Non-commutative Gravity and the Einstein-Van der Waals Equation of State
NNon-commutative Gravity and the Einstein-Van der Waals Equation of State
Simon Moolman ∗ NITheP, School of Physics, and Mandelstam Institute for Theoretical Physics,University of the Witwatersrand, Johannesburg, WITS 2050, South Africa
Dated: September 22, 2018
Abstract
A calculation by Jacobson [1] strongly implies that the field equations which describegravity are emergent phenomena. In this paper, the method is extended to the caseof a non-commutative spacetime. By making use of a non-commutative version of theRaychaudhuri equation, a new set of non-commutative Einstein equations is derived.The results demonstrate that it is possible to use spacetime thermodynamics to workwith non-commutative gravity without the need to vary a non-commutative action. ∗ [email protected] a r X i v : . [ h e p - t h ] M a y Introduction
In four dimensions, black hole solutions to the Einstein equations are determined solelyby mass, electric charge and angular momentum [2]. In hindsight, this provides the firstsmall hint that the behaviour of black holes is in some way similar to that of classicalthermodynamics. When describing a classical gas, it is not feasible to keep an exhaustivelist of all positions and velocities of all the gas particles. We can restrict ourselves to a fewvariables and still compute meaningful physical quantities. Similarly, a star can be describedby many physical variables, but after its collapse we are forced into using only three.This similarity is, of course, nothing more than a hint. Only with the identificationof the area of a black hole with entropy and the establishment of the four laws of blackhole mechanics [2][3], did black hole thermodynamics become something which could bemeaningfully explored and questioned. This means, however, that an entirely valid questionwould be to ask whether the analagous behaviour signifies something deeper.The evidence for spacetime as a thermodynamic system grew when, in 1995, Jacobson[1] brought the idea of black hole thermodynamics full circle by showing that the Einsteinequations can be derived as an equation of state. The assumption that needs to be madeis that entropy and the area of causal horizons are still equal, up to some multiplicativeconstant. This is not an unreasonable assumption since entropy measures information andcausal horizons hide information from observers in space time.The Einstein equations stipulate how the geodesics of spacetime bend in response to thepresence of matter. However, these field equations need not be assumed from the outset.Simply insist that for some matter crossing a causal horizon, δQ = T dS holds ( δQ beingthe amount of energy moving across the horizon, T being the Unruh temperature and dS being the associated increase in entropy of the universe on the other side of the causalhorizon). From the only initial assumption, δQ = T dS = ηdA since dS = ηdA . What thisdemonstrates is that you only need assume the entropy-area relation to show that matterwill bend the geodesics in a spacetime. It will be shown below that these simple argumentswill lead to the Einstein equations. This is a thermodynamic derivation of the equations ofGeneral Relativity and it shows that they are, in fact, an equation of state.If the Einstein equations truly are an equation of state, this begs the question of whetheror not other equations of state exist. While it is possible to arrive at other equations of stateby changing the entropy functional δS = κδA , this is not the approach which will be followedin this paper. Rather, the reasoning will be analogous to that of classical thermodynamics.2nstead of making assumptions about the thermodynamic functionals, instead we change theassumptions about the physics of the system in question.An example of this is seen in moving from the ideal gas equation P V = nRT (1)to the Van der Waals equation (cid:18) P + n aV (cid:19) ( V − nb ) = nRT. (2)The ideal gas law does not allow for a system of gas particles to interact, nor for the particlesto have any size. By giving the particles size and the ability to interact, no broad statementabout thermodynamics is made. All that is made is a change in the assumptions aboutthe microstructure of the thermodynamical system. Additionally, no information about thekinetic theory of gases nor their statistical mechanics was needed to derive the Van derWaals equation. This surprising paucity of information shows that it might be possible toderive a new spacetime equation of state with relatively simple assumptions about spacetimemicrostructure.An assumption that is tempting to make is that there is a minimum length in spacetime.To see a simple reason why, recall that entropy can be written as S = k ln Ω (3)where Ω is the number of states accessible to the system. Since Ω is an integer, the value of S cannot be continuous - it can only take on certain values. Note, however, that in classicalgeometry the area of a black hole is continuous and therefore the entropy given by S = A / is also continuous. This seems like a contradiction but we do similar work in statisticalmechanics. We treat variables semiclassically and use them as if they were continuous, butwe know from the microscopic nature of the theory that they are in fact quantized.The above might be a simple argument but it is nonetheless forceful and reason enoughto look for different field equations of spacetime in which area quantization is enforced fromthe beginning. In this paper we will attempt to enforce the area quantization by requiringthat the spacetime coordinates obey the non-commutative relation [ x i , x j ] = iθ ij . Once non-commutativity is imposed, the work of Jacobson will be used to show how a non-commutativeversion of the Einstein Field Equations can be derived.3n section 2, Jacobson’s argument is reviewed. Section 3 begins by addressing concernsover imposing non-commutativity on spacetime and then provides information on the math-ematics needed to work with functions on a non-commutative manifold. After this, a non-commutative version of the Raychaudhuri equation is derived and used to repeat Jacobson’smethod for the case of non-commutative manifold. Let us briefly recapitulate the argument of [1]. Assume that there is some acceleratingobserver in a spacetime, the field equations of which are not specified at the beginning.Since the observer has a causal horizon and there are not yet any field equations, we are freeto specify how the generators of the horizon behave when matter crosses them. If we assumethat the area of the horizon changes proportionally to the entropy of any matter crossingthe horizon, the Raychaudhuri equation can be used to calculate the change in area of thehorizon. If this done, we are shown in [1] that the Einstein equations come out as a result.To see how the Einstein equations can be interpreted as an equation of state, pick a point p in spacetime and make the approximation that the space around p is, locally, Minkowskian.Now choose a small patch of 2-surface which contains p and call this patch O .Figure 1: Minkowski space containing O and p. Choose one side of the boundary as the past of O. B of the past of the patch O. Close to the point p, this boundary is a congruence of null geodesics orthogonal to O .This congruence constitutes the causal horizon which will be studied in the derivation.Figure 3: The null geodesics orthogonal to O . We can use the Raychaudhuri equation tostudy these geodesics and hence the behaviour of the the patch O. Choose the patch so that the expansion and shear of the congruence vanish close to p .It is always possible to do this. Note that the rotation vanishes due to the fact that thecongruence is normal to the 2-surface patch. By making this construction, a local Rindlerhorizon has been defined around p . There are local Rindler horizons in all null directionsaround a spacetime point due to the fact an observer can accelerate in any direction.Since an approximately flat region of spacetime exists near our point p and around thepatch O , the spacetime there will have all the usual Poincare symmetries. This makes5t possible to find an approximate Killing field χ a generating Lorentz boosts which areorthogonal to O and which vanish at O. Suppose now that some stress energy tensor T ab is defined in the spacetime. The flow of energy orthogonal to the patch O will be given by T ab χ a .Now choose χ a to be future pointing to the inside past of our patch O . The energy fluxto the past of the patch will then be δQ = ˆ H T ab χ a d Σ b (4)where the integral is over the generators of the inside past horizon of O .Figure 4: The heat flux across the boundary B. This picture illustrates that heat (in whateverform) is moving across the causal horizon B into an area of spacetime from which a null raycan reach the Rindler observer.Call the tangent vector to the horizon generators k a and let λ be the affine parameterwhich vanishes at the patch O and is negative to the past of O. This implies χ a = − κλk a (5)and that the small patch of directed surface area is d Σ a = k a dλdA (6)where the dA is an infinitesimal piece of horizon. Rename the energy flux to heat flux, ascan be done in ordinary thermodynamics, and write it as:6 Q = − κ ˆ H λT ab k a k b dλdA (7)Making use of the entropy-area rule from black hole thermodynamics makes it possible tosay that the entropy of this heat flux is associated with a small change in the area of thehorizon: dS = ηdA. (8)It is best to leave the constant of proportionality undetermined for now - the rest of thederivation is insensitive to this. A small patch of cross-sectional area of the null horizongenerators is given by: δA = ˆ H θdλdA. (9)Figure 5: The horizon generators expand as the entropy moves across the horizon.Recall that the thermodynamic relations being used are: δQ = T dS = ηδA. (10)Use the Unruh temperature as the temperature in the above, then set − κ ˆ H λT ab k a k b dλdA = ˆ H θdλdA (11)and employ the Raychaudhuri equation for null surfaces. Recall that the congruence has beenchosen to give vanishing expansion and shear, so that the Raychaudhuri equation reduces to7 θdλ = − R ab k a k b (12)and upon integration, θ integrates to − λR ab k a k b . This tells us that (11) becomes − κ ˆ H λT ab k a k b dλdA = ˆ H − λR ab k a k b dλdA (13)and we get to T ab k a k b = (cid:18) (cid:126) η π (cid:19) R ab k a k b (14)for all null k a . Given g ab k a k b = 0 for null k a , the metric tensor can be added into (14) forfree T ab k a k b = (cid:18) (cid:126) η π (cid:19) R ab k a k b + f g ab k a k b (15)Where f is some undetermined function. Since the expressions above are true for any null vector, the result is: T ab = (cid:18) (cid:126) η π (cid:19) R ab + f g ab . (16)Making use of the fact that T ab is divergence-free and the contracted Bianchi identity tospecify f = − R + Λ , where Λ is a constant, this gives [1]: (cid:18) π (cid:126) η (cid:19) T ab = R ab − Rg ab + Λ g ab (17)Jacobson’s calculation brings spacetime thermodynamics full circle. By studying the be-haviour of back holes, one can infer that the solutions to the Einstein equations encode thearea-entropy relationship. By assuming that the entropy-area relationship exists, one canthen get back to the Einstein equations. More importantly, by demonstrating that thereexists a thermodynamic derivation of the equations of general relativity, the result demon-strates that the geometrical variables of gravity (and not just the black hole parameters) canbe treated as thermodynamic variables. Care must be taken when it comes to making assumptions about the microstructure ofspacetime - simply placing spacetime on a lattice will break diffeomorphism invariance. A8ore subtle and perhaps more useful choice is to rather state that coordinates on spacetimedo not commute: (cid:2) x i , x j (cid:3) = iθ ij . (18)This statement has a strong analogue in quantum mechanics. By thinking of phase space asa manifold, the quantum mechanical coordinates ˆ x, ˆ p on the phase space manifold obey thecommutation relation [ˆ x, ˆ p ] = i (cid:126) (19)which demonstrates that the geometry of the phase space manifold is, in fact, noncommu-tative. This setup is desirable because in both cases it gives us cells of a definite area, butdoes so without forcing either manifold onto a lattice.What is desired is a way to implement these commutation relations which still allows usto perform calculations in a straightforward manner. The approach which will be followed inthis paper is to replace the ordinary multiplication of functions with the Moyal star productmultiplication. This star product represents the deformation of a classical, commutativetheory (the algebra of smooth functions on the manifold) in the sense that it turns itscommutative product into a non-commutative product. Several steps need to be taken before we can arrive at a new set of field equations. First,ordinary multiplication must be replaced the Moyal star product. This is an operation whichhas the desirable properties of associativity, bilinearity and the Leibniz rule. For scalars a, b : ( f (cid:63) g ) (cid:63) h = f (cid:63) ( g (cid:63) h ) (20) af (cid:63) bg = abf (cid:63) g (21) ∂ µ ( f (cid:63) g ) = ( ∂ µ f ) (cid:63) g + f (cid:63) ( ∂ µ g ) . (22)In order to take advantage of the above, it will be best to use the tetrad formalism ofgeneral relativity. To set notation, we write: 9 µν e µa e νb = η ab (23) g µν = e aµ e bν η ab . (24)The Christoffel symbols and the spin connection are related by ω aµ b = e aν e λb Γ νµλ − e λb ∂ µ e aλ . (25)The spin connection obeys its own transformation law ω a (cid:48) µ b (cid:48) = Λ a (cid:48) a Λ b (cid:48) b ω aµ b − Λ b (cid:48) c ∂ µ Λ a (cid:48) c . (26)which resembles the transformation law of the connection of a gauge-invariant theory: A A (cid:48) µ B (cid:48) = O A (cid:48) A O BB (cid:48) A Aµ B − O CB (cid:48) ∂ µ O A (cid:48) C . (27)The last step in the mathematical setup is to make use of the Sieberg-Witten map [4].This is a way to map between a gauge theory living on a commutative manifold and thatsame gauge theory living on a noncommutative manifold. For R µ with coordinates x i , imposecoordinate noncommutativity by saying that the coordinates obey the algebra: (cid:2) x i , x j (cid:3) = iθ ij where θ is real. We want to use this to deform the algebra of functions living on R n to anoncommutative algebra such that f (cid:63) g = f g + i θ ij ∂ i f ∂ j g + O ( θ ) . (28)The unique solution to this problem is [4]: f ( x ) (cid:63) g ( x ) = exp (cid:104) i θ ij ∂∂α i ∂∂β j (cid:105) f ( x + α ) g ( x + β ) | α = β =0 = f g + i θ ij ∂ i f ∂ j g + O ( θ ) . (29)If the functions f and g are matrix-valued functions, then the star product becomes thetensor product of matrix multiplication with the star product of functions as just defined.For a commutative gauge theory, the gauge transformations and field strength are writtenas 10 λ A i = ∂ i + i [ λ, A i ] , (30) F ij = ∂ i A j − ∂ j A i − i [ A i , A j ] (31)and δ λ F ij = i [ λ, F ij ] . (32)For a non-commutative gauge theory, we apply the same formulae for the gauge trans-formation law and the field strength, except that the matrix multiplication is defined by thestar product. If the gauge parameter is ˆ λ the gauge transformations and field strength ofnon-commutative Yang-Mills theory are: ˆ δ ˆ λ ˆ A i = ∂ i ˆ λ + i ˆ λ (cid:63) ˆ A i − i ˆ A i (cid:63) ˆ λ (33) ˆ F ij = ∂ i ˆ A j − ∂ j ˆ A i − i ˆ A i (cid:63) ˆ A j + i ˆ A j (cid:63) ˆ A i (34) ˆ δ ˆ λ ˆ F ij = i ˆ λ (cid:63) ˆ F ij − i ˆ F ij (cid:63) ˆ λ. (35)To first order in θ, these expressions are: ˆ δ ˆ λ ˆ A i = ∂ i ˆ λ − θ kl ∂ k ˆ λ∂ l ˆ A i + O ( θ ) (36) ˆ F ij = ∂ i ˆ A j − ∂ j ˆ A i − θ kl ∂ k ˆ A i ∂ l ˆ A j + O ( θ ) (37) ˆ δ ˆ λ ˆ F ij = − θ kl ∂ k ˆ λ∂ l ˆ F ij + O ( θ ) . (38)It now remains to find a mapping from ordinary gauge fields A to non-commutativegauge fields ˆ A which are local to any finite order in θ. It is also necessary to impose therequirement that if two ordinary gauge fields A and A (cid:48) are equivalent by an ordinary gaugetransformation U = exp ( iλ ) , then the noncommutative gauge fields ˆ A and ˆ A (cid:48) will be gaugeequivalent by a non-commutative gauge transformation U = exp ( i ˆ λ ) . Note that ˆ λ dependson A and λ. N , so that the gauge parameters are N × N matrices.It is necessary to find a mapping between commutative and non-commutative fields suchthat ˆ A ( A ) + ˆ δ ˆ λ ˆ A ( A ) = ˆ A ( A + δ λ A ) (39)where the variables λ and ˆ λ are infinitesimal. This ensures that if A undergoes a transformby λ, then the transformation of ˆ A by ˆ λ is equivalent. This forces ordinary fields whichare gauge-equivalent to be mapped to non-commutative gauge fields which are also gauge-equivalent. Working to first order in θ and write ˆ A = A + A (cid:48) ( A ) and ˆ λ ( λ, A ) = λ + λ (cid:48) ( λ, A ) . Thus, we expand (40) as A (cid:48) i ( A + δ λ A ) − A (cid:48) i ( A ) − ∂ i λ (cid:48) − i [ λ (cid:48) , A i ] − i [ λ, A (cid:48) i ]= − θ kl ( ∂ k λ∂ l A i + ∂ l A i ∂ k λ ) + O ( θ ) (40)where all the products appearing in the above are ordinary matrix products. Equation (19)is solved by ˆ A i ( A ) = A i + A (cid:48) i ( A ) = A i − θ kl { A k , ∂ l A i + F li } + O ( θ ) (41)and ˆ λ i ( λ, A ) = λ + λ (cid:48) i ( λ, A ) = λ + 14 θ ij { ∂ i λ, A j } + O ( θ ) . (42)The gauge strength is then written as ˆ F ij = F ij + 14 θ kl (2 { F ik , F jl } − { A k , D l F ij + ∂ l F ij } ) + O ( θ ) . (43)Equations (42), (43) and (44) illustrate what a gauge theory would look like if it were tolive on a noncommutative manifold and they do so in terms of what the field looks like on aregular, commutative manifold. There now exists a new noncommutative theory expressedentirely in terms of expressions and functions we already know.So far, the work has only been done to first order. To work to higher orders in θ, considermapping the field ˆ A ( θ ) to ˆ A ( θ + δθ ) . The only property of the (cid:63) − product that one needs tocheck in order to see that (43) and (44) satisfy (40) is δθ ij ∂∂θ ij ( f (cid:63) g ) = i δθ ij ∂f∂x i (cid:63) ∂g∂x j (44)when θ = 0 . Since this is true for any value of θ, it is possible to write down formulas whichtell us how ˆ A ( θ ) and ˆ λ ( θ ) change when θ is varied. These are:12 ˆ A i ( θ ) = δθ kl ∂∂θ kl ˆ A i ( θ )= − δθ kl (cid:104) ˆ A k (cid:63) (cid:16) ∂ l ˆ A i + ˆ F li (cid:17) + (cid:16) ∂ l ˆ A i + ˆ F li (cid:17) (cid:63) ˆ A k (cid:105) (45) δ ˆ λ ( θ ) = δθ kl ∂∂θ kl ˆ λ ( θ )= − δθ kl ( ∂ k λ (cid:63) A l + A l (cid:63) ∂ k λ ) (46)and δ ˆ F ij ( θ ) = δθ kl ∂∂θ kl ˆ F ij ( θ )= − δθ kl (cid:104) F ik (cid:63) ˆ F jl + 2 ˆ F jl (cid:63) ˆ F ik − ˆ A k (cid:63) (cid:16) ˆ D l ˆ F ij + ∂ l ˆ F ij (cid:17) − (cid:16) ˆ D l ˆ F ij + ∂ l ˆ F ij (cid:17) (cid:63) ˆ A k (cid:105) . (47)Work by Chamseddine [5] shows how the Sieberg-Witten map can transform the quanti-ties in the tetrad formalism into the same quantities living on a noncommutative manifold.The key lies in introducing the gauge fields ˆ ω ABµ which are subject to: ˆ ω AB † µ ( x, θ ) = − ˆ ω ABµ ( x, θ ) (48) ˆ ω ABµ ( x, θ ) r ≡ ˆ ω ABµ ( x, − θ ) = − ˆ ω ABµ ( x, θ ) . (49)Expanding these fields in terms of θ :ˆ ω ABµ ( x, θ ) = ω ABµ − iθ νρ ω ABµνρ + . . . (50)These new fields are related to the old fields via the Sieberg-Witten map: ˆ ω ABµ ( ω ) + δ ˆ λ ˆ ω ABµ ( ω ) = ˆ ω ABµ ( ω + δ λ ω ) (51)where ˆ g = e ˆ λ and the infinitesimal transformation of of ω ABµ is given by δ λ ω ABµ = ∂ µ λ AB + ω ACµ λ CB − λ AC ω CBµ . (52)The deformed fields are given by the same expression, but with matrix multiplicationreplaced by the star product: δ ˆ λ ˆ ω ABµ = ∂ µ ˆ λ AB + ˆ ω ACµ (cid:63) ˆ λ CB − ˆ λ AC (cid:63) ˆ ω CBµ . (53)13he above equation can be solved to all orders in θ and the result is [5]: δ ˆ ω ABµ = i θ νρ (cid:110) ˆ ω ν , (cid:63) ∂ ρ ˆ ω µ + ˆ R ρµ (cid:111) AB . (54)This result is necessary because it will be used to find the higher-order corrections to thedeformed tetrads. Even though it is now possible to expand and solve the above equation,we have not yet determined how ˆ e aµ is related to the undeformed field, since it is not agauge field. To proceed, treat ˆ e aµ as the gauge field of the translational generator of theinhomogeneous Lorentz group obtained by contracting SO (4 , to ISO (3 , . To do this,define the SO (4 , gauge field ω ABµ with the strength R ABµν = ∂ µ ω ABν + ∂ ν ω ABµ + ω ACµ ω CBν + ω ACν ω CBµ (55)where A = a, . Now define ω a µ = ke aµ so that we get R abµν = ∂ µ ω abν − ∂ ν ω abµ + ω acµ ω cbν + k (cid:0) e aµ e bν − e aν e bµ (cid:1) (56)and R a µν = kT aµν = k (cid:0) ∂ µ e aν − ∂ ν e aµ + ω acµ e cν − ω acν e cµ (cid:1) . (57)Now perform the contraction by taking k → and impose the condition T aµν = 0 so that ω abµ can be solved for in terms of e aµ . For the deformed case, write ˆ ω a µ = ke aµ and ˆ ω µ = k ˆ φ µ . It is not necessary impose ˆ T aµν = 0 because φ u drops out when k → . This gives the deformed tetrad to second order as [5] ˆ e aµ = e aµ − i θ νρ (cid:0) ω acµ ∂ ρ e cµ + (cid:0) ∂ ρ ω acµ + R acρµ (cid:1) e cν (cid:1) + θ νρ θ κσ (cid:0) { R σν , R µρ } ac e cκ − ω acκ (cid:0) D ρ R cdσµ + ∂ ρ R cdσµ (cid:1) e dν − { ω ν , ( D ρ R σµ + ∂ ρ R σµ ) } ad e dκ − ∂ σ { ω ν , ( ∂ ρ ω µ + R ρµ ) } ac e cκ − ω acκ ∂ σ (cid:0) ω cdν ∂ ρ e dµ + (cid:0) ∂ ρ ω cdµ + R cdρµ (cid:1) e dν (cid:1) + ∂ ν ω acκ ∂ ρ ∂ σ e cµ − ∂ ρ (cid:0) ∂ σ ω acµ + R acσµ (cid:1) ∂ nu e cκ − { ω ν , ( ∂ ρ ω κ + R ρκ ) } ac ∂ σ e cµ − (cid:0) ∂ σ ω acµ + R acσµ (cid:1) (cid:0) ω cdν ∂ ρ e dκ + (cid:0) ∂ ρ ω cdκ + R cdρκ (cid:1) e dν (cid:1)(cid:1) . (58)There now exists a deformed spin connection entirely in terms of normal geometric variablesas well as a deformed tetrad entirely in terms of normal geometric variables. These morebasic indexed objects can be used in calculations to build up more complex tensor objects.For example, it can be shown that the expansion of ˆ R ABµν in terms of θ ˆ R abµν = R abµν + iθ ρτ R abµνρτ + θ ρτ θ κσ R abµνρτκσ . (59)14as coefficients [5] R abµν = ∂ µ ω abν + ∂ ν ω abµ + ω acµ ω cbν + ω acν ω cbµ R abµνρτ = ∂ µ ω abνρτ + ω acµ ω cbνρτ + ω acµρτ ω cbν − ∂ ρ ω acµ ∂ τ ω cbν − µ ↔ νR abµνρτκσ = ∂ µ ω abνρτκσ + ω acµ ω cbνρτκσ + ω acµρτκσ ω cbν − ω acµρτ ω cbνκσ − ∂ ρ ∂ κ ω acµ ∂ τ ∂ σ ω cbν − µ ↔ ν. (60)These tetrad calculations contribute only indirectly to the derivation of a new equationof state. It is necessary to move back to the tensors of normal general relativity and theRaychaudhuri equation to derive a new equation of state. The first step to take is to find an expression for the deformed Riemann tensor which appearsin the Raychaudhuri equation. Equation (60) will not suffice because it mixes tetrad indiceswith coordinate-frame indices whereas a Riemann tensor with only coordinate-frame indicesis necessary. The classical expression R σµνρ = ∂ ν Γ σµρ − ∂ ρ Γ σµν + Γ σαν Γ αµρ − Γ σαρ Γ αµν (61)can be replaced with ˆ R σµνρ = ∂ ν ˆΓ σµρ − ∂ ρ ˆΓ σµν + ˆΓ σαν (cid:63) ˆΓ αµρ − ˆΓ σαρ (cid:63) ˆΓ αµν (62)and then expanded to higher orders in the noncommutative parameter. To accomplish this,note that we take the classical expression Γ νµλ = e νa ∂ µ e aλ + e νa e bλ ω aµb (63)and turn it into the deformed relation ˆΓ νµλ = ˆ e νa (cid:63) ∂ µ ˆ e aλ + ˆ e νa (cid:63) ˆ e bλ (cid:63) ˆ ω aµb . (64)Expressions already exist for ˆ e νa and ˆ ω aµb so these can be used to find the second-ordercorrections to the Christoffel symbols. Begin by expanding to second order: ˆΓ νµλ = Γ νµλ − iθ xy Γ νµλxy + θ xy θ pq Γ νµλxypq . (65)By explicitly calculating the terms in (65) just like was done for (60), the results up to secondorder are 15 νµλ = e νa ∂ µ e aλ + e νa e bλ ω aµb , (66) Γ νµλxy = − e νa ∂ µ e aλxy − e νaxy ∂ µ e aλ − ∂ x e νa ∂ y ∂ µ e aν − e bλxy e νa ω aµb + e νaxy e bλ ω aµb ∂ x e νa ∂ y e bλ ω aµb + e νa e bλ ω aµbxy − ∂ x e va e bλ ∂ y ω aµb , (67)and Γ νµλxypq = + e va ∂ µ e aλxypq − e vaxy ∂ µ e aλpq + e vaxypq ∂ µ e aλ − ∂ x e va ∂ y ∂ µ e aνpq − ∂ x e νapq ∂ y ∂ µ e aν − ∂ x ∂ p e va ∂ y ∂ q ∂ µ e aν + e bλxypq e va ω aµb − e νaxy e bλpq ω aµb + e νaxypq e bλ ω aµb − ∂ y e bλpq ∂ x e νa ω aµb − ∂ x ∂ p e νa ∂ y ∂ q e bλ ω aµb + e bλxy e νa ω aµbpq + e νaxy e bλ ω aµbpq + ∂ x e νa ∂ y e bλ ω aµbpq + e νa e bλ ω aµbxypq − ∂ x e bλpq e νa ∂ y ω aµb − ∂ x e νapq e bλ ∂ y ω aµb − ∂ x ∂ p e νa ∂ q ∂ y e bλ ω aµb − ∂ x ∂ p e νa e bλ ∂ y ∂ p ω aµb . (68)Encouragingly, this matches to first order. Every factor that appears in (67), (68) and (69)has already been calculated. So even though expressing (37) in terms of classical indexquantities would be tedious, it would be straightforward.Now that there are expressions for the Christoffel symbols on a deformed manifold, it ispossible to calculate (63). The second order expansion is: ˆ R νµδλ = R νµδλ + iθ ρτ R νµδλρτ + iθ ρτ θ κσ R νµδλρτκσ . (69)By performing similar calculations used to reach (61), it can be shown that that the requiredcoefficients in (70) are R νµδλ = ∂ δ Γ νµλ − ∂ µ Γ νδλ + Γ αµλ Γ ναδ + µ ↔ δ (70) R νµδλρτ = − ∂ δ Γ νµλρτ + ∂ µ Γ νδλρτ − Γ αµλ Γ ναδρτ − Γ αµλρτ Γ ναδ + ∂ ρ Γ αµν ∂ τ Γ ναδ + µ ↔ δ (71)and 16 νµδλρτκσ = ∂ δ Γ νµλρτκσ − ∂ µ Γ νδλρτκσ + Γ αµλ Γ ναδρτκσ +Γ αµλρτ Γ ναδκσ + ∂ ρ Γ αµν ∂ τ Γ ναδκσ + ∂ ρ Γ αµνκσ ∂ τ Γ ναδ − ∂ ρ ∂ κ Γ αµλ ∂ τ ∂ σ Γ ναδ + µ ↔ δ. (72)Computing the terms in the deformed Riemann tensor is a necessary step to take, but itmust be noted that the Riemann tensor appears in the Raychaudhuri equation due to thefact that covariant derivatives do not commute. Up until this point, it has not been madeclear what a covariant derivative would look like if it operated on a manifold with a minimumlength. It is therefore necessary that a prescription is developed for a covariant derivativewhich reproduces (35) when its anti-commutator is calculated.In the classical case, ∇ ν V ρ = ∂ ν V ρ + Γ ρνσ V σ (73)and when a covariant derivative acts on a tensor, a Christoffel symbol is introduced for eachindex; positive for up indices and negative for down indices. I propose that a deformedcovariant derivative acts in almost exactly the same way: ˆ ∇ ν V ρ = ∂ ν ˆ V ρ + ˆΓ ρνσ (cid:63) ˆ V σ (74)All that is necessary is to keep the partial derivative and use the deformed Christoffel symbolwith a star product instead of regular multiplication. The true test is to take the star productanti-commutator and see what results. To redo the standard calculation for the Riemanntensor, begin with (cid:104) ˆ ∇ δ ˆ ∇ µ − ˆ ∇ µ ˆ ∇ δ (cid:105) (cid:63) X v (75)from which follows (cid:104) ˆ ∇ δ ˆ ∇ µ − ˆ ∇ µ ˆ ∇ δ (cid:105) (cid:63) X v = ∂ δ (cid:16) ˆ ∇ µ X v (cid:17) − ˆΓ αδµ (cid:63) (cid:16) ˆ ∇ α X v (cid:17) + ˆΓ vδσ (cid:63) (cid:16) ˆ ∇ µ X σ (cid:17) − ∂ µ (cid:16) ˆ ∇ δ X v (cid:17) + ˆΓ αµδ (cid:63) (cid:16) ˆ ∇ α X v (cid:17) − ˆΓ vµσ (cid:63) (cid:16) ˆ ∇ δ X σ (cid:17) . (76)We should strictly be writing ˆ X v and ˆΓ vασ (cid:63) X σ , however it is not necessary to do this becausethe X v are only being used to keep track of indices. By expanding (77) and making use ofthe distributive property of the (cid:63) -product, the result will be17 ˆ ∇ δ ˆ ∇ µ − ˆ ∇ µ ˆ ∇ δ (cid:105) (cid:63) X v = ∂ δ ∂ µ X v + ∂ δ ˆΓ vµα X α − ˆΓ αδµ (cid:63) ∂ α X v − ˆΓ αδµ (cid:63) ˆΓ vασ X σ + ˆΓ vδσ (cid:63) ∂ µ X σ + ˆΓ vδσ (cid:63) ˆΓ σµα X α − ∂ µ ∂ δ X v + − ∂ µ ˆΓ vδα X α + ˆΓ αµδ (cid:63) ∂ α X v +ˆΓ αµδ (cid:63) ˆΓ vασ X σ − ˆΓ vµσ (cid:63) ∂ δ X σ − ˆΓ vµσ (cid:63) ˆΓ σδα X α . (77)At this point in the classical derivation the no-torsion condition is assumed. Analogouslythe assumption that ˆΓ ijk = ˆΓ ikj will be made. By doing this, unwanted terms are eliminatedfrom (78) and the result which appears is (cid:104) ∂ δ ˆΓ vµλ − ∂ µ ˆΓ vδλ + ˆΓ vδσ (cid:63) ˆΓ σµλ − ˆΓ vµσ (cid:63) ˆΓ σδλ (cid:105) X ν = ˆ R vλδµ (78)which is exactly what is needed. Since it is possible to show that the Leibniz rule holds forthis new deformed covariant derivative, all the ingredients to derive a deformed Raychaudhuriequation are present.Given that the derivation of the Raychaudhuri is known, it is only necessary to changemultiplication to the (cid:63) -product and add hats to show that the quantitites which appear arethe deformed quantities.There is an important point to make regarding the vectors which appear in the derivation.Since they are deformed versions of the original vectors, the substitution ξ a → ˆ ξ a can bemade. At no point will the explicit form of the ˆ ξ a vectors be calculated. This is because thevectors will eventually fall out of the expression for the Einstein Field Equations when thedeformed analogue of Jacobson’s derivation is computed.Perform a slightly simplified derivation by writing ˆ B µν = ˆ ∇ ν (cid:63) ˆ ξ µ = ∂ ν ˆ ξ µ + ˆΓ ανµ (cid:63) ˆ ξ α (79)and define the deformed expansion, shear and twist as ˆ ϑ = ˆ B µν (cid:63) ˆ h µν , (80) ˆ σ µν = ˆ B ( µν ) −
12 ˆ ϑ (cid:63) ˆ h µν (81)and ˆ ω µν = ˆ B [ µν ] . (82)18ote that from now on it will be prudent to use the variable ϑ instead of θ for the ex-pansion variable in order to avoid confusing it with the non-commutative parameter. TheRaychaudhuri equation is then derived as ˆ ξ α (cid:63) ˆ ∇ α ˆ B µν = ˆ ξ α (cid:63) ˆ ∇ ν ˆ ∇ α ˆ ξ µ + ˆ R δανµ (cid:63) ˆ ξ α (cid:63) ˆ ξ δ (83) = ˆ ∇ ν (cid:16) ˆ ξ α (cid:63) ˆ ∇ α ˆ ξ µ (cid:17) − (cid:16) ˆ ∇ ν ˆ ξ α (cid:17) (cid:63) (cid:16) ˆ ∇ α ˆ ξ µ (cid:17) + ˆ R δανµ (cid:63) ˆ ξ α (cid:63) ˆ ξ δ (84) = ˆ B αν (cid:63) ˆ B µα + ˆ R δανµ (cid:63) ˆ ξ α (cid:63) ˆ ξ δ . (85)Which allows the trace to be taken, ultimately giving the relation: ddλ ˆ ϑ = −
12 ˆ ϑ (cid:63) ˆ ϑ − ˆ σ µν (cid:63) ˆ σ µν + ˆ ω µν (cid:63) ˆ ω µν + ˆ R αµ (cid:63) ˆ ξ α (cid:63) ˆ ξ µ . (86)Equation (87) is a highly desirable expression. It shows that the “hats and stars” pre-scription carries over to the Raychaudhuri equation unchanged. Now it is possible to takethe deformed Raychaudhuri equation and use it to repeat Jacobson’s calculation. It is notnecessary to repeat the conceptual explanation; so start with δQ = − κ ˆ H λ ˆ T ab (cid:63) ˆ k a (cid:63) ˆ k b dλdA (87)and feel justified doing this integral, for the reasons outlined in [3]. Now make use of thedeformed Raychaudhuri equation to get − κ ˆ H λ ˆ T ab (cid:63) ˆ k a (cid:63) ˆ k b dλdA = ˆ H − λ ˆ R ab (cid:63) ˆ k a (cid:63) ˆ k b dλdA. (88)Removing the integrals from (89) gives ˆ T ab (cid:63) ˆ k a (cid:63) ˆ k b dλdA = ˆ R ab (cid:63) ˆ k a (cid:63) ˆ k b dλdA. (89)and by requiring that the definition for a null surface still holds: ˆ g ab (cid:63) ˆ k a (cid:63) ˆ k b = 0 . (90)This means that it is possible to recover ˆ T ab (cid:63) ˆ k a (cid:63) ˆ k b dλdA = (cid:16) ˆ R ab + ˆ g ab (cid:17) (cid:63) ˆ k a (cid:63) ˆ k b dλdA (91)19hich becomes ˆ T ab = ˆ R ab + ˆ g ab . (92)The Bianchi identities carry through as before, since only indices are being contracted, whichwill give the relation: (cid:18) π (cid:126) η (cid:19) ˆ T ab = ˆ R ab −
12 ˆ
R (cid:63) ˆ g ab + Λˆ g ab . (93)The result in (94) is pleasing and simple, but additional work is required to expand theexpressions to second order in the non-commutative parameter and to interpret the non-commutative stress-energy tensor.The first issue is not terribly problematic, since it is possible to work as before andevaluate (94) term-by-term. The Ricci tensor would be expressed as ˆ R µλ = ˆ R νµνλ = ˆ R µ λ + ˆ R µ λ + ˆ R µ λ + ˆ R µ λ, (94)from which it is possible to calculate the deformed Ricci scalar. The star product-term wouldbe ˆ R (cid:63) g µν = ˆ R ˆ g µν + i θ ij ∂ i ˆ R∂ j ˆ g µν − θ ij θ kl ∂ i ∂ k ˆ R∂ j ∂ l ˆ g µν (95)and an expression for the metric can be written as ˆ g µν = η mn ˆ e mµ (cid:63) ˆ e nν .Additionally, there is a way to deal with a stress-energy tensor living on a non-commutativespace. The idea is to proceed as in [6] and say that ˆ T µν is the stress-energy tensor of a mas-sive field living on a non-commutative manifold. If we assume that it is a massive scalar fieldwhich is the gravitational source, then we can expand ˆ T µν in powers of the non-commutativeparameter [6]: ˆ T µν = ( ∂ µ φ (cid:63) ∂ ν φ + ∂ ν φ (cid:63) ∂ µ φ ) − η µν ( ∂ α φ (cid:63) ∂ α φ − m φ (cid:63) φ )= T µν + η µν m l θ αβ θ σρ ∂ α ∂ σ φ∂ β ∂ ρ φ. (96)By doing this, we can take an explicit form of the stress-energy tensor, calculate the non-commutative corrections and match them, term-by-term, to the corrections to the geometricvariables. This, then, provides a complete description for calculating a full theory of deformedgravity. 20 Discussion
Obtaining non-commutative versions of the Raychaudhuri and Einstein equations is im-portant for a few reasons. Firstly, they demonstrate that it is possible to reproduce theRaychaudhuri equation on a noncommutative spacetime using the Sieberg-Witten map. Al-though this was only a means to an end in this paper, a noncommutative Raychaudhuriequation can be used independently of the study of spacetime thermodynamics.It was stated earlier that putting spacetime on a lattice would break diffeomorphism in-variance and it is legitimate to ask whether imposing non-commutativity truly does preserveinvariance. As was shown in [5], it is possible to use the work of Kontsevich [7] to retain theuse of the star product, but change its definition to accommodate for the fact that, underdiffeomorphisms, θ ij becomes a function of coordinates. This redefining of the star productmight change the appearance of some power series expansions, but more crucially it showsthat it is possible to use the star product and also preserve diffeomorphism invariance.The derivation of a noncommutative version of the Einstein field equations is also novelwhen compared to other attempts to derive noncommutative spacetime field equations. Thenormal procedure is to attempt to vary some action containing a tensor quantity like ˆ R .The variation of the Einstein-Hilbert action may be straightforward but varying the non-commutative analogue becomes exceedingly difficult due to the huge number of terms. Itwas possible to derive field equations in this paper with relative ease because Jacobson’sapproach does not require the variation of an action.The thermodynamic Van der Waals equation provides a better fit to reality because itincorporates important physical phenomena that the ideal gas law neglects. There is strongtheoretical justification for a minimum length in spacetime so it is hoped that by incorpo-rating this into the Einstein equations that we can match reality ever more closely. This isclearly desirable, but the fact remains that we are still doing spacetime thermodynamics. Itremains unclear how to use our current knowledge to go beyond spacetime thermodynamicsand start building a theory of spacetime statistical mechanics.There are important directions to take the work in future. One would be to start verifyingand interpreting solutions to (93). It would be unsatisfying to simply know that solutionsexist - to truly appreciate the behaviour of non-commutative solutions it would be necessaryto calculate the non-commutative corrections to a few orders. Another direction to takethe work would be to to use Kontsevich’s work [7] to calculate corrective terms which areguaranteed to not break diffeomorphism invariance.21 cknowledgements This work was supported by DST/NRF under a South African Research Chair InitiativeGrant. This work formed part of a Masters of Science in Physics thesis at the University ofthe Witwatersrand.I am grateful to V. Jejjala for his extensive helpful comments on the drafts of this paper.
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