The classical theorems A Local approach Local Topology Change In General Relativity J. Benavides March 10, 2012 This is a very weak argument as was pointed by C. Isham “General relativity is about space-time, and quantum theory tends to involve quantum fluctuations in things. Therefore, if you talk about quantum gravity, there might be some sort of fluctuation in something to do with space-time. It’s that sort of level of argument.” The classical theorems A Local approach Topology Fluctuations In the 1950’s John Archibald Wheeler claimed that at small scales (about 10−35 meters) quantum fluctuations can become relevant in the structure of the spacetime breaking down the smooth structure of spacetime in some kind of “foaminess” where the topology and the differential structure of spacetime can change dramatically. The classical theorems A Local approach Topology Fluctuations In the 1950’s John Archibald Wheeler claimed that at small scales (about 10−35 meters) quantum fluctuations can become This is a very weak argument as relevant in the structure of the was pointed by C. Isham “General spacetime breaking down the relativity is about space-time, and smooth structure of spacetime in quantum theory tends to involve some kind of “foaminess” where quantum fluctuations in things. the topology and the differential Therefore, if you talk about structure of spacetime can change quantum gravity, there might be dramatically. some sort of fluctuation in something to do with space-time. It’s that sort of level of argument.” General Relativity describes spacetime as a four dimensional Lorentzian manifold (M; g). Thus, globally, the topology of the spacetime manifold is fixed. However within this manifold we can consider spacelike regions as evolving entities, we can ask for example: can a simply connected spacelike region, (like, apparently, are the regions we live in), become a multiply connected one?. The classical theorems A Local approach Can the topology of spacetime change? Nevertheless a more natural question arise: Are topology changing processes allowed by General Relativity? The classical theorems A Local approach Can the topology of spacetime change? Nevertheless a more natural question arise: Are topology changing processes allowed by General Relativity? General Relativity describes spacetime as a four dimensional Lorentzian manifold (M; g). Thus, globally, the topology of the spacetime manifold is fixed. However within this manifold we can consider spacelike regions as evolving entities, we can ask for example: can a simply connected spacelike region, (like, apparently, are the regions we live in), become a multiply connected one?. - Geometrically there is no restriction, given any two closed (compact without boundary) 3-manifolds S1, S2 there exist a compact manifold N, such that @N = S1 [ S2 (Any two orientable closed 3-manifolds are cobordant, this classical result was originally proved by Thom in 1954). - The fact that N has to admit a Lorentzian Metric, in such a way that S1 and S2 are spacelike, does not impose any restriction in the topology of S1 and S2 as was proved by Misner in 1963. -However Geroch proved that if we impose that the spacetime does not contain closed timelike curves (i.e. that time travel is forbidden) ∼ then S1 = S2. The classical theorems A Local approach The Classical Theorems In 1966 R. Geroch consider the next problem: suppose that spacetime contains two spacelike hypersurfaces S1, S2 connected by a compact region N, is there any relation between the topology of the hypersurfaces? - The fact that N has to admit a Lorentzian Metric, in such a way that S1 and S2 are spacelike, does not impose any restriction in the topology of S1 and S2 as was proved by Misner in 1963. -However Geroch proved that if we impose that the spacetime does not contain closed timelike curves (i.e. that time travel is forbidden) ∼ then S1 = S2. The classical theorems A Local approach The Classical Theorems In 1966 R. Geroch consider the next problem: suppose that spacetime contains two spacelike hypersurfaces S1, S2 connected by a compact region N, is there any relation between the topology of the hypersurfaces? - Geometrically there is no restriction, given any two closed (compact without boundary) 3-manifolds S1, S2 there exist a compact manifold N, such that @N = S1 [ S2 (Any two orientable closed 3-manifolds are cobordant, this classical result was originally proved by Thom in 1954). -However Geroch proved that if we impose that the spacetime does not contain closed timelike curves (i.e. that time travel is forbidden) ∼ then S1 = S2. The classical theorems A Local approach The Classical Theorems In 1966 R. Geroch consider the next problem: suppose that spacetime contains two spacelike hypersurfaces S1, S2 connected by a compact region N, is there any relation between the topology of the hypersurfaces? - Geometrically there is no restriction, given any two closed (compact without boundary) 3-manifolds S1, S2 there exist a compact manifold N, such that @N = S1 [ S2 (Any two orientable closed 3-manifolds are cobordant, this classical result was originally proved by Thom in 1954). - The fact that N has to admit a Lorentzian Metric, in such a way that S1 and S2 are spacelike, does not impose any restriction in the topology of S1 and S2 as was proved by Misner in 1963. The classical theorems A Local approach The Classical Theorems In 1966 R. Geroch consider the next problem: suppose that spacetime contains two spacelike hypersurfaces S1, S2 connected by a compact region N, is there any relation between the topology of the hypersurfaces? - Geometrically there is no restriction, given any two closed (compact without boundary) 3-manifolds S1, S2 there exist a compact manifold N, such that @N = S1 [ S2 (Any two orientable closed 3-manifolds are cobordant, this classical result was originally proved by Thom in 1954). - The fact that N has to admit a Lorentzian Metric, in such a way that S1 and S2 are spacelike, does not impose any restriction in the topology of S1 and S2 as was proved by Misner in 1963. -However Geroch proved that if we impose that the spacetime does not contain closed timelike curves (i.e. that time travel is forbidden) ∼ then S1 = S2. Note that the above theorem does not mention the Einstein Equations. There exists numerous solutions of the Einstein equations which contain closed timelike curves, then a natural question is: if we drop the hypothesis about closed timelike curves, do the Einstein equations constrain in any way the topology of S1 and S2 in the above theorem. The classical theorems A Local approach Geroch’s Theorem Let (M; g) a 4-dimensional time oriented Lorentzian manifold without closed timelike curves, S1; S2 ⊂ M closed spacelike hypersurfaces ∼ and N ⊂ M a compact region such that @N = S1 [ S2; then S1 = S2. The classical theorems A Local approach Geroch’s Theorem Let (M; g) a 4-dimensional time oriented Lorentzian manifold without closed timelike curves, S1; S2 ⊂ M closed spacelike hypersurfaces ∼ and N ⊂ M a compact region such that @N = S1 [ S2; then S1 = S2. Note that the above theorem does not mention the Einstein Equations. There exists numerous solutions of the Einstein equations which contain closed timelike curves, then a natural question is: if we drop the hypothesis about closed timelike curves, do the Einstein equations constrain in any way the topology of S1 and S2 in the above theorem. Tipler’s Theorem Let N, S1 and S2 as above. If the Einstein equations hold on N ∼ for some kind of normal matter then S1 = S2. The classical theorems A Local approach In 1977 Tipler proved that if we drop the chronology hypothesis but we assume that spacetime (M; g) satisfies the Einstein equations for some kind of “realistic matter distribution” then, ∼ taking S1, S2 and N as above, we can prove again that S1 = S2. The classical theorems A Local approach In 1977 Tipler proved that if we drop the chronology hypothesis but we assume that spacetime (M; g) satisfies the Einstein equations for some kind of “realistic matter distribution” then, ∼ taking S1, S2 and N as above, we can prove again that S1 = S2. Tipler’s Theorem Let N, S1 and S2 as above. If the Einstein equations hold on N ∼ for some kind of normal matter then S1 = S2. The classical theorems A Local approach A local approach Considering S1 and S2 in the above theorems as compact manifolds without boundary is a very strong hypothesis about the geometry of the spacetime. Therefore a more natural question would be: can the topology of a local spacelike region of spacetime change? The classical theorems A Local approach Definition (Causal Tube) Let (M; g) be a time orientable spacetime, and let N ⊂ M such that @N is a compact hypersurface which satisfies @N = S1 [ S2 [ T ; where the Si are disjoint closed spacelike manifolds, @Si is homeomorphic to the two sphere S2 and T is a causal surface that connects @S1 with @S2 and is homeomorphic to @Si × [0; 1], supposing by convention that @S2 is in the future of @S1. Furthermore we suppose that every future causal vector field V in M tangent to T points inside N in S1 and outside N in S2. If N satisfies all this conditions we will call it a causal tube. The classical theorems A Local approach Using the above definition a local version of Geroch’s theorem can be easily proved, this was originally given by S. Hawking in 1991: Theorem (Local Geroch’s Theorem) Let N be a compact causal tube, If causality holds within N ∼ ∼ then N = S1 × [0; 1], therefore S1 = S2 The classical theorems A Local approach Proof -Let V be a smooth future directed timelike vector field on N tangent to T .