Sheaves of Structures in Physics: From Relativistic Causality to the Foundations of Quantum Mechanics
|
|
- Jade Hutchinson
- 5 years ago
- Views:
Transcription
1 Sheaves of Structures in Physics: From Relativistic Causality to the Foundations of Quantum Mechanics J. Benavides
2 The Stability of the Global Hyperbolic Property and Cosmic Censor Whether black holes are fundamental phenomena related to the gravitational collapse of massive objects or simply a particular feature of certain kind of gravitational collapse models, is one of the more important open problems in general relativity.
3 The Stability of the Global Hyperbolic Property and Cosmic Censor Whether black holes are fundamental phenomena related to the gravitational collapse of massive objects or simply a particular feature of certain kind of gravitational collapse models, is one of the more important open problems in general relativity. Cosmic Censorship Hypothesis (CCH) The complete gravitational collapse of a body results in the formation of a black hole rather than a naked singularity.
4 The Stability of the Global Hyperbolic Property and Cosmic Censor Whether black holes are fundamental phenomena related to the gravitational collapse of massive objects or simply a particular feature of certain kind of gravitational collapse models, is one of the more important open problems in general relativity. Cosmic Censorship Hypothesis (CCH) The complete gravitational collapse of a body results in the formation of a black hole rather than a naked singularity. The fundamental principle acting on the formation of black holes is normally associated to a process acting to conserve classical determinism to some degree.
5 Weak Cosmic Censorship (WCC) In physical reasonable spacetimes the black hole exterior region is globally hyperbolic.
6 Strong Cosmic Censorship (SCC) Every physical reasonable spacetime is globally hyperbolic. Weak Cosmic Censorship (WCC) In physical reasonable spacetimes the black hole exterior region is globally hyperbolic.
7 Deficiencies of WCC and SCC The definition of the black hole exterior region makes use of curves that escape to spacetime infinity= no local description. The SCC does not give a definition of black holes. The SCC discards as physical reasonable any spacetime where causality is violated.
8 Deficiencies of WCC and SCC The definition of the black hole exterior region makes use of curves that escape to spacetime infinity= no local description. The SCC does not give a definition of black holes. The SCC discards as physical reasonable any spacetime where causality is violated. Can we obtain a satisfactory formulation of the CCH within the classical relativistic formalism? do WCC or SCC capture the fundamental property related to the formation of event horizons?
9 Deficiencies of WCC and SCC The definition of the black hole exterior region makes use of curves that escape to spacetime infinity= no local description. The SCC does not give a definition of black holes. The SCC discards as physical reasonable any spacetime where causality is violated. Can we obtain a satisfactory formulation of the CCH within the classical relativistic formalism? do WCC or SCC capture the fundamental property related to the formation of event horizons? We can try to answer these questions studying stability properties of the global hyperbolic property.
10 The Stability of the Global Hyperbolic Property The Stability of the Global Hyperbolic Property The global hyperbolic property A global hyperbolic spacetime (M, g) is a causal spacetime such that for every p, q M, J + (p) J (q) is a compact set.
11 The Stability of the Global Hyperbolic Property The Stability of the Global Hyperbolic Property The global hyperbolic property A global hyperbolic spacetime (M, g) is a causal spacetime such that for every p, q M, J + (p) J (q) is a compact set. Interval Topology Let Lor(M) be the space of Lorentzian metrics over M, given g, g Lor(M), we say that g < g if for every p M, the causal cone of g in T pm is contained in the timelike cone of g, we say that g g if for every p M, the causal cone of g in T pm is contained in the causal cone of g. The intervals (g, g ) = {h Lor(M) : g < h < g }, with g, g Lor(M), generate a topology in Lor(M)
12 The Stability of the Global Hyperbolic Property A property of the spacetime (M, g) is said to be stable with respect to the interval topology if there exists an interval (h, h ) of the interval topology such that g (h, h ) and for all g (h, h ) the property holds for the spacetime (M, g ).
13 The Stability of the Global Hyperbolic Property A property of the spacetime (M, g) is said to be stable with respect to the interval topology if there exists an interval (h, h ) of the interval topology such that g (h, h ) and for all g (h, h ) the property holds for the spacetime (M, g ). Theorem Global hyperbolicity is stable in the interval topology.
14 The Stability of the Global Hyperbolic Property A property of the spacetime (M, g) is said to be stable with respect to the interval topology if there exists an interval (h, h ) of the interval topology such that g (h, h ) and for all g (h, h ) the property holds for the spacetime (M, g ). proof Let h be a complete Riemannian metric over M. Fix m M, let K 0 = m and define inductively K n+1 = J + (B h (m, r n )) J (B h (m, r n )), with r n > r n such that K n IntB h (m, r n ). Theorem Global hyperbolicity is stable in the interval topology.
15 The Stability of the Global Hyperbolic Property K n is compact for all n. n=0 K n = M, For every compact set C there exists n such that C K n, For n 0, J + (K n ) J (K n ) intk n+1,
16 The Stability of the Global Hyperbolic Property K n is compact for all n. n=0 K n = M, For every compact set C there exists n such that C K n, For n 0, J + (K n ) J (K n ) intk n+1, Then we construct a family of metrics {g n } n s.t. g n > g, g n+1 < g n, g n+1 = g n on K n and J g + n (K n ) Jg n (K n ) intk n+1.
17 The Stability of the Global Hyperbolic Property K n is compact for all n. n=0 K n = M, For every compact set C there exists n such that C K n, For n 0, J + (K n ) J (K n ) intk n+1, Then we construct a family of metrics {g n } n s.t. g n > g, g n+1 < g n, g n+1 = g n on K n and J g + n (K n ) Jg n (K n ) intk n+1. Let g 0 > g be sucht that (M, g 0 ) is stably causal. Suppose we have constructed g n, since = J + (K n+1 ) J (K n+1 ) K n+2 = g<g <g n (J + g (K n+1 ) J g (K n+1 ) K n+2 ), g < g n+1 < g n such that J + (K g n+1 n+1 ) J g n+1(k n+1 ) K n+2 = and J + (K g n+1 n+1 ) J g n+1(k n+1 ) IntK n+2. Consider χ : M R continuous s.t. 0 χ 1, χ = 0 on K n and χ = 1 on M \ intk n+1. Let g n+1 = χg n+1 + (1 χ)g n.
18 The Stability of the Global Hyperbolic Property K n is compact for all n. n=0 K n = M, For every compact set C there exists n such that C K n, For n 0, J + (K n ) J (K n ) intk n+1, Then we construct a family of metrics {g n } n s.t. g n > g, g n+1 < g n, g n+1 = g n on K n and J g + n (K n ) Jg n (K n ) intk n+1. Let g 0 > g be sucht that (M, g 0 ) is stably causal. Suppose we have constructed g n, since = J + (K n+1 ) J (K n+1 ) K n+2 = g<g <g n (J + g (K n+1 ) J g (K n+1 ) K n+2 ), g < g n+1 < g n such that J + (K g n+1 n+1 ) J g n+1(k n+1 ) K n+2 = and J + (K g n+1 n+1 ) J g n+1(k n+1 ) IntK n+2. Consider χ : M R continuous s.t. 0 χ 1, χ = 0 on K n and χ = 1 on M \ intk n+1. Let g n+1 = χg n+1 + (1 χ)g n. Define h > g on M such that h (p) = g n(p) on K n, h g 0 is stably causal, J + h (p) J h (q) is closed p, q and J + h (p) J h (q) J + h (K n) J h (K n) K n+1 Then (M, h ) is globally hyperbolic.
19 The Stability of the Global Hyperbolic Property The stability of other familiar properties as the energy conditions, or the existence of Hawking-Penrose singularities follow straightforward from the regularity of the differential structure. On the other hand, the stability of global hyperbolicity is highly non-trivial.
20 The Stability of the Global Hyperbolic Property The stability of other familiar properties as the energy conditions, or the existence of Hawking-Penrose singularities follow straightforward from the regularity of the differential structure. On the other hand, the stability of global hyperbolicity is highly non-trivial. Can we expect the existence of generic properties within the relativistic formalism able to capture collapsing gravitational scenarios and the formation of event horizons,and that imply the validity of the global hyperbolic property?
21 The Stability of the Global Hyperbolic Property The stability of other familiar properties as the energy conditions, or the existence of Hawking-Penrose singularities follow straightforward from the regularity of the differential structure. On the other hand, the stability of global hyperbolicity is highly non-trivial. Can we expect the existence of generic properties within the relativistic formalism able to capture collapsing gravitational scenarios and the formation of event horizons,and that imply the validity of the global hyperbolic property? Or should we take seriously counterexamples of the SCC, particularly those which contain causality violations? Denying the physical importance of these kind of examples we are taking a position that is very reminiscent to the position assumed during the first years of general relativity in confront to solutions of the Einstein equations which predicted the formation of singularities.
22 Causality Violations and Quantum Geometry Causality violations as black holes are one of the most intriguing predictions of classical relativistic formalism. These phenomena have been normally considered as unphysical mainly because, within the classical formalism, causality violations seem to impose more initial data constraints than those normally required in causality respecting spacetimes.
23 Causality Violations and Quantum Geometry Causality violations as black holes are one of the most intriguing predictions of classical relativistic formalism. These phenomena have been normally considered as unphysical mainly because, within the classical formalism, causality violations seem to impose more initial data constraints than those normally required in causality respecting spacetimes. Even if these kind of initial constraints have not been observed so far, these do not provide a conclusive answer respect to the unphysical character of causality violations.
24 Causality Violations and Quantum Geometry Causality violations as black holes are one of the most intriguing predictions of classical relativistic formalism. These phenomena have been normally considered as unphysical mainly because, within the classical formalism, causality violations seem to impose more initial data constraints than those normally required in causality respecting spacetimes. Even if these kind of initial constraints have not been observed so far, these do not provide a conclusive answer respect to the unphysical character of causality violations. Then it results important to establish whether violations of causality are indeed in real conflict with some fundamental physical principles or otherwise causality violating regions as those predicted by Kerr s solution have to be considered seriously, possibly as a fundamental hint towards a quantum gravity theory.
25 Computational Networks and The Classical Time Travel Paradoxes Computational Networks and The Classical Time Travel Paradoxes The study of physical effects of closed timelike curves via computational networks allows to abstract away most of the geometry of such situations avoiding the technicalities associated to strange objects linked to these violations (e.g singularities or wormholes) and at the same time it allows to include quantum mechanics, something not possible yet in the geometrical approach due to the lack of a satisfactory quantum gravity theory.
26 Computational Networks and The Classical Time Travel Paradoxes Computational Networks and The Classical Time Travel Paradoxes The study of physical effects of closed timelike curves via computational networks allows to abstract away most of the geometry of such situations avoiding the technicalities associated to strange objects linked to these violations (e.g singularities or wormholes) and at the same time it allows to include quantum mechanics, something not possible yet in the geometrical approach due to the lack of a satisfactory quantum gravity theory. The models obtained, even if constituted just of finite particles travelling along fixed trajectories and interacting only at short range, do not result restrictive because such models include the class of classical and quantum computational networks
27 Computational Networks and The Classical Time Travel Paradoxes Computational Networks and The Classical Time Travel Paradoxes The study of physical effects of closed timelike curves via computational networks allows to abstract away most of the geometry of such situations avoiding the technicalities associated to strange objects linked to these violations (e.g singularities or wormholes) and at the same time it allows to include quantum mechanics, something not possible yet in the geometrical approach due to the lack of a satisfactory quantum gravity theory. The models obtained, even if constituted just of finite particles travelling along fixed trajectories and interacting only at short range, do not result restrictive because such models include the class of classical and quantum computational networks The Turing Principle (Deutsch 1985) Every finitely realizable physical system can be perfectly simulated* by a universal model computing machine operating by finite means
28 Computational Networks and The Classical Time Travel Paradoxes Computational Networks and The Classical Time Travel Paradoxes The study of physical effects of closed timelike curves via computational networks allows to abstract away most of the geometry of such situations avoiding the technicalities associated to strange objects linked to these violations (e.g singularities or wormholes) and at the same time it allows to include quantum mechanics, something not possible yet in the geometrical approach due to the lack of a satisfactory quantum gravity theory. The models obtained, even if constituted just of finite particles travelling along fixed trajectories and interacting only at short range, do not result restrictive because such models include the class of classical and quantum computational networks The Turing Principle (Deutsch 1985) Every finitely realizable physical system can be perfectly simulated* by a universal model computing machine operating by finite means * A computing machine C is capable of perfect simulating a physical system S, if there exists a program π(s) for C that renders C computationally equivalent to S.
29 Computational Networks and The Classical Time Travel Paradoxes A spacetime bounded network can be simplified without altering its computational character, or in other words obtaining an alternative network where the outputs are the same function of their inputs, in the following way:
30 Computational Networks and The Classical Time Travel Paradoxes Localize all the Interactions in Gates. Closed temporal trajectories on carriers are replaced by trajectories that goes first in the ambiguous future of all gates, the with a single negative delay in the ambiguous past and then forward to the future (S 2 ). A spacetime bounded network can be simplified without altering its computational character, or in other words obtaining an alternative network where the outputs are the same function of their inputs, in the following way: Replace the particles travelling in the network by sets of particles which carry a 2-state internal degree of freedom (the carriers of the bits)
31 Computational Networks and The Classical Time Travel Paradoxes Localize all the Interactions in Gates. Closed temporal trajectories on carriers are replaced by trajectories that goes first in the ambiguous future of all gates, the with a single negative delay in the ambiguous past and then forward to the future (S 2 ). A spacetime bounded network can be simplified without altering its computational character, or in other words obtaining an alternative network where the outputs are the same function of their inputs, in the following way: Replace the particles travelling in the network by sets of particles which carry a 2-state internal degree of freedom (the carriers of the bits) Figure: Chronology violating network.
32 Computational Networks and The Classical Time Travel Paradoxes Grandfather Paradox (Classical Perspective) We can understand the grandfather paradox as a particle that travels around a closed timelike curve arriving back before its departure preventing itself from setting out.
33 Computational Networks and The Classical Time Travel Paradoxes Grandfather Paradox (Classical Perspective) We can understand the grandfather paradox as a particle that travels around a closed timelike curve arriving back before its departure preventing itself from setting out. The Values 1 or 0 represent whether the simulated particle is on that carrier or not respectively. The first two bits represent two possible trajectories that the particle can follow. Before going on their separate ways the particles carrying the bits find an older version of the right one which acts on them in the next way, x x 1 y x y x y 1 y i.e if they meet the older version of the particle they change the value of their bit changing the trajectory of the simulated particle, otherwise they do not change their value.
34 Computational Networks and The Classical Time Travel Paradoxes Grandfather Paradox (Classical Perspective) We can understand the grandfather paradox as a particle that travels around a closed timelike curve arriving back before its departure preventing itself from setting out. The Values 1 or 0 represent whether the simulated particle is on that carrier or not respectively. The first two bits represent two possible trajectories that the particle can follow. Before going on their separate ways the particles carrying the bits find an older version of the right one which acts on them in the next way, x x 1 y x y x y 1 y i.e if they meet the older version of the particle they change the value of their bit changing the trajectory of the simulated particle, otherwise they do not change their value. Consistency Condition: x y 1 = y, then x = 1. The particle has to take the left
35 Local Causality Violations and Singularities Local Causality Violations and Singularities The above analysis arise from the false premise that classical physics is approximately true near causality violating regions.
36 Local Causality Violations and Singularities Local Causality Violations and Singularities The above analysis arise from the false premise that classical physics is approximately true near causality violating regions. Causal Tube N in (M, g) N = S 1 S 2 T, is compact The S i are disjoint spacelike manifolds, S i = S2, T = S i [0, 1], timelike S 2 I + ( S 1 ), every future causal vector field V in M tangent to T points inside N in S 1 and outside N in S 2, I N (S 1) = = I + N (S 2)
37 Local Causality Violations and Singularities Theorem Let (M, g) be a spacetime where Einstein s equation and the weak energy condition hold. Let N be a causal tube in M which contains a chronology violating region and such that on every null geodesic σ past inextendible contained in N, there exists a point where R(σ, σ ) 0. Then N is non compact.
38 Local Causality Violations and Singularities Theorem Let (M, g) be a spacetime where Einstein s equation and the weak energy condition hold. Let N be a causal tube in M which contains a chronology violating region and such that on every null geodesic σ past inextendible contained in N, there exists a point where R(σ, σ ) 0. Then N is non compact. This result can be interpreted as evidence of the formation of singularities in the region N or as the incapacity of the classical formalism to give a local description of the formation of chronology violating regions. From this theorem follows that classical (non-quantum) physics cannot give an accurate description close to causality violating regions.
39 Local Causality Violations and Singularities Sketch of the Proof Let D + N (S 1) be the set of points p such that every inextendible past causal curve in N from p intersects S 1 and H + N (S 1 ) = D + N (S 1) I N (D+ N (S 1 ))
40 Local Causality Violations and Singularities Sketch of the Proof Since N contains a chronology violating region, there exists a closed timelike curve contained in N, then N D + N (S 1) and H + N (S 1) is not empty. Let D + N (S 1) be the set of points p such that every inextendible past causal curve in N from p intersects S 1 and H + N (S 1 ) = D + N (S 1) I N (D+ N (S 1 ))
41 Local Causality Violations and Singularities Sketch of the Proof Let D + N (S 1) be the set of points p such that every inextendible past causal curve in N from p intersects S 1 and Since N contains a chronology violating region, there exists a closed timelike curve contained in N, then N D + N (S 1) and H + N (S 1) is not empty. The weak energy condition and the fact that Ric(σ, σ ) 0 at some point on every past null geodesic σ contained in N imply that for every null geodesic generator γ of H + N (S 1), Ric(γ, γ ) 0 on every point and is not zero at some point. H + N (S 1 ) = D + N (S 1) I N (D+ N (S 1 ))
42 Local Causality Violations and Singularities Sketch of the Proof Let D + N (S 1) be the set of points p such that every inextendible past causal curve in N from p intersects S 1 and H + N (S 1 ) = D + N (S 1) I N (D+ N (S 1 )) Since N contains a chronology violating region, there exists a closed timelike curve contained in N, then N D + N (S 1) and H + N (S 1) is not empty. The weak energy condition and the fact that Ric(σ, σ ) 0 at some point on every past null geodesic σ contained in N imply that for every null geodesic generator γ of H + N (S 1), Ric(γ, γ ) 0 on every point and is not zero at some point. From this follows that H + N (S 1) is a closed non-compact subset of N
43 Local Causality Violations and Singularities Sketch of the Proof Let D + N (S 1) be the set of points p such that every inextendible past causal curve in N from p intersects S 1 and H + N (S 1 ) = D + N (S 1) I N (D+ N (S 1 )) Since N contains a chronology violating region, there exists a closed timelike curve contained in N, then N D + N (S 1) and H + N (S 1) is not empty. The weak energy condition and the fact that Ric(σ, σ ) 0 at some point on every past null geodesic σ contained in N imply that for every null geodesic generator γ of H + N (S 1), Ric(γ, γ ) 0 on every point and is not zero at some point. From this follows that H + N (S 1) is a closed non-compact subset of N Then N is non-compact.
44 Causality Violations: The quantum Approach Causality Violations: The Quantum Approach Since classical physics cannot give an appropriate description near causality violating regions, we require quantum computational networks to simulate the behaviour of particles close to such regions. This means:
45 Causality Violations: The quantum Approach Causality Violations: The Quantum Approach Since classical physics cannot give an appropriate description near causality violating regions, we require quantum computational networks to simulate the behaviour of particles close to such regions. This means: 1 To consider the carrier of the bits (qubits) as a quantum physical systems which non trivial observables are Boolean (i.e. observables which have just two eigenvalues). 2 The inputs/outputs of quantum gates are not necessarily elements of the computational basis (i.e. they are not necessarily eigenstates of the tensor product of the observables). 3 Computational Gates are unitary transformations over composed n-qubit systems.
46 Causality Violations: The quantum Approach A general Chronology Violating Network The initial joint state of the m + 2n bits is given by ρ 12 ρ 3,. The final joint state is given by U(ρ 12 ρ 3 )U, where U is an arbitrary Unitary interaction. Let ρ 12 = ρ 1 ρ 2 H 1 H 2 be a density operator representing the initial state of the non time travelling bits and of the younger versions of the time travelling bits. Let ρ 3 be the density operator representing the initial state of the older versions of the time travelling bits.
47 Causality Violations: The quantum Approach A general Chronology Violating Network The initial joint state of the m + 2n bits is given by ρ 12 ρ 3,. The final joint state is given by U(ρ 12 ρ 3 )U, where U is an arbitrary Unitary interaction. Consistency Condition: Tr 1,3 (U(ρ 12 ρ 3 )U ) = ρ 3, (1) i.e. the n older time travelling bits enter the gate in the same state as the younger bits leave it Let ρ 12 = ρ 1 ρ 2 H 1 H 2 be a density operator representing the initial state of the non time travelling bits and of the younger versions of the time travelling bits. Let ρ 3 be the density operator representing the initial state of the older versions of the time travelling bits.
48 Causality Violations: The quantum Approach A general Chronology Violating Network The initial joint state of the m + 2n bits is given by ρ 12 ρ 3,. The final joint state is given by U(ρ 12 ρ 3 )U, where U is an arbitrary Unitary interaction. Consistency Condition: Tr 1,3 (U(ρ 12 ρ 3 )U ) = ρ 3, (1) i.e. the n older time travelling bits enter the gate in the same state as the younger bits leave it Let ρ 12 = ρ 1 ρ 2 H 1 H 2 be a density operator representing the initial state of the non time travelling bits and of the younger versions of the time travelling bits. Let ρ 3 be the density operator representing the initial state of the older versions of the time travelling bits. Theorem (Deutsch) The equation 1 admits a solution ρ 3 for every initial state ρ 12 and every interaction U.
49 Causality Violations: The quantum Approach A general Chronology Violating Network The initial joint state of the m + 2n bits is given by ρ 12 ρ 3,. The final joint state is given by U(ρ 12 ρ 3 )U, where U is an arbitrary Unitary interaction. Consistency Condition: Tr 1,3 (U(ρ 12 ρ 3 )U ) = ρ 3, (1) i.e. the n older time travelling bits enter the gate in the same state as the younger bits leave it Let ρ 12 = ρ 1 ρ 2 H 1 H 2 be a density operator representing the initial state of the non time travelling bits and of the younger versions of the time travelling bits. Let ρ 3 be the density operator representing the initial state of the older versions of the time travelling bits. Theorem (Deutsch) The equation 1 admits a solution ρ 3 for every initial state ρ 12 and every interaction U. Quantum theory does not impose initial data constrains
50 Causality Violations: The quantum Approach The Grandfather Paradox Revisited If the input state is the one which classically leads to a paradoxical situation, i.e. ρ 12 = 0 1, we obtain that equation 1 admits as unique solution ρ 3 = 1 2 I Thus, the density operator of the two output bits is Tr 3 (U(ρ 12 ρ 3 )U ) = 1 ( ). 2 Therefore, the probability that there will be two particles in the output, or zero particles in the output is 1/2 in both cases, and the probability to obtain exactly one single particle, as we will expect based on our classical intuition, is zero! The quantum state history of the GF paradox 1 The original input state of the particle is that one that models the particle taking the trajectory that will take it back in time if nothing intervened, 2 The particle will meet an older version of itself in a mixed state ρ 3 = 1 I of being 2 present and absent, 3 The younger version enters the gate in a mixed state of being prevented or not from going back in time, 4 Finally in the unambiguous future we find the mixed state which is a state of there being two particles present and none particles present.
51 Causality Violations: The quantum Approach The Grandfather Paradox Revisited If the input state is the one which classically leads to a paradoxical situation, i.e. ρ 12 = 0 1, we obtain that equation 1 admits as unique solution ρ 3 = 1 2 I Thus, the density operator of the two output bits is Tr 3 (U(ρ 12 ρ 3 )U ) = 1 ( ). 2 Therefore, the probability that there will be two particles in the output, or zero particles in the output is 1/2 in both cases, and the probability to obtain exactly one single particle, as we will expect based on our classical intuition, is zero! The quantum state history of the GF paradox 1 The original input state of the particle is that one that models the particle taking the trajectory that will take it back in time if nothing intervened, 2 The particle will meet an older version of itself in a mixed state ρ 3 = 1 I of being 2 present and absent, 3 The younger version enters the gate in a mixed state of being prevented or not from going back in time, 4 Finally in the unambiguous future we find the mixed state which is a state of there being two particles present and none particles present. What happened to the modelled particle?
52 Causality Violations: The quantum Approach The GF paradox in the statistical interpretation of QM When the older and younger version encounter each other, the older is in a mixed state of being absent and being present. Under the statistical interpretation this means that in one half of the elements of the ensemble, that the quantum state represents, there was not encounter and in the other half there was an encounter. In those elements of the statical ensemble where there was not an encounter, the younger version travels back in time and experiences the encounter in the same event of the ensemble where there was not encounter leading to the same classical paradox, By consequence the statistical interpretation require some constrains on the original states as in the classical case, but we saw that the quantum formalism does not impose such constraints.
53 Causality Violations: The quantum Approach The GF paradox in the statistical interpretation of QM When the older and younger version encounter each other, the older is in a mixed state of being absent and being present. Under the statistical interpretation this means that in one half of the elements of the ensemble, that the quantum state represents, there was not encounter and in the other half there was an encounter. In those elements of the statical ensemble where there was not an encounter, the younger version travels back in time and experiences the encounter in the same event of the ensemble where there was not encounter leading to the same classical paradox, By consequence the statistical interpretation require some constrains on the original states as in the classical case, but we saw that the quantum formalism does not impose such constraints. The GF paradox in the Deutsch-Everett interpretation In all the universes the particle approaches the chronology violating region in a state which would take it back in time (i.e. ρ 12 = 01 ), but just in half of the universes the particle follows the time back travelling trajectory, because in half of the universes there is an encounter with a particle that looks like an older version of the particle (this follows from the fact that ρ 3 = 1 I), which changes the trajectory of 2 the younger version to avoid the time travel trajectory. After that, both versions live together in the unambiguous future. In the other half of the universes there is no encounter and the particle takes the time travelling trajectory, but travelling back to the past of one of the other half universes where the encounter happens. This is consistent with the the density operator of the output bits Tr 3 (U(ρ 12 ρ 3 )U ) = 1 ( ). 2
54 Towards a Quantum Geometry Towards a Quantum Geometry: The Deutsch-Everett Interpretation If given an observable ˇB with eigenvectors b 1,..., b n, the state vector ψ(t) is expressed in the basis formed by the eigenvectors of ˇB as: ψ(t) = α 1 b α m b m. Then a measurement of B at time t will give as a result one of the eigenvalues b i with probability α i 2 respectively, and none other result that is not one of the eigenvalues is obtained.
55 Towards a Quantum Geometry Towards a Quantum Geometry: The Deutsch-Everett Interpretation If given an observable ˇB with eigenvectors b 1,..., b n, the state vector ψ(t) is expressed in the basis formed by the eigenvectors of ˇB as: ψ(t) = α 1 b α m b m. Then a measurement of B at time t will give as a result one of the eigenvalues b i with probability α i 2 respectively, and none other result that is not one of the eigenvalues is obtained. Measurement Problem The quantum formalism is not able to distinguish the actual result of measurement from all the possible results.
56 Towards a Quantum Geometry Towards a Quantum Geometry: The Deutsch-Everett Interpretation If given an observable ˇB with eigenvectors b 1,..., b n, the state vector ψ(t) is expressed in the basis formed by the eigenvectors of ˇB as: ψ(t) = α 1 b α m b m. Then a measurement of B at time t will give as a result one of the eigenvalues b i with probability α i 2 respectively, and none other result that is not one of the eigenvalues is obtained. Measurement Problem The quantum formalism is not able to distinguish the actual result of measurement from all the possible results. Deutsch-Everett Everett 1957: QM is consistent with the idea that all possible results of the measurement actually happen, being the single universe where we perceive one single outcome part of a larger structure of many-worlds where the different outcomes happen. The multiverse is a set with a measure, whose elements are maximal sets of observables with definite values that correspond to different universes or different histories. In this context the terms α i 2 associated to the expression ψ(t) = α 1 b α m b m represent the values of the measure of the sets of universes where the observable ˇB assume the value b i respectively.
57 Towards a Quantum Geometry Different sets of compatible observables determine different expressions of the state vector ψ(t), these different forms to express the state vector correspond to different foliations of the multiverse in the same sense that a region of spacetime can be foliated by spacelike surfaces in different ways.
58 Towards a Quantum Geometry Different sets of compatible observables determine different expressions of the state vector ψ(t), these different forms to express the state vector correspond to different foliations of the multiverse in the same sense that a region of spacetime can be foliated by spacelike surfaces in different ways. Each universe in any foliation is associated to a classical system, which corresponds to the classical physical world where we see the measuring apparatus taking one unique value.
59 Towards a Quantum Geometry Different sets of compatible observables determine different expressions of the state vector ψ(t), these different forms to express the state vector correspond to different foliations of the multiverse in the same sense that a region of spacetime can be foliated by spacelike surfaces in different ways. Each universe in any foliation is associated to a classical system, which corresponds to the classical physical world where we see the measuring apparatus taking one unique value. The universes interact via interference phenomena, but such interactions are suppressed at the classical level described by classical physics.
60 Towards a Quantum Geometry The Geometry of The Grand Father Paradox Classical Evolution Zone Interference Zone
61 Towards a Quantum Geometry The Geometry of The Grand Father Paradox Classical Evolution Zone Interference Zone There is not yet a sound mathematical model of the quantum multiverse, but it is clear that if such model exists, it is probably the key towards a quantum gravity theory.
62 Sheaves of Structures Sheaves of structures (Motivation) Figure: Galilean Spacetime Galilean Spacetime A topological space X =temporal line For each x X there exists a structure A x constituted by: -A universe E x formed by snapshots of extended objects in time. - Functions f x 1, f x 2,... and relations R x 1, R x 2,..., that give the instantaneous properties of extended objects. The different worlds E x attach in an extended universe E, in such a way that the different functions and relations attach in a continuous way. A x = (E x, R x 1, R x 2..., f x 1, f x 2...)
63 Sheaves of Structures Sheaves of Structures (Definition) Definition Given a fix type of structures τ = (R 1,..., f 1,..., c 1,...) a sheaf of τ-structures A over a topological space X is given by:
64 Sheaves of Structures Sheaves of Structures (Definition) Definition Given a fix type of structures τ = (R 1,..., f 1,..., c 1,...) a sheaf of τ-structures A over a topological space X is given by: a-) A sheaf (E, p) over X( i.e a local homeomorphism).
65 Sheaves of Structures Sheaves of Structures (Definition) Definition Given a fix type of structures τ = (R 1,..., f 1,..., c 1,...) a sheaf of τ-structures A over a topological space X is given by: a-) A sheaf (E, p) over X( i.e a local homeomorphism). b-) For each x X, a τ-structure A x = (E x, R x 1, Rx 2..., f x 1,..., cx 1,...), where E x = p 1 (x) (the fiber that could be empty) is the universe of the τ-structure A x, and the following conditions are satisfied:
66 Sheaves of Structures Sheaves of Structures (Definition) Definition Given a fix type of structures τ = (R 1,..., f 1,..., c 1,...) a sheaf of τ-structures A over a topological space X is given by: a-) A sheaf (E, p) over X( i.e a local homeomorphism). b-) For each x X, a τ-structure A x = (E x, R1 x, Rx 2..., f 1 x,..., cx 1,...), where E x = p 1 (x) (the fiber that could be empty) is the universe of the τ-structure A x, and the following conditions are satisfied: i. R A = x R x is open in x E x n seeing as subspace of E n, where R is an n-ary relation symbol. ii. f A = x f x : x E x m x E x is a continuous function, where f is an m-parameter function symbol. iii. h : X E such that h(x) = c x, where c is a constant symbol, is continuous.
67 Sheaves of Structures Sheaf Logic (Motivation) A sheaf of structures is a space extended over the base space X of the sheaf as Galilean spacetime extends over time. The elements of this space are not the points of E but the sections of the sheaf conceived as extended objects. The single values of these sections represent just point-wise descriptions of the extended object. As the objects of a sheaf of structures are the sections of the sheaf, the logic which governs them should define when a property for an extended object holds in a point of its domain of definition.
68 Sheaves of Structures Sheaf Logic (Motivation) A sheaf of structures is a space extended over the base space X of the sheaf as Galilean spacetime extends over time. The elements of this space are not the points of E but the sections of the sheaf conceived as extended objects. The single values of these sections represent just point-wise descriptions of the extended object. As the objects of a sheaf of structures are the sections of the sheaf, the logic which governs them should define when a property for an extended object holds in a point of its domain of definition. Contextual Truth Paradigm If a property for an extended object holds in some point of its domain then it has to hold in a neighbourhood of that point.
69 Sheaf Logic Sheaf Logic, Point-Wise Semantics Sheaf Logic (Definition) A x ϕ[σ 1,..., σ n ] exists U neighbourhood of x such that for all y U, A y ϕ[σ 1 (y),..., σ n (y)]. A x (ϕ ψ)[σ 1,..., σ n ] Exists U neighbourhood of x such that for all y U if A y ϕ[σ 1 (y),..., σ n (y)] then A y ψ[σ 1 (y),..., σ n (y)]. A x vϕ(v, σ 1,..., σ n ) exists U neighbourhood of x such that for all y U and all σ defined in y, A y ϕ[σ(y), σ 1 (y),..., σ n (y)].
70 Sheaf Logic Sheaf Logic, Local Semantics Given an open subset U X and sections defined over U, we say that a proposition about these sections holds in U if it holds at each point in U or in other words: A U ϕ[σ 1,..., σ n] x U, A x ϕ[σ 1,..., σ n]
71 Sheaf Logic Sheaf Logic, Local Semantics Given an open subset U X and sections defined over U, we say that a proposition about these sections holds in U if it holds at each point in U or in other words: A U ϕ[σ 1,..., σ n] x U, A x ϕ[σ 1,..., σ n] Kripke-Joyal Semantics A U (ϕ ψ)[σ 1,..., σ n] there exist open sets V, W such that U = V W, A V ϕ[σ 1,..., σ n] and A W ψ[σ 1,..., σ n]. A U vϕ(v, σ 1,..., σ n) there exists {U i } i an open cover of U and µ i sections defined on U i such that A Ui ϕ[µ i, σ 1,..., σ n] for all i. A U vϕ(v, σ 1,..., σ n) for any open set W U and µ defined on W, A W ϕ(µ, σ 1,..., σ n).
72 Sheaf Logic The logic just defined can be seen as a multivalued logic with truth values that variate over the Heyting algebra of the open sets of the base space X.
73 Sheaf Logic The logic just defined can be seen as a multivalued logic with truth values that variate over the Heyting algebra of the open sets of the base space X. Let σ 1,..., σ n be sections of a sheaf A defined over an open set U, we define the truth value of a proposition ϕ in U as: [[ϕ(σ 1,..., σ n )]] U := {x U : A x ϕ[σ 1,..., σ n ]} (2) [[ϕ(σ 1,..., σ n )]] U is an open set, thus we can define a valuation as a topological valuation on formulas: T U : ϕ [[ϕ(σ 1,..., σ n )]] U.
74 Sheaf Logic The logic just defined can be seen as a multivalued logic with truth values that variate over the Heyting algebra of the open sets of the base space X. Let σ 1,..., σ n be sections of a sheaf A defined over an open set U, we define the truth value of a proposition ϕ in U as: [[ϕ(σ 1,..., σ n )]] U := {x U : A x ϕ[σ 1,..., σ n ]} (2) [[ϕ(σ 1,..., σ n )]] U is an open set, thus we can define a valuation as a topological valuation on formulas: T U : ϕ [[ϕ(σ 1,..., σ n )]] U. The definition of the logic allows to define the value of the logic operators in terms of the operations of the algebra of open sets.
75 Quantum Set Theory Variable Sets Using the comprehension axiom in classical set theory, given a proposition ϕ(x) and a set A, we can construct a set B such that x B if and only if x A and ϕ(x) is truth for x, or in other words, B = {x A : ϕ(x)}.
76 Quantum Set Theory Variable Sets Using the comprehension axiom in classical set theory, given a proposition ϕ(x) and a set A, we can construct a set B such that x B if and only if x A and ϕ(x) is truth for x, or in other words, B = {x A : ϕ(x)}. Consider the following proposition: ϕ(x) x is an even number greater or equal than 4 and x can be written as the sum of two prime numbers.
77 Quantum Set Theory Variable Sets Using the comprehension axiom in classical set theory, given a proposition ϕ(x) and a set A, we can construct a set B such that x B if and only if x A and ϕ(x) is truth for x, or in other words, B = {x A : ϕ(x)}. Consider the following proposition: ϕ(x) x is an even number greater or equal than 4 and x can be written as the sum of two prime numbers. "Now and here" the following is valid in our temporal sheaf Now and Here ({x N : ϕ(x)} = {x N : x 4 (x is even )}), because now and here we do not know if the Goldbach conjecture is valid.
78 Quantum Set Theory Variable Sets Instead of conceiving sets as absolute entities, we can conceive them as variable structures which variate over our Library of the states of knowledge.
79 Quantum Set Theory Variable Sets Instead of conceiving sets as absolute entities, we can conceive them as variable structures which variate over our Library of the states of knowledge. It is natural then to conceive the set of nodes where our states of Knowledge variates as nodes in a partial order or points in a topological space, that can represent, for instance, the causal structure of spacetime.
80 Quantum Set Theory Variable Sets Instead of conceiving sets as absolute entities, we can conceive them as variable structures which variate over our Library of the states of knowledge. It is natural then to conceive the set of nodes where our states of Knowledge variates as nodes in a partial order or points in a topological space, that can represent, for instance, the causal structure of spacetime. Our states of Knowledge will be then structures that represent the sets as we see them in our nodes. Therefore from each node we will see arise a cumulative Hierarchy of variable sets, which structure will be conditioned by the perception of the variable structures in the other nodes that relate to it. Or more precisely.
81 The valuation V over the open sets constitute an exact presheaf of structures, the respective sheaf of germs V X constitute the cumulative hierarchy of variable sets. Quantum Set Theory Definition (The Hierarchy of Variable Sets) Let X be an arbitrary topological space. Given U Op(X) we define inductively: V 0 (U) = V α+1 (U) ={f : Op(U) P(V α(w )) : 1. If W U then f (W ) V α(w ), W U 2. If V W U, then for all g f (W ), g Op(V ) f (V ), 3. Given {U i } i an open cover of U and g i f (U i ) such that g i op(ui U j )= g j op(ui U j ) for any i, j, there exists g f (U) such that g op(ui )= g i for all i} V λ (U) = α<λ V α(u) if λ is a limit ordinal, V (U) = α On V α(u).
82 Quantum Set Theory For each U Op(X) the set V (U) is a set of functions defined over Op(U) which values for W Op(U) are functions over Op(W ) which values for V Op(W ) are functions over Op(V ) and so on.
83 Quantum Set Theory For each U Op(X) the set V (U) is a set of functions defined over Op(U) which values for W Op(U) are functions over Op(W ) which values for V Op(W ) are functions over Op(V ) and so on. the relation f U g (i.e. U f g) f g(u), i.e. that respect to the context U, f belongs to g if and only if f g(u) as classical sets.
84 Quantum Set Theory For each U Op(X) the set V (U) is a set of functions defined over Op(U) which values for W Op(U) are functions over Op(W ) which values for V Op(W ) are functions over Op(V ) and so on. the relation f U g (i.e. U f g) f g(u), i.e. that respect to the context U, f belongs to g if and only if f g(u) as classical sets. Theorem For any topological space X V X ZF
85 a â(u) defines an embedding of V in V (U) for any open set U X. Quantum Set Theory For each U Op(X) the set V (U) is a set of functions defined over Op(U) which values for W Op(U) are functions over Op(W ) which values for V Op(W ) are functions over Op(V ) and so on. the relation f U g (i.e. U f g) f g(u), i.e. that respect to the context U, f belongs to g if and only if f g(u) as classical sets. Theorem For any topological space X V X ZF V V (U) To each classical set a we can associate a constant set â(u) : Op(U) V (W ) â(u)(w ) = { b(u) Op(W ) : b a}. W U
86 Quantum Set Theory Using this embedding, it can be proved that: N(U) = N(U), Z(U) = Ẑ(U), Q(U) = Q(U) for any open set U X
87 Quantum Set Theory Using this embedding, it can be proved that: N(U) = N(U), Z(U) = Ẑ(U), Q(U) = Q(U) for any open set U X These tools provide a mechanism to construct new mathematical universes over arbitrary topological spaces. If we find a topological space able to capture the essence of quantum logic this will provide a mathematical quantum universe that will probably improves our understanding of quantum mechanics.
88 Quantum Set Theory Quantum Variable Sets Foliations Let U be an abelian Von Neumann subalgebra of the algebra of operators of the Hilbert space of a quantum system. Each self-adjoint operator Ǎ U admits a spectral decomposition in U, i.e a family of projections {ˇP r } r R U such that Ǎ = rd ˇP r
89 Quantum Set Theory Quantum Variable Sets Foliations Let U be an abelian Von Neumann subalgebra of the algebra of operators of the Hilbert space of a quantum system. Each self-adjoint operator Ǎ U admits a spectral decomposition in U, i.e a family of projections {ˇP r } r R U such that Ǎ = rd ˇP r We perceive the quantum system through an abelian Von Neumann frame of observables in analogous way as we perceive classical spacetime through an inertial frame which determines a particular foliation of a spacetime region.
90 Quantum Set Theory Quantum Variable Sets The base space=the space of Histories The Gelfand spectrum S A of U is the space of positive linear functions σ : U C of norm 1 such that σ(ab) = σ(a)σ(b) for all A, B U. These are the histories because when restricted to the self-adjoint operators of U they become valuations. A valuation is a function λ from the set of self-adjoint operators B sa (U) on U to the real numbers, λ : B sa (H) R, which satisfies: 1.λ(Ǎ) belongs to the spectrum of Ǎ, for all Ǎ B sa(u) 2.λ(ˇB) = f (λ(ǎ)) whenever ˇB = f (Ǎ) with f : R R a continuous function.
91 Quantum Set Theory Quantum Variable Sets The Topology: Similar Histories Interfere Consider a self-adjoint operator Ǎ U such that Ǎ = N a n ˇPn is the spectral representation of Ǎ in U. Fix m such that 1 m N; and λ S U such that λ(ǎ) = am, then λ(ˇp m) = 1. n=1
92 Quantum Set Theory Quantum Variable Sets The Topology: Similar Histories Interfere Consider a self-adjoint operator Ǎ U such that Ǎ = N a n ˇPn n=1 is the spectral representation of Ǎ in U. Fix m such that 1 m N; and λ S U such that λ(ǎ) = am, then λ(ˇp m) = 1. Thus if Ǎ represents a physical observable, we have that in all the histories λ such that λ(ˇp m) = 1, the physical observable A assumes the value a m. Therefore, given a projection ˇP P(U) the set P = {λ S U : λ(ˇp) = 1} (3) is a context of histories which are similar in the sense that some physical observables assume the same values or the values satisfy the same inequalities in each history.
93 Quantum Set Theory Quantum Variable Sets The Topology: Similar Histories Interfere Consider a self-adjoint operator Ǎ U such that Ǎ = N a n ˇPn n=1 is the spectral representation of Ǎ in U. Fix m such that 1 m N; and λ S U such that λ(ǎ) = am, then λ(ˇp m) = 1. Thus if Ǎ represents a physical observable, we have that in all the histories λ such that λ(ˇp m) = 1, the physical observable A assumes the value a m. Therefore, given a projection ˇP P(U) the set P = {λ S U : λ(ˇp) = 1} (3) is a context of histories which are similar in the sense that some physical observables assume the same values or the values satisfy the same inequalities in each history. {P}ˇP P(U) defines a topology in S U.
Black Holes and Thermodynamics I: Classical Black Holes
Black Holes and Thermodynamics I: Classical Black Holes Robert M. Wald General references: R.M. Wald General Relativity University of Chicago Press (Chicago, 1984); R.M. Wald Living Rev. Rel. 4, 6 (2001).
More informationInitial-Value Problems in General Relativity
Initial-Value Problems in General Relativity Michael Horbatsch March 30, 2006 1 Introduction In this paper the initial-value formulation of general relativity is reviewed. In section (2) domains of dependence,
More informationExotica or the failure of the strong cosmic censorship in four dimensions
Exotica or the failure of the strong cosmic censorship in four dimensions Budapest University of Technology and Economics Department of Geometry HUNGARY Stará Lesná, Slovakia, 20 August 2015 The meaning
More informationCosmic Censorship Conjecture and Topological Censorship
Cosmic Censorship Conjecture and Topological Censorship 21 settembre 2009 Cosmic Censorship Conjecture 40 years ago in the Rivista Nuovo Cimento Sir Roger Penrose posed one of most important unsolved problems
More information16. Time Travel 1 1. Closed Timelike Curves and Time Travel
16. Time Travel 1 1. Closed Timelike Curves and Time Travel Recall: The Einstein cylinder universe is a solution to the Einstein equations (with cosmological constant) that has closed spacelike curves.
More informationThe cosmic censorship conjectures in classical general relativity
The cosmic censorship conjectures in classical general relativity Mihalis Dafermos University of Cambridge and Princeton University Gravity and black holes Stephen Hawking 75th Birthday conference DAMTP,
More informationQuantum Information Types
qitd181 Quantum Information Types Robert B. Griffiths Version of 6 February 2012 References: R. B. Griffiths, Types of Quantum Information, Phys. Rev. A 76 (2007) 062320; arxiv:0707.3752 Contents 1 Introduction
More informationExact Solutions of the Einstein Equations
Notes from phz 6607, Special and General Relativity University of Florida, Fall 2004, Detweiler Exact Solutions of the Einstein Equations These notes are not a substitute in any manner for class lectures.
More informationStability and Instability of Black Holes
Stability and Instability of Black Holes Stefanos Aretakis September 24, 2013 General relativity is a successful theory of gravitation. Objects of study: (4-dimensional) Lorentzian manifolds (M, g) which
More informationCosmic Censorship. Emily Redelmeier (student number ) May 1, 2003
Cosmic Censorship Emily Redelmeier (student number 990740016) May 1, 2003 1 Introduction One of the fundamental differences between the relativistic universe and the Newtonian one is the behaviour of information.
More informationLecture notes 1. Standard physics vs. new physics. 1.1 The final state boundary condition
Lecture notes 1 Standard physics vs. new physics The black hole information paradox has challenged our fundamental beliefs about spacetime and quantum theory. Which belief will have to change to resolve
More informationOLIVIA MILOJ March 27, 2006 ON THE PENROSE INEQUALITY
OLIVIA MILOJ March 27, 2006 ON THE PENROSE INEQUALITY Abstract Penrose presented back in 1973 an argument that any part of the spacetime which contains black holes with event horizons of area A has total
More informationAn Overview of Mathematical General Relativity
An Overview of Mathematical General Relativity José Natário (Instituto Superior Técnico) Geometria em Lisboa, 8 March 2005 Outline Lorentzian manifolds Einstein s equation The Schwarzschild solution Initial
More informationA Brief Introduction to Mathematical Relativity
A Brief Introduction to Mathematical Relativity Arick Shao Imperial College London Arick Shao (Imperial College London) Mathematical Relativity 1 / 31 Special Relativity Postulates and Definitions Einstein
More informationSpeed limits in general relativity
Class. Quantum Grav. 16 (1999) 543 549. Printed in the UK PII: S0264-9381(99)97448-8 Speed limits in general relativity Robert J Low Mathematics Division, School of Mathematical and Information Sciences,
More information14. Black Holes 2 1. Conformal Diagrams of Black Holes
14. Black Holes 2 1. Conformal Diagrams of Black Holes from t = of outside observer's clock to t = of outside observer's clock time always up light signals always at 45 time time What is the causal structure
More informationarxiv:gr-qc/ v1 31 Jul 2001
SINGULARITY AVOIDANCE BY COLLAPSING SHELLS IN QUANTUM GRAVITY 1 arxiv:gr-qc/0107102v1 31 Jul 2001 Petr Hájíček Institut für Theoretische Physik, Universität Bern, Sidlerstrasse 5, 3012 Bern, Switzerland.
More informationRelativistic simultaneity and causality
Relativistic simultaneity and causality V. J. Bolós 1,, V. Liern 2, J. Olivert 3 1 Dpto. Matemática Aplicada, Facultad de Matemáticas, Universidad de Valencia. C/ Dr. Moliner 50. 46100, Burjassot Valencia),
More informationGravitation: Tensor Calculus
An Introduction to General Relativity Center for Relativistic Astrophysics School of Physics Georgia Institute of Technology Notes based on textbook: Spacetime and Geometry by S.M. Carroll Spring 2013
More informationSingularities and Causal Structure in General Relativity
Singularities and Causal Structure in General Relativity Alexander Chen February 16, 2011 1 What are Singularities? Among the many profound predictions of Einstein s general relativity, the prediction
More informationIs Spacetime Hole-Free?
Is Spacetime Hole-Free? John Byron Manchak Abstract Here, we examine hole-freeness a condition sometimes imposed to rule out seemingly artificial spacetimes. We show that under existing definitions (and
More informationA Remark About the Geodesic Principle in General Relativity
A Remark About the Geodesic Principle in General Relativity Version 3.0 David B. Malament Department of Logic and Philosophy of Science 3151 Social Science Plaza University of California, Irvine Irvine,
More informationCausal Structure of General Relativistic Spacetimes
Causal Structure of General Relativistic Spacetimes A general relativistic spacetime is a pair M; g ab where M is a di erentiable manifold and g ab is a Lorentz signature metric (+ + +::: ) de ned on all
More informationCausality and Boundary of wave solutions
Causality and Boundary of wave solutions IV International Meeting on Lorentzian Geometry Santiago de Compostela, 2007 José Luis Flores Universidad de Málaga Joint work with Miguel Sánchez: Class. Quant.
More informationSo the question remains how does the blackhole still display information on mass?
THE ZERO POINT NON-LOCAL FRAME AND BLACKHOLES By: Doctor Paul Karl Hoiland Abstract: I will show that my own zero point Model supports not only the no-hair proposals, but also the Bekenstein bound on information
More informationGeometric inequalities for black holes
Geometric inequalities for black holes Sergio Dain FaMAF-Universidad Nacional de Córdoba, CONICET, Argentina. 3 August, 2012 Einstein equations (vacuum) The spacetime is a four dimensional manifold M with
More informationSingularity formation in black hole interiors
Singularity formation in black hole interiors Grigorios Fournodavlos DPMMS, University of Cambridge Heraklion, Crete, 16 May 2018 Outline The Einstein equations Examples Initial value problem Large time
More informationGeneral Relativity in AdS
General Relativity in AdS Akihiro Ishibashi 3 July 2013 KIAS-YITP joint workshop 1-5 July 2013, Kyoto Based on work 2012 w/ Kengo Maeda w/ Norihiro Iizuka, Kengo Maeda - work in progress Plan 1. Classical
More informationarxiv:gr-qc/ v1 8 Nov 2000
New properties of Cauchy and event horizons Robert J. Budzyński Department of Physics, Warsaw University, Hoża 69, 00-681 Warsaw, Poland arxiv:gr-qc/0011033v1 8 Nov 2000 Witold Kondracki Andrzej Królak
More informationTOPOS THEORY IN THE FORMULATION OF THEORIES OF PHYSICS
TOPOS THEORY IN THE FORMULATION OF THEORIES OF PHYSICS August 2007 Chris Isham Based on joint work with Andreas Doering Theoretical Physics Group Blackett Laboratory Imperial College, London c.isham@imperial.ac.uk
More informationare Banach algebras. f(x)g(x) max Example 7.4. Similarly, A = L and A = l with the pointwise multiplication
7. Banach algebras Definition 7.1. A is called a Banach algebra (with unit) if: (1) A is a Banach space; (2) There is a multiplication A A A that has the following properties: (xy)z = x(yz), (x + y)z =
More information1 The role of gravity The view of physics that is most generally accepted at the moment is that one can divide the discussion of the universe into
1 The role of gravity The view of physics that is most generally accepted at the moment is that one can divide the discussion of the universe into two parts. First, there is the question of the local laws
More informationIncompatibility Paradoxes
Chapter 22 Incompatibility Paradoxes 22.1 Simultaneous Values There is never any difficulty in supposing that a classical mechanical system possesses, at a particular instant of time, precise values of
More informationIs spacetime hole-free?
Gen Relativ Gravit (2009) 41:1639 1643 DOI 10.1007/s10714-008-0734-1 RESEARCH ARTICLE Is spacetime hole-free? John Byron Manchak Received: 24 September 2008 / Accepted: 27 November 2008 / Published online:
More informationBLACK HOLES (ADVANCED GENERAL RELATIV- ITY)
Imperial College London MSc EXAMINATION May 2015 BLACK HOLES (ADVANCED GENERAL RELATIV- ITY) For MSc students, including QFFF students Wednesday, 13th May 2015: 14:00 17:00 Answer Question 1 (40%) and
More informationDef. 1. A time travel spacetime is a solution to Einstein's equations that admits closed timelike curves (CTCs).
17. Time Travel 2 Def. 1. A time travel spacetime is a solution to Einstein's equations that admits closed timelike curves (CTCs). Def. 2. A time machine spacetime is a time travel spacetime in which the
More informationQUANTUM FIELD THEORY: THE WIGHTMAN AXIOMS AND THE HAAG-KASTLER AXIOMS
QUANTUM FIELD THEORY: THE WIGHTMAN AXIOMS AND THE HAAG-KASTLER AXIOMS LOUIS-HADRIEN ROBERT The aim of the Quantum field theory is to offer a compromise between quantum mechanics and relativity. The fact
More informationTHE INITIAL VALUE FORMULATION OF GENERAL RELATIVITY
THE INITIAL VALUE FORMULATION OF GENERAL RELATIVITY SAM KAUFMAN Abstract. The (Cauchy) initial value formulation of General Relativity is developed, and the maximal vacuum Cauchy development theorem is
More informationAsymptotic Behavior of Marginally Trapped Tubes
Asymptotic Behavior of Marginally Trapped Tubes Catherine Williams January 29, 2009 Preliminaries general relativity General relativity says that spacetime is described by a Lorentzian 4-manifold (M, g)
More informationGlobal properties of solutions to the Einstein-matter equations
Global properties of solutions to the Einstein-matter equations Makoto Narita Okinawa National College of Technology 12/Nov./2012 @JGRG22, Tokyo Singularity theorems and two conjectures Theorem 1 (Penrose)
More informationLocalizing solutions of the Einstein equations
Localizing solutions of the Einstein equations Richard Schoen UC, Irvine and Stanford University - General Relativity: A Celebration of the 100th Anniversary, IHP - November 20, 2015 Plan of Lecture The
More informationQuantum versus classical probability
Quantum versus classical probability Jochen Rau Goethe University, Frankfurt arxiv:0710.2119v2 [quant-ph] Duke University, August 18, 2009 Reconstructing quantum theory: In search of a physical principle
More informationAccelerated Observers
Accelerated Observers In the last few lectures, we ve been discussing the implications that the postulates of special relativity have on the physics of our universe. We ve seen how to compute proper times
More informationParticle and photon orbits in McVittie spacetimes. Brien Nolan Dublin City University Britgrav 2015, Birmingham
Particle and photon orbits in McVittie spacetimes. Brien Nolan Dublin City University Britgrav 2015, Birmingham Outline Basic properties of McVittie spacetimes: embedding of the Schwarzschild field in
More informationOn the origin of probability in quantum mechanics
On the origin of probability in quantum mechanics Steve Hsu Benasque, September 2010 Outline 1. No Collapse quantum mechanics 2. Does the Born rule (probabilities) emerge? 3. Possible resolutions R. Buniy,
More informationMalament-Hogarth Machines
Malament-Hogarth Machines John Byron Manchak May 22, 2017 Abstract We show a clear sense in which general relativity allows for a type of machine which can bring about a spacetime structure suitable for
More informationThe following definition is fundamental.
1. Some Basics from Linear Algebra With these notes, I will try and clarify certain topics that I only quickly mention in class. First and foremost, I will assume that you are familiar with many basic
More informationEinstein Toolkit Workshop. Joshua Faber Apr
Einstein Toolkit Workshop Joshua Faber Apr 05 2012 Outline Space, time, and special relativity The metric tensor and geometry Curvature Geodesics Einstein s equations The Stress-energy tensor 3+1 formalisms
More informationChapter 5. Density matrix formalism
Chapter 5 Density matrix formalism In chap we formulated quantum mechanics for isolated systems. In practice systems interect with their environnement and we need a description that takes this feature
More information3. The Sheaf of Regular Functions
24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice
More informationarxiv:gr-qc/ v5 14 Sep 2006
Topology and Closed Timelike Curves II: Causal structure Hunter Monroe International Monetary Fund, Washington, DC 20431 (Dated: September 25, 2018) Abstract Because no closed timelike curve (CTC) on a
More informationAdvanced Study, 9 Adela Court, Mulgrave, Victoria 3170, Australia
ON OLBERS PARADOX Vu B Ho Advanced Study, 9 Adela Court, Mulgrave, Victoria 3170, Australia Email: vubho@bigpond.net.au Abstract: In this work we discuss a possibility to resolve Olbers paradox that states
More informationAn Introduction to Quantum Computation and Quantum Information
An to and Graduate Group in Applied Math University of California, Davis March 13, 009 A bit of history Benioff 198 : First paper published mentioning quantum computing Feynman 198 : Use a quantum computer
More informationarxiv: v2 [gr-qc] 25 Apr 2016
The C 0 -inextendibility of the Schwarzschild spacetime and the spacelike diameter in Lorentzian geometry Jan Sbierski April 26, 2016 arxiv:1507.00601v2 [gr-qc] 25 Apr 2016 Abstract The maximal analytic
More informationCausal Set Theory as a Discrete Model for Classical Spacetime
Imperial College London Department of Theoretical Physics Causal Set Theory as a Discrete Model for Classical Spacetime Author: Omar F. Sosa Rodríguez Supervisor: Prof. Fay Dowker Submitted in partial
More informationCausal completeness of probability theories results and open problems
Causal completeness of probability theories results and open problems Miklós Rédei Department of Philosophy, Logic and Scientific Method London School of Economics and Political Science Houghton Street
More information(1) Tipler s cylinder quickly collapses into singularity due to gravitational. (2) existence of infinitely long objects is very questionable.
1. Introduction In this talk I m going to give an overview of my Master thesis which I have finished this year at University of South Africa under the supervision of professor Bishop. First I will give
More informationTime in Bohmian Mechanics
Perimeter Institute, 28 September 2008 BM as Axioms For Quantum Mechanics Bohmian Mechanics is Time Symmetric BM in Relativistic Space-Time 1. Bohmian Mechanics BM as Axioms For Quantum Mechanics Bohmian
More informationLattice Theory Lecture 4. Non-distributive lattices
Lattice Theory Lecture 4 Non-distributive lattices John Harding New Mexico State University www.math.nmsu.edu/ JohnHarding.html jharding@nmsu.edu Toulouse, July 2017 Introduction Here we mostly consider
More informationMeasurement Independence, Parameter Independence and Non-locality
Measurement Independence, Parameter Independence and Non-locality Iñaki San Pedro Department of Logic and Philosophy of Science University of the Basque Country, UPV/EHU inaki.sanpedro@ehu.es Abstract
More informationCOSMOLOGICAL TIME VERSUS CMC TIME IN SPACETIMES OF CONSTANT CURVATURE
COSMOLOGICAL TIME VERSUS CMC TIME IN SPACETIMES OF CONSTANT CURVATURE LARS ANDERSSON, THIERRY BARBOT, FRANÇOIS BÉGUIN, AND ABDELGHANI ZEGHIB Abstract. In this paper, we investigate the existence of foliations
More informationFrom Bohmian Mechanics to Bohmian Quantum Gravity. Antonio Vassallo Instytut Filozofii UW Section de Philosophie UNIL
From Bohmian Mechanics to Bohmian Quantum Gravity Antonio Vassallo Instytut Filozofii UW Section de Philosophie UNIL The Measurement Problem in Quantum Mechanics (1) The wave-function of a system is complete,
More informationNull Cones to Infinity, Curvature Flux, and Bondi Mass
Null Cones to Infinity, Curvature Flux, and Bondi Mass Arick Shao (joint work with Spyros Alexakis) University of Toronto May 22, 2013 Arick Shao (University of Toronto) Null Cones to Infinity May 22,
More informationWhat is wrong with the standard formulation of quantum theory?
What is wrong with the standard formulation of quantum theory? Robert Oeckl Centro de Ciencias Matemáticas UNAM, Morelia Seminar General Boundary Formulation 21 February 2013 Outline 1 Classical physics
More informationHolographic Special Relativity:
Holographic Special Relativity: Observer Space from Conformal Geometry Derek K. Wise University of Erlangen Based on 1305.3258 International Loop Quantum Gravity Seminar 15 October 2013 1 Holographic special
More informationINVESTIGATING THE KERR BLACK HOLE USING MAPLE IDAN REGEV. Department of Mathematics, University of Toronto. March 22, 2002.
INVESTIGATING THE KERR BLACK HOLE USING MAPLE 1 Introduction IDAN REGEV Department of Mathematics, University of Toronto March 22, 2002. 1.1 Why Study the Kerr Black Hole 1.1.1 Overview of Black Holes
More informationSpace, time, Spacetime
Space, time, Spacetime Marc September 28, 2011 1 The vanishing of time A? 2 Time ersatz in GR special relativity general relativity Time in general relativity Proper in GR 3 4 5 gravity and Outline The
More informationThe Twin Paradox in Static Spacetimes and Jacobi Fields
The Twin Paradox in Static Spacetimes and Jacobi Fields Leszek M. Sokołowski Abstract The twin paradox of special relativity formulated in the geometrical setting of general relativity gives rise to the
More informationarxiv:gr-qc/ v1 17 Mar 2005
A new time-machine model with compact vacuum core Amos Ori Department of Physics, Technion Israel Institute of Technology, Haifa, 32000, Israel (May 17, 2006) arxiv:gr-qc/0503077 v1 17 Mar 2005 Abstract
More informationCausal Sets: Overview and Status
University of Mississippi QGA3 Conference, 25 Aug 2006 The Central Conjecture Causal Sets A causal set is a partially ordered set, meaning that x, y, z x y, y z x z x y, y x x = y which is locally finite,
More informationThe Conservation of Matter in General Relativity
Commun. math. Phys. 18, 301 306 (1970) by Springer-Verlag 1970 The Conservation of Matter in General Relativity STEPHEN HAWKING Institute of Theoretical Astronomy, Cambridge, England Received June 1, 1970
More informationWORLD THEORY CRISTINEL STOICA
WORLD THEORY CRISTINEL STOICA Abstract. In this paper a general mathematical model of the World will be constructed. I will show that a number of important theories in Physics are particularizations of
More informationA Boundary Value Problem for the Einstein Constraint Equations
A Boundary Value Problem for the Einstein Constraint Equations David Maxwell October 10, 2003 1. Introduction The N-body problem in general relativity concerns the dynamics of an isolated system of N black
More informationEffect of Monopole Field on the Non-Spherical Gravitational Collapse of Radiating Dyon Solution.
IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn:2319-765x. Volume 10, Issue 1 Ver. III. (Feb. 2014), PP 46-52 Effect of Monopole Field on the Non-Spherical Gravitational Collapse of Radiating
More information(iv) Whitney s condition B. Suppose S β S α. If two sequences (a k ) S α and (b k ) S β both converge to the same x S β then lim.
0.1. Stratified spaces. References are [7], [6], [3]. Singular spaces are naturally associated to many important mathematical objects (for example in representation theory). We are essentially interested
More informationone tries, the metric must always contain singularities. The point of this note is to give a simple proof of this fact in the case that n is even. Thi
Kinks and Time Machines Andrew Chamblin, G.W. Gibbons, Alan R. Steif Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW, England. We show that it is not possible
More informationFree probability and quantum information
Free probability and quantum information Benoît Collins WPI-AIMR, Tohoku University & University of Ottawa Tokyo, Nov 8, 2013 Overview Overview Plan: 1. Quantum Information theory: the additivity problem
More informationA CONSTRUCTION OF TRANSVERSE SUBMANIFOLDS
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLI 2003 A CONSTRUCTION OF TRANSVERSE SUBMANIFOLDS by J. Szenthe Abstract. In case of Riemannian manifolds isometric actions admitting submanifolds
More informationSome global properties of neural networks. L. Accardi and A. Aiello. Laboratorio di Cibernetica del C.N.R., Arco Felice (Na), Italy
Some global properties of neural networks L. Accardi and A. Aiello Laboratorio di Cibernetica del C.N.R., Arco Felice (Na), Italy 1 Contents 1 Introduction 3 2 The individual problem of synthesis 4 3 The
More informationarxiv: v1 [gr-qc] 3 Nov 2013
Spacetime Singularities Pankaj S. Joshi 1, 1 Tata Institute of Fundamental Research, Homi Bhabha road, Colaba, Mumbai 400005, India arxiv:1311.0449v1 [gr-qc] 3 Nov 2013 We present here an overview of our
More informationABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY
ABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY ARDA H. DEMIRHAN Abstract. We examine the conditions for uniqueness of differentials in the abstract setting of differential geometry. Then we ll come up
More informationBlack Holes and Thermodynamics. Law and the Information Paradox
Black Holes and Thermodynamics I. Classical Black Holes Robert M. Wald II. The First Law of Black Hole Mechanics III. Quantum Black Holes, the Generalized 2nd Law and the Information Paradox Black Holes
More information3.1 Transformation of Velocities
3.1 Transformation of Velocities To prepare the way for future considerations of particle dynamics in special relativity, we need to explore the Lorentz transformation of velocities. These are simply derived
More informationBlack Hole Physics. Basic Concepts and New Developments KLUWER ACADEMIC PUBLISHERS. Valeri P. Frolov. Igor D. Nbvikov. and
Black Hole Physics Basic Concepts and New Developments by Valeri P. Frolov Department of Physics, University of Alberta, Edmonton, Alberta, Canada and Igor D. Nbvikov Theoretical Astrophysics Center, University
More informationLecture XIV: Global structure, acceleration, and the initial singularity
Lecture XIV: Global structure, acceleration, and the initial singularity Christopher M. Hirata Caltech M/C 350-17, Pasadena CA 91125, USA (Dated: December 5, 2012) I. OVERVIEW In this lecture, we will
More informationExcluding Black Hole Firewalls with Extreme Cosmic Censorship
Excluding Black Hole Firewalls with Extreme Cosmic Censorship arxiv:1306.0562 Don N. Page University of Alberta February 14, 2014 Introduction A goal of theoretical cosmology is to find a quantum state
More informationOn the topology of black holes and beyond
On the topology of black holes and beyond Greg Galloway University of Miami MSRI-Evans Lecture UC Berkeley, September, 2013 What is the topology of a black hole? What is the topology of a black hole? Hawking
More informationBlack Holes: Complementarity vs. Firewalls
Black Holes: Complementarity vs. Firewalls Raphael Bousso Center for Theoretical Physics University of California, Berkeley Strings 2012, Munich July 27, 2012 The Question Complementarity The AMPS Gedankenexperiment
More informationThree Descriptions of the Cohomology of Bun G (X) (Lecture 4)
Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and
More informationThe Structure of the Multiverse
David Deutsch Centre for Quantum Computation The Clarendon Laboratory University of Oxford, Oxford OX1 3PU, UK April 2001 Keywords: multiverse, parallel universes, quantum information, quantum computation,
More informationAre naked singularities forbidden by the second law of thermodynamics?
Are naked singularities forbidden by the second law of thermodynamics? Sukratu Barve and T. P. Singh Theoretical Astrophysics Group Tata Institute of Fundamental Research Homi Bhabha Road, Bombay 400 005,
More informationCAUSAL SET APPROACH TO DISCRETE QUANTUM GRAVITY
CAUSAL SET APPROACH TO DISCRETE QUANTUM GRAVITY S. Gudder Department of Mathematics University of Denver Denver, Colorado 80208, U.S.A. sgudder@du.edu Abstract We begin by describing a sequential growth
More informationThe Cosmic Censorship Conjectures in General Relativity or Cosmic Censorship for the Massless Scalar Field with Spherical Symmetry
The Cosmic Censorship Conjectures in General Relativity or Cosmic Censorship for the Massless Scalar Field with Spherical Symmetry 1 This page intentionally left blank. 2 Abstract This essay describes
More informationClosed Universes, de Sitter Space and Inflation
Closed Universes, de Sitter Space and Inflation Chris Doran Cavendish Laboratory Based on astro-ph/0307311 by Lasenby and Doran The Cosmological Constant Dark energy responsible for around 70% of the total
More informationStochastic Quantum Dynamics I. Born Rule
Stochastic Quantum Dynamics I. Born Rule Robert B. Griffiths Version of 25 January 2010 Contents 1 Introduction 1 2 Born Rule 1 2.1 Statement of the Born Rule................................ 1 2.2 Incompatible
More informationThe complexity of classification problem of nuclear C*-algebras
The complexity of classification problem of nuclear C*-algebras Ilijas Farah (joint work with Andrew Toms and Asger Törnquist) Nottingham, September 6, 2010 C*-algebras H: a complex Hilbert space (B(H),
More informationClosed Timelike Curves
Closed Timelike Curves Bryan W. Roberts December 5, 2008 1 Introduction In this paper, we explore the possibility that closed timelike curves might be allowed by the laws of physics. A closed timelike
More informationMalament-Hogarth Machines
Malament-Hogarth Machines John Byron Manchak [Forthcoming in The British Journal for the Philosophy of Science.] Abstract We show a clear sense in which general relativity allows for a type of machine
More informationKnot Physics: Entanglement and Locality. Abstract
Knot Physics: Entanglement and Locality C. Ellgen (Dated: July 18, 2016) Abstract We describe entanglement and locality in knot physics. In knot physics, spacetime is a branched manifold. The quantum information
More informationThe Einstein field equation in terms of. the Schrödinger equation. The meditation of quantum information
The Einstein field equation in terms of the Schrödinger equation The meditation of quantum information Vasil Penchev Bulgarian Academy of Sciences: Institute for the Study of Societies of Knowledge vasildinev@gmail.com
More information