Gravity is derived from an entropic action coupling matter fields with geometry. The fundamental idea is to relate the metric of Lorentzian spacetime to a density matrix and to describe the matter fields topologically according to a DiracKhler description as the direct sum of a zeroform a oneform and a twoform. While the geometry of spacetime is defined by its metric the matter fields can be used to define an alternative metric the metric induced by the matter fields which geometrically describes the interplay between spacetime and matter. The proposed entropic action is the quantum relative entropy between the metric of spacetime and the metric induced by the matter fields. The modified Einstein equations obtained from this action reduce to the Einstein equations with zero cosmological constant in the regime of low coupling. By introducing the Gfield which acts as a set of Lagrangian multipliers the proposed entropic action reduces to a dressed EinsteinHilbert action with an emergent small and positive cosmological constant only dependent on the Gfield. The obtained equations of modified gravity remains second order in the metric and in the Gfield. A canonical quantization of this field theory could bring new insights into quantum gravity while further research might clarify the role that the Gfield could have for dark matter.
['General Relativity and Quantum Cosmology', 'Condensed Matter - Statistical Mechanics', 'High Energy Physics - Theory', 'Quantum Physics']
Gravity from entropy
{'Gravity from entropy': 'Ginestra Bianconi 1, ∗  \n1 School of Mathematical Sciences, Queen Mary University of London, London, E1 4NS, United Kingdom  \nGravity is derived from an entropic action coupling matter fields with geometry. The fundamental idea is to relate the metric of Lorentzian spacetime to a density matrix and to describe the matter fields topologically, according to a Dirac-Kahler description, as the direct sum of a zero-form, a one-form and a two-form. While the geometry of spacetime is defined by its metric, the matter fields can be used to define an alternative metric, the metric induced by the matter fields, which geometrically describes the interplay between spacetime and matter. The proposed entropic action is the quantum relative entropy between the metric of spacetime and the metric induced by the matter fields. The modified Einstein equations obtained from this action reduce to the Einstein equations with zero cosmological constant in the regime of low coupling. By introducing the G-field , which acts as a set of Lagrangian multipliers, the proposed entropic action reduces to a dressed Einstein-Hilbert action with an emergent small and positive cosmological constant only dependent on the G-field. The obtained equations of modified gravity remains second order in the metric and in the G-field. A canonical quantization of this field theory could bring new insights into quantum gravity while further research might clarify the role that the G-field could have for dark matter.', 'I. INTRODUCTION': 'The relation between general relativity, statistical mechanics and information theory is a central research topic in theoretical physics. The interest in the subject has its roots in the discovery that black holes have an entropy [1, 2] and emit Hawking radiation [3]. Recently, important results have been obtained relating information theory, entanglement entropy [4-6] and gravity [7-13] involving the holographic principle [14-16], the entanglement properties of quantum field theory and the theory of von Neumann algebras [17-19].  \nThese results define a very active research direction [20] indicating that the quest for an ultimate gravitational theory based on information theory and statistical mechanics is ongoing. A comprehensive statistical mechanics approach to gravity is expected to give rise to modified Einstein equations [21-23] that on the one side can be testable experimentally [24, 25] while on the other side can bring important conceptual insights into the ultimate theory for black holes [26], dark matter [27] and quantum gravity [28-33].  \nIn this work, a continuum modified theory of gravity based on a statistical mechanics action is considered. This theory treats the metric at each point of spacetime as a density matrix, this central idea being inspired by the use of von Neumann algebras in explaining entanglement in field theories [17, 18]. While the geometry of spacetime is defined through its associated metric, the interplay between matter fields and geometry is captured by the metric induced by the matter fields which describes how the matter fields effectively curves spacetime. Embracing a statistical mechanics approach to gravity, this work proposes an action based on the quantum entropy between the metric and the metric induced by the matter fields. Thus the proposed action is meant to fully  \ndescribe how matter curves geometry and how geometry affects the matter fields. A crucial aspect of the presented theory is the adoption of a topological (Dirac-Kahler like [34, 35]) description of bosonic matter fields. Note that the extension of Dirac-Kahler and staggered fermions formalism to bosonic particles is gaining increasing interest in lattice gauge theory [36, 37] and in network theory as well [38]. These bosonic matter fields are described as the direct sum of a 0-form, a 1-form, and a 2-form defined on the Riemannian manifold describing spacetime. Moreover the metric induced by the topological matter fields is expressed in terms of the Hodge-Dirac operator [39]. From this statistical mechanics approach to gravity we derive the modified Einstein equations by introducing an auxiliary field associated to gravity which we call G-field . The introduction of this new field is justified as it acts as a set of Lagragian multipliers enforcing linear constraints on metric induced by the matter fields. In this way the G-field extends the popular use of Lagrangian multipliers in f ( R ) theories [21, 23]. Given the particular entropic structure of the action, the modified Einstein equations take a very simplified expression. The gravitational part of the action takes the form of a dressed Einstein-Hilbert action in which we observe an emergent positive cosmological constant that depends exclusively on the G-field.  \nThis work greatly expands on previous results [38] obtained in the discrete setting by the same author. On one side, here a continuum and fully Lorentz invariant theory is proposed. This progress is based on the development of the suitable mathematical framework to define the Lorentz invariant entropy and cross-entropy between the metric of the spacetime and the metric induced by the matter fields. On the other side, here the relation of this statistical mechanics/information theory action with the Einstein-Hilbert action [40, 41] is established defining a clear connection to gravity. Two fundamental aspects of this work that are not present in Ref. [38], are relevant to clearly relate this approach to gravity. First, this work considers a local theory, defining the entropy of the met-  \nric at each point of spacetime, while the previous work considers the entropy associated to the full metric of the higher-order network. The present local theory allows a closer connection to gravity and constitutes a step forward to establish the connection between this approach and the quantum theory of entanglement [17]. Secondly, by adopting the continuum limit, in this work the intrinsic difficulty related to the definition of the curvature of networks, simplicial and cell complexes is avoided.  \nTo keep the discussion concise the focus is here exclusively on scalar (bosonic) matter fields, and their topological generalizations while in Ref. [38] the theory covers also fermionic matter fields and gauge fields. Further extensions of the proposed local framework to Dirac and gauge fields [38, 42, 43] in the continuum or in the discrete setting are left for future investigations.  \nThis work is structured as follows. In Sec. II we provide the motivation of the proposed theory and we discuss preliminary results on an instructive warm-up scenario. In Sec. III we outline the proposed theoretical framework, we postulate the entropic action for gravity and we derive the corresponding modified Einstein equations. In Sec.IV we provide the concluding remarks. The paper also includes three Appendices discussing possible extensions of the proposed theoretical framework, and providing the mathematical background and all the necessary details regarding the notation used Sec. III .', 'A. Eigenvalues and logarithm of rank 2-tensors': 'Spacetime is described by a torsion free, d -dimensional Riemannian manifold K associated with a Lorentzian metric g µν of signature {-1 , 1 , , 1 . . . , 1 } and a metric compatible Levi-Civita connection Γ σ νµ determining the covariant derivative ∇ µ . In order to formulate our statistical mechanics and information theory action for gravity, we need first to define the eigenvalues the logarithm of rank-2 tensors ˆ G . To this end we first define the eigenvalues λ and eigenvectors V ( λ ) ν of the covariant tensor ˆ G of elements ˆ G µν in a Lorentz invariant way. These satisfy the eigenvalue problem  \nˆ G µν [ V ( λ ) ] ν = λV ( λ ) µ . (1)  \nWe say that a rank-2 tensor is positively defined if all its eigenvalues are positive. We notice that this definition of the eigenvalue of a rank-2 tensor reduces to the definition of the eigenvalue of the matrix ˆ G g -1 as Eq.(1) can be rewritten as  \nˆ G µσ g σν [ V ( λ ) ] ν = λV ( λ ) µ . (2)  \nOne striking consequence of this definition is that the eigenvalues of the metric g µν are all identically equal to one.  \nAssuming that the tensor ˆ G µν is positively defined, we define the logarithm of this tensor as  \n[ln( ˆ G )] µν = V ( λ ) µ V ( λ ) ν ln( λ ) . (3)  \nand the inverse of a tensor as  \n[ ˆ G -1 ] µν = [ V ( λ ) ] µ [ V ( λ ) ] ν λ -1 . (4)  \nIt follows that if the tensor ˆ G is invertible, the logarithm of the inverse of a positively defined tensor is given by  \n[ln( ˆ G -1 ] µν = -[ V ( λ ) ] µ [ V ( λ ) ] ν ln( λ ) . (5)  \nFinally we define the trace of a rank two tensor as the sum of its eigenvalues, i.e.  \nTr ˆ G = ∑ λ λ. (6)  \nThus the trace of a rank-2 tensor can be also calculated as usual in tensor calculus, as the trace of the matrix ˆ G g -1 , i.e.  \nTr ˆ G = Tr M ˆ G g -1 = ˆ G µν g µν . (7)  \nIn this first section we are interested exclusively on ranktwo tensors that are metrics between 1-forms. In the subsequent paragraphs and in Appendix C we will extended the notion of eigenvalues also to metric matrices between two-forms represented by rank-4 tensors. Such an approach will be shown in Appendix C to be general and applicable to metric tensors between two n -forms with n of any order.', 'B. Warm-up scenario: Entropic action': "Before we develop our theory, let us consider an instructive warm-up scenario that will help justify the theory that we will present in the following. Having defined the logarithm of positively defined rank-2 tensors, we are now in the position to define their Lorentz invariant quantum entropy H for metric associated to 1-forms. This is inspired by the expression of the Von Neumann entropy albeit we do not require the tensor to have trace one at every point in spacetime. We consider positive definite, invertible metric for 1-forms, represented by the covariant tensor ˆ G of rank two. We define the entropy H of ˆ G as  \nH = Tr ˆ G ln ˆ G -1 = ˆ G µν [ln ˆ G -1 ] νµ = -∑ λ ln λ. (8)  \nNote that the problem of defining eigenvalues and entropy of tensors is a topic of intense research and similar definitions are been provided for instance in the theory of elasticity [44] and in applied tensor analysis [45] (although not defined in a Lorentz invariant way). Recalling that the metric g for 1-forms has all the eigenvalues equal  \nto one, it follows that the entropy of of the metric g is null, i.e.  \nH = Tr g ln g -1 = 0 . (9)  \nThe fundamental assumption of the present theory is that spacetime is endowed with two metrics: the metric g fully determining the spacetime geometry and the metric induced by the matter fields G that fully capture the interplay between matter fields and geometry. Leaving the detailed discussion about the metric induced by the matter fields to the next paragraph, we postulate that the action should explicitly express the relation between these two metrics and their reciprocal coupling. By embracing a statistical mechanics approach we consider the Lagrangian L given by the quantum relative entropy between the metric g and the metric G induced by the matter field defined as  \nL = -Tr g ln g -1 +Tr g ln G -1 . (10)  \nUsing H = 0 we thus obtain:  \nL = Tr g ln G -1 . (11)  \nWe observe that since g has all the eigenvalues equal to the identity, and Eq.(2) holds for ˆ G = G , the Lagrangian L can be also be expressed as  \nL ≡ Tr M ln G g -1 = -∑ µ ln( µ ) , (12)  \nwhere with Tr M ln G g -1 we indicate the usual trace of the logarithm of the matrix ln G g -1 and with µ the eigenvalues of the 2-tensor G as defined in Eq.(2).  \nAn important point here is that the Lorentz invariant definition of the quantum relative entropy given by Eq.(12) requires that the matrix G has to be invertible as well as the matrix g . Thus this is another difference with respect to the strictly speaking density matrices that can be semi-definite positive. We are now in the position to consider an action S associated to the Lagrangian L , as given by  \nS = ∫ d r √ | -g |L , (13)  \nwhere | -g | indicates the absolute value of the determinant of g .  \nA series of observation are here in place to justify our interpretation of the metrics matrices as density matrices. These considerations are important for developments that might lead to a full quantum theory upon canonical quantization. As remarked previously we treat the metric matrices as density matrices with some important differences, including the fact that we require that they are invertible and we do not require that they have unitary trace. The requirement of treating exclusively invertible metrics g and G is dictated by our desire to  \ndefine the entropy in a Lorentz invariant way. In fact here we desire to treat the metric G on the same footing of its inverse G -1 . It might be argued that also our relaxation of the requirement to have density matrices of trace one, has similar roots. In fact if we require G to have unitary trace its inverse in general will not have a unitary trace.  \nThe relaxation of the condition to have unitary trace density matrices is related to the theory of quantum entanglement and the use of von Neumann algebras [17, 18]. Physically, we assume that the tensor ˆ G that is fully characterizing the geometry of the entire manifold, is defined at each point of p of the manifold K . It might be tempting to describe this geometry as an uncountable tensor product of the metrics defined at each point p . Thus, unless we integrate over a suitable measure, the trace associated to the entire metric is not defined. However, at each point p the metric ˆ G is associated to a finite trace, and to a finite entropy H which depends on p . This interpretation of the metric matrices ˜ G is in line with the physical understanding of von Neumann algebras [17, 18], although the establishment of a mathematical rigorous relation of the present approach with von Neumann algebras is beyond the scope of this work and will be the subject of subsequent publications.  \nThe interpretation of the metric matrix g as a density operator, also requires addressing which is the underlying quantum statistical ensemble in this physical interpretation. Although it is not our goal here to develop a full quantum theory, an indication of the underlying quantum interpretation might be provided by the introduction of quantum forms or q-forms (see Appendix A). These qforms might represent the quantum state of geometry and correspond to a geometric version of q-bits. Thus the metric ˆ G can be interpreted as a 'renormalizable density matrix' acting on the Hilbert space of q-forms.", 'C. Warm-up scenario: scalar matter fields': "We now apply this warm-up scenario in presence of scalar matter fields. We consider the complex valued scalar matter field ϕ ( x ) ∈ C with x ∈ M and we consider the d -dimensional manifold immersed in K ⊗ C defined by the points ( x , ϕ ( x )). The metric G induced on M by this matter fields is given by  \nG = g + α M , (14)  \nwhere α is a real positive parameter, M is the rank-2 tensor of elements  \nM µν = ( ∇ µ ¯ ϕ )( ∇ ν ϕ ) , (15)  \nwhere here and in the following ¯ ˙ indicates complex conjugation. Since G should be adimensional it is convenient to work in the units ℏ = c = 1 and to put α = α ' ℓ d P where ℓ P is the Planck length and α ' is adimensional.  \nWe observe that in the limit in which the field ϕ is real, i.e. ϕ ( x ) ∈ R and the metric flat and Euclidean, i.e. g µν = δ µν , the metric induced by the matter fields G given by Eq.(14) reduces to the first fundamental form of Gauss for the d manifold immersed in R d +1 defined by the set of points ( x , ϕ ( x )). For the underlying mathematical treatment of the metric induced by real scalar fields we refer the interested reader to Ref. [46].  \nLet us define the scalar product | ∇ ϕ | 2 as  \n| ∇ ϕ | 2 = ∇ µ ¯ ϕg µν ∇ ν ϕ. (16)  \nWe observe that the inverse G -1 of the induced metric G has metric given by  \n[ G -1 ] µν = g µν -α M µν 1 + α | ∇ ϕ | 2 . (17)  \nAdopting this notation we observe that the logarithm of the induced metric ln G and the logarithm of ln G -1 are given by  \nln G = f ( | ∇ ϕ | 2 ) G , ln G -1 = -f ( | ∇ ϕ | 2 ) G -1 , (18)  \nwhere  \nf ( w ) = ln(1 + αw ) w . (19)  \nwhere f ( w ) → α for | αw | ≪ 1. With this choice of the metric induced by the matter field we obtain that that the Lagrangian (11) reads  \nL = -ln(1 + α | ∇ ϕ | 2 ) . (20)  \nWe observe that in the limit α |∇ ϕ | → 0 then we have L → α | ∇ ϕ | 2 , i.e. we recover the Lagrangian corresponding to the massless Klein-Gordon equation.  \nBy minimizing the action S with respect to the field ϕ and to the metric g we obtain the Euler-Lagrange equations of motion:  \n∇ µ h ( |∇ ϕ | 2 ) g µν ∇ ν ϕ = 0 , (21)  \nwhere h ( w ) is given by  \nh ( w ) = α 1 + αw . (22)  \nIn the limit αw → 0 we have h ( w ) → α and the equation for the scalar field reduces to the simple massless Klein-Gordon equation. As we have already discussed previously, one might wonder whether our choice to take directly the metric matrix and interpret it as a 'renormalizable density matrix' is the most proper one, and whether it would be better to instead renormalize directly g and G with their trace. As an additional possible answer to this important question, in Appendix B we investigate what would be the corresponding equation of motion in this warm-up scenario. The result of this  \nsimple exercise points out that if we renormalize g and G with their trace at each point p of the manifold, we will not obtain the Klein-Gordon equation in the linear approximation, in part justifying our approach inspired from von Neumann algebras.  \nBy putting equal to zero the variation of the action δ S = 0 with respect to the metric g , we get  \nδ S = -h ( |∇ ϕ | 2 ) M µν -1 2 L g µν = 0 . (23)  \nIn empty spacetime M = 0 and L = 0, thus this equation is automatically satisfied independently on the value of g . Thus in this limit, the metric g is not determined by the action.  \nLet us now make some remarks about this warm-up derivation. We proposed a statistical mechanics framework that is very inspiring as we get the massless KleinGordon equation as the outcome of the minimization of a quantum entropy action for low coupling, i.e. 0 < α |∇ ϕ | 2 ≪ 1. However this approach has two important limitations. The first limitation is that the KleinGordon equation does not contain the mass term. The second limitation is that in absence of matter fields the metric is not determined. In the following we provide a more comprehensive framework to address these two limitations. This framework can be related to gravity as it gives rise to modified Einstein equations that reduce to the Einstein equations for low coupling.", 'A. Topological matter fields and their associated metrics': "In order to derive gravity from our entropic action we need to consider the topological bosonic matter field. The topological bosonic matter field is a type of Dirac-Kahler [34, 35] boson given by the direct sum of a 0-form, a 1form and a 2-form. Topological bosonic fields are receiving increasing attention in discrete theories developed in network [38] and lattice gauge theories [36, 37]. Taking into consideration topological bosonic fields will allow us to introduce in the metric induced by the matter fields, terms depending on the mass of the bosonic field. Thus, in this way, we address the first limitation of the warm-up scenario that we have presented above.  \nIn order to address the second limitation of the warmup scenario discussed previously, we include in the expression of the metric induced by the matter field, also terms depending directly on the curvature of the manifold. These terms will be expressed in terms of the Ricci scalar R , the Ricci (covariant) tensor R of elements R µν and the Riemann tensor of elements R µνρσ .  \nWe define the topological fields | Φ ⟩ as the direct sum between a zero (complex valued) form ϕ and a (complex  \nvalued) one-form ω µ dx µ , and a (complex valued) twoform ζ µν dx µ ∧ dx ν , with ζ µν = -ζ νµ i.e.  \n| Φ ⟩ = ϕ ⊕ ω µ dx µ ⊕ ζ µν dx µ ∧ dx ν , (24)  \nand its conjugate topological field ⟨ Φ | as  \n⟨ Φ | = ¯ ϕ ⊕ ¯ ω µ dx µ + ¯ ζ µν dx µ ∧ dx ν . (25)  \nThe considered metric matrix ˜ g is defined as the direct sum of the metric matrix for 0-forms (the identity), the metric matrix g among 1-forms introduced previously, and the metric g (2) for 2-forms given by  \ng (2) = g µν g ρσ ( dx µ ∧ dx ν ) ⊗ ( dx ρ ∧ dx σ ) , (26)  \nor, exploiting the anti-symmetry of the 2-forms,  \ng (2) = [ g (2) ] µνρσ ( dx µ ∧ dx ν ) ⊗ ( dx ρ ∧ dx σ ) , (27)  \nwith  \n[ g (2) ] µνρσ = 1 2 ( g µρ g νσ -g µσ g νρ ) . (28)  \nIt then follows that ˜ g is given by  \n˜ g = 1 ⊕ g µν dx ν ⊗ dx ν (29) ⊕ [ g (2) ] µνρσ ( dx µ ∧ dx ν ) ⊗ ( dx ρ ∧ dx σ ) .  \nThe scalar product among topological fields is defined as  \n⟨ Φ | Φ ⟩ = | ϕ | 2 + ¯ ω µ ω µ + ¯ ζ µν ζ µν , (30)  \nwhere ω µ ω µ = ω ν g νµ ω µ , while the outer product is given by  \n| Φ ⟩ ⟨ Φ | = ¯ ϕϕ ⊕ ( ¯ ω µ ω ν dx µ ⊗ dx ν ) ⊕ ¯ ζ µν ζ ρσ ( dx µ ∧ dx ν ) ⊗ ( dx ρ ∧ dx σ ) . (31)  \nIndicating here with d the differential operator and with δ the codifferential operator, we define the Dirac operator D as D = δ + d , that is restricted to the space of topological bosons. Thus we define the action of D over | Φ ⟩ as  \nD | Φ ⟩ = -∇ µ ω µ ⊕ ( ∇ µ ϕ -∇ ρ ζ ρµ ) dx µ ⊕∇ µ ω ν dx µ ∧ dx ν . (32)  \nThe metric ˜ G induced by the topological matter field will have structure similar to ˜ g that we can write in full generality as  \n˜ G = G (0) +[ G (1) ] µν dx µ ⊗ dx ν (33) ⊕ [ G [2] ] µνρσ ( dx µ ∧ dx ν ) ⊗ ( dx ρ ∧ dx σ ) . (34)  \nProceeding as in the previous case we might wish to define the metric induced by the topological field as  \n˜ G = ˜ g + α ( D | Φ ⟩ ⟨ Φ | D ) . (35)  \nIn our interpretation of the metric as a quantum density matrix, this would correspond to density matrix corresponding to a pure state D | Φ ⟩ . However, if the induced metric ˜ G is interpreted as a density matrix, it is natural to add further terms in the metric induced by the topological matter field ˜ G . These terms will depend on the topological field and the geometry of the space which will allow us to describe mixed states. First, we introduce in ˜ G a term proportional to | Φ ⟩ ⟨ Φ | . Specifically we introduce a term ( m 2 + ξR ) | Φ ⟩ ⟨ Φ | where R is the Ricci scalar and ξ might include the case of conformal coupling ξ = 1 / 6. This term can be also interpreted as a projector. Secondly, we introduce a term depending explicitly on the curvature of the manifold. Since ˜ G involves the metric for 0-forms, 1-forms and 2-forms on equal footing, it is natural to consider a further term involving ˜ R given by the direct sum of the Ricci scalar R , the Ricci tensor of elements R µν and the Riemann tensor of elements R µνρσ , i.e.  \n˜ R = R ⊕ ( R µν dx µ ⊗ dx ν ) ⊕ R µνρσ ( dx µ ∧ dx ν ) ⊗ ( dx ρ ∧ dx σ ) . (36)  \nIncluding ˜ R into the metric ˜ G will allow as to describe more general metric matrices that are not decomposable into sum of projectors.  \nFrom these considerations, it follows that in the induced metric ˜ G , we will substitute the term D | Φ ⟩ ⟨ Ψ | D with ˜ M defined as  \n˜ M = D | Φ ⟩ ⟨ Φ | D +( m 2 + ξR ) | Φ ⟩ ⟨ Φ | . (37)  \nAs we will see in the following paragraph this choice will allow us to effectively overcome the first limitation of the warm-up scenario and to recover the Klein-Gordon equation in curved spacetime in full. Furthermore we consider also the additional term proportional to ˜ R and we postulate that the metric ˜ G induced by the geometry and the matter fields is given by  \n˜ G = ˜ g + α ˜ M -β ˜ R . (38)  \nAs we will see in the following the addition of the term proportional to ˜ R will allow also to overcome the second limitation of the warm-up scenario. Note that here α, β are positive constants. In particular, since we require ˜ G to be adimensional, we need to consider α = α ' ℓ d P and β = β ' ℓ 2 P where ℓ P is the Planck length and α ' , β ' are adimensional in the units ℏ = c = 1.  \nPossibly this approach could be extended to include higher-forms. However, for simplicity, we consider here only topological matter fields formed by the direct sum between a 0-form, a 1-form and a 2-form as this the minimal choice that will allow us to include in the action the Ricci scalar, the Ricci and the Riemann tensor explicitly.  \nFIG. 1. Schematic representation of this theoretical framework.The metric induced by the matter field ˜ G affects the metric of the manifold ˜ g and vice versa the metric of the manifold affects the metric induced by the matter field. The considered Lagrangian is given by the quantum relative entropy between the topological metric ˜ g of spacetime and the topological metric induced by the topological matter fields ˜ G . Since, by definition, the entropy of the metric vanishes identically, the quantum relative entropy in the Lagrangian L reduces to a single term: the quantum cross-entropy.  \n<!-- image -->", 'B. Entropic topological and geometrical action': "We propose a statistical mechanics action formulated in terms of the quantum relative entropy between the metric ˜ g and the metric ˜ G induced by the topological matter fields. In order to define our action we need to extend the notion of eigenvalues to metric tensors between 2-forms. This will allow to define the entropy and the quantum relative entropy in this novel framework. For a detailed discussion of this mathematical background see Appendix C that constitutes also the reference for our notation. First of all we observe that the entropy associated to the metric ˜ g remains zero. Indeed we define ˜ H as  \n˜ H = Tr˜ g ln ˜ g -1 = 1ln1 + Tr g ln g -1 = Tr g (2) ln g -1 (2) = 0 , (39)  \nwhere we have used Eq.(9) and the identity derived in the Appendix C (in Eq.(C39)),  \nTr g (2) ln g -1 (2) = 0 , (40)  \nThis equation is a direct consequence of the fact that not only g but also g (2) , has all the eigenvalues equal to one.  \nWe are now in the position to consider the Lagrangian given by the quantum relative entropy between ˜ g and ˜ G ,  \nL := -Tr˜ g ln ˜ g -1 +Tr˜ g ln ˜ G -1 (41)  \n(see Figure 1 for an illustration of the physical model beyond the choice of this Lagrangian). Since we have H = 0 we obtain  \nL = Tr˜ g ln ˜ G -1 . (42)  \nBy treating separately the contributions of G (0) G (1) and G (2) we get the explicit expression for the Lagrangian L given by  \nL := ln[ G (0) ] -1 +Tr g ln[ G (1) ] -1 +Tr g (2) ln[ G (2) ] -1 . (43)  \nSince the metric matrices g ( n ) have all their eigenvalues equal to one, using the notation developed in Appendix C we can express this Lagrangian as well as  \nL := -Tr F ln ˜ G ˜ g -1 = -ln[ G (0) ] -Tr F ln[ G (1) ] g -1 -Tr F ln[ G (2) ] -1 [ g (2) ] -1 . (44)  \nThe resulting statistical mechanics action S associated to the Lagrangian L is given by  \nS = ∫ d r √ | -g |L . (45)  \nThis action defines a modified theory of gravity. In the linearised limit α ' ≪ 1, β ' ≪ 1 this action reduces to the Einstein-Hilbert action with zero cosmological constant [40, 41] coupled with the scalar topological field. Indeed in this limit L reduces to  \nL = 3 βR -α ⟨ Φ | D 2 | Φ ⟩ -α ( m 2 + ξR ) ( | ϕ | 2 + ¯ ω µ ω µ + ¯ ζ µν ζ µν ) . (46)  \nwhere  \n⟨ Φ | D 2 | Φ ⟩ = |∇ ϕ | 2 + |∇ µ ω µ | 2 + ∣ ∣ ∣ ∇ σ ¯ ω σµ ∣ ∣ ∣ 2 + ∣ ∣ ∣ ϵ µνρ ∇ µ ω ν ∣ ∣ ∣ 2 . (47)  \nIn the interesting limit in which ω µ = 0 and ζ µν = 0 and the bosonic matter field is only scalar, Eq.(46) reduces to the widely studied Einstein-Hilbert action with zero cosmological constant coupled with the scalar field [48],  \nL = 3 βR -α |∇ ϕ | 2 -α ( m 2 + ξR ) | ϕ | 2 . (48)  \nAs already anticipated, this implies, among the other things, that this framework overcomes the first and the second limitations of the warm-up scenario. In fact in this way we recover the Klein-Gordon Lagrangian in curved spacetime in full. Moreover, by adding to ˜ G the term proportional to the curvature ˜ R we solve the second limitation of the warm-up scenario, and we obtain that the entropic action allows for the full determination of the metric in the vacuum.", 'C. Introduction of G-field ˜ G and the field ˜ Θ': "In order to investigate the property of the modified gravity emerging from the entropic action, here we introduce the auxiliary G-field ˜ G and the auxiliary field ˜ Θ . In  \nthis way we will be able to show in the following that the proposed entropic action can be interpreted as the sum of a dressed Einstein-Hilbert action with an emergent positive cosmological constant only depending on the G-field ˜ G and a matter action.  \nAs mentioned before we can express the Lagrangian L given by Eq.(42) as  \nL = -Tr F ln ˜ G ˜ g -1 . (49)  \nThis Lagragian is nonlinear in ˜ G and thus in ˜ R .  \nSeveral modified gravity actions are questioned because they give rise to theories with derivatives of the metric higher than two. Such theories can be affected by the Ostrogradsky instability, however non-linear theories not suffering from this pathology are also known, most notably the f ( R ) theories [21, 23]. Thus an important question is whether the proposed theory is also affected by the Ostrogradsky instability or rather, it generalizes and extends the realm of viable modified gravity theories beyond the f ( R ) theories. In particular the Ostrogradsky instability can arise when the Lagrangian depends on derivatives of order higher than two of the fields, and thus, in the gravitational setting on higher powers of the curvature ˜ R .  \nIn order to tackle this question we observe that the Lagrangian L has an easy expression as it is only dependent on the product ˜ G ˜ g -1 . Thus, by introducing an auxiliary field ˜ Θ and imposing the constraints  \n˜ G ˜ g -1 = ˜ Θ . (50)  \nwith Lagrangian multipliers ˜ G (constituting another auxiliary field) that we will call the G-field , we can reduce our theory to a theory driven by the Lagrangian ˜ L given by  \n˜ L = -Tr F ln ˜ Θ -Tr F ˜ G ( ˜ G ˜ g -1 -˜ Θ ) . (51)  \nThis Lagrangian is now linear in ˜ G ˜ g -1 , thus by introducing the auxiliary fields and giving a physical meaning to them as encoding for new particles, we have reduced our theory to a theory linear in ˜ R . The fields ˜ Θ and ˜ G are given by  \n˜ Θ = Θ (0) ⊕ Θ (1) ⊕ Θ (2) , ˜ G = G (0) ⊕ G (1) ⊕ G (2) , (52)  \nwith Θ (1) having elements [ Θ (1) ] ν µ , Θ (2) having elements [ Θ (2) ] ρσ µν and G (1) having elements [ G (1) ] ν µ , G (2) having elements [ G (2) ] ρσ µν . The term W , of the Lagrangian ˜ L that is linear in ˜ G ˜ g -1 , can be expressed as  \nW = Tr F ˜ G ( ˜ G ˜ g -1 -˜ Θ ) = 2 ∑ n =0 W n , (53)  \nwith  \nW 0 = G (0) ( G (0) -Θ (0) ) , W 1 = [ G (1) ] µ ρ ( [ G (1) ] µν g νρ -[Θ (1) ] ρ µ ) , W 2 = [ G (2) ] µν ηθ ( [ G (2) ] µνρσ [ g (2) ] ρσηθ -[Θ (2) ] ηθ µν ) . (54)  \nNote that the use Lagrange multipliers in the present theory extends the equivalence between the f ( R ) theories and Brans-Dicke Theory achieved through a Legendre transformation [21, 23]. However using an extensive number of Lagrange multipliers might be considered a dangerous mathematical operation. For the sake of simplicity here we work under the assumption that ˜ L is equivalent to L . Whereas it might be proven that L is not equivalent to ˜ L we assume that our true theory is given by ˜ L which we might always consider as a 'canonical' version of L .  \nThe resulting statistical mechanics action ˜ S associated to the Lagrangian ˜ L is given by  \n˜ S = ∫ d r √ | -g | ˜ L . (55)  \nWe are now in the position to derive the equations for the matter fields Φ, the metric g µν and the fields ˜ Θ and ˜ G by considering the variation of ˜ L with respect to these fields.", 'D. Equation of motion for the matter fields': 'Here we consider the action ˜ L defined in Eq.(51) depending linearly on ˜ G which is encoding for the matter fields. By using the definition of ˜ G given by Eq.(38), we can derive the equation of motion of the matter fields by considering the variation with respect to ⟨ Φ | , getting,  \nD ˜ g -1 G D | Φ ⟩ +˜ g -1 G ( m 2 + ξR ) | Φ ⟩ = 0 . (56)  \nHere ˜ g G , given by  \n˜ g G = ˜ G -1 g, (57)  \ncan be interpreted as the dressed metric which affects the matter fields. Specifically, g can be seen as a bare metric and the G-field ˜ G -1 can be seen as a dressing of this metric that gives rise to the dressed metric ˜ g G given by Eq.(57).', 'E. Modified gravity': 'We now turn to the equations for modified gravity considering variation of our action ˜ S with respect to the g and the fields ˜ Θ and ˜ G .', '1. Variation with respect to ˜ G': "The variation with respect to the fields ˜ G enforces the constraints in Eq.(50) which, by using the expression for ˜ G given by Eq.(38), can be expressed explicitly as  \n˜ Θ = ˜ I + α ˜ M ˜ g -1 -β ˜ R ˜ g -1 , (58)  \nwhere ˜ I is the (topological) identity. Since α = α ' ℓ 4 P and β = β ' ℓ 2 P in the regime of low energies and low curvatures, we have that ˜ Θ is a small perturbation of the identity ˜ I .  \nNote that these are equations determine the relation between the fields ˜ Θ the curvature ˜ R and the matter field determining ˜ M . We can write these equations by separating the contribution corresponding to the metric for 0-forms, 1-forms and 2-forms obtaining  \nΘ (0) = 1 + αM (0) -βR, [Θ (1) ] ν µ = δ ν µ + α [ M (1) ] µρ g ρν -βR µρ g ρν , [Θ (2) ] ⃗ν ⃗ µ = δ ⃗ ρ ⃗ µ + α [ M (2) ] ⃗ µ⃗ρ [ g (2) ] ⃗ ρ⃗ν -βR ⃗ µ⃗ρ [ g (2) ] ⃗ ρ⃗ν ,  \nwhere we have indicated a pair of indices with a vector symbol, e.g. ⃗ µ . Since the trace of the Ricci tensor as well as the trace of the Riemann tensor, are both equal to the Ricci scalar, upon performing the trace of these expressions, we get some constraints on the trace of the fields Θ ( n ) , i.e.  \n( d n ) -Tr F Θ ( n ) + α Tr F M ( n ) g -1 ( n ) = βR, (59)  \nfor any n ∈ { 0 , 1 , 2 } .", '2. Variation with respect to ˜ Θ': 'Thanks to the logarithmic nonlinearity in ˜ L the variation of the action ˜ S with respect to ˜ Θ simply reads  \n˜ Θ -1 = ˜ G . (60)  \nThus we obtain that the field ˜ Θ corresponds to the inverse of the dressing G-field ˜ G . Using Eqs.(58) we find that the equation of motion for the G-field ˜ G is given by  \n˜ G -1 = ˜ I + α ˜ M ˜ g -1 -β ˜ R ˜ g -1 , (61)  \nand we can simply eliminate the field ˜ Θ , from the action. In this way the considered action ˜ S can be decomposed into two terms as  \n˜ S = β S G + α S M , (62)  \nwith  \nS G = ∫ √ | -g |L G d x , S M = ∫ √ | -g |L M d x . (63)  \nThe Lagrangians L G and L M appearing in the above definition of S G and S M can be expressed as  \nL G = ( R G -2Λ G ) , L M = -M G . (64)  \nwhere R G is the dressed Ricci scalar and M G includes all dependence with the matter fields | Φ ⟩ while Λ G is the emergent positive cosmological constant. Specifically we have  \nR G = Tr F ˜ g -1 G ˜ R , M G = Tr F ˜ g -1 G ˜ M , Λ G = 1 2 β ( ˜ G -˜ I -Tr F ln ˜ G ) . (65)  \nFrom this reformulation of the action ˜ S we see that the Lagrangian L G has a the structure of a dressed EinsteinHilbert action and depends on the metric and the G-field ˜ G . In particular the Ricci scalar R is substituted to the dressed Ricci scalar R G and the role of the cosmological constant is played by Λ G that is non-negative and only dependent on the G-field ˜ G . Whenever this field is close to the identity, e.g. ˜ G ≃ ˜ I + ˜ ϵ we obtain that Λ G is positive but very small, i.e. its first non-trivial term is quadratic in ˜ ϵ . Thus the emergent positive cosmological constant Λ G is small in this limit and only determined by the G-field. Finally we recover as already noticed, that the action ˜ S reduces to the Einstein-Hilbert with zero cosmological constant in the low coupling limit.', '3. Variation with respect to g': 'The modified Einstein equations are obtained by performing the variation of the action ˆ S with respect to g . Let us define the stress-energy tensor T in the usual way with elements T µν given by  \n-1 √ | -g | δ S M dg µν = T µν . (66)  \nWith this notation we can expressed the modified Einstein equations as  \nR G ( µν ) -1 2 g µν ( R G -2Λ G ) + D ( µν ) = κ T ( µν ) , (67)  \nwhere κ = α/β , ( µν ) indicates symmetrization of the indices, R G µν are the elements or the dressed Ricci tensor given by  \nR G µν = G (0) R µν +[ G (1) ] ρ µ R ρν +2[ G (2) ] ρ 1 ρ 2 µη g ση R ρ 1 ρ 2 νσ , (68)  \nand D µν are the elements depending on second derivatives of the G-field ˜ G given by  \nD µν = ( ∇ ρ ∇ ρ g µν -∇ µ ∇ ν ) G (0) +2 ∇ η ∇ ρ [ G (2) ] µρνη + 1 2 ∇ ρ ∇ ρ [ G (1) ] µν + 1 2 ∇ ρ ∇ η [ G (1) ] ρη g µν -∇ ρ ∇ ν [ G (1) ] ( ρµ ) . (69)  \nIt follows that the modified Einstein equations involve only second derivatives of the metric and second derivatives of the field ˜ G . However these equations might have more solutions than the Einstein equations. A detailed study of the solutions and the viability of these equations is beyond the scope of this work and will be the subject of future investigations.', '4. Discussion': 'In summary the introduction of the G-field turns the proposed entropy action Eq.(42) into the action Eq.(62) involving a dressed Einstein-Hilbert action and a dressed matter action given by Eq.(63) depending on the matter and the gravity dressed Lagrangians Eq.(64) respectively. The gravitational dressed action L G displays an emergent positive cosmological constant Λ G only dependent on the G-field and given by Eq.(65). The equations of motion associated to this action involve solving Eqs.(56) for the matter fields, and Eqs. (61) and Eqs. (67) for the metric and the G-field. All these equations involve at most second derivatives of the fields.', 'IV. CONCLUSIONS': 'This work proposes a modified theory of gravity emerging from statistical mechanics and information theory action. The fundamental idea of this theory is to associate the metric to a density matrix. In particular two metrics are discussed: the metric of spacetime that fully defines its geometry and the metric induced by matter-fields that are effectively curving the space. The interplay between geometry of spacetime and matter fields is explicitly captured by the proposed action given by the quantum relative entropy between the metric and the metric induced by the matter fields.  \nHere a Lorentzian theory consistent with this fundamental physical interpretation of gravity is formulated. In order to do this we have built the necessary mathematical background to define the entropy and the crossentropy of metric tensors between two n -forms in a Lorentzian well defined way. A modified gravity is emerging from this framework when the matter fields are described by topological Dirac-Kahler bosons formed by the direct sum between a 0-form, a 1-form and a 2-form and the induced metric also depends on the curvature of spacetime. The modified Einstein equations reduce to the Einstein equations in the regime of low coupling. By introducing the G-field we obtain the modified Einstein equations and the equations of motion for the matter and the G-field. From this theory it emerges that the proposed entropic action takes the form of a sum between a dressed Einstein-Hilbert action and a matter action. Interesting the dressed Einstein-Hilbert action displays an emergent positive cosmological constant that only depends on the G-field. Moreover, thanks to the intro-d  \nction of the G-field, the equations of modified gravity remains second order in the metric, in the matter and in the G-field.  \nThe interpretation of the metric as a density matrix is shown to be very useful, albeit we have deformed a bit the notion of density matrices relaxing the constraints of having unitary trace, and requiring the existence of the inverse. These choices are motivated here by the necessity to have a Lorentzian invariant theory. Moreover this generalization of the notion of density matrices points out to possible connections of this approach to von Neumann algebras and the theory of entanglement in quantum field theories [17] that will be explored in future works.  \nIn conclusion, we hope that this approach can help identify the deep connections between gravity and statistical physics. Given the interpretation of the metric as a density matrix our hope is that this approach will also be instrumental to formulate new approaches to quantum gravity. Finally future investigation might explore the role of the the G-field in dark matter. Future directions in this line of research involve also the investigation of the proposed entropic action under the renormalization group, and possible connections with phenomenology and experimental results.  \nThis work was partially supported by a grant from the Simons Foundation. The author would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Hypergraphs: Theory and Applications, where work on this paper was undertaken. This work was supported by EPSRC grant EP/R014604/1.', 'Appendix A: Quantum interpretation: q-forms': "In order to outline a possible quantum interpretation of our theory we consider here a toy quantum example that extends the notion of a q-bits to q-forms. We assume that q-forms belong to the Hilbert space H (1) p defined at point p of coordinates x of the considered d -dimensional manifold K . This Hilbert space includes as states all q-forms or more specifically 'quantum 1-forms' | ω p ⟩ defined at p . A given choice of coordinate systems for our manifold define a basis for the Hilbert space H (1) p with basis | dx µ ⟩ with µ ∈ { 1 , 2 , . . . , d } .  \nThus quantum 1-form | ω ⟩ is given by  \n| ω p ⟩ = ω µ ( x ) | dx µ ⟩ , (A1)  \nwhere ω µ ( x ) is the probability amplitude that the q-form is along dx µ . A possible quantum interpretation of the metrics for 1-forms g and G , is then as a 'renormalizable density matrices' for q-forms.  \nThis construction of q-forms can be similarly extended to higher-order q-forms such as q-forms of order 2. These latter q-forms live in the Hilbert space H (2) p of with basis  \n| dx µ ∧ dx µ ⟩ with µ, ν ∈ { 1 , 2 , . . . , d } and µ < ν with  \n| dx µ ∧ dx µ ⟩ = | dx µ ⟩ | dx ν ⟩ - | dx ν ⟩ | dx µ ⟩ . (A2)  \nConsequently the q -form | ζ p ⟩ can be expressed as  \n| ζ p ⟩ = ζ µν ( x ) | dx µ ∧ dx µ ⟩ . (A3)  \nAccordingly, a possible quantum interpretation of the metric for 2-forms g (2) or G (2) , defined in Sec. III is then as 'renormalizable density matrices' for q-forms of order 2.  \nIt is to note that a full topological theory of q-forms involving at the same time 0, 1 and 2 q-forms will need to be compatible with the uncertainty principle. Thus this theory might involve non trivial fundamental assumptions on the quantum topological q-forms which are beyond the scope of this work.", 'Appendix B: The effect of renormalizing the metric matrices pointwise': "Let assume for the moment that instead of considering the metric matrices g and G , as 'renormalizable density matrices' at point p , we normalize g and G at each point in space. In this scenario we would consider the density matrices ρ g and σ G given by  \nρ g = g Tr g , σ G = G Tr G . (B1)  \nThe first observation that we make is that with this choice, we cannot any longer treat ρ g and ρ -1 g on the same footing, as we do for g and g -1 as ρ g will have unitary trace while ρ -1 g will not.  \nAnother difference emerges if we want to consider the dynamics of the warm-up scenario in which we use ρ g and σ G instead of directly g and G . Indeed the Lagrangian L defined in Eq.(10) would turn into ˆ L  \nˆ L = Tr ρ g ln ρ -1 g -Tr ρ g ln σ -1 G , (B2)  \nwhere the Lorentz invariant trace for rank-2 tensor is defined as in Sec. II A. Let us now investigate in the warm-up example, the consequence of renormalizing the metric matrices in this way. We observe that since g has all eigenvalue equal to one, its trace is simply Tr g = d , thus  \nρ g = g d , (B3)  \nand the entropy ˆ H becomes non zero and given by ln d . In fact, using H = Tr g ln g -1 = 0 we have  \nˆ H = Tr ρ g ln( ρ g ) -1 = ln d. (B4)  \nIf we take G , the metric induced by the matter field as in Eq.(14), we have than that its trace is given by  \nTr G = d + α |∇ ϕ | 2 . (B5)  \nUsing these results it can be easily shown that we can express the Lagrangian ˆ L using the normalized density matrices ρ g and σ G and given by Eq.(B2) as a function of L given by Eq. (10), i.e.  \nˆ L = 1 d L +ln ( 1 + α d |∇ ϕ | 2 ) . (B6)  \nUsing the Eq.(20) to express L we thus obtain  \nˆ L = -1 d ln ( 1 + α |∇ ϕ | 2 ) +ln ( 1 + α d |∇ ϕ | 2 ) . (B7)  \nIf follow that for α ' ≪ 1 we obtain that the first nontrivial term is given by  \nˆ L = d -1 2 d 2 α 2 ( |∇ ϕ | 2 ) 2 . (B8)  \nIn other words, in this limit we will not recover the zeromass Klein-Gordon action as we obtain with the action L using directly g and G instead of ρ g and σ G . Therefore this exercise provides us an hint of the relevance to use the metric matrices as 'renormalizable density matrices'.", 'Appendix C: Eigenvalues and entropy for metrics between 1-forms and for metric between n -forms': 'In this Appendix our goal is to use algebraic geometry [49] to define the eigenvalues of the metrics between n -forms and their associated entropy. In order to establish the foundation of this treatment let us revisit the treatment outlined in Sec. II for defining the eigenvectors and the associated entropy of the metric for 1-forms and treat on the same footing metrics tensor between pair of n -forms with arbitrary n .', 'a. Eigenvalues of metric matrices G ( n )': 'A one form can be written as  \nω = ω ρ dx ρ (C1)  \nand the metric for among two of such 1-forms can be written as  \nG (1) = [ G (1) ] µν dx µ ⊗ dx ν , (C2)  \nwhere [ G (1) ] µν is Hermitian. The contraction between the metric G (1) and the generic one form ω is defined as  \nG (1) · ω = ˆ ω = ˆ ω µ dx µ , (C3)  \nwhere, by definition  \nˆ ω µ := [ G (1) ] µν ω ρ ⟨ dx ν , dx ρ ⟩ , (C4)  \nwhere here and in the following ⟨ , ⟩ indicates the inner product between n -forms. Thus for canonical 1-forms we have  \n⟨ dx ν , dx ρ ⟩ = g νρ . (C5)  \nInserting this expression in Eq.(C4) we get,  \nˆ ω µ = [ G (1) ] µν g νρ ω ρ . (C6)  \nSummarizing we have shown that according to our definition we have  \nG (1) · ω = [ G (1) ] µν g νρ ω ρ dx µ . (C7)  \nThe eigenvalue problem of a metric G (1) among 1-forms is then defined as  \nG (1) · ω ≡ λω, (C8)  \nor equivalently in matrix form as  \n[ G (1) ] µν g νρ ω ρ = λω µ . (C9)  \nWe are now in the position to extent the formalism to the metricx G (2) among two forms of order two. The generic two form ζ is defined as  \nζ = ζ ηθ dx η ∧ dx θ , (C10)  \nwhere ζ ηθ = -ζ θη , i.e. where ζ ηθ is antisymmetric. The metric G (2) for two-forms is defined as  \nG (2) = [ G (2) ] µνρσ ( dx µ ∧ dx ν ) ⊗ ( dx ρ ∧ dx σ ) . (C11)  \nwhere G (2) is antisymmetric in the first 2 indices and in the last 2 indices and Hermitian under the exchange of the first 2 indices with the second 2 indices. Following a similar line of reasoning used to define the eigenvalue of the metric G (1) we define the contraction  \nG (2) · ζ = ˜ ζ = ˆ ζ µν dx µ ∧ dx ν , (C12)  \nwhere, by definition ˆ ζ µν is defined as  \nˆ ζ µν := 1 2 [ G (2) ] µνρσ ζ ηθ 〈 dx ρ ∧ dx σ , dx η ∧ dx θ 〉 . (C13)  \nPerforming the inner products we thus get  \nˆ ζ µν = 1 2 [ G (2) ] µνρσ ζ ηθ ( g ρη g σθ -g ρθ g ση ) = [ G (2) ] µνρσ [ g (2) ] ρσηθ ζ ηθ , (C14)  \nwhere we have used the definition of g (2) given in Eq.(28) that we rewrite here for convenience,  \n[ g (2) ] µνρσ = 1 2 ( g µρ g νσ -g µσ g νρ ) . (C15)  \nSummarizing we have shown that according to our definition we have  \nG (2) · ζ = [ G (2) ] µνρσ [ g (2) ] ρσηθ ζ ηθ dx µ ∧ dx ν . (C16)  \nIn total analogy with the previous case, the eigenvalue problem associated to the metric G (2) among 2-forms, is defined as  \nG (2) · ζ ≡ λζ, (C17)  \nor equivalently,  \n[ G (2) ] µνρσ [ g (2) ] ρσηθ ζ ηθ = λζ µν . (C18)  \nLet us now consider the particular case in which G (2) = g (2) . In this case the eigenvalue problem Eq.(C18) reads  \n1 4 ( g µρ g νσ -g µσ g νρ )( g ρη g σθ -g ρθ g ση ) ζ ηθ = λζ µν .  \nwhich has solution  \nλ = 1 , (C19)  \nfor any arbitrary 2-form associated to the antisymmetric tensor ζ µν . Thus the degeneracy of the eigenvalue is given by d ( d -1) / 2.  \nThe treatment of the eigenvalues associated to the metrics tensors G ( n ) between n -forms can be performed in a straightforward way, following similar steps. Let us observe that the generic n -form is given by  \nζ = ζ ν 1 ν 2 ...ν n dx ν 1 ∧ dx ν 2 , ∧ . . . ∧ dx ν n . (C20)  \nwhere ζ ν 1 ν 2 ...ν n is antisymmetric, while the generic expression for G ( n ) is given by  \nG ( n ) = [ G ( n ) ] µ 1 µ 2 ...µ n ν 1 ν 2 ...ν n ( dx µ 1 ∧ dx µ 2 ∧ . . . dx µ n ) ⊗ ( dx ν 1 ∧ dx ν 2 , ∧ . . . ∧ dx ν n ) , (C21)  \nwhere G ( n ) is antisymmetric in the first n indices and in the last 2 indices and Hermitian under the exchange of the first n indices with the second n indices.  \nFor instance, it is instructive to discuss explicitly the form of the metric g ( n ) induced on n -forms by g . This can be immediately shown to be given by  \nwith  \ng ( n ) = n ∏ i =1 g µ i ρ i ( dx µ 1 ∧ dx µ 2 ∧ . . . dx µ n ) ⊗ ( dx ν 1 ∧ dx ν 2 , ∧ . . . ∧ dx ν n ) , (C22)  \nor, more explicitly, using the antisymmetric properties of  \nthe n forms, as  \ng ( n ) = [ g ( n ) ] µ 1 µ 2 ...µ n ν 1 ν 2 ...ν n ( dx µ 1 ∧ dx µ 2 ∧ . . . dx µ n ) ⊗ ( dx ν 1 ∧ dx ν 2 , ∧ . . . ∧ dx ν n ) , (C23)  \n[ g ( n ) ] µ 1 µ 2 ...µ n ν 1 ν 2 ...ν n = 1 n ! δ ρ 1 ρ 2 ...ρ n ν 1 ,ν 2 ...ν n n ∏ i =1 g µ i ρ i . (C24)  \nHere and in the following we will use the notation [ g ( n ) ] ⃗ µ⃗ν of indicating the element of this tensor where ⃗ µ = ( µ 1 µ 2 . . . µ n ) and ⃗ν = ( ν 1 ν 2 . . . ν n ) indicate n -tuples  \nof indices. Using an analogous notation, we define the scalar product between the generic metric G ( n ) between pairs of n -forms and the n form ζ as as  \nG ( n ) · ζ = ˆ ζ, (C25)  \nwith  \nˆ ζ ⃗ µ := 1 n ! [ G ( n ) ] ⃗ µ⃗ν ζ ⃗ ρ ( dx µ 1 ∧ dx µ 2 ∧ . . . dx µ n ) ⟨ dx ν 1 ∧ dx ν 2 , ∧ . . . ∧ dx ν n , dx ρ 1 ∧ dx ρ 2 ∧ . . . dx ρ n ⟩ . (C26)  \nThus the eigenvalue problem for G ( n ) can be written as  \n[ G ( n ) ] ⃗ µ⃗ν [ g ( n ) ] ⃗ν⃗ρ ζ ⃗ ρ = λζ ⃗ µ , (C27)  \nwhere here and in the following we have indicated with ⃗ µ ⃗ρ and ⃗η tuples of n indices. From this definition of the eigenvector of a metric between n -forms it follows immediately that the metric tensor g ( n ) as all eigenvalues λ equal to one, i.e.  \nλ = 1 , (C28)  \nfor any arbitrary value of n , with 0 ≤ n ≤ d .  \nb. Flattening of the metric tensors G ( n ) and reduction to a matrix eigenvalue problem  \nTo define the eigenvalues associated to the metric tensor G (1) between 1-forms, we can use a matrix formalism and define  \n[ N (1) ] ν µ := [ G (1) ] µρ g ρν . (C29)  \nThe eigenvalues of the tensor G (1) are defined as the eigenvalues of the matrix N (1) . Note that now the multiplication of this latter matrix with itself preserves Lorentz invariance.  \nThe eigenvalue problem for 2-forms Eq.(C18) can be as well written in matrix form, by considering the m × m  \nflattened matrices G F (2) and g F (2) with m = d ( d -1) / 2 representing the dimension of the basis for the 2-forms defined on the d dimensional manifold K . For instance, taking the basis { dx µ ∧ dx ν } with µ < ν the flattened matrices G F (2) and g M (2) will have respectively matrix elements  \n[ G F (2) ] µν ; ρσ = 2[ G (2) ] µνρσ , [ g F (2) ] µν ; ρσ = 2[ g (2) ] µνρσ .  \nA practical example might be helpful to illustrate this construction. Assuming that the manifold K is d = 3 dimensional and has three coordinates 0 , 1 , 2, the flattened matrix g F (2) associated to g (2) , defined in the basis dx 0 ∧ dx (1) , dx (0) ∧ dx (2) and dx (1) ∧ dx (2) is given by  \ng F (2) =   g 00 g 11 -g 01 g 10 g 00 g 12 -g 02 g 10 g 01 g 12 -g 02 g 11 g 00 g 21 -g 01 g 20 g 00 g 22 -g 02 g 20 g 01 g 22 -g 02 g 21 g 10 g 21 -g 11 g 20 g 10 g 22 -g 12 g 20 g 11 g 22 -g 12 g 21    \nUsing these flattened matrices we then construct the matrix N (2) of matrix elements  \n[ N (2) ] ηθ µν := [ G F (2) ] µνρσ [ g F (2) ] ρσηθ , (C30)  \nwhere µ < ν and η < θ . Note that now the usual matrix multiplication of N (2) with itself preserves Lorentz invariance. The eigenvalues of this matrix N (2) are equal to the eigenvalue of G (2) defined in Eq.(C18).  \nSimilarly metric tensors G ( n ) between n -forms can be flattened by considering a basis of independent n -forms.  \n.  \nFor instance a basis for these n forms can be given by the canonical n forms  \ndx µ 1 ∧ dx µ 2 ∧ . . . ∧ dx µ n , (C31)  \nwith µ 1 < µ 2 < . . . < µ n . This canonical basis is formed by m = ( d n ) forms.  \nOn this basis we can define the m × m flattened matrices  \n[ G F (2) ] ⃗ µ ; ⃗ν = n ![ G (2) ] ⃗ µ ; ⃗ν , [ g F (2) ] ⃗ µ ; ⃗ν = n ![ g (2) ] ⃗ µ ; ⃗ν .  \nStarting from these flattened matrices we define the matrix N ( n ) of elements  \n[ N ( n ) ] ⃗ η ⃗ µ := [ G F ( n ) ] ⃗ µ⃗ρ [ g F ( n ) ] ⃗ ρ⃗η . (C32)  \nAs we have seen for n = 1 and n = 2 now these matrices can be multiplied preserving Lorentz invariance, and the eigenvalues of these matrices correspond to the eigenvalues of the metric tensor G ( n ) .', 'c. Trace of the metric tensors G ( n )': "The trace of the metric tensors G ( n ) is defined as the trace of the matrix N ( n ) or equivalently as  \nTr G ( n ) = Tr M N ( n ) . (C33)  \nFor ease of notation in the following we will also define indicate this trace with  \nTr F G ( n ) g -1 ( n ) = Tr G ( n ) = Tr M N ( n ) . (C34)  \nFrom the above definition of N ( n ) it follows that our definition of the trace is the usual definition of the trace of  \n- [1] Jacob D Bekenstein. Black holes and entropy. Physical Review D , 7(8):2333, 1973.\n- [2] Jacob D Bekenstein. Generalized second law of thermodynamics in black-hole physics. Physical Review D , 9(12):3292, 1974.\n- [3] Stephen W Hawking. Particle creation by black holes. Communications in Mathematical Physics , 43(3):199220, 1975.\n- [4] Shinsei Ryu and Tadashi Takayanagi. Aspects of holographic entanglement entropy. Journal of High Energy Physics , 2006(08):045, 2006.\n- [5] Tatsuma Nishioka, Shinsei Ryu, and Tadashi Takayanagi. 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