Derivation of relativistic Burgers equation on de Sitter background

Derivation of relativistic Burgers equation on de Sitter background BAVER OKUTMUSTUR Middle East Technical University Department of Mathematics 06800 Ankara TURKEY baver@metu.edu.tr TUBA CEYLAN Middle East Technical University Department of Mathematics 06800 Ankara TURKEY ceylanntuba@gmail.com Abstract: Recently several versions of relativistic Burgers equations have been derived on different spacetime geometries by the help of Lorentz invariance property and the Euler system of relativistic compressible flows on the related backgrounds. The concerning equations on Minkowski flat and Schwarzshild spacetimes are obtained in the article [6] where the finite volume approximations and numerical calculations of the given models are presented in detail. On the other hand a similar work on the Friedmann Lemaître Robertson Walker FLRW geometry is described in [3]. In this paper, we consider a family member of FLRW spacetime so-called de Sitter background, introduce some important features of this spacetime with its metric and derive the relativistic Burgers equation on it. The Euler system of equations on de Sitter spacetime can be found by a known process by the help of Christoffel symbols and tensors for perfect fluids. We applied the usual techniques used in [3, 6] to derive relativistic Burgers equations from the Euler system on de Sitter background. By the help of finite volume method on curved spacetimes, we examined the numerical illustrations of the given model in the last part. Key Words: relativistic Burgers equations, de Sitter metric, Euler system, spacetime, finite volume approximation 1 Introduction The classical Burgers equation with zero viscosity is one of the simplest nonlinear, hyperbolic partial differential equation model which is important in a variety of applications such as modeling fluid dynamics, turbulence and shocks. It is also the simplest conservation law and formulated by t u + x u 2 /2 0, u ut, x, t > 0. In this paper, we derive relativistic versions of Burgers equations on de Sitter spacetime and by finite volume method, we study the discretization of these equations on the given background. The finite volume method is an important discretization technique for partial differential equations, especially those that arise from conservation laws. This method is the ideal method for computing discontinuous solutions arising in compressible flow and because of the conservation law, they give physically correct weak solutions. In the present paper, we propose a finite volume approximation for general balance laws of hyperbolic partial differential equations, and we apply finite volume method to the derived relativistic Burgers equations on 1 + 1 dimensional de Sitter spacetime. We consider the following class of nonlinear hyperbolic balance laws div ω T v Sv, 1 on an n + 1 dimensional spacetime denoted by M that is endowed with a volume form ω, where the unknown function v is a scalar field, div ω is the divergence operator, T v is the flux vector field and Sv is the scalar field source term represented on M. The spacetime M is assumed to be foliated by hypersurfaces, that is, M t 0 H t, where each slice H t is an n-dimensional spacelike manifold with initial slice H 0. Then the class of nonlinear hyperbolic equations 1 gives a scalar model on which one can analyse numerical methods. We refer the reader to the follow-up works by LeFloch and collaborators [1, 6, 7, 8] for details. The inspiration of the relativistic Burgers equation on de Sitter spacetime comes from the papers [3, 6]. We take into account the relativistic Euler equations on a given curved background M α T αβ 0, 2 where T αβ is the energy-momentum tensor of perfect fluids. The relativistic Burgers equations are derived from 2 by imposing pressure to be zero in this system. The derived equations satisfy the Lorentz invariance property and in the limit case, one obtains the classical non relativistic Burgers equation. In [6], ISBN: 978-1-61804-258-3 41

M is taken to be 1+1 dimensional Minkowski flat and Schwarzshild spacetimes; whereas in [3], the curved spacetime is considered to be the Friedmann- Lemaitre-Robertson-Walker FLRW spacetime of 1 + 1 dimension. In this paper we consider our background to be de Sitter spacetime which belongs to the family of the FLRW geometry. The metric of both FLRW and de Sitter backgrounds are solutions of Einsteins field equations. Our objective is to derive relativistic Burgers equation on de Sitter background with numerical results by finite volume approximations. An outline of the article is as follows. We introduce basic features of de Sitter geometry and its metric in the first part. The derivation of Euler equations via Christoffel symbol is the second part. The next issue will be deriving the particular cases of the relativistic Burgers equations depending on a constant parameter Λ. Then we present the finite volume approximations, and finally we terminate the article with numerical experiments by comparing the classical and relativistic Burgers equations depending on different values of cosmological constant. 2 de Sitter backround The de Sitter background is a Lorentzian manifold which is a cosmological solution to the Einsteins field equations and it plays an important role in physical cosmology. It shares crucial features with Minkowski spacetime whereas its physical interpretation is quite complicated. The de Sitter spacetime has a constant curvature and its metric contains a cosmological constant Λ. The particular case Λ 0 in the de Sitter metric gives the Minkowski metric. The corresponding metric for a 3 + 1 dimensional de Sitter spacetime in terms of the proper time t, the corresponding radial r and angular θ and ϕ coordinates is formulated by g 1 Λr 2 dt 2 + 1 1 Λr 2 dr2 +r 2 dθ 2 +sin 2 θdϕ 2, 3 where the non-zero covariant components and the corresponding contravariant components of its diagonal matrix are formulated respectively by and g 00 1 Λr 2, g 11 1 1 Λr 2, g 22 r 2, g 33 r 2 sin 2 θ, g 00 1 Λr 2 1, g11 1 Λr 2, g 22 1, g 33 1 r 2 r 2 sin 2 θ, with g ik g kj δ i j, where δj i is the Kronecker s delta function. Christoffel symbols for de Sitter background In order to obtain Euler equations on de Sitter background, firstly Christoffel symbols should be calculated. Christoffel symbols are denoted by Γ µ αβ. Applying the formula Γ µ αβ 1 2 gµν ν g αβ + β g αν + α g βν, 4 where the terms α, β, µ, ν {0, 1, 2, 3}, each term of the Christoffel symbols can be easily calculated. As an example Γ 0 00 1 2 g00 g 00 + g 00 + g 00 0, Γ 0 01 1 2 g00 g 01 g 00 + g 10 1 2 g00 1 g 00 1 1 2 Λr 2 1 1 1 Λr 2 1 1 2 Λr 2 1 2Λr Λr Λr 2 1. Repeating this process, the non-zero terms of the Christoffel symbols are obtained as Γ 0 01 Λr Λr 2 1, Γ0 10 Γ 1 11 Λr 1 Λr 2 Γ 1 00 ΛrΛr 2 1, Γ 1 22 rλr 2 1, Γ 1 33 rλr 2 1 sin 2 θ, Γ 2 12 Γ 2 21 Γ 3 13 Γ 3 31 1 r, Γ 2 33 sin θ cos θ, Γ 3 23 Γ 3 32 cot θ. All remaining terms are zero. Derivation of Burgers equation The purpose is to find tensors for perfect fluids on de Sitter background to obtain the Euler equations. We consider our spacetime to be 1 + 1 dimensional and the solutions to the Euler equations depend only on the time variable t and radial variable r, so that the angular components vanish. Thus we have u α u 0 t, r, u 1 t, r, 0, 0 with u α u α 1, by which we obtain u α u α u 0 u 0 + u 1 u 1 g 00 u 0 u 0 + g 11 u 1 u 1. ISBN: 978-1-61804-258-3 42

It follows that To begin with β 0 in 9, we have 1 g 00 u 0 2 + g 11 u 1 2, 1 1 Λr 2 u 0 2 + 1 1 Λr 2 u1 2. 5 We consider our 3 + 1 dimensional coordinate as x 0, x 1, x 2, x 3 ct, r, 0, 0. Next we define the velocity component v as u 1 c v : 1 Λr 2 u 0. 6 To use 5 and 6, we obtain u 0 and u 1 as u 0 2 c 2 1 Λr 2 c 2 v 2, u1 2 v2 1 Λr 2 c 2 v 2. 7 In order to find the tensors, we need the formula of energy momentum tensor for perfect fluids, that is, T αβ ρc 2 + p u α u β + p g αβ. 8 Then by the help of 7 and 8, the tensors can be derived easily. The first term T 00 is T 00 ρc 2 + pu 0 u 0 + pg 00 c 2 ρc 2 + p 1 Λr 2 c 2 v 2 + p Λr 2 1 ρc 4 + pv 2 c 2 v 2 1 Λr 2. Analogously the remaining terms are obtained as follows T 01 T 10 cvρc2 +p c 2 v 2, T 11 c2 1 Λr 2 v 2 ρ+p c 2 v 2 T 22 p, T 33 p r 2 r 2 sin 2 θ, T 02 T 03 T 12 T 13 T 20 T 21 T 23 T 30 T 31 T 32 0. 3 Zero pressure Euler system on de Sitter background We intend to first derive the Euler system and then impose the pressure to be zero in the concerning equations. To this aim, we use the equation 2, that can be rewritten as α T αβ + Γ α αγt γβ + Γ β αγt αγ 0. 9 α T α0 + Γ α αγt γ0 + Γ 0 αγt αγ 0. After substituting α, γ {0, 1, 2, 3} in this relation, it follows that T 00 + Γ 0 00T 00 + Γ 0 00T 00 + Γ 0 01T 10 + Γ 0 01T 01 + Γ 0 02T 20 + Γ 0 02T 02 + Γ 0 03T 30 + Γ 0 03T 03 T 10 + Γ 1 10T 00 + Γ 0 10T 10 + Γ 1 11T 10 + Γ 0 11T 11 + Γ 1 12T 20 + Γ 0 12T 12 + Γ 1 13T 30 + Γ 0 13T 13 + 2 T 20 + Γ 2 20T 00 + Γ 0 20T 20 + Γ 2 21T 10 + Γ 0 21T 21 + Γ 2 22T 20 + Γ 0 22T 22 + Γ 2 23T 30 + Γ 0 23T 23 + 3 T 30 + Γ 3 30T 00 + Γ 0 30T 30 + Γ 3 31T 10 + Γ 0 31T 31 + Γ 3 32T 20 + Γ 0 32T 32 + Γ 3 33T 30 + Γ 0 33T 33 0. We continue by letting β 1 in 9, that is, α T α1 + Γ α αγt γ1 + Γ 1 αγt αγ 0, and putting α, γ {0, 1, 2, 3} we obtain the following equation T 01 + Γ 0 00T 01 + Γ 1 00T 00 + Γ 0 01T 11 + Γ 1 01T 01 + Γ 0 02T 21 + Γ 1 02T 02 + Γ 0 03T 31 + Γ 1 03T 03 T 11 + Γ 1 10T 01 + Γ 1 10T 10 + Γ 1 11T 11 + Γ 1 11T 11 + Γ 1 12T 21 + Γ 1 12T 12 + Γ 1 13T 31 + Γ 1 13T 13 + 2 T 21 + Γ 2 20T 01 + Γ 1 20T 20 + Γ 2 21T 11 + Γ 1 21T 21 + Γ 2 22T 21 + Γ 1 22T 22 + Γ 2 23T 31 + Γ 1 23T 23 + 3 T 31 + Γ 3 30T 01 + Γ 1 30T 30 + Γ 3 31T 11 + Γ 1 31T 31 + Γ 3 32T 21 + Γ 1 32T 32 + Γ 3 33T 31 + Γ 1 33T 33 0. Finally, by substituting the Christoffel symbols and the calculated values of tensors for perfect fluids into the Euler system in 1+1 dimensional de Sitter back- ISBN: 978-1-61804-258-3 43

ground, we get the simplified form of the equations as ρc 4 + v 2 p 1 Λr 2 c 2 v 2 + 2Λr cvρc 2 + p Λr 2 1 c 2 v 2 cvρc 2 + p c 2 v 2 + Λr cvρc 2 + p 1 Λr 2 c 2 v 2 + Λr cvρc 2 + p 1 Λr 2 c 2 v 2 + 1 cvρc 2 + p r c 2 v 2 + 1 cvρc 2 + p r c 2 v 2 0, cvρc 2 + p c 2 v 2 + ΛrΛr 2 ρc 4 + v 2 p 1 1 Λr 2 c 2 v 2 + Λr c 2 1 Λr 2 ρv 2 + p Λr 2 1 c 2 v 2 c 2 1 Λr 2 ρv 2 + p c 2 v 2 + 2Λr c 2 1 Λr 2 ρv 2 + p 1 Λr 2 c 2 v 2 + 1 c 2 1 Λr 2 ρv 2 + p r c 2 v 2 + rλr 2 1 p r 2 + 1 c 2 1 Λr 2 ρv 2 + p r c 2 v 2 + rλr 2 1 sin 2 p θ r 2 sin 2 θ 0. 10 Now it is the time to impose p 0 in the above relations 10 in order to get the following two equations so called the zero pressure Euler system on 1 + 1 dimensional de Sitter background, that is, c 1 Λr 2 c 2 v 2 2v + rc 2 v 2 0, cv c 2 v 2 v c 2 v 2 v 2 1 Λr 2 c 2 v 2 Λrc2 c 2 v 2 + Λrv2 c 2 v 2 + 21 Λr2 v 2 rc 2 v 2 0. 11 4 The relativistic Burgers equation on de Sitter background The purpose of this section is to deduce our main equation by the help of the zero pressure Euler equations 11 obtained in the previous section and investigate the static and spatially homogeneous solutions if exist. We combine the first and second equations of 11 to derive the following single equation, t v+1 Λr 2 r v2 2 +Λrv c2 2v 2 0, 12 which is the desired relativistic Burgers equation on de Sitter background. Here Λ is the cosmological constant and c is the speed of light which is a positive parameter. Depending on various values of Λ, not only the equation 12, but also the metric 3 take different forms. We already stated that, the particular case of the de Sitter metric 3 for Λ 0 is Minkowski metric. In addition, by plugging Λ 0 in 12, one can recover the classical Burgers equation t v + r v2 2 0. This is a common property shared by relativistic equations. The limit cases of the relativistic Euler and Burgers equations on Schwarzschild and FLRW spacetimes have the same feature. For more detail, we refer the reader to the following papers [3, 6]. Static and spatially homogeneous solutions The equation 12 can be written as t v + r 1 Λr 2 v2 Λrv 2 v + c 2. 13 2 We investigate the t independent solutions to 13. We consider r 1 Λr 2 v2 Λrv 2 v + c 2, 14 2 by using the change of variable X : 1 Λr 2 and Y : 2v 2 + c 2, we obtain X dy 2 dx Y c 2 2 Y. It follows after the integration that the static solutions can be implicitly formulated as ln2v 2 + c 2 v + 1 1 v 1 arctan 4 2 c 2 2 1 c 2 16 2 1 16 2 ln1 Λr 2 + C 15 where C is a constant. It can be observed as a remark that, there is no need for searching spatially homogeneous r independent solutions since the equation has always r dependency. t v Λrv 2 v + c 2 ISBN: 978-1-61804-258-3 44

5 Finite Volume schemes We consider the hyperbolic balance law 1 in 1+1 dimensional spacetime M. Following the article [1], we formulate the finite volume approximations for the equation 1 on the given background, that is, 1 div ω T v ωt 0 v, x ωt 1 v, x ω Sv, x, where T 0, T 1 are flux fields, S is the source term and ω is the weight function. Supposing that the spacetime is prescribed in coordinates t, r, we consider the finite volume method over each grid cells [t n, t n+1 ] [r j 1/2, r j+1/2 ] with t n n t. In order to obtain the finite volume scheme, we integrate both sides of the equation 1 over each grid cell given above, use Stokes theorem on the boundary and finally take approximations of the numerical flux functions. Consequently, a general formula of the finite volume scheme for the given balance law on local coordinates reads as ω j T n+1 j ω j T n j λ ω j+1/2 Q n j+1/2 ω j 1/2Q n j 1/2 + tsv n j ω j, where Q n j±1/2 and T n j T vj n are the approximations of the numerical flux functions, S is the approximation of the source term and λ t/ r. For the details of triangulations and discretization of finite volume approximations on a given curved spacetime, we address the reader to the papers by LeFloch et al [1, 6, 7]. 6 Numerical Experiments Lax Friedrichs scheme for the Burgers equation on de Sitter background In this part numerical experiments are illustrated for the model derived on de Sitter background. The behaviours of initial single shocks and rarefactions are examined in the applications. We use Lax Friedrichs scheme for finite volume approximations of the derived equation on de Sitter spacetime by taking r [0, 1] with various data of cosmological constant Λ. A standard CFL condition is assumed to be satisfied. After normalization using c 1, the main equation 12 reads as t v +1 Λr 2 r v2 2 +Λrv 1 2v2 0, 16 then rewriting the source term on the right hand side of the equation 16, we get t v + 1 Λr 2 r v2 2 Λrv 1 2v2. 17 The corresponding finite volume scheme for this model is v n+1 j where 1 2 vn j 1 + v n j+1 T 2 X bn j+1g n j+1 b n j 1g n j 1 + T S n j, 18 S n j Λr n j v n j 1 2v n j 2, b n j 1 Λr n j 2, fv v 2, g n j 1 fv n j 1, g n j+1 fv n j+1. In the following figures, we compare the classical Burgers equation with the relativistic one that is derived on de Sitter background. We notice that the classical Burgers equation is a particular case of our model where Λ 0. We investigate the attitudes of an imposing shock and rarefaction depending on diverse values of Λ on the scheme. In the graphs the red line represents Λ 0 and green line represents Λ ±1. The rarefactions are examined in Figure 1 and 3, whereas, shocks are examined in Figure 2 and 4. In these experiments we observe that the equations for Λ ±1, that is the relativistic case, shows similar properties with the classical Burgers equation. However, the curves related to λ ±1 move away from the curve of classical Burgers equation by the time. One can realize from the given tests that, for Λ > 0 the graph of the model extends in upward direction and for Λ < 0 in downward direction. One can also observe that there is an effect of the cosmological constant Λ on the speed of the movement of the curves. As long as the absolute value of Λ becomes larger, the green curve corresponding to the relativistic one goes further away from the red one corresponding to the classical one in a faster manner. References: [1] P. Amorim, P. G. LeFloch, and B. Okutmustur, Finite volume schemes on Lorentzian manifolds, Communications in Mathematical Sciences Volume 6, Number 4 2008, 1059 1086. [2] M. Ben-Artzi and P. G. LeFloch, The well posedness theory for geometry compatible hyperbolic conservation laws on manifolds, Ann. Inst. H. Poincare Anal. Non Linéaire, 24 2007, pp. 9891008 ISBN: 978-1-61804-258-3 45

Figure 1: The numerical solutions given by the Lax Friedrichs scheme with a rarefaction for Λ 1. Figure 3: The numerical solutions given by the Lax Friedrichs scheme with a rarefaction for Λ 1. Figure 2: The numerical solutions given by the Lax Friedrichs scheme with a shock for Λ 1. Figure 4: The numerical solutions given by the Lax Friedrichs scheme with a shock for Λ 1. ISBN: 978-1-61804-258-3 46

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