Local Topology Change in General Relativity

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). - 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

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

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

The classical theorems A Local approach

Proof -Let V be a smooth future directed timelike vector field on N tangent to T . Suppose λ is one integral curve of V which start at S1 and does not have a future endpoint in S2. λ ∩ S1 = ∅ = λ ∩ T . -Take a sequence of points {p } in λ, such that p < p and for n n∈N n n+1 0 0 every point p of λ there exist a m such that p < pm. Since N is compact the sequence converge to a point p ∈ N. + -Since I (p) is an open set, taking pn close enough to p, the portion of λ + in the future of pn should intersect I (p). Thus there exists a point q on the − − curve λ such that p ∈ I (q), then the family {I (r)}r∈λ is an open cover of the compact set λ. − − -Take a finite family I (r1), ..., I (rn) that cover λ, let rj the futuremost of the points that defines the finite family, then since rj ∈ λ there exists − another point ri in the family such that rj ∈ I (ri ) but we have also that − ri ∈ I (rj ) violating causality. The classical theorems A Local approach

The classical theorems A Local approach

Local Tipler’s Theorem

To give a local version of Tipler’s theorem a lot of work is needed, we have to redeﬁne some of the standard concepts of the global methods in general relativity and give the respective generalizations of standard theorems. + + − + HN (S) = DN (S) − IN (DN (S)) + + + = {q ∈ DN (S): IN (q) ∩ DN (S) 6= ∅}, − + where IN (DN (S)) is the set of points which are future chronological related + + with DN (S) within N and IN (q) is deﬁned analogously.

The classical theorems A Local approach

Local Tipler’s Theorem

To give a local version of Tipler’s theorem a lot of work is needed, we have to redeﬁne some of the + standard concepts of the global methods in general DN (S1) is the set of points p such that relativity and give the respective generalizations of standard theorems. every inextendible causal curve from p within N intersects S1. The classical theorems A Local approach

Local Tipler’s Theorem

+ + − + HN (S) = DN (S) − IN (DN (S)) + + + = {q ∈ DN (S): IN (q) ∩ DN (S) 6= ∅}, − + where IN (DN (S)) is the set of points which are future chronological related + + with DN (S) within N and IN (q) is deﬁned analogously. Lemma + Let N be a causal tube, if HN (S1) is non empty and compact then the + null geodesic generators of HN (S1) are geodesically complete in the past direction.

The classical theorems A Local approach

Lemma + Let N be a causal tube, if HN (S1) is non empty then from each point + of HN (S1) there is a past inextendible null geodesic without conjugate + points that is entirely contained in HN (S). We will call this curves the + null geodesic generators of HN (S). Theorem + Let N be a causal tube such that HN (S1) is non empty. Suppose that + 0 0 for every null geodesic generator σ of HN (S1), Ric(σ , σ ) > 0 for + + some point in HN (S1), then HN (S1) is non compact.

The classical theorems A Local approach

Lemma + Let N be a causal tube, if HN (S1) is non empty and compact then the + null geodesic generators of HN (S1) are geodesically complete in the past direction. The classical theorems A Local approach

Lemma + Let N be a causal tube, if HN (S1) is non empty and compact then the + null geodesic generators of HN (S1) are geodesically complete in the past direction.

Theorem + Let N be a causal tube such that HN (S1) is non empty. Suppose that + 0 0 for every null geodesic generator σ of HN (S1), Ric(σ , σ ) > 0 for + + some point in HN (S1), then HN (S1) is non compact. Proof. Again consider V a smooth future timelike vector ﬁeld on N tangent to T and suppose there exists an integral curve from S1 that does not reach S2, from Geroch’s theorem we know there exists a closed timelike curve within N. From the existence of a closed timelike curve + + within N, we conclude N 6= DN (S1) and HN (S1) is not empty. A realistic matter distribution imply that Ric(σ0, σ0) > 0 everywhere on + + HN (S1) for every null geodesic generator σ of HN (S1).Thus by the + + above theorem HN (S1) is non compact.On the other hand HN (S1) is + a closed subset of the compact set N, then HN (S1) is compact, a contradiction. Then every integral curve of V from S1 reaches S2, then S1 is homeomorphic to S2 and N is homeomorphic to S1 × [0, 1].

The classical theorems A Local approach

Theorem (Local Tipler Theorem) Let (M, g) be a time oriented spacetime where Einstein’s equations for a “normal matter distribution” are satisﬁed. Let N be a compact causal tube in M. Then N is diffeomorphic to S1 × [0, 1] from Geroch’s theorem we know there exists a closed timelike curve within N. From the existence of a closed timelike curve + + within N, we conclude N 6= DN (S1) and HN (S1) is not empty. A realistic matter distribution imply that Ric(σ0, σ0) > 0 everywhere on + + HN (S1) for every null geodesic generator σ of HN (S1).Thus by the + + above theorem HN (S1) is non compact.On the other hand HN (S1) is + a closed subset of the compact set N, then HN (S1) is compact, a contradiction. Then every integral curve of V from S1 reaches S2, then S1 is homeomorphic to S2 and N is homeomorphic to S1 × [0, 1].

The classical theorems A Local approach

Proof. Again consider V a smooth future timelike vector ﬁeld on N tangent to T and suppose there exists an integral curve from S1 that does not reach S2, A realistic matter distribution imply that Ric(σ0, σ0) > 0 everywhere on + + HN (S1) for every null geodesic generator σ of HN (S1).Thus by the + + above theorem HN (S1) is non compact.On the other hand HN (S1) is + a closed subset of the compact set N, then HN (S1) is compact, a contradiction. Then every integral curve of V from S1 reaches S2, then S1 is homeomorphic to S2 and N is homeomorphic to S1 × [0, 1].

The classical theorems A Local approach

Proof. Again consider V a smooth future timelike vector ﬁeld on N tangent to T and suppose there exists an integral curve from S1 that does not reach S2, from Geroch’s theorem we know there exists a closed timelike curve within N. From the existence of a closed timelike curve + + within N, we conclude N 6= DN (S1) and HN (S1) is not empty. Thus by the + + above theorem HN (S1) is non compact.On the other hand HN (S1) is + a closed subset of the compact set N, then HN (S1) is compact, a contradiction. Then every integral curve of V from S1 reaches S2, then S1 is homeomorphic to S2 and N is homeomorphic to S1 × [0, 1].

The classical theorems A Local approach

Proof. Again consider V a smooth future timelike vector ﬁeld on N tangent to T and suppose there exists an integral curve from S1 that does not reach S2, from Geroch’s theorem we know there exists a closed timelike curve within N. From the existence of a closed timelike curve + + within N, we conclude N 6= DN (S1) and HN (S1) is not empty. A realistic matter distribution imply that Ric(σ0, σ0) > 0 everywhere on + + HN (S1) for every null geodesic generator σ of HN (S1). + On the other hand HN (S1) is + a closed subset of the compact set N, then HN (S1) is compact, a contradiction. Then every integral curve of V from S1 reaches S2, then S1 is homeomorphic to S2 and N is homeomorphic to S1 × [0, 1].

The classical theorems A Local approach

Proof. Again consider V a smooth future timelike vector ﬁeld on N tangent to T and suppose there exists an integral curve from S1 that does not reach S2, from Geroch’s theorem we know there exists a closed timelike curve within N. From the existence of a closed timelike curve + + within N, we conclude N 6= DN (S1) and HN (S1) is not empty. A realistic matter distribution imply that Ric(σ0, σ0) > 0 everywhere on + + HN (S1) for every null geodesic generator σ of HN (S1).Thus by the + above theorem HN (S1) is non compact. Then every integral curve of V from S1 reaches S2, then S1 is homeomorphic to S2 and N is homeomorphic to S1 × [0, 1].

The classical theorems A Local approach

Proof. Again consider V a smooth future timelike vector ﬁeld on N tangent to T and suppose there exists an integral curve from S1 that does not reach S2, from Geroch’s theorem we know there exists a closed timelike curve within N. From the existence of a closed timelike curve + + within N, we conclude N 6= DN (S1) and HN (S1) is not empty. A realistic matter distribution imply that Ric(σ0, σ0) > 0 everywhere on + + HN (S1) for every null geodesic generator σ of HN (S1).Thus by the + + above theorem HN (S1) is non compact.On the other hand HN (S1) is + a closed subset of the compact set N, then HN (S1) is compact, a contradiction. The classical theorems A Local approach

Proof. Again consider V a smooth future timelike vector ﬁeld on N tangent to T and suppose there exists an integral curve from S1 that does not reach S2, from Geroch’s theorem we know there exists a closed timelike curve within N. From the existence of a closed timelike curve + + within N, we conclude N 6= DN (S1) and HN (S1) is not empty. A realistic matter distribution imply that Ric(σ0, σ0) > 0 everywhere on + + HN (S1) for every null geodesic generator σ of HN (S1).Thus by the + + above theorem HN (S1) is non compact.On the other hand HN (S1) is + a closed subset of the compact set N, then HN (S1) is compact, a contradiction. Then every integral curve of V from S1 reaches S2, then S1 is homeomorphic to S2 and N is homeomorphic to S1 × [0, 1]. (Work in progress)

The classical theorems A Local approach

As before the existence of a compact Lorentzian interpolating manifold does not impose limitations between the topology of the cobordant manifolds (i.e. the hypothesis of the local theorems are not redundant), it is possible to develop a cobordism theory of manifolds with spherical boundary and it is possible to show that any two 3-manifolds with spherical boundary are cobordant. The existence of a Lorentz metric follow in the same way as in the closed manifold case. The classical theorems A Local approach

The classical theorems A Local approach

Can the topology of spacetime change? (Again) The above theorems tell us that if the topology of a spacelike region change the interpolating region has to be non-compact. This result has been normally interpreted as the impossibility of topology change in realistic scenarios but the results are just telling us that a local topology change process cannot be described within the formalism of classical general relativity unless some of the hypothesis of the theorems can be violated. One way to do this is to assume the existence of exotic matter which can violate the standard energy conditions, allowing topology change processes which derive in the formation of wormholes. Analogous problems appear in the context of black holes and singularities. If we try to modify the theory to capture the singularities in the manifold formalism, local characteristics related with Cauchy determinism break down.

The classical Continuum implies Cauchy determinism If we want a theory which capture black holes singularities, closed timelike curves and multiple histories determinism we cannot use our classical ruler i.e the real numbers, complex numbers, manifolds etc. We need a new conception of the continuum to capture these notions. This implies that we need to overcome classical logic based formalism.

GR+black holes+“Time travel”+ Quantum Mechanics > Classical Logic + Set theory

The classical theorems A Local approach

However the presence of closed timelike curves break down the local classical predictability based on the Cauchy problem conception of determinism. The classical Continuum implies Cauchy determinism If we want a theory which capture black holes singularities, closed timelike curves and multiple histories determinism we cannot use our classical ruler i.e the real numbers, complex numbers, manifolds etc. We need a new conception of the continuum to capture these notions. This implies that we need to overcome classical logic based formalism.

GR+black holes+“Time travel”+ Quantum Mechanics > Classical Logic + Set theory

The classical theorems A Local approach

However the presence of closed timelike curves break down the local classical predictability based on the Cauchy problem conception of determinism. Analogous problems appear in the context of black holes and singularities. If we try to modify the theory to capture the singularities in the manifold formalism, local characteristics related with Cauchy determinism break down. GR+black holes+“Time travel”+ Quantum Mechanics > Classical Logic + Set theory

The classical theorems A Local approach

GR+black holes+“Time travel”+ Quantum Mechanics > Classical Logic + Set theory The classical theorems A Local approach

The Biggest Revolution black holes+“Time travel”+ Quantum Mechanics= Intuitionistic Logic + Set Theory The classical theorems A Local approach

The Biggest Revolution black holes+“Time travel”+ Quantum Mechanics= Intuitionistic Logic + Set Theory