General Relativistic Wormhole Connections from Planck-Scales

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On the other hand, QM in an EPR pair cannot be described independently of the other presents non-locality and the well-known probabilistic behav- states [25, 27]. The ER = EPR equivalence is valid if there ior from the deterministic equations that rule the quanta. are no traversable wormhole solutions that do not require the This contrast between GR and QG can find a fusion in the violation of the strong and/or weak energy conditions [31–34], simple heuristic approach formulated by Susskind and Malda- they may instead behave as quantum communication channels cena who, starting from the quantum mechanical language, set between the quantum fields there defined [35]. an hypothetical equivalence between non-traversable worm- The ER = EPR conjecture was initially formulated in hole connections of two (or more) particles or events in the gravity/gauge theory equivalence between Anti-de Sitter space–time through Einstein–Rosen (ER) bridges and entan- (AdS) space–times and conformal field theories (CFTs) by gled states (the idea that wormholes and flux tubes can play Maldacena (gauge/gravity duality) within a relationship be- a role in quantum mechanics and quantum field theory is tween the entanglemententropy of a set of black holes and the not new, in particular for systems with electric and/or mag- cross-section area of ER bridges connecting them. AdS space– netic charges and their renormalization has earlier work in times represent an elegant solution of Einstein’s equations [10, 11]), and the quantum properties of the “spooky action with negative curvature where the outer boundary is a surface at distance” of Einstein–Podolski–Rosen (EPR) states [12– and, in the CFT, correspondence quanta can interact and gen- 15]. The ER=EPR equivalence was first defined in Anti-de- erate the holographic universe there contained. The AdS/CFT Sitter (AdS) space–times in their equivalence with CFTs [16– correspondence provides a complete non-perturbative defini- 20]. In other words, EPR entangled particles are supposed tion of gravity with quantum field theory, extending this cor- to be equivalent to connections obtained through ER worm- respondence also to space–time scenarios of quantum gravity holes involving the concept of entanglement entropy to de- where the asymptotic behavior of the space–time is that of scribe these many-body quantum state/wormhole connections, AdS space–time. The AdS/CFT correspondence plays a key even if the ER = EPR equivalenceis moreevident with monog- role in the calculations of strong coupled quantum field theo- amous entangled pairs [21]. Spacetime is supposed to emerge ries. When the boundary theory is strongly coupled, the bulk from quantum entanglement, as discussed in [22] where, from theory is weakly coupled, and vice versa. A strongly coupled some examples where gauge theory/gravity duality is valid, field theory can have an AdS dual gravity description weakly one finds that the emergence of space–time is related to the coupled and therefore calculable and vice versa. As an exam- quantum entanglement of the degrees of freedom present in ple, if the curvature of the AdS space increases, the gravita- these quantum systems. Superpositions of quantum states cor- tional coupling becomes stronger and the boundary coupling responding to disconnected space–times can give rise to states is weaker. that are interpreted in terms of classically connected space– To extend ER = EPR conjecture to space–times different times. In this vision, gravity can be also interpreted as an from the Anti-de Sitter solution, one has to investigate how entropic force, a thermodynamic property of physical systems much ER = EPR depends strictly on AdS/CFT correspon- defined in an holographic scenario: gravity and space–time dence and from the properties of wormholes also in de Sitter connectionsare emergentphenomenafrom the degrees of free- (dS) space–times. First of all we must consider that AdS/CFT dom of a physical system encoded in an holographicboundary is a structural correspondence between bulk and screen, but or to emerge from a background-freeapproach by using quan- does not contain in itself any specific indication of the possi- tum entanglement [23, 24]. At all effects, one can conclude ble dynamics of wormhole formation. Wormholes require a that space–time is built with the quantum information shared cosmological scenario. For example, it is plausible that the between EPR states that are equivalently connected with an wormholes were formed in the initial chaotic phases of the ER wormhole. universe with a rate similar to that of the formation of mini While ER wormholes are classical solutions of GR, a local black holes (BHs), with very specific traces as regards the deterministic theory, quantum entanglement, is instead one of event horizon, as discussed in [36–38]. This aspect is deci- the most intriguing quantum physical aspects of nature charac- sive because all the problems related to the quantum aspect of terized by non-locality and the stochastic properties of quan- the wormholes imply a cosmological background capable of tum mechanics. Entanglement occurs when a pair of particles providing a plausible scenario for their existence described by (or a group of quanta) is generated in a way that the quan- the Ryu–Takanayagi’s entropy that relates the entanglement tum state of each particle of the pair cannot be described in- entropy in CFT and the geometry of AdS space–times. The dependently of the state of the others even if they are sepa- Ryu–Takanayagi formula is a generalization of the BH en- rated by large distances. For a deeper insight see [25–30]. In tropy formula by Bekenstein–Hawking [39, 40] to a whole ER = EPR, causality is not violated. ER bridges do not vi- class of holographictheories [41, 42] where gravitational mod- olate causality because of the topological censorship, which els with dimension D are dual to a gauge theory in dimension forbids ordinary traversable wormholes; EPR states, instead, D − 1. prevent causality violations because of the properties of entan- In these recent years, the interest in the maximum symme- gled states described by Bell’s inequalities—no information is try properties of de Sitter’s space gave this structure a new transferred between the two entangled states during the wave- centrality with respect to AdS. One of the most relevant prob- function collapse of the entangled pair as each quantum state lems was to project a hologram of a quantum particle that 3 lives in the infinite future of AdS, which makes it difficult the quantum entanglement of degrees of freedom in the cor- to describe real-time space in holographic terms. In particu- responding conventional quantum system, building up space– lar, the main classes of essential results must be mentioned time with quantum entanglement. The ER = EPR equivalence here: the CPT Universe [43] and the numerous results on the suggests that space–time and gravity may emerge from the non-locality in dS space–times [44–48]. Of relevant impor- degrees of freedom of the field theory. On the other hand, tance is the so-called “uplifting” technique by Dong et al. [49] space–time becomes the optimal way to build entanglement where two Anti-de Sitter space–times are transformed into starting from wormhole connections. At Planck scales, the a de Sitter space–time. The uplifting changes the curvature Planck area is defined as the area by which the surface of a of two “saddle-shaped” AdS space–times that, once warped, Schwarzschild black hole increases when in the black hole is are glued together along their rims and turned into a “bowl- injected one bit of information. In a Riemannian manifold shaped” dS space–time via entanglement or more general two- (M,g) the scalar curvature in an (n − 1) hyperplane relates throated Randall–Sundrum systems [50, 51] and the CFTs rel- GR with the entanglement of quantum states in an arbitrary ative to both hemispheres become coupled with each other. Hilbert space without reference to AdS=CFT or any other In this way one forms a single quantum system that is holo- holographic boundary construction. graphically dual to the entire spherical de Sitter space, defined on its boundary located at a finite distance away. The tech- nique of uplifting two AdS into a dS permits us to modify the Einstein’s Equations in the Neighborhood of an Event curvature in a more general way than that offered by the set of local transformations obtained through Wick rotations that Einstein’s equations are the core of GR—they describe can only act locally—the curvature changes everywhere by in- gravity in terms of the curvature of space–time. Spacetime troducing extra fields whose energy density acts as an extra geometry and the metric tensor gik are determined from Ein- source of curvature to landscape the AdS space–time into a stein’s equations, given the distribution of energy, mass and dS one. The cosmological constant in the bulk space is then momentum in space–time encoded in the stress–energy tensor transformed from negative to positive and the holographic pro- Tik. jection of the space–time into its boundariesis changed. Some In our approach, by assuming that Einstein’s equations re- examples are reported by Silverstein and Polchinski [52] or in main valid down to the Planck scale, we find that the connec- Vasiliev’s higher-spin gravity—in AdS, the boundary theory is tion between events are achieved through wormhole connec- an O(N)-vector field theory, while in dS space it becomes an tions, avoiding the gravitational collapse and the presence of Sp(N) scalar field theory, where N is the number of the vector singularities at Planck scales. To this aim, we adopt the ap- (scalar) fields of the boundary theories [53, 54]. proach by Schwinger in the analysis of classical fields [60]— In this work, we analyze Einstein’s equations in a finite vol- to determine the properties of a field, one cannot measure the ume of space–time down to the Planck scale, finding worm- field in a point, otherwise, because of the equivalence princi- hole connections that avoid the singularity problem and an ple, one finds only a local Minkowskian space–time tangent indetermination relationship that involves the Riemann curva- to the manifold (M,g) in the given point event. ture. This finds application to the ER = EPR conjecture—in Following Schwinger, from his studies on electromagnetic this framework, geometry behaves as a geodesic tensor net- theory [60, 61], the analysis of a classical field must be made work that defines the quantum state properties of a fundamen- in a neighborhood of the event. This approach is clearly valid tal quantum state of a given metric [55] and a virtual graviton in electromagnetism and antenna theory: when sensing the exchange becomes equivalent to entanglement to which one electromagnetic field with an antenna, one cannot measure in can apply the concept of Penrose’s decoherence of a quantum a point the fluctuations of the electric field. The antenna must state [56]. In this crossing between locality of GR and the have a finite length in space and the field must be measured emergence of non-locality of QM as in [57, 58], where de Sit- in a finite time interval to be revealed. As happens for any an- ter space–time is taken as the geometric structure of vacuum, tenna, as well as for the gravitational field, one has to consider the analysis of Einstein’s equations can provide an additional a finite length or a finite hypervolume in which to determine support to the ER = EPR conjecture extended from AdS to the properties of the field. For the same reasons, one can- dS [59] and to locally Euclidean space–times. This can be not deduce the properties of the gravitational field in a single interpreted as the route to ER = EPR from general relativity. point and at a given time because of the equivalence principle, as discussed in the free-falling particle paradox [62]. Let us find the main properties of the gravitational field WORMHOLE CONNECTIONS DOWN TO PLANCK when a finite length L of measure is fixed down to the Planck SCALES FROM EINSTEIN’S EQUATIONS scale. Given a Riemannian manifold (M,g), where M is the manifold and g the metric tensor, g ∈⊗2T˙ of tensorial order In the ER = EPR scenario, wormholeconnections are funda- 2, (written as g(2)), Einstein’s equations are mental in the building of space–time. Consider an entangled quantum system. The emergence of space–time in terms of 1 R − Rg + Λg = 8πT (1) ER connections, in the gravity picture, is intimately related to ik 2 ik ik ik 4

lm where Λ is the cosmological constant and Rik = g Rlmik and ik lm R = g g Rlmik represent the tensorial and scalar space–time 2 g 1 1 2 curvature terms obtained from the Riemann tensor Riklm that, R (g,L) ∼ ∆ ∆g(g)− + ∆g(g)− (3) (4) L2 in the tensorial index notation, takes the usual well-known 

 form [63] ∆ −1 2 Γn Γp Γn Γp where the term g (g) {kl,im−km,il} = kl im − km il rep- 2 2 2 2 resents the affine connection
and ∆ g−1∆g is the second 1 ∂ glm ∂ gkl ∂ gil ∂ gkm Riklm = + − − + derivatives of the metric tensor with respect to the coordi- 2 ∂xk∂xl ∂xi∂xm ∂xk∂xm ∂xi∂xl 
∂ 2 g ∆ ∆ g2 ∆ ∆ −1 n p n p nates, being g = 2 k l g = 2 k l g(g) . The Ricci + gnp Γ Γ − Γ Γ (2) kl L L kl im km il −1
−2
tensor and scalar are R(2) ∼ g R(4) and R ∼ g R(4), the −1 the tensor is an element of the rank-four tensors R ∈⊗4T˙ Einstein tensor is G = R(4)(g) and Einstein’s equations are iklm Λ in the cotangent bundle T˙ of the manifold (M,g) that we will G + g = T , where T is the energy–momentum tensor. By introducing a characteristic length L, if Einstein’s equa- indicate with the symbol R 4 , where 4 is the tensorial index. ( ) tions hold down to the Planck scales, from the basic formula- The gravitational field has the fundamental property that, tion of the Riemann tensor and Einstein equations we find that any body, independently from their mass, moves in the same the field equations, integrated and averaged over a 3D space- way. This is described by the strong equivalence principle, like hypersurface σ with unit normal vector n ∼ g−1/2, obey which suggests that gravity is a geometrical quantity and one an indetermination relationship that recalls Heisenberg’s. In- cannot measure the gravitational field in an event, as the field stead of focusing on the more general energy–tensor quantity becomes locally Galilean and diagonalizable, and the energy (or the momentumvector), we consider for the sake of simplic- of the field cannot be uniquely defined. ity the scalar proper energy E, averaged over a proper volume In a neighborhood of a given event, space–time is built by L3, which is given by the integral of the energy momentum chains of events and observers. The building of space–time is tensor over a given proper volume element of a space-like 3D– obtained by causally transferring information encoded locally hypersurface. This leads to the following formulation of the in one event of the field to form events and coincidences of proper energy averaged over the given proper volume events described by punctual (point-to-point) correlationsin a 2 four-dimensional manifold (M,g). g 2 E gΛ E¯ R L ∆ ∆g g −1 ∆g g −1 Because of the equivalence principle, these observables h − i = ∼ (4) = ( ) + ( ) . L   must be generated within a volume of finite spatial extent

(4) from a given spatial length L and propagated—througha chain If we rescale this relationship down to the Planck scale Lp, of events—in the form of four-volumetric densities (or in by defining the light crossing time as τ = L and the Planck geometrical sub-varieties) in the four-dimensional manifold Time τp, the Einstein equations retain their validity down to (M,g) to a remotely located finite region of space–time of like- the Planck scale, even if metric fluctuations over a scale larger wise finite spatial/temporal extent—the observation (hyper- than Lp can occur. We find that these fluctuations can give rise )volume V ⊆ M over which they are volume integrated into to a relationship observables, allowing the information carried by them to be 2 2 τp 2 E¯ × τ Lp E¯ × τ L extracted and decoded. More specifically, we will use the lo- = = R (g,L) (5) τ 2 (4) cal split of 3 + 1 in space and time where we will consider    h¯   L   h¯  g the integration of the field properties over a three-dimensional that holds down to the Planck scales. Fixing a characteristic volume V = L3. The volumetric density of every gravitational spatial scale (or time), the relationship in Equation (5) cor- observable carried by the gravitational field is a linear combi- responds to the introduction of fluctuations of the averaged nation of quantities that are second order (quadratic/bilinear) quantity over L3 of the proper energy E¯. If we set E¯ = ∆E∗ in the metric of the field, and/or of the derivatives of the field. and τ = ∆t, we can write Equation (5) in a more familiar To obtain these quantities one must integrate the density of a Heisenberg relationship that involves the Riemann tensor and conserved quantity e.g., over a given 3D space-like hypersur- the contribution from the dark energy face σ or over a four-dimensional interval, characterized by a given finite length L; there, the field equations are integrated 2 2 τ L2 h¯ L2 and averaged to obtain the field observables we need to ana- ∆E∗ × ∆t = h¯ R (g,L)= R (g,L) τ g2 (4) g2 L (4) lyze the properties of space–time down to the Planck scales.  p   p  (6) Let us consider a Lorentzian manifold as example, with that at Planck scales becomes cosmological constant Λ and then introduce a characteristic 2 length L. This quantity is the proper length associated to the Lp 2 ∆E∗ × ∆t = h¯ R (g,L)= h¯ ∆ ∆g(g)−1 + ∆g(g)−1 generic coordinate variation ∆x written in terms of the metric 2 (4) g   tensor g, viz., L ∼ g1/2∆x. The Riemann tensor is then writ-

(7) ten in terms of the metric variations ∆g, the covariant metric where ∆E∗ = ∆E +∆g Λ+g∆Λ averaged on the volume L3 of tensor g and the contrarvariant one, g−1 the 3D space-like hypersurface σ. 5

The Energy of the Gravitational Field be provided by a mixed state between the two entangled pairs with that of the virtual graviton. The question is what is the Dark energy and other different vacua are parameterized by correct perspective? the cosmological constant Λ. When Λ > 1, the equations de- From Einstein’s equations, the observable averaged total en- scribe an AdS space–time. The gravitational fluctuations are ergy of a metric fluctuation over the volume V on a scale L, 2 mainly expressed by the affine connection term ∆g(g)−1 becomes τ for any space–time. To describe the energy of the gravita-
−1 E ∼ ∆ ∆g(g) Ep + Eg + EΛ (11) tional field, which is not defined as a global and conserved τ  V p
quantity, one has to introduce pseudotensorial quantities de- and is made with the energy of geometry and vacuum and scribing the energy trapped non-locally in the geometry. One energy of interaction expressed in terms of gradients of the example is the non-symmetric Einstein pseudotensor, which geometry fluctuations, second order derivatives of the metric is constructed exclusively from the metric tensor and its first tensor, as in the Riemann tensor that make the connection be- derivatives but is not suitable for our purposes. Instead, the tween observers. Landau–Lifshitz pseudotensor tik [63] permits us to write for the integrated non-local gravitational energy Eg that includes the contribution of the cosmological constant in terms of the Planck-Scale Wormhole Connections curvature tensor. This quantity is quadratic in the connection and, for a general covariant component of the pseudotensor, In a local neighborhood of a given event xi , one per- 3 { }0 averaged on the volume V = L one obtains forms a discrete infinite denumerable 3+1 local slicing of the Eg + EΛ space–time with time steps a Planck time unit. To the initial g−1 (t + Λg) ∼ (8) i V=L3 L3 event {x }0 corresponds the slice N = 0. The N−th slice corre- sponds to the time τN = (N +1)τp, building up a symbolic dy- where EΛ = g−1ΛgL3 is the energy associated to the value of namics of space–time events. The energy of the gravitational the cosmological constant and to dark energy. Considering perturbation in the N−th slice is that Ep (Eg + EΛ) ∼ (12) ∆g 2 N N + 1 |g| g−2 (t + Λg) ∼ , (9)  ∆x  the total energy is instead this relationship leads to a background curvature with fluctua- −1 Ep EN ∼ (N + 1) ∆ ∆g(g) Ep + (13) tions having wavelength λ = L that can be interpreted as con- V,N N + 1
nections between events due to an exchange of virtual gravi- for N → 0 the energy fluctuation becomes EN → Ep for which, λ λ tons with wavelength and energy h¯/ or, in the ER = EPR by definition, EN τN ∼ h¯ and the metric tidal fluctuations tend scenario, to the connection through an ER wormhole, to zero—space–time at Planck lengths is homogeneous and isotropic and the local geometry depends only on the energy 2 Eg + EΛ τp Eg + EΛ ∆g(g)−1 ∼ = (10) of fluctuations in space–time and from the energy of the cos- V=L3 L τ E D
E   p mological constant. This because ∆g/g → 1 are both on the where Ep is Planck’s energy. In the ER = EPR hypothesis, order of the Planck scale. This is the reason why the field does these energy fluctuations would be considered as equivalent to not diverge and no singularities are present. the connection between two entangled events separated by the For N → ∞ the dominant energy is that of tidal fluctuations distance L, giving a paradoxical meaning to the exchange of a at scales larger than Lp accompanied with that of the cosmo- virtual graviton in terms of entanglement connections between logical constant when integrated over the metric and remains events like in an emergent gravity scenario. as a constant function over the volume of integration; Eg be- Following the already cited works by De Witt and the comes instead negligible. This means that for a process con- classical QG interpretation found in the literature [1–5], this necting two events lasting a time τ, the amount of energy does term would describe a virtual graviton exchange between two not entirely contribute to the vacuum energy but it is partially events within a space–time connection. On the other hand, spent in geometry in this process, involving the dark energy this term—that can be also interpreted in terms of a worm- contribution expressed by the cosmological constant Λ. Re- hole connection between the two events—with the exchange calling Heisenberg principle from Equation (5), the larger is of at least 1 qbit of information (in the ER = EPR conjecture) the energy fluctuation, the smaller results the space/time inter- would correspond to the entanglement of two particles. If ER val fluctuation. wormholes are equivalent to a monogamous connection be- In the neighborhood of Planck scales, when N > 0, the cur- tween the two events [21], as realized through a virtual gravi- vature of space–time remains finite and the Riemann tensor ton exchange, one could state that entanglement of EPR states can be written as 2 can derive from the exchanges of virtual gravitons between Ep τp 2 g R (g,L) ∼ (14) two events. From another perspective, entangled states should (4) τ 2 h¯   L 6 and the Ricci scalar is Heisenberg’s uncertainty principle for the momentum p and the position x that includes the existence of a characteristic 1 Ep τp 2 R(g,L) ∼ (15) length L such as the Planck scale, or any other scale typical L2 h¯ τ p   that can be found in certain quantum gravity models, is given 2 that for L → Lp we have R(g,L) → 1/Lp with the result that at by [64, 65] Planck scales there is no singularity in the curvature and the h¯ gravitational radius becomes ∆x = ∆x + ∆x ≥ + k ∆p (17) QM GR 2∆p τ 2 E Lp Rg = 2 (16) the existence of a minimum interval in space–time is revealed h¯ L by a deviation from the classical term due to quantum mechan- that is written as R 2L , which corresponds to elementary g = p ics only, ∆xQM. The quantum gravity term, ∆GR, due to the ex- wormhole connections at the Planck scale and finding a triv- istence of a characteristic length Lp and to the propertiesof the ial equivalence with the corresponding Penrose diagrams. Di- gravitational field, can be characterized instead by a parameter rectly from Einstein’s equations we find that, at Planck scales, k, a constant characteristic of the quantum theory of gravita- the singularities expected from quantum gravity can be inter- tion here considered. To give an example, in a string theory preted in terms of wormhole connections between the events, scenario, k = αY, where α is the string tension and Y a con- as required in the ER = EPR conjecture and obtain an in- stant that depends on the theory. In our case, following [66– determination relationship shown in Equation (6) involving 2 68], one can find that k = 2Lp/h¯. From our calculations that the Riemann tensor and geometry fluctuations. In this view, ∗ involve the Riemann tensor, we find that ∆xQG = 2Ep/∆E wormhole and equivalent EPR connections can also be for- ∆ ∆ ∗ 2 and thus p = hE¯ p/ E Lp, a term that includes the effects of mally equivalent to an exchange of a virtual graviton at scales dark energy in the term ∆E∗ too. larger than Planck scale, whilst any group of superimposed We write now the Heisenberg relationship for sets of states below Planck scales, instead, will be indistinguishable N−particle entangled states. Following [69–73], consider first and therefore entangled. a couple of entangled particles with positions x1 and x2 and momenta p1 and p2, respectively. For N = 2, the classical indetermination principle is Tests for the ER = EPR Conjecture h¯ 2 ∆(x ,x )2 = ∆(x )2 + ∆(x )2 × ∆(p )2 + ∆(p )2 ≥ . If we suppose the validity of the ER = EPR conjecture, 1 2 QM 1 2 1 2 4 the geometry fluctuations present at Planck scales may be     (18) revealed with quantum entanglement. By applying the inde- In the simplest case, where ∆(x1) = ∆(x2) = ∆xe and termination relationship that involves the Riemann tensor ex- ∆(p1)= ∆(p2)= ∆pe, the uncertainty relationship becomes 2 2 2 pressed in Equations (6) and (7), we argue that one can ob- (∆xe) (∆pe) ≥ h¯ . For N identical entangled states, the ex- tain information about the fluctuations of space–time and de- tended indetermination principle becomes termine whether a characteristic scale like the Planck’s one is 2 2 present, as expected in QG. 2 2 N h¯ (∆xe) (∆pe) ≥ (19) If space–time is discrete, its discreteness is expected to be 4 characterized by a typical scale of space and/or time: There and when we include the effects of the gravitational field one exist a minimum time interval t and a minimum length L p p obtains where wormholes connections—equivalentto entangled states 2 ∆ between two or more regions of space–time—connect differ- Nh¯ 2NLp p ∆x = ∆(x ,x ) + ∆(x ,x ) ≥ + . (20) ent events or space–times. If events/space–times are con- 1 2 QM 1 2 GR 2∆p h¯ nected with intervals smaller than Lp and tp, they would be en- tangled and actually be the same event or the same space–time. By assuming that Einstein’s equations retain their validity Their quantum superposition can exist and collapse after a fi- down to the Planck scales and that wormhole connections nite time interval and the properties of wormhole connections represent the building blocks of the physics of the gravita- are reflected in the properties of the corresponding EPR states tional field at and below Planck scales (tp and Lp), ER = also when they connect events at scales larger than the Planck EPR links connecting any space–time (or event) with a dif- scale. This scenario is different from Penrose’s assumptions ference smaller than tp and Lp mean that different space– [56], where space–time is thought to be continuous and the times and events are physically identical, and then in prin- quantum superposition of space–times result unfeasible lead- ciple undetectable and entangled. Instead, in a region with ing to the gravitational collapse. radius R, the spatial difference of two space–times/events is ∆ 2 ∆ ∗ Noe we propose to test the ER = EPR scenario by using the L = 2Lp E /hc¯ , and the difference of their corresponding Heisenberg uncertainty principle applied to pairs (or groups) space–times is the difference of the proper spatial sizes of the of entangled particles and including the additional indetermi- regions occupied by them and the time of the wavefunction ∗ 2 nation introducedby quantum gravity effects. The generalized collapse is on the order of τc ∼ 2hE¯ p/(∆E ) . 7

If the properties of ER = EPR links remain valid from the unavoidably introduce an uncertainty in the result. There is Planck up to the macroscopic scales, where entanglement can no need to have any preferred reference frame for the wave- be observed in the lab, the term ∆xQG in the Heisenberg re- function collapse of entangled states. If an experimenter tries lationship expressed in Equation (20) is expected to reveal to determine the exact reference frame where the wavefunc- the properties of the wormhole structure of space–time from tion collapsed, the measurement process will unavoidably in- a deep analysis of the wavefunction collapse of an entangled troduce an uncertainty that would make impossible the identi- pair. In other words, the deviation from the quantity ∆xQM of fication of the “exact” reference frame. Obviously, if one can the classical Heisenberg principle would reveal the fuzziness determine the order of the measurement in a shared reference space–time or, better, of the point-by-point identification of frame it can result like that in certain reference frames and, the spatial section of the two events/space–times, better evi- instead, indeterminate in the other reference frames [75, 76]. dent with a set of a large numberof N entangled quanta like a Schr¨odinger cat. From the point of view of relativistic quantum information DISCUSSION AND CONCLUSIONS discipline, entanglement and wormholes are expected to cre- ate space–time and entanglement events (and space–times) It is one of the great merits of Albert Einstein to have in- through quantum information—information that emerges vestigated the possibility of a multiple-connected space–time from the connection of quantum bits. In fact, from a quantum- and in theoretical physics there is a long tradition of studying computational interpretation of space–time entanglement,ina quantum behavior in spaces of this type [77]. These lines of foliation of space–time, the quantum fluctuations of the met- research have progressively merged into the quantum study ric present on the slice n can be interpreted as wormhole con- of wormholes, assuming a decisive relevance not only for the nections between one Planckian pixel in the slice n with that study of the structure of the GR and its cosmological impli- one present in the n−1 slice. Following [74], the holographic cations, but has given the question a decisive configuration principle suggests that such a geometricalconnection is space– as regards the relations of ”coexistence” peaceful ”between time entanglement. If not entangled, following Penrose’s argu- QM and GR. In this work we proposed a formal technique mentation only the quantum superposition of two space–times for the study of the quantum effects of a wormhole within the with a difference larger than the minimum sizes can not exist, conjecture ER = EPR. We then considered different scenarios and should collapse instantaneously. If they are connected by from the original Susskind and Maldacena one, in particular an ER wormhole they should obey the indetermination rela- those related to the dS space, which seems to be a much more tionship expressed in Equation (20). promising ground for the study of the emergence of classical To verify possible additional anomalies in the indetermi- information starting from a quantum background where time nation principle introduced by the ER = EPR conjecture one is not defined [57, 78–80]. may instead want to consider to measure the time/energy en- These reflections suggest that an effective generalization tangled states and study the time of collapse as a function of of the physical meaning of ER = EPR requires a different their energy differences. This may explain why the wavefunc- and more complex philosophy on the emergence of phys- tion collapse of an EPR pair is not always instantaneous, as it ical space–time as a holographic “settlement” of tempera- may depend on the geometry fluctuations. Moreover, one has ture/energy scales, and the use of well-known techniques in to also consider the effects introduced by the presence of the QFT [81, 82]. cosmological constant, of the information encoded and shared In other words, these scenarios suggest that the idea of tran- between the entangled quantum states and their relationship sition of the metric suggested by Sacharov may be the most with the gravitational information entropy that go beyond the “natural” way to characterize non-locality in a metric formal- purpose of the present work. ism. The assumption of ER = EPR would be only one of Of course an experimenter has to consider that EPR states the aspects of a more general phenomenon of Raum–Zeit– depend on the choice of reference frames and that Bell’s in- Materie production starting from a non-local Euclidean back- equalities are preserved in certain reference frames only, and ground through quantum computation procedures. The ob- should also consider the effects of simultaneity and include in servable part of space time would therefore, in a rather literal the experiment the additional macroscopic effects induced by sense, result in a thin layer of ice emerging from an ocean of the gravitationalfield at large scales in the presence of massive non-locality and the extension of ER = EPR conjecture to Eu- bodies. As an example, simultaneity is responsible for the un- clidean non-locality may extend its domain from the original certainty of the ordering of non-local wavefunction collapse AdS/CFT scenario. when the relativistic effects cannot be neglected. In any case, Finally, we suggest readers consider the conjecture ER = if a time measurement performed with an entangled pair of EPR within the scenario of de Sitter’s projective cosmology, photons is seen as simultaneous in one shared reference frame, described by Hartle-Hawking boundary conditions as Nucle- then the result of this measure can be considered simultaneous ation by Sitter Vacuum [58]. In this cosmological approach to all measuring observers who do not share a reference frame. one can define the localization conditions in time of the parti- The inversion of the temporal order due to simultaneity is im- cles starting from an Euclidean pre-space that models a non- possible to determine, the attempt to measure this effect will local phase. Using the Bekenstein relation, it is possible to 8

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