Spacetime Structure & Symmetry from Entanglement

Prespacetime Journal | April 2016 | Volume 7 | Issue 5 | pp. xxx-xxx

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Crowell, L. B., Spacetime Structure & Symmetry from Entanglement Entropy

Article

Spacetime Structure & Symmetry from Entanglement Entropy Lawrence B. Crowell 1 Abstract In this article the relationship between quantum mechanics and general relativity is examined according to vacuum polarization and flow. The entanglement of states on either side of a boundary results in equations that give geodesic deviation. This is derived by looking at regions on either side of the boundary in a sheaf construction. This results in a relationship between quantum mechanics, unitarity and the Petrov classification of spacetimes. This points to a generalization of the equivalence principle and relationships between inertial and accelerated frames.

1

Entropy force and entanglement as gravity

There is a growing recognition that gauge fields have a duality with gravitation, and further indications that gravitation may in fact be quantum mechanics in a disguised form. It has been demonstrated that entropy force along a displacement a derives gravitation [1]. In addition spacetime is argued to be constructed from the entanglement of states [2]. These developments were take to its complete limit by stating that general relativity and quantum mechanics are ultimately the same [3]. These three papers form a triad of potential new physics. The entropy force F~ = T ∇S is derived from the first law of thermodynamics and the work-energy theorem of elementary mechanics dE = F~ ·d~r = T dS [1]. The energy and temperature are related by the equipartition rule E = 12 N kT , and the energy bound by a holographic screen or horizon is E = M c2 . It is easy to illustrate for a black hole mass is the ADM mass. The area of the black hole is A = 4πr2 and the number of bits on the horizon is N =

A GM 2 = 4π 2 4Lp ~c

The change in entropy on a small particle near the screen is δS = 2πk. This is used with the Compton wavelength spread of a mass unit relative to the screen ∆x = ~/mc so the entropy force is F = T

∆S mc = 2πkT . ∆x ~

The Unruh temperature is kT =

1 ~g 2π c

which we put into the equation for the force F = mg This unit of entropy on one qubit is equal to the entropy of the black hole divided by the number of qubits N that applied to the black hole entropy gives mg =

GM m . r2

1 Correspondence: Lawrence B. Crowell, PhD, Alpha Institute of Advanced Study, 2980 FM 728 Jefferson, TX 75657 and 11 Rutafa Street, H-1165 Budapest, Hungary. Tel.: 1-903-601-2818 Email: [email protected].

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This is a derivation of Newton’s gravity force based on thermodyanmics[1]. The entropy force refers to a small mass, or a Planck mass qubit, that is moving relative to the holographic screen. We may think of this as a small deviation in the holographic screen. There is some controversy over this, which can be reviewed at [4]. The entropy force of gravity suggests that gravitation could be constructed from entanglement entropy. We may think of thermal entropy as similar to entanglement entropy, but for two entangled qubits one of the qubits is irretrievably unobservable or lost. This happens when a qubit enters a black hole, but we recognize the other qubit is not entangled with the black hole. Thermal entropy is then entanglement entropy with additional loss of information on the part of the observer. The entropy force of gravity shares a relationship with concept of building up spacetime with entanglements, and spacetime emerges from quantum mechanics.

2

Entanglement entropy as geodesic flow and spacetime symmetry

Given a boundary between two regions, a pure fabrication, we find that close to this boundary on either side are entangled pairs of virtual particles. This approach was used by Leonard Susskind in a Stanford lecture [5] The closer we examine the region around the boundary we find higher frequency modes entangled on either side. The diagram below illustrates how the small bubbles with small wave lengths have a high probability of containing a vacuum mode entangled with a mode in a nearby small bubble across the boundary. Similarly for the medium size bubbles this holds with longer wavelengths and this scales up as we look away from the boundary.

The first law of thermodynamics gives dE = T

∂S dt = − T ∇S · dx, ∂t

where the last equation is the entropy force. In a similar vein this is done with SA = − kT rρA log(ρA ).

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The Schrodinger equation i~

∂ ~2 2 |ψi = H|ψi = − ∇ |ψi + V |ψi ∂t 2m

is used for a free particle V = 0. The time derivative of the density matrix is i~ This is further i~ for J a current J =

~2 ∂ρ = − (|∇2 ψihψ| − |ψih∇2 ψ|). ∂t 2m

∂ρ ~2 = − ∇ · (|∇ψihψ| − |ψih∇ψ|) = i~∇ · J ∂t 2m

i~ 2m (|∇ψihψ|

− |ψih∇ψ|). The derivative of the entropy S = − kT rρlog(ρ) is ∂ρ ∂S = − kT r (1 + log(ρ)) = kT r[∇ · J(1 + log(ρ))] (2.1) ∂t ∂t

or for a large number of states with diag ρ = [ 12 , ... 12 ] n × n matrix with log(ρ) ∼ nlog(2) >> 1 ∂S ∼ − nk log(2)∇ · J = − nk∇ · J, ∂t where the last equality is for base 2 logarithm pertaining to qubits. The entropy force equation, dE = − nkT (∇ · J)dt = − T ∇S · dx, is now expressed according to the gradient of the current, nk(∇ · J) = − ∇S ·

dx . dt

The gradient of the entropy is ∇S = − kT r[∇ρ(1 + log(ρ))] ' − nk∇ρ The current is J = J + + J − , for J + = (i~/2m)|∇ψihψ| and J − = entropy force equation becomes the pair of equations ∇ · J± ±

2mi ± dx J · = 0, ~ dt

− (i~/2m)|ψih∇ψ|, and the

(2.2)

which are similar to a geodesic equations for J ± ∝ p± or ±dx/dt. With this recognition is is not hard to generalize this to a four-dimension as a geodesic-like equation. Consider this differential equation in a full covariant form. In order to derive geodesic deviation equations consider a sheaf (pre-sheaf) bundle construction over the different sized bubbles[6]. We generalize the physics to a manifold of four dimensions. The virtual momenta at the scale of the ith bubble is pµi = dxµi /dt and the j th bubble is pνj = dxνj /dt, here mass m = 1 and the difference is Fijµ = pµi − pµj . We make the expansion for the momentum in the ith region by its relative motion j th region pµi = pµi0 +

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∂pµ ν δx ∂xνj j




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and there is a similar expression for pµi . The difference may be expressed according to a field strength tensor with pµi − pµj = pµi0 − pµj0 + (∂νj pµi δxνj − ∂νi pµj δxνi ) The antisymmetry in the exchange in the i and j indices is an exchange of coordinates that is compensated by the exchange of the µ and ν indices µ Fijµ = Fij0 + ∂jν pµi δxνj − ∂jµ pνi δxνj . µ Now set Fij0 = 0 as the flat space case. This momentum is a covariant momentum pµi → pµi + gAµi and we have ν] Fijµ = ∂ [ν Aij δxνj .

The continuity equation between two bubbles gives us the pair of geodesic-like equations +µ ∂µ δJij + Ji+µ pµi + Jj+µ pµj = 0 −µ ∂µ δJij − J −µi pµi − J −jµ pµj = 0.

The geodesic-like equations are ±µ ±µ σ ∂ν δJij ± Fνσ δJij δpij = 0. µ

(2.3)

ρ

The gauge-like term Fνσ is curvature with Rµρνσ e ∧ e = Fνσ This equation is seen to be a geodesic deviation equation. This illustrates that given a partition in a region of space there is a relative acceleration between regions with different scales of virtual modes. The spherical symmetry of flat space means it may be partitioned in arbitrary ways, and homogeneity means the partition may be placed anywhere. Consequently the sum over all possible partitions cancel out and there is no general geodesic motion. In this case there will be no curvature and geodesic deviation of flow. This is different for a partition of space or spacetime at an event horizon or with respect to some polarization of the vacuum. In this case is curvature and geodesic deviation of flow. For a spacetime with a vacuum polarization this provides preferred orientations for such partitions with nonzero net geodesic deviations. The Riemann curvature is given by the Weyl tensor plus Ricci tensor and Ricci scalar terms, Riemann = Weyl + Ricci, where the Ricci terms are zero for a vacuum solution. The Weyl curvature is the main determinant of vacuum polarizations, where directions of polarization are determined by eigenvectors which are null[7]. The eigenvectors are the eigenvectors of the Weyl tensor so that Cαβµν = C(Uα0 , Uβ00 , Uµ000 , Uν0000 ). For a gravitational wave there are four degenerate vectors, for a black hole there are two pairs of them and there are other spaces with more complex description. The Petrov classification defines eigen-bivector U αβ such that 12 C αβ µν U µν = λU αβ . These bivectors may be composed from null vectors so that U αβ = [U α , U β ]. The current equations are obtained from the well known quantum expression i~∂t φ = [H, ρ]. We may then assign the right hand side of equation ±µ ±µ 3 with an eigenvalue ∂ν δJij = λδJij δpνij . In the vacuum case we may then see that ± Cµανβ U α U 0β = λδJµij Uν ,

(2.4)

±µ where for the vacuum state for massless bosons Jij is a null vector and the U α , related to the momentum pµ for massless bosons, are also null. We then have for a vacuum solution a geodesic deviation equation that is a form of the Weyl tensor eigenvalue equation.

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3

Symmetry properties of spacetime as connected to unitary quantum mechanics

This ties the quantum vacuum to the spacetime classification scheme by the Weyl tensor. This section starts with a presentation of a basic overview of the Petrov classification and then how unitary symmetries of quantum physics can underlie it. In the Newman-Penrose formalism [8] the null vectors are expressed as bivectors, Xαβ = − x[α yβ] , Yαβ = z[α yβ] , Zαβ = y[α y¯β] − z[α xβ] for xα , yα , zα null vectors and yα = 2−1/2 (θ + iφ)α , where clearly y¯α yα = 1. The other null vectors satisfy xµ yµ = − 1 which defines the metric gαβ = − xα zβ + zα xβ + yα y¯β + y¯α yβ . The Weyl curvature tensor is then expressed according to combinations of bivectors Cαβµν = Ψ0 Xαβ Xµν + Ψ1 (Xαβ Zµν + Zαβ Xµν ) + Ψ2 (Yαβ Xµν + Xαβ Yµν + Zαβ Zµν ), + Ψ3 (Vαβ Wµν + Wαβ Vµν ) + Ψ4 Yαβ Yµν for Ψi Weyl scalars. The scalar Ψ2 gives the vacuum in a central source, Ψ0 , Ψ4 are for ingoing and outgoing transverse modes and Ψ1 , Ψ2 are for ingoing and outgoing longitudinal modes. Now consider the geodesic deviation equation accordingly. Differentials of bivectors U αβ = U α U β define differences in currents and momentum. ∇α U αβ = (∇α U [α )U β] + U [α ∇α U β] where we identify the vectors U α as the momentum differences for null vacuum modes. This gives ∇α U αβ = 2U α Fαµ U µ U β = 2U α Cαµγν U µ U β U 0γ U 0ν . The term U [α ∇α U β] defines the propagation equations in the Newman-Penrose formalism. The Petrov classification system pairs indices together [9] so that C A B = C µν αβ : (µν) ↔ A : (23) ↔ 1, (31) ↔ 2, (12) ↔ 3, (01) ↔ 4, (02) ↔ 5, (03) ↔ 6 The Newman-Penrose transport equation may be written for an operator ∇A = U µ ∇ν , for the appropriate correspondence of indices, as ∇A U A = 2FA U A or ∇A U A = 2CAB U A U B . The Petrov matrix P A B = M A B + iN A B for M A B ↔ C µν αβ and N A B ↔ ∗ C µν αβ , for ∗ the Hodge star operator on upper indices ∗C µν αβ = 12 µνσρ Cσραβ . These matrices can be seen as 3 × 3 matrices. This matrix obeys 1 1 P 3 + [(T r P )2 − (T r P 2 )]P − (T r P 3 )I = 0, (3.1) 2 3

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for det P = 13 T r P 3 , that is the Cayley-Hamilton theorem. This gives the eigenvalues of the matrices for det P − λ I = 0. The system in this way is similar to the Jordan-von Neumann-Wigner quantum formalism. This leads to some speculation that gravitation or quantum gravity embeds in a larger field theory that is a Hermitian symmetric space formed from J 3 (O) system of octonions. In this setting the field theory resulting from the quotient H = J 3 (O)/K, for K a divisor group, is associative with exceptional symmetries. The classification system with T ype I U[α Cβ]µν[γ Uσ] U µ U ν = 0 T ype II U[α Cβ]µνγ U µ U ν = 0 → Cβµνγ U µ U ν = λUα Uβ 0 0 T ype D U[α Cβ]µνγ U µ U ν = 0 → Cβµνγ U µ U ν = λUα Uβ , ∗ Cβµνγ U µ U ν = λ0 U[α Uβ]

T ype III U[α Cβ]µνγ U µ = 0

,

(3.2)

T ype N Cβµνγ U µ = 0 T ype O Cβµνγ U = 0 is an algebraic system. The Weyl tensor obeys hC(U, V )X, Y i = −hC(V, U )X, Y i = −hC(U, V )Y, Xi, The four vectors written according to a Lie algebra in a Vierbein construction, Ua = Eα Uaα permits these symmetries to be Lie algebraic. Here the spacetme indices are Latin letters and the Lie algebraic root vectors Eα are Greek. This parameterizes spacetime vectors according to internal Lie algebraic elements. The root vectors E α obey [Eα , Eβ ] = Nαβ Eα+β We may then use this to express the type II and D solutions with Cabcd U b U c = λUa Ud as Cabcd U b U c = σ[Eα , Eβ ]Uaα Ubβ [Eγ , Eδ ]Ucγ Udδ U b U c = σNαβ Eα+β Nγδ Eγ+δ Uaα Ubβ Ucγ Udδ U b U c = σNαβ Nγδ Eα+β Eγ+δ Eβ Eγ Uaα Udδ = λEα Eδ Uaα Udδ . Here σ is a factor to be determined for a specific problem, and there is the Einstein summation over the Lie algebraic indices. Now commute boths sides of this equation with E−α and then with E−δ so that σNαβ Nγδ [[Eα+β Eγ+δ Eβ Eγ , E−α ], E−δ ] = λHα Hδ Under the representation R the co-root vectors Hi spanning the Cartan subalgebra give the weights Hi |µi , Ri = µi |µi , Ri, and the eigenvalue λ is now expressed entirely according to Lie algebraic information. This example illustrates how the spacetime categories can be analyzed with Lie algebras. A general set of commutators would be assigned according to type I, and further restrictions gives the other Petrov types. The transition between types could be seen as a form of quantum transition. This would be seen with the coalescence of two black holes. This is a case where two type D solutions, or sets of algebraic operators, transition into a type D plus type N solutions in the asymptotic domain. The correspondence with quantum mechanics means this is equivalent to a unitary evolution of a quantum system. The CayleyHamilton theorem further indicates how this can embed into the quantum mechanics of the Freudenthal system proposed by Jordan, von Neumann and Wigner.

4

Underlying principles with the equivalence between entanglement and gravitation

The physical picture is of vacuum flow, where spacetime with some Petrov symmetry besides type O, conformally flat spacetime, has some directed flow of vacuum modes. These modes flow towards an

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event horizon, at least as viewed by an exterior observer, and the horizon forms an attractor. In the euclideanized picture the horizon is a fixed point on the manifold, and is then an attractor point for these flows. These flows clearly have a type of renormalization group property, with the scale of these regions increasing the further away from the boundary region. In a black hole this boundary region is the event horizon. This renormalization group flow is similar to that presented for an accelerated probe that is lowered to the event horizon[10]. This generic scheme is identical to the RG flow of the vacuum. The two approaches to RG flow illustrate a general form of the equivalence principle[11]. An observer on a local inertial frame finds a constant vacuum, there is no change in the structure of the vacuum. However, this equivalence principle result is an approximation. An infalling observer witnesses a polarization of the vacuum due to the tidal acceleration across the extent of the frame. As the frame falls towards the object the tidal acceleration increases, which changes the polarization of the vacuum. This is similar to the case for an accelerated frame that approaches the Rindler or Schwarzschild horizon. The case where the tidal force is zero is the pure inertial condition. A small constant tidal force across a small inertial frame means a force is required to keep two particles at a constant distance from each other. Due to the radial nature of a central gravity field this is a transient condition. Similarly two objects on an accelerated frame will not remain at a constant distance from each other. A force is required to keep their mutual distance constant. Further this force must increase and diverges as one mass passes across the Rindler horizon relative to the other. We may compare this to a path integral, where a classical free particle follows a geodesic. However, only the most expected path is inertial and the other paths deviate from the geodesic, and are thus accelerated nongeodesic paths. This is not surprising, for the Ehrenfest theorem taught in undergraduate quantum mechanics courses illustrates how Newton’s laws are derived as the expectation average of the time development of a quantum system. For a free particle Newton’s first law tells us the path of a particle is a straight line with a constant velocity; it also informs us that the optimal frame to observe physics is an inertial frame. However, we can see that this is a matter of quantum expectations, and further that geodesics are the result of vacuum flow. There is a net flow of the vacuum when there is some polarization of the vacuum. With the connection between vacuum flow and geodesic deviation we then have an underlying equivalence between inertial and accelerated frames. It has been found there are difficulties with reconciling the equivalence principle with unitarity of quantum physics. The entanglement of Hawking radiation means the entanglement entropy of a black hole grows to exceed the Bekenstein entropy bound. This violation of this bound further means the maximal entropy between two states in a qubit, a quantum monogamy, is violated. Susskind illustrates an equivalency between quantum mechanics and general relativity finds that the third state is actually identical to a state interior to a black hole by means of wormholes[3]. This EP = EP R approach may have some relationship with the construction of spacetime from quantum entanglements, for the wormhole construction is essentially an entanglement mechanism. Received March 22, 2016; Accepted March 26, 2016

References [1] E. P. Verlinde, ”On the Origin of Gravity and the Laws of Newton,” JHEP 1104 029 (2011) [2] M. Van Raamsdonk, ”Building up spacetime with quantum entanglement,” Gen. Rel. Grav. 42 2323-2329 (2010) and Int. J. Mod. Phys. D19 2429-2435 (2010) http://arxiv.org/pdf/1005.3035v1

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[3] Leonard Susskind, ”ER=EPR, GHZ, and the Consistency of Quantum Measurements,” http://arxiv.org/abs/1412.8483 [4] S. M. Carroll, G. N. Remmen, ”What is the Entropy in Entropic Gravity?,” CALT-TH2015-038, http://arxiv.org/abs/1601.07558 [5] ER = EPR or What is behind the horizons of black holes, recorded Stanford lecture, https : //www.youtube.com/watch?v = uiGE tV Qu5o [6] R. S. Ward, R. O. Wells, Twistor Geometry and Field Theory, Cambridge University Press, Cambridge [7] M.A.H. MacCallum, ”Classification of spaces defining gravitational fields”. General Relativity and Gravitation 32, 8, 16611663 (2000). [8] E. T. Newman, R. Penrose ”An Approach to Gravitational Radiation by a Method of Spin Coefficients”. J. Math. Physics, 3, 3, 566768, (1962). [9] V. Barrera-Figueroa, J. Lpez-Bonilla, R. Lpez-Vzquez S. Vidal-Beltrn, ”Algebraic Classification of the Weyl Tensor,” Prespacetime Journal, 7, 3, 445-661 (2016) [10] L. B. Crowell, ”Rindler Frames with dg/dt > 0, Unruh Radiation and RG Flow,” Prespacetime Journal, 6, 6 457-556 (2015) [11] L. B. Crowell, ”Topology of States on a Black Hole Event Horizon,” EJTP 12, IYL15-34 (2015)

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