Computational Aspects of Computational . . . Physical Models Based - PowerPoint PPT Presentation

Introduction: Main . . . Symmetry of . . . Towards a General . . . A New Model: . . . Computational Aspects of Computational . . . Physical Models Based on Euclidean Space: Proof Berwald-Moore-based From Euclidean to . . . Finsler Geometry: From Minkowski to . . . General Discussion Computational Complexity Towards Analyzing . . . Title Page and Specifics of ◭◭ ◮◮ Relativistic Celestial Mechanics ◭ ◮ Testing Page 1 of 24 Vladik Kreinovich Department of Computer Science Go Back University of Texas at El Paso Full Screen El Paso, Texas 79968, USA emails vladik@utep.edu Close Quit

Introduction: Main . . . Symmetry of . . . 1. Introduction: Main Ideas Behind Special Relativity Towards a General . . . • Special relativity started with two principles. A New Model: . . . Computational . . . • Relativity principle: all inertial motions are physically Euclidean Space: Proof equivalent. From Euclidean to . . . • Additional idea: all physical velocities are limited by From Minkowski to . . . the speed of light c : | � v | ≤ c . Discussion • Special relativity: an event e = ( t, x ) can causally in- Towards Analyzing . . . fluence an event e ′ = ( t ′ , x ′ ) ( e � e ′ ) if it is possible, Title Page – starting at location x at moment t , ◭◭ ◮◮ – reach location x ′ at moment t ′ , ◭ ◮ – while traveling at speed | � v | ≤ c . Page 2 of 24 • Resulting formula: d ( x, x ′ ) Go Back ≤ c t ′ − t Full Screen Close Quit

Introduction: Main . . . Symmetry of . . . 2. Symmetry of Relativity Theory Towards a General . . . • Reminder: e = ( t, x ) � e ′ = ( t ′ , x ′ ) ⇔ d ( x, x ′ ) A New Model: . . . ≤ c . t ′ − t Computational . . . • Resulting formula: t ′ > t and c 2 · ( t ′ − t ) 2 − d 2 ( x, x ′ ) ≥ 0. Euclidean Space: Proof From Euclidean to . . . • Kinematic causality relation: generated by moving bod- From Minkowski to . . . ies (with non-zero rest mass), for which | � v | < c. Discussion • Formula: t ′ > t and c 2 · ( t ′ − t ) 2 − d 2 ( x, x ′ ) > 0. Towards Analyzing . . . • Symmetry: the future cone is homogeneous : Title Page – for every three events e, e ′ , e ′′ for which e ≺ e ′ and ◭◭ ◮◮ e ≺ e ′′ , ◭ ◮ – there exists a causality-preserving transformation Page 3 of 24 that transforms ( e, e ′ ) into ( e, e ′′ ). Go Back Full Screen Close Quit

Introduction: Main . . . Symmetry of . . . 3. From Special Relativity to More Physical Curved Space-Times Towards a General . . . A New Model: . . . • Causality (reminder): Computational . . . e = ( t, x ) � e ′ = ( t ′ , x ′ ) ⇔ d ( x, x ′ ) Euclidean Space: Proof ≤ c. t ′ − t From Euclidean to . . . • Einstein and Minkowski proposed the pseudo-Euclidean From Minkowski to . . . Minkowski metric Discussion s 2 (( t, x ) , ( t ′ , x ′ )) = c 2 · ( t ′ − t ) 2 − d 2 ( x, x ′ ) . Towards Analyzing . . . Title Page • Fact: this metric forms the basis of the current Riemannian- ◭◭ ◮◮ geometry-based physical theories of space-time. ◭ ◮ • Problems: from the physical viewpoint, there are still Page 4 of 24 many problems with current space-time models. Go Back • Result: search for more general geometrical models. Full Screen • Reasonable idea: search for a basic space-time model which is different from Minkowski space. Close Quit

Introduction: Main . . . Symmetry of . . . 4. Towards a General Symmetric Space-Time Model Towards a General . . . • Objective: find a general basic space-time model. A New Model: . . . Computational . . . • Main physical requirement: basic symmetries: Euclidean Space: Proof – shift-invariance, From Euclidean to . . . – scale-invariance, From Minkowski to . . . – homogeneous (= satisfies relativity principle). Discussion • Mathematical description: a “flat” space-time ( R n ) in Towards Analyzing . . . which the causality relation is: Title Page – shift-invariant: e ≺ e ′ ⇔ e + e ′′ ≺ e ′ + e ′′ ; ◭◭ ◮◮ – scale-invariant: e ≺ e ′ ⇔ λ · e ≺ λ · e ′ ; and ◭ ◮ – homogeneous: Page 5 of 24 ∗ for every three events e, e ′ , e ′′ for which e ≺ e ′ Go Back and e ≺ e ′′ , Full Screen ∗ there exists a causality-preserving transforma- tion that transforms ( e, e ′ ) into ( e, e ′′ ). Close Quit

Introduction: Main . . . Symmetry of . . . 5. Classification of Symmetric Space-Time Models Towards a General . . . • Reminder: we are looking for orderings ≺ of R n which A New Model: . . . are: Computational . . . Euclidean Space: Proof – shift- and scale-invariant and – homogeneous on the future cone { e ′ : e ≺ e ′ } . From Euclidean to . . . From Minkowski to . . . • Classification theorem (A.D. Alexandrov): each such Discussion space-time is Towards Analyzing . . . Title Page – either a Minkowski space, – or a Cartesian product X 1 × . . . × X n of Minkowski ◭◭ ◮◮ spaces of smaller dimension: ◭ ◮ ( x 1 , . . . , x n ) ≺ ( x ′ 1 , . . . , x ′ n ) ⇔ ( x 1 ≺ x ′ 1 ) & . . . & ( x n ≺ x ′ n ) . Page 6 of 24 • 4-D example: the product of 4 subspaces R is R 4 with Go Back the ordering x ≺ x ′ iff x i < x ′ i for all i . Full Screen Close Quit

Introduction: Main . . . Symmetry of . . . 6. A New Model: Symmetries and Metrics Towards a General . . . • Ordering (reminder): x ≺ x ′ iff x i < x ′ i for all i . A New Model: . . . Computational . . . • General symmetries: x i → f i ( x i ). Euclidean Space: Proof • Symmetries consistent with shift and scaling: From Euclidean to . . . x i → a i · x i + b i . From Minkowski to . . . • From ordering to a metric τ ( x, x ′ ) – requirements: Discussion – shift-invariant: τ ( x, x ′ ) = τ ( x + x ′′ , x ′ + x ′′ ); Towards Analyzing . . . Title Page – scale-invariant: τ ( λ · x, λ · x ′ ) = λ · τ ( x, x ′ ); ◭◭ ◮◮ – homogeneous: under ◭ ◮ def T ( x 1 , . . . , x n ) = ( a 1 · x 1 + b 1 , . . . , a 1 · x 1 + b 1 ) , Page 7 of 24 we have τ ( T ( x ) , T ( x ′ )) = c ( T ) · τ ( x, x ′ ). Go Back � 4 � 1 / 4 • Conclusion: τ ( x, x ′ ) = ( x ′ � i − x i ) . Full Screen i =1 Close • Comment: this is Berwald-Moore metric. Quit

Introduction: Main . . . Symmetry of . . . 7. From the Basic Space-Time to General Space-Time Models Towards a General . . . A New Model: . . . • Traditional basis: Minkowski metric. Computational . . . Euclidean Space: Proof • Traditional extension: spaces which are locally isomor- From Euclidean to . . . phic to the Minkowski metric – pseudo-Riemannian spaces. From Minkowski to . . . Discussion • New basis: Berwald-Moore metric. Towards Analyzing . . . • Natural idea: consider physical models of space-time Title Page based on this metric. ◭◭ ◮◮ • To be more precise: models based on the Finsler spaces ◭ ◮ which are locally isomorphic to this metric. Page 8 of 24 Go Back Full Screen Close Quit

Introduction: Main . . . Symmetry of . . . 8. Relation of Berwald-Moore Coordinates to Usual Physical Coordinates Towards a General . . . A New Model: . . . • Berwald-Moore coordinates: a i for which Computational . . . � 3 � 1 / 4 Euclidean Space: Proof � τ ( a, a ′ ) = ( a ′ i − a i ) . From Euclidean to . . . From Minkowski to . . . i =0 Discussion • Usual physical coordinates: x 0 = c · t and x i . Towards Analyzing . . . • Relation – example: Title Page a 0 = x 0 + 1 3 · ( x 1 + x 2 + x 3 ) , a 1 = x 0 + 1 ◭◭ ◮◮ √ √ 3 · ( x 1 − x 2 − x 3 ) , ◭ ◮ a 2 = x 0 + 1 3 · ( − x 1 + x 2 − x 3 ) , a 3 = x 0 + 1 √ √ 3 · ( − x 1 − x 2 + x 3 ) . Page 9 of 24 Go Back Full Screen Close Quit

Introduction: Main . . . Symmetry of . . . 9. Computational Complexity of Prediction Problems Towards a General . . . • First problem: A New Model: . . . Computational . . . – how the change in geometry Euclidean Space: Proof – affects the computational complexity of the corre- From Euclidean to . . . sponding predictions. From Minkowski to . . . • Euclidean space (result): Discussion – if we want to know all the distances with a given Towards Analyzing . . . Title Page accuracy ε > 0, – then it is sufficient to find all the coordinates x i of ◭◭ ◮◮ all the events with a similar accuracy O ( ε ). ◭ ◮ Page 10 of 24 Go Back Full Screen Close Quit

Introduction: Main . . . Symmetry of . . . 10. Euclidean Space: Proof Towards a General . . . • Result (reminder): A New Model: . . . Computational . . . – to know all the distances with a given accuracy Euclidean Space: Proof ε > 0, From Euclidean to . . . – it is sufficient to find all the coordinates x i of all From Minkowski to . . . the events with a similar accuracy O ( ε ). Discussion • Proof: Towards Analyzing . . . – triangle inequality implies that Title Page | d ( x, y ) − d ( x ′ , y ′ ) | ≤ d ( x, x ′ ) + d ( y, y ′ ); ◭◭ ◮◮ – for Euclidean metric, we have ◭ ◮ d ( x, x ′ ) ≤ | x 1 − x ′ 1 | + . . . + | x n − x ′ n | ; Page 11 of 24 – hence, if | x i − x ′ i | ≤ ε and | y i − y ′ Go Back i | ≤ ε , then d ( x, x ′ ) ≤ n · ε , d ( y, y ′ ) ≤ n · ε , and Full Screen | d ( x, y ) − d ( x ′ , y ′ ) | ≤ 2 n · ε. Close Quit