with countable base is metrizable. Go Back Full Screen

  3. Causality: A Reminder Urysohn’s Problem: . . . 2. Extension to Space-Times: Urysohn’s Problem Space-Time Models: . . . Space-Time Analog of . . . • Fact: a few years before that, in 1919, Einstein’s GRT Space-Time Analogs . . . has been experimentally confirmed. How the (Non- . . . • Corresponding structure: topological space with an or- Constructive . . . der (casuality). Constructive . . . • Urysohn’s problem: extend his lemma and metrization Constructive Space- . . . theorem to (causality-)ordered topological spaces. Constructive Space- . . . Auxiliary Results • Tragic turn of events: Urysohn died in 1924. Symmetries: A . . . • Follow up: Urysohn’s student Vadim Efremovich; Efre- Acknowledgments movich’s student Revolt Pimenov; Pimenov’s students. Title Page • Other researchers: H. Busemann (US), E. Kronheimer ◭◭ ◮◮ and R. Penrose (UK). ◭ ◮ • Result: by the 1970s, space-time versions of Uryson’s Page 3 of 16 lemma and metrization theorem have been proven. Go Back Full Screen

  4. Causality: A Reminder Urysohn’s Problem: . . . 3. Causality: A Reminder Space-Time Models: . . . Space-Time Analog of . . . ✻ t Space-Time Analogs . . . x = − c · t x = c · t ❅ � How the (Non- . . . ❅ � ❅ � Constructive . . . ❅ � ❅ � Constructive . . . ❅ � ❅ � Constructive Space- . . . ❅ � ❅ � Constructive Space- . . . ❅ � ❅ � Auxiliary Results ❅ � ✲ x Symmetries: A . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 16 Go Back Full Screen

  5. Causality: A Reminder Urysohn’s Problem: . . . 4. Urysohn’s Problem: Remaining Issues Space-Time Models: . . . Space-Time Analog of . . . • Main issue: the 1970s results are not constructive. Space-Time Analogs . . . • Why this is important: we want useful applications to How the (Non- . . . physics. Constructive . . . • What we have now: theoretical existence of a pseudo- Constructive . . . metric. Constructive Space- . . . Constructive Space- . . . • What we need: an algorithm generating such a metric Auxiliary Results based on the empirical causality. Symmetries: A . . . • Also: we need a physically relevant constructive de- Acknowledgments scription of a causality-type ordering relation. Title Page • Our objective: ◭◭ ◮◮ – to propose such a description, and ◭ ◮ – to prove constructive space-time versions of the Uryson’s Page 5 of 16 lemma and metrization theorem. Go Back Full Screen

  6. Causality: A Reminder Urysohn’s Problem: . . . 5. Space-Time Models: Reminder Space-Time Models: . . . Space-Time Analog of . . . • Theoretical relation: (transitive) causality a � b . Space-Time Analogs . . . a ≈ a , � • Problem: events are not located exactly: � b ≈ b . How the (Non- . . . • Practical relation: kinematic casuality a ≺ b . Constructive . . . Constructive . . . • Meaning: every event in some small neighborhood of b Constructive Space- . . . causally follows a , i.e., b ∈ Int( a + ). Constructive Space- . . . • Properties of ≺ : ≺ is transitive; a �≺ a ; Auxiliary Results ∀ a ∃ a, a ( a ≺ a ≺ a ); a ≺ b ⇒ ∃ c ( a ≺ c ≺ b ); Symmetries: A . . . Acknowledgments a ≺ b, c ⇒ ∃ d ( a ≺ d ≺ b, c ); b, c ≺ a ⇒ ∃ d ( b, c ≺ d ≺ a ) . Title Page • Alexandrov topology: with intervals as the base: ◭◭ ◮◮ def ( a, b ) = { c : a ≺ c ≺ b } . ◭ ◮ def ≡ b ∈ a + . • Description of causality: a � b Page 6 of 16 • Additional property: b ∈ a + ⇔ a ∈ b − . Go Back Full Screen

  7. Causality: A Reminder Urysohn’s Problem: . . . 6. Space-Time Analog of a Metric Space-Time Models: . . . Space-Time Analog of . . . • Traditional metric: a function ρ : X × X → R + 0 s.t. Space-Time Analogs . . . ρ ( a, b ) = 0 ⇔ a = b ; How the (Non- . . . Constructive . . . ρ ( a, b ) = ρ ( b, a ); Constructive . . . ρ ( a, c ) ≤ ρ ( a, b ) + ρ ( b, c ) . Constructive Space- . . . • Physical meaning: the length of the shortest path be- Constructive Space- . . . tween a and b . Auxiliary Results • Kinematic metric: a function τ : X × X → R + Symmetries: A . . . 0 s.t. Acknowledgments τ ( a, b ) > 0 ⇔ a ≺ b ; Title Page a ≺ b ≺ c ⇒ τ ( a, c ) ≥ τ ( a, b ) + τ ( b, c ) . ◭◭ ◮◮ ◭ ◮ • Physical meaning: the longest (= proper) time from event a to event b . Page 7 of 16 • Explanation: when we speed up, time slows down. Go Back Full Screen

  8. Causality: A Reminder Urysohn’s Problem: . . . 7. Space-Time Analogs of Urysohn’s Lemma and Metriza- Space-Time Models: . . . tion Theorem Space-Time Analog of . . . Space-Time Analogs . . . • Main condition: the kinematic space is separable , i.e., How the (Non- . . . there exists a countable dense set { x 1 , x 2 , . . . , x n , . . . } . Constructive . . . • Condition of the lemma: X is separable, and a ≺ b . Constructive . . . • Lemma: ∃ a cont. � -increasing f-n f ( a,b ) : X → [0 , 1] Constructive Space- . . . s.t. f ( a,b ) ( x ) = 0 for a �≺ x and f ( a,b ) ( x ) = 1 for b � x . Constructive Space- . . . Auxiliary Results • Relation to the original Urysohn’s lemma: f ( a,b ) sepa- rates disjoint closed sets − a + and b + . Symmetries: A . . . Acknowledgments • Condition of the theorem: ( X, ≺ ) is a separable kine- Title Page matic space. ◭◭ ◮◮ • Theorem: there exists a continuous metric τ which gen- ◭ ◮ erates the corresponding relation ≺ . Page 8 of 16 • Corollary: τ also generates the corresponding topology. Go Back Full Screen

  9. Causality: A Reminder Urysohn’s Problem: . . . 8. How the (Non-Constructive) Space-Time Metriza- Space-Time Models: . . . tion Theorem Is Proved Space-Time Analog of . . . Space-Time Analogs . . . • First lemma: for every x , there exists a ≺ -monotonic How the (Non- . . . function f x : X → [0 , 1] for which f x ( b ) > 0 ⇔ x ≺ b . Constructive . . . � ∞ 2 − i · f ( x,y i ) ( b ). • Proof: ∃ y i ց x ; take f x ( b ) = Constructive . . . i =1 Constructive Space- . . . • Second lemma: for every x , there exists a ≺ -monotonic Constructive Space- . . . function g x : X → [0 , 1] for which g x ( a ) > 0 ⇔ a ≺ x . Auxiliary Results • Proof: similar. Symmetries: A . . . Acknowledgments • Resulting metric: for a countable everywhere dense se- Title Page quence { x 1 , x 2 , . . . , x n , . . . } , take ◭◭ ◮◮ ∞ � 2 − i · min( g x i ( a ) , f x i ( b )) . τ ( a, b ) = ◭ ◮ i =1 Page 9 of 16 Go Back Full Screen

  10. Causality: A Reminder Urysohn’s Problem: . . . 9. Constructive Causality: What Does It mean? Space-Time Models: . . . Space-Time Analog of . . . • How to find causality: we send a signal at event a : Space-Time Analogs . . . – if this signal is detected at b , then a � b ; How the (Non- . . . – if this signal is not detected at b , then a �� b . Constructive . . . Constructive . . . • Practical problem: we can only locate an event with a Constructive Space- . . . certain accuracy. Constructive Space- . . . • Result: we have 3 options: Auxiliary Results – if the signal is detected in the entire vicinity of b , Symmetries: A . . . then a ≺ b ; Acknowledgments Title Page – if no signal is detected in the entire vicinity of b , then a �� b ; ◭◭ ◮◮ – in all other cases, we do not know. ◭ ◮ • Conclusion: we have relations ≺ n corr. to increasing Page 10 of 16 location accuracy, so a ≺ b ⇔ ∃ n ( a ≺ n b ) . Go Back Full Screen

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