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Eddington-Born-Infeld Action and the dark Side of General Relativity Maximo banados Abstract We review a recent proposal to describe dark matter and da based on an Eddington-Born-Infeld action. The theory is successful in he evolution of the expansion factor as well as galactic flat rotation curves tions and the CMB spectra are currently under study. This paper in writter u Claudio Bunster on the occasion of his 60th birthday Marc Henneaux remarked in his lecture at Claudio's Fest (Valdivia Chile, January 2008) that working with him one learns to be brave. Claudio's tuition has beer articularly important for me over the last 2 years. I have been working on an idea that looked crazy at first sight and still looks pretty mad today. I have not had the chance to talk to Claudio about these ideas and I have not sent him the papers I have written 2, 3] because I know he will not read them! In this short contribution in his honor I attempt to shoot two birds with one stone, hoping that the bird will not be me. I shall review a project whose aim is to provide a candidate for dark matter and dark energy, and whose seed relies on the study of general relativity around the de- generate field guv =0. The project started as a purely formal idea, but it immediately succeeded in reproducing some of the known phenomenological curves associated to dark matter and dark energy. I thus decided to purse the idea to the end. In this contribution I shall concentrate on the original motivation based on studying general relativity near guv=0[3]. This controversial motivation is now not needed because an action implementing most of the ideas is available [2]. However, I believe that exploring physics at guv =0 is an attractive idea, certainly not new, and perhap necessary to understand the origin of the Universe The first step comes from the first order vielbein formulation of general relativity equations of motion abadr e= o M. Barados P. Universidad Catolica de Chile, Av. Vicuna Mackenna 4860. Santiago, Chile e-mail: mbandos @uccl M. Henneaux, J Zanelli(eds ) Quantum Mechanics of Fundamental Systems: The Quest 27 for Beauty and Simplicity, DOI 10.1007/978-0-387-87499-9-4 C Springer Science+ Business Media LLC 2009
Eddington–Born–Infeld Action and the Dark Side of General Relativity Maximo Ba ´ nados ˜ Abstract We review a recent proposal to describe dark matter and dark energy based on an Eddington–Born–Infeld action. The theory is successful in describing the evolution of the expansion factor as well as galactic flat rotation curves. Fluctuations and the CMB spectra are currently under study. This paper in written in honor of Claudio Bunster on the occasion of his 60th birthday. Marc Henneaux remarked in his lecture at Claudio’s Fest (Valdivia Chile, January 2008) that working with him one learns to be brave. Claudio’s tuition has been particularly important for me over the last 2 years. I have been working on an idea that looked crazy at first sight and still looks pretty mad today. I have not had the chance to talk to Claudio about these ideas and I have not sent him the papers I have written [2, 3] because I know he will not read them! In this short contribution in his honor I attempt to shoot two birds with one stone, hoping that the bird will not be me. I shall review a project whose aim is to provide a candidate for dark matter and dark energy, and whose seed relies on the study of general relativity around the degenerate field gμν = 0. The project started as a purely formal idea, but it immediately succeeded in reproducing some of the known phenomenological curves associated to dark matter and dark energy. I thus decided to purse the idea to the end. In this contribution I shall concentrate on the original motivation based on studying general relativity near gμν = 0 [3]. This controversial motivation is now not needed because an action implementing most of the ideas is available [2]. However, I believe that exploring physics at gμν = 0 is an attractive idea, certainly not new, and perhaps necessary to understand the origin of the Universe. The first step comes from the first order vielbein formulation of general relativity. An intriguing solution to the equations of motion, εabcdRabec = 0 (1) εabcdTaec = 0, (2) M. Ba˜nados P. Universidad Cat´olica de Chile, Av. Vicu˜na Mackenna 4860, Santiago, Chile e-mail: mbandos@uc.cl M. Henneaux, J. Zanelli (eds.), Quantum Mechanics of Fundamental Systems: The Quest 27 for Beauty and Simplicity, DOI 10.1007/978-0-387-87499-9 4, c Springer Science+Business Media LLC 2009
M. Bardos The existence of this solution has not gone unnoticed, e. g, [9, 10, 13 ]. Its most salient property is that it preserves the full set of diffeomorphisms and for this reason it is often called theunbroken' ground state of general relativity. This point was stressed in[9]were a symmetry breaking transition from e=0 to e#0, the big-bang, was suggested. Topological transitions in this formulation were studied in [10] Now, a key aspect of this solution is the fact that the spin connection is left undetermined. The above equations of motion are supposed to determine both a and wab. However, since e=0 kills both(1)and( 2), the spin connection becomes a random field. I The first step towards accepting ea=0 as a solution is to understand the nature of the spin connection at that point. If one first solves the algebraic equation for the torsion expressing wwe-lde as a function of e the same problem reappears in a different way. One may try to recover the solution ea=0 by a limit e-0. The connection wee-lde has the structure g and can be anything. This is equivalent to the statement that w becomes random at e=0. one does learn something with this exercise though. The structure g appears provided both e" and its derivative vanish at all points. This tells us that the limit e-0 cannot be associated to time evolution( where e=0 would occur at some particular time to). Trying to understand the big-bang as a transition from zero metric into non-zero metric raises complicated issues on the role of causality Before the metric is created there is no causality at all [9] The problem we shall attack in this paper is the arbitrariness of the connection at e=0. Our approach will consist on adding a new interaction to the action such that as e-0, the spin connection does not go random but continues to be controlled by second order field equations Interestingly there are no too many terms one could add. For reasons we shall ex plain in a moment, it is necessary to go back to the metric formulation. Consider the following Palatini action including a new term that depends only on the connection, /[g, r]=/vg(g Ruv(r)+A)+KVIRuv(r Here Ruv is the Ricci tensor, which only depends on the connection, not the metric The new term is known as Eddington theory. The constant K is a coupling con stant which in principle should be small enough such that this action is not in This is related to another feature of e =0. The leading term of the action e, w]-fEabcRabefed is cubic with respect to w=e=0, eabcddwabe e. Thus, around e"=0 there is no quadratic term to expand, and no linearized theory can be defined. Of course the action can be expanded around the "broken solution ef= af with a well-defined linearized theory, but the interactions become non-renormalizable. In three dimensions this problem does not occur because the action has one less power of e and the quantum theory can be explored much further [13] Note that in particular the limit may be a smooth differentiable function. In that case the curvature is well-defined. In particular Ruy exists at the limit while R Ruy does not. Not surprisingly, metric invariants are not good objects to characterize the g=0 phase
28 M. Ba˜nados is ea = 0. (3) The existence of this solution has not gone unnoticed, e.g., [9,10,13]. Its most salient property is that it preserves the full set of diffeomorphisms and for this reason it is often called the ‘unbroken’ ground state of general relativity. This point was stressed in [9] were a symmetry breaking transition from ea = 0 to ea = 0, the big-bang, was suggested. Topological transitions in this formulation were studied in [10]. Now, a key aspect of this solution is the fact that the spin connection is left undetermined. The above equations of motion are supposed to determine both aa and wab. However, since ea = 0 kills both (1) and (2), the spin connection becomes a random field.1 The first step towards accepting ea = 0 as a solution is to understand the nature of the spin connection at that point. If one first solves the algebraic equation for the torsion expressing w ∼ e−1∂e as a function of ea the same problem reappears in a different way. One may try to recover the solution ea = 0 by a limit ea → 0. The connection w e−1∂e has the structure 0 0 and can be anything.2 This is equivalent to the statement that w becomes random at ea = 0. One does learn something with this exercise though. The structure 0 0 appears provided both ea and its derivative vanish at all points. This tells us that the limit ea → 0 cannot be associated to time evolution (where ea = 0 would occur at some particular time t0). Trying to understand the big-bang as a transition from zero metric into non-zero metric raises complicated issues on the role of causality. Before the metric is created there is no causality at all [9]. The problem we shall attack in this paper is the arbitrariness of the connection at ea = 0. Our approach will consist on adding a new interaction to the action such that, as ea → 0, the spin connection does not go random but continues to be controlled by second order field equations. Interestingly there are no too many terms one could add. For reasons we shall explain in a moment, it is necessary to go back to the metric formulation. Consider the following Palatini action including a new term that depends only on the connection, I[g,Γ ] = √g(gμν Rμν(Γ ) +Λ) +κ |Rμν(Γ )| . (4) Here Rμν is the Ricci tensor, which only depends on the connection, not the metric. The new term is known as Eddington theory. The constant κ is a coupling constant which in principle should be small enough such that this action is not in 1 This is related to another feature of ea = 0. The leading term of the action I[e,w] ∼ εabcdRabeced is cubic with respect to w = e = 0, εabcd dwabeced . Thus, around ea = 0 there is no quadratic term to expand, and no linearized theory can be defined. Of course the action can be expanded around the ‘broken’ solution ea μ = δa μ with a well-defined linearized theory, but the interactions become non-renormalizable. In three dimensions this problem does not occur because the action has one less power of ea and the quantum theory can be explored much further [13]. 2 Note that in particular the limit may be a smooth differentiable function. In that case the curvature is well-defined. In particular Rμν exists at the limit while R = gμν Rμν does not. Not surprisingly, metric invariants are not good objects to characterize the g = 0 phase
Eddington-Borm-Infeld Action and the Dark Side of General Relativity contradiction with well-known experiments. It is interesting to note the uniqueness of this term. In the absence of a metric, Eddington's functional is the only density with the correct weight to respect diffeomorphism invariance. Note that Eddingtons action cannot be defined in the first order tetrad formalism. The SO(3, 1)curvature Rbuv(w)cannot be traced to produce a two index object, without using eu. This is in contrast with the GL(4)curvature R Buy(r)whose trace RB,(r)is a tensor and ndependent from the metric. The attractive feature of the action()is that if the metric was not present, then he first two terms are not present and the dynamics is governed by Eddingtons ction. In this sense we have produced an action whose dynamics is well-defined even if the metric is switched off However, it is now a simple exercise to prove that the action(4)does not pro- duce any interesting new effects. Actually, this was already known to Eddington. What happens is that the Einstein-Hilbert action with a cosmological term is dual to Eddingtons action [8]. In other words, the Eddington term in(4)only renormal izes Newtons constant This can be seen as follows. Consider the Palatini action for gravity with a cos- mological constant Ipl, r]=/v8(g"VRuv(r)-2A) (5) It is well known that upon eliminating the connection using its own equation of motion one arrives at the usual second order hilbert action 图=/√(8)-24 It is less well-known but also true [8]that, if A f0, then the metric can also be eliminated by using its own equations. The variation aoHr=0 yields - Ruv(r) Since this is an algebraic relation for guy it is legal to replace it back in the the action obtaining Eddingtons functional 门=/V(n (8) row here the prescription from the guv=0. Another way to see this is 和mm he metric inverse g", at least for d>2. 4 This duality is of course well know to Claudio, and in fact the first time I heard about it was
Eddington–Born–Infeld Action and the Dark Side of General Relativity 29 contradiction with well-known experiments. It is interesting to note the uniqueness of this term. In the absence of a metric, Eddington’s functional is the only density with the correct weight to respect diffeomorphism invariance. Note that Eddington’s action cannot be defined in the first order tetrad formalism. The SO(3,1) curvature Ra bμν(w) cannot be traced to produce a two index object, without using ea μ. This is in contrast with the GL(4) curvature Rα βμν(Γ ) whose trace Rβ ν(Γ ) is a tensor and independent from the metric. The attractive feature of the action (4) is that if the metric was not present, then the first two terms are not present and the dynamics is governed by Eddington’s action.3 In this sense we have produced an action whose dynamics is well-defined even if the metric is switched off. However, it is now a simple exercise to prove that the action (4) does not produce any interesting new effects. Actually, this was already known to Eddington. What happens is that the Einstein–Hilbert action with a cosmological term is dual to Eddington’s action [8]. In other words, the Eddington term in (4) only renormalizes Newton’s constant. This can be seen as follows. Consider the Palatini action for gravity with a cosmological constant, IP[g,Γ ] = √g(gμν Rμν(Γ )−2Λ) (5) It is well known that upon eliminating the connection using its own equation of motion one arrives at the usual second order Hilbert action IH[g] = √g(R(g)−2Λ). (6) It is less well-known but also true [8] that, if Λ = 0, then the metric can also be eliminated by using its own equations.4 The variation δ I δgμν = 0 yields, gμν = 1 ΛRμν(Γ ). (7) Since this is an algebraic relation for gμν it is legal to replace it back in the the action obtaining Eddington’s functional IE[Γ ] = 2 Λ det(Rμν). (8) 3 We borrow here the prescription from the tetrad formalism: εabcdRabeced ∼ √ggμν Rμν vanishes if ea ∼ gμν → 0. Another way to see this is by noticing that the volume element √g scales faster than the metric inverse gμν , at least for d > 2. 4 This duality is of course well know to Claudio, and in fact the first time I heard about it was from him
M. Bardos In the terminology of dualities, the action (5) is called the Parent action, while the Einstein-Hilbert action(6) and Eddingtons action( 8)are its daughters. IH and IE are said to be dual to each other, and in many respects they are equivalent [6, 8 Summarizing, the action(4) can be understood as general relativity interact ing with its own dual field. By a set of duality transformations we can transform the whole action(4)into standard general relativity with a new coupling constant (Starting from(4)one eliminate the metric and to get Eddington action twice. Then apply a new transformation to get back to Einstein-Hilbert) An important note of caution is in order here. The equivalence between the Einstein-Hilbert and Eddington actions is true provided guv is not degenerate. For degenerate fields they do represent different dynamics, and in fact, only Eddington action is well-defined in that case. The reason we shall not consider these cases at, at the end of the day, we are interested in non-degenerate metrics anyway. Our guiding principle is to uncover what sort of modifications would be necessary to incorporate guv =0 as an allowed state. But, the physics phenomena we shall be nterested do require a non-degenerate metric We shall now recall an analogy with condensed matter physics, suggesting a different interpretation for Eddingtons action, which will truly depart from pure general relativity Within standard general relativity, we have observed that if the metric is removed then the spin connection goes random. This looks very similar to a set of spins, at T>T, in the presence of an external magnetic field. As the field is removed the spins go random. However, if the temperature is below its critical value T<Tc, then the external field can be removed and the spins retain their ordered state. The crucial property of spins which makes this possible is their self interaction. Eddington's action has a similar role. For the action (4), as the metric is removed, the connection continues to be described by a well-defined set of equations. 5 Now, spins also exhibit the opposite phenomena, namely spontaneous"ma netization. If T< T then the random disordered state is unstable and decay spontaneously into a broken ordered state. No external field is needed to trigger this phenomena. Let us imagine that the connection in general relativity exhibits a similar phenomena. That is, without introducing a metric, we assume that a con- nection can exists and be described by some well-defined equations. We shall call this connection Cg. This field is fully independent from the metric. Obviously,the only action consistent with general covariance is again Eddingtons theory, 1[C]=/VIKuv(C) S At this point we treat the metric or tetrad as an extemal field which can be switched on and off,as a mathematician would do. On a first approximation one does not look at the Maxwell equations governing the external field but simply assume that it can be controlled at will. We have assumed the same with the metric, treating it as an external field. A full action governing the coupled system will be displayed below
30 M. Ba˜nados In the terminology of dualities, the action (5) is called the Parent action, while the Einstein–Hilbert action (6) and Eddington’s action (8) are its daughters. IH and IE are said to be dual to each other, and in many respects they are equivalent [6, 8]. Summarizing, the action (4) can be understood as general relativity interacting with its own dual field. By a set of duality transformations we can transform the whole action (4) into standard general relativity with a new coupling constant. (Starting from (4) one eliminate the metric and to get Eddington action twice. Then apply a new transformation to get back to Einstein–Hilbert.) An important note of caution is in order here. The equivalence between the Einstein–Hilbert and Eddington actions is true provided gμν is not degenerate. For degenerate fields they do represent different dynamics, and in fact, only Eddington’s action is well-defined in that case. The reason we shall not consider these cases is that, at the end of the day, we are interested in non-degenerate metrics anyway. Our guiding principle is to uncover what sort of modifications would be necessary to incorporate gμν = 0 as an allowed state. But, the physics phenomena we shall be interested do require a non-degenerate metric. We shall now recall an analogy with condensed matter physics, suggesting a different interpretation for Eddington’s action, which will truly depart from pure general relativity. Within standard general relativity, we have observed that if the metric is removed then the spin connection goes random. This looks very similar to a set of spins, at T > Tc, in the presence of an external magnetic field. As the field is removed the spins go random. However, if the temperature is below its critical value T < Tc, then the external field can be removed and the spins retain their ordered state. The crucial property of spins which makes this possible is their self interaction. Eddington’s action has a similar role. For the action (4), as the metric is removed, the connection continues to be described by a well-defined set of equations.5 Now, spins also exhibit the opposite phenomena, namely spontaneous “magnetization.” If T < Tc then the random disordered state is unstable and decays spontaneously into a broken ordered state. No external field is needed to trigger this phenomena. Let us imagine that the connection in general relativity exhibits a similar phenomena. That is, without introducing a metric, we assume that a connection can exists and be described by some well-defined equations. We shall call this connectionCμ αβ. This field is fully independent from the metric. Obviously, the only action consistent with general covariance is again Eddington’s theory, I[C] = |Kμν(C)| (9) 5 At this point we treat the metric or tetrad as an external field which can be switched on and off, as a mathematician would do. On a first approximation one does not look at the Maxwell equations governing the external field but simply assume that it can be controlled at will. We have assumed the same with the metric, treating it as an external field. A full action governing the coupled system will be displayed below