Why Ordering Relations Causality: Brief History Kinematic Spaces: . . . First Result: . . . Kinematic Spaces: Symmetry: a . . . Motivations from de Vries Algebras: . . . Relation Between . . . Space-Time Physics, Main Product Operations . . . Main Result Mathematical Results, Home Page Algorithmic Results and Title Page Challenges, and Possible ◭◭ ◮◮ ◭ ◮ Relation to de Vries Algebras Page 1 of 32 Go Back Vladik Kreinovich and Francisco Zapata Department of Computer Science, University of Texas Full Screen El Paso, Texas, USA, vladik@utep.edu, fazg74@gmail.com Close Quit
Why Ordering Relations Causality: Brief History 1. Why Ordering Relations Kinematic Spaces: . . . • Traditionally, in physics, space-times are described by First Result: . . . (pseudo-)Riemann spaces, i.e.: Symmetry: a . . . de Vries Algebras: . . . – by smooth manifolds Relation Between . . . – with a tensor metric field g ij ( x ). Product Operations . . . • However, in several physically interesting situations Main Result smoothness is violated and metric is undefined: Home Page – near the singularity (Big Bang), Title Page – at the black holes, and ◭◭ ◮◮ – on the microlevel, when we take into account quan- ◭ ◮ tum effects. Page 2 of 32 • In all these situations, what remains is causality � – Go Back an ordering relation. Full Screen • Geometers H. Busemann, R. Pimenov, physicists E. Kro- nheimer, R. Penrose: a theory of kinematic spaces. Close Quit
Why Ordering Relations Causality: Brief History 2. Causality: Brief History Kinematic Spaces: . . . • In Newton’s physics, signals can potentially travel with First Result: . . . an arbitrarily large speed. Symmetry: a . . . de Vries Algebras: . . . • Let a = ( t, x ) denote an event occurring at the spatial Relation Between . . . location x at time t . Product Operations . . . • Then, an event a = ( t, x ) can influence an event a ′ = Main Result ( t ′ , x ′ ) if and only if t ≤ t ′ . Home Page • The fundamental role of the non-trivial causality rela- Title Page tion emerged with the Special Relativity (SRT). ◭◭ ◮◮ • In SRT, the speed of all the signals is limited by the ◭ ◮ speed of light c . Page 3 of 32 • As a result, a = ( t, x ) � a ′ = ( t ′ , x ′ ) if and only if t ′ ≥ t and d ( x, x ′ ) Go Back ≤ c , i.e.: t ′ − t Full Screen � c · ( t ′ − t ) ≥ 1 ) 2 + ( x 2 − x ′ 2 ) 2 + ( x 3 − x ′ ( x 1 − x ′ 3 ) 2 . Close Quit
Why Ordering Relations Causality: Brief History 3. Causality: A Graphical Description Kinematic Spaces: . . . ✻ t First Result: . . . x = − c · t x = c · t Symmetry: a . . . ❅ � ❅ � de Vries Algebras: . . . ❅ � ❅ � Relation Between . . . ❅ � ❅ � Product Operations . . . ❅ � ❅ � Main Result ❅ � Home Page ❅ � ❅ � ❅ � ✲ Title Page x ◭◭ ◮◮ ◭ ◮ Page 4 of 32 Go Back Full Screen Close Quit
Why Ordering Relations Causality: Brief History 4. Importance of Causality Kinematic Spaces: . . . • In the original special relativity theory, causality was First Result: . . . just one of the concepts. Symmetry: a . . . de Vries Algebras: . . . • Its central role was revealed by A. D. Alexandrov (1950) Relation Between . . . who showed that in SRT, causality implied Lorenz group: Product Operations . . . • Every order-preserving transforming of the corr. partial Main Result ordered set is linear, and is a composition of: Home Page – spatial rotations, Title Page – Lorentz transformations (describing a transition to ◭◭ ◮◮ a moving reference frame), and ◭ ◮ – re-scalings x → λ · x (corresponding to a change of Page 5 of 32 unit for measuring space and time). Go Back • This theorem was later generalized by E. Zeeman and is known as the Alexandrov-Zeeman theorem . Full Screen Close Quit
Why Ordering Relations Causality: Brief History 5. When is Causality Experimentally Confirmable? Kinematic Spaces: . . . • In many applications, we only observe an event b with First Result: . . . some accuracy. Symmetry: a . . . de Vries Algebras: . . . • For example, in physics, we may want to check what Relation Between . . . is happening exactly 1 second after a certain reaction. Product Operations . . . • However, in practice, we cannot measure time exactly, Main Result so, we observe an event occurring 1 ± 0 . 001 sec after a . Home Page • In general, we can only guarantee that the observed Title Page event is within a certain neighborhood U b of the event b . ◭◭ ◮◮ • Because of this uncertainty, the only possibility to ex- ◭ ◮ perimentally confirm that a can influence b is when Page 6 of 32 a ≺ b ⇔ ∃ U b ∀ � b ∈ U b ( a � � b ) . Go Back • In topological terms, this means that b is in the interior Full Screen K + a of the closed cone C + a = { c : a � c } . Close Quit
Why Ordering Relations Causality: Brief History 6. Kinematic Orders Kinematic Spaces: . . . • In physics, a ≺ b correspond to influences with speeds First Result: . . . smaller than the speed of light. Symmetry: a . . . de Vries Algebras: . . . • There are two types of objects: Relation Between . . . – objects with non-zero rest mass can travel with any Product Operations . . . possible speed v < c but not with the speed c ; Main Result – objects with zero rest mass (e.g., photons) can travel Home Page only with the speed c , but not with v < c . Title Page • Thus, ≺ correspond to causality by traditional (kine- ◭◭ ◮◮ matic) objects. ◭ ◮ • Because of this: Page 7 of 32 – the relation ≺ is called kinematic causality , and Go Back – spaces with this relation ≺ are called kinematic Full Screen spaces . Close Quit
Why Ordering Relations Causality: Brief History 7. Kinematic Spaces: Towards a Description Kinematic Spaces: . . . • To describe space-time, we thus need a (pre-)ordering First Result: . . . relation � (causality) and topology (= closeness). Symmetry: a . . . de Vries Algebras: . . . • Natural continuity: for every event a and for every neighborhood U a , there exist a − ≺ a and a ≺ a + . Relation Between . . . Product Operations . . . • Natural topology: every neighborhood U a contains an Main Result open interval ( a ′ , a ′′ ) = { b : a ′ ≺ c ≺ a ′′ } . Home Page • Natural idea: a motion with speed c is a limit of mo- Title Page tions with speeds v < c when v → c . ◭◭ ◮◮ • Resulting description of � in terms of ≺ : C + = K + � � ◭ ◮ and C − = K − , i.e., b � a ⇔ ∀ U b ∃ � � b ∈ U b & � b b ≻ a . Page 8 of 32 b ≺ b ′′ hence • For U b = ( b ′ , b ′′ ), when b ≺ b ′′ , we get a ≺ � Go Back a ≺ b ′′ . Full Screen • Thus, a � b ⇔ ∀ c ( b ≺ c ⇒ a ≺ c ) . Close Quit
Why Ordering Relations Causality: Brief History 8. Resulting Definition Kinematic Spaces: . . . • A set X with a partial order ≺ is called a kinematic First Result: . . . space if is satisfies the following conditions: Symmetry: a . . . de Vries Algebras: . . . ∀ a ∃ a − , a + ( a − ≺ a ≺ a + ); Relation Between . . . ∀ a, b ( a ≺ b → ∃ c ( a ≺ c ≺ b )); Product Operations . . . ∀ a, b, c ( a ≺ b, c → ∃ d ( a ≺ d ≺ b, c )); Main Result ∀ a, b, c ( b, c ≺ a → ∃ d ( b, c ≺ d ≺ a )) . Home Page • We take a topology generated by intervals Title Page ( a, b ) = { c : a ≺ c ≺ b } . ◭◭ ◮◮ • A kinematic space is called normal if ◭ ◮ b ∈ { c : c ≻ a } ⇔ a ∈ { c : c ≺ b } . Page 9 of 32 Go Back • For a normal kinematic space, we denote b ∈ { c : c ≻ a } by a � b . Full Screen • It is proven that a ≺ b � c and a � b ≺ c imply a ≺ c . Close Quit
Why Ordering Relations Causality: Brief History 9. First Result: Reconstructing ≺ from � Kinematic Spaces: . . . • We consider separable kinematic spaces. First Result: . . . Symmetry: a . . . • We say that a space is complete if every � -decreasing de Vries Algebras: . . . bounded sequence { s n } has a limit, i.e., ∧ s n . Relation Between . . . • Lemma. If every closed intervals { c : a � c � b } is Product Operations . . . compact, then the space is complete. Main Result • If two complete separable normal kinematic orders ≺ Home Page and ≺ ′ on X lead to the same closed order � = � ′ , then Title Page ≺ = ≺ ′ . ◭◭ ◮◮ ◭ ◮ • Let S e denote the set of all � -decreasing sequences s = { s n } for which ∧ s n = e . Page 10 of 32 • For s, s ′ ∈ S e , we define s ≥ s ′ ⇔ ∀ n ∃ m ( s n � s ′ m ); Go Back then: Full Screen a ≻ b ⇔ ∃ e � b ∃ s = { s n } ∈ S e ( s is largest in S e & s 1 = a ) . Close Quit
Recommend
More recommend
