Infinities and CosmologyCambridge, U.K.

George Ellis : Infinities of age and size (including global topol-ogy issues)There are two aspects of the concept of inifinity. The "good" one which isassociated with the idealisation of a very large number. The formalizationof this idea is expressed mathematically in the concept of limit using ε andδ. In fact in this type of formulation the concept of infinity is establish byan effective infinity, just meaning that there is a number big enough thatmake some difference arbitrarily small. Ellis argued that this kind of infinityshould be the only way the infinity should be used in physics. He supportedhis position by commenting about several physical and biological systemsthat are finite. For example, the number of states in the brain, the combi-nations of the genetic code, the number of words in a language.

The "bad" aspect of infinity is their paradoxical character. For example,the equation ∞ + 1 = ∞. This properties of infinity should not be used inempirical science. Also, the assumption of the physical existence of infinityis dubious. If something is infinite how could we ever know. There is noexperiment (because all of them are in a finite region of spacetime) thatwould be able to falsify it.

Finally, Ellis talked about the universe as a system. He strongly believesthat the universe must be finite in space and time. Although, there is nostrong evidence to support that, some predictions can be made if one assumea finite space. In addition, he suggested that any good physics should not relyin arguments that does not quantify the infinity via ε and δ. He encouragedthe audience to think in two directions :

– Develop mathematical tools for global properties. All our equations arelocal in character and to deal with questions about size (compactness)this equations are no longer relevant.

– Replace infinity in physical arguments for a very large number andfind all the relevant parameter to establish the bound.

Anthony Aguirre : Infinite and finite spacetimesAccording to the eternal inflation model there might be an infinite numberof universes coexisting with us. This kind of idea is discuss by Everett in his

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idea of the multiverse. The basic assumption is the existence of a primordialfield that in the past produced inflation.

The origin of the potential of that field is currently unknown. Whatis done is one guesses the functional form of the potential, use some codeto simulate the physics given by that type of inflation and finally comparewith current data. Moreover, we got a intriguing view about the form of theuniverse if we allow a functional form with multiple minima, slow transitionsand think in the possibility of quantum tunnelling. In fact the form is fractallike and there are infinite bubbles of inflation that contains whole universes.Physically, one can measure some cosmological signature that gives proof ofthis other universes.

This challenges Ellis’s views. Infinity might not be measurable but maybethere are good reasons to assume it exists.

Michael Douglas : Can we test the string theory landscape ?The success of QFT with the ideas of supersymmetry and gauge theorieshelp to develop string theory. Douglas stated that currently string theorydeserves a dominant place in our current thinking in search of a fundamentaltheory. He used the fact that string theory develop many interesting andunexpected mathematical results with the fact that the theory is currentlyempirically consistent with all available data.

Moreover, the claim that string theory is not falsifiable is false. In prin-ciple, there are predictions for energies for "string oscillations". If the exper-iments could reach these energies the theory could be tested. The problemof testability is practical not philosophical.

Then the problem of multiple solution (the string theory landscape) wastreated. First, it was pointed that multiple solutions are far more commonthat one usually expect. As an example, he mentioned the metastable statesin quantum chemistry. This states arise as a combination of multielectronatoms interacting due to repulsive and attractive forces that creates com-plex interactions with slow transitions between minima. Roughly one canestimate the number of metastable minima as :

number of minima = Nd (1)

where d is the dimension of the configuration space and N the number ofvalues that the variable can take.

Moreover, the metastable landscape of string theory can be used to sug-gest models that "predict" the value of the cosmological constant or give areason why it is the case that it is very near zero but not zero. Specificallythe anthropic solution can be realized. Bousso and Polchinski’s used the ideaof flux compactification on a Calabi-Yau manifold to build their argument.

Mihalis Dafermos : Singularities and cosmic censorship in gen-eral relativity

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The Cauchy problem for Einstein’s Equations is a highly non linear sec-ond order hyperbolic problem in partial differential equations. Since thebeginning of the 20th century many conceptual issues have been studiedfor example, the existence of initial data, the notion of global hyperbolicityand the unique maximal Cauchy development. All the work done was some-how used in the fundamental theorem of General Relativity that states thefollowing :

Theorem 1 Let (Σ, g,K) be a smooth vacuum initial data set. There existsa unique smooth spacetime (M, g) such that

– Ric(g)=0– (M, g) is globally hyperbolic with Cauchy surface Σ with induced firstfundamental form g and second fundamental form K.

– Any smooth spacetime with those properties isometrically embeds intoM .

Similar theorems can be proven for suitable coupled Einstein-matter sys-tems.

Dafermos then proceeded for clarity to restrict the solutions to asymp-totically flat data. This data is the data that correspond to isolated gravi-tational fields. The reason for this is that the metric at "infinity" is flat andthere is a compact set C such that the spacelike surface Σ\C is diffeomor-phic to R\B where B is a closed 3 − sphere. The notion at infinity is a"good" one since we only mean that outside a certain big enough 3− spherethe metric is arbitrarily close to the flat one.

The Schwarzschild solution is an asymptotically flat solution that causedconfusion in the sense to define the space M where the solution lives. Fromthe perspective of the Cauchy problem, we can define the spacetime Mas the maximal development. However, we have to allow for two compactset in the initial data. The study of this spacetime is useful as a proxyfor Oppenheimer-Snyder solution. The Oppenheimer-Snyder solution is thesolution of an initially homogeneous dust ball surrounded by vacuum. Thisspacetime will coincide with the Schwarzschild outside the support of matter.

One of the most striking facts of the Schwarzschild solution is the stableincompleteness of certain geodesics. At first, it was suspected that the highdegree of symmetry of the solution was the cause for this. However, Penroseproved that this was not the case and it was a generic property. The cos-mic censorship conjectures are statements to characterize this pathologicalbehaviour.

The first thing to notice is that the past of future null infinity has anon trivial complement, the so called "black hole" region and all incompletegeodesics must enter this region. In this sense, the observers at infinity can’tsee the incompleteness. The conjecture called the weak cosmic conjecturepostulates that the incompleteness of future null infinity only exist for a

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set of initial data of measure zero. Notice, that there is no reference to theexistence of a black hole in this formulation.

The second property is the curvature pathology that exists in the solu-tion. This accounts what happen to a geodesic that enter into the black holefor all the observers in the classical sense. However, there are spacetime likeKerr where initial data doesn’t account for all observers. There is a CauchyHorizon. The breakdown of uniqueness is due to the fact of the non globalhyperbolic nature of the spacetime. So the so called strong cosmic censorshipstates that there aren’t stable Cauchy Horizons.

Dafermos finally talked about the status of the conjectures. He com-mented that reducing the problem to a 2-dimensional case is more tractablebecause of very nice properties that arise in the hyperbolic behaviour of thepartial differential equations. In this sense, he continued saying that spheri-cal symmetric spacetimes are this type of spacetime. Both conjectures wereproven in this setting for the asymptotically flat initial data for sphericalsymmetric vacuum solutions.

For the Einstein-scalar field system there are initial data that each pos-sibility is realized. There are spacetime that don’t satisfy weak but satisfystrong cosmic censorship, there are spacetimes which violate the conjecturesthe other way around and there are spacetimex that don’t satisfy any ofthose. But, the pathological behaviour is not stable so the conjectures re-main valid.

There are also interesting facts we have learned from the conjectures.First there is a theorem that states that if certain amount of energy is con-centrated in an angular regions of the light cones then trapped surfaces willform. This is different from the naive idea that black holes form becausematter condensed in spatial spherical regions. Another property that now isknown is that generically the boundary of the maximal Cauchy developmentgenerated by singularities is not going to be spacelike (as in Schwarzschild)but rather is going to be null.

Mark Hogarth : Infinite computations and spacetimeThe notion of computability is strongly attached to the idea of a Turingmachine. Hogarth argued that an idea of computability different from thatcan be possible. The idea is to use the spacetime structure based on the ideaof the twin paradox. According to relativity, two observers can be reunitedand measure different proper times. In this case for example we can leave aTuring machine on Earth and travel. When we return we will have a lot oftime in calculations even if we just travel shortly.

If we extend further the idea and ask for a Malament-Hogarth spacetime(a spacetime with an infinitely long worldline in the past of some event) wecan perform an infinite number of calculations. For example, the Goldbachconjecture could be solve using this kind of setting. We could leave a Turingmachine in the infinitely long worldline and give the instruction that if there

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is a counterexample send a signal to the future point of that worldline. Thenwhen we arrive to that point we will have a counterexample or the fact thatthe conjecture is true. A concrete example of this kind of spacetimes areKerr, Anti-DeSitter and Resisner-Norstrom. Unfortunately, all of the M-Hspacetime violate Cosmic Censorship.

Simon Saunders : At home and at sea in an infinite universe :Newtonian and Machian theories of motionNewton assume in the Principia that the centre of the world is the Sun andthen proceed to deduct Kepler’s law. Nowadays, we know the Sun is not thecentre of the Universe. So if we assume a finite sphere bigger even than theobservable universe with homogeneous density distribution of matter andwe use Newton laws to calculate the gravitational force feel by the rest. Wereach the conclusion that we should be pulled into space because the forcefrom outer space is bigger than the gravitational force of the earth. So weshould feel that absolute acceleration. Saunders pointed that the answer forthe dilemma rest in another Newton’s observation. The observation statethat parallel gravitational fields are unobservable. So considering the lengthof the centre of the sphere, the whole solar system will feel a parallel forceso it would be unnoticeable.

Now, Galilean spacetime is assume to be the spacetime of NewtonianMechanics. In light of the previous idea, this can’t be the case. Galileanspacetime is able to distinguish between absolute accelerations (accelera-tions with respect to the static space). Saunders argued that the correctspacetime for Newtonian mechanics is given by considering the differenceof the equations given by Newton’s law. In that case the group symmetriesof the system is another spacetime different from Galilean spacetime. How-ever, it is necessary to give up the conservation of momentum. So in thecorrect spacetime for the Principia only relative angles of spatial vectors arephysically meaningful.

That means Galilean spacetime is not the correct spacetime for classicalmechanics. The correct reference frame in Newtonian mechanics are thosenon-rotaing only. Notice that it is not necessary to be inertial. In the partof Machianism, Saunder mentioned how this posture has been driven a lotfurther by attempts to construct dynamics using only relative configura-tions. Saunders concluded that Galilean spacetime postulate a realist viewof spacetime, Machian views try to get rid of any spacetime structure andthat Newtonian spacetime is something in between.

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