2 The wave equation in the finite element formalism

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1 1 Introduction In flat space-time, the constant speed wave equation ( 2 2 t )φ = ρ (1) is simple enough, but when we let space-time curve, the equation becomes very complex. This is especially the case for the vector wave equation, which I will not have the time to consider in this project, but also, to a lesser degree, for the scalar equation. However, space can always be approximated as flat locally, and so as long as one looks at a small enough element of space, it is sufficient to solve the flat-space equations in that region. The goal if this project is to see if this approach combined with the finite element method satisfactory solves the scalar wave equation for the Schwarzschild space-time. 2 The wave equation in the finite element formalism Using the Galerkin method to minimize the residual for each element, we get Ω e d x j (φ j ( 2 N j )N i φ j N j N i ) = ρn i d x (2) Ω e M φ + Kφ =b (3) with M i j = Ω e d xn i N j, K i j = Ω e d x( N i ) ( N j ) and b i = Ω e d xρn i. The surface term has been set equal to zero here, since I will enforce zero boundary conditions here. Since I will be solving the wave equation the open space around a black hole, there shouldn t really be any boundary - space should extent infinitly. The simulation, however, has to be finite, and so there must be a boundary with appropriate conditions. Simply enforcing zero derivative and zero value at the border will not give correct behavior; in this case the field will be reflected with opposite phase from the border. This is clearly wrong: the border is just a random line in space, after all. What is needed is non-reflecting boundary conditions. These are very hard to implement correctly, but a common approximation is to extend the area considered slightly, and introduce a damping term (a term proportional to φ) in the new area only. If the damping isn t too dramatic (sudden damping would be akin to a wall, and cause reflection), a wave headed out of the area will be gradually damped to negligibility before being reflected by the boundary. 1

2 The analytical equation with a simple damping term becomes 2 tφ =ρ + 2 φ β φ (4) which leads to the discrete equation M φ + B φ + Kφ =b (5) with B i j = Ω e d xβn i N j. We can now use finite difference in time, and get ( t 2 b =M φ l+1 2φ l +φ l 1) + t ( 2 B φ l+1 φ l 1) + t 2 Kφ l φ l+1 =2φ l φ l 1 + M + t 2 B }{{} A 1 (6) ( ) [B + tk]φ l + Bφ l 1 + tb t Using lumped M and B, A becomes diagonal, and so this should be efficient to solve. (7) 3 Test of non-reflection boundary conditions I modified the fem/wave1 program to include a source and damping term. The damping variable beta is a scalar field set to inside the main area of the simulation, and increasing linearly to a given value at the end of the extended area. To make any reflections more apparent, I let the source exist only a t =, generating a wave that, ideally, would be sharp and circular, with both outside and inside the circle, propagating at constant speed from the source. I generated a homogenous circular grid t1.grid with 1 elements and 1 border nodes using the command makegrid --batch -m PreproStdGeom -g DISK r= degrees=36 \ -p e=elmt3n2d nel=1 resol=1 --casename nreftest" solving it for 4 time steps of size.1 with the input file nreftest.i. It turns out that the initial wave propagates outwards while dispersing, leaving a trail of concentric, weaker, circular waves. The effect of the damping can be measured by measuring the amplitude of the reflected waves as they converge on the center. Varying the damping parameter by orders of magnitude around 1, I found β max = 1 to be ideal, with a reflected 1 wave of amplitude about 1 of the original wave. Figure 1 to 4 illustrate this. Be ware that the scale on these figures is not the same. 2

3 u at time= e 17 Figur 1: Flat-space wave equation with damping outside r = 1. t = 1. u at time= Figur 2: Flat-space wave equation with damping outside r = 1. t = 11. 3

4 u at time= e e Figur 3: Flat-space wave equation with damping outside r = 1. t = 1. u at time= e Figur 4: Flat-space wave equation with damping outside r = 1. t = 1. 4

5 u at time= e Figur 5: Pulse after 4 timeunits in big grid with dense center but coarse otherwise. 3.1 Other ways of simulating infinite space Another way of simulating infinite space (which is the point of nonreflecting boundary conditions), is to make a huge but coarse grid, and gradually refine it towards the middle, where the interesting area is. I have not studied this alternative closely, but a test with a coarse grid of originally 1 elements with radius 1, refined at radii 25, 2 and shows that there is quite a bit of reflection even with this model. Figure 5 shows how a pulse starting in the center at t = looks like at t = 4. It seems like more of the wave has been reflected than what is still continuing outwards. I will stick with damping in the following. 4 Introducing space-time curvature Until now, the finite element simulation has assumed that space is flat, that is, that every element has time running at the same speed, and space being stretched equally. In the Schwarzschild metric, this is not the case. 5

6 The metric (for 2D space with rectangular coordinates) is given by g = 1 µ 2 r 2 µr 2 x 2 + µy 2 xy R r, r = x 2 + y 2, xy R r µx 2 + y 2 µ = 1 R r (8) This metric can be seperated into a time and space part, ( g tt = 1 R ) r γ = 1 ( x 2 + µy 2 xy R ) µr 2 r xy R r µx 2 + y 2 γ 1 = 1 ( µx 2 + y 2 xy R ) r 2 r xy R r x 2 + µy 2 (9) (1) (11) The flat-space method used earlier should still be valid at the element level, but a correction is needed to stitch these correctly together. As seen from the global Schwarzschild system, the locally flat coordinate system of each element is compressed or streteched corresponding to a linear coordinate transformation dx i = Λ i i dxi, with Λ 1i i Λ 1 j j γ {}}{ i j = γ i j, where marked coordinates belong the locally flat system. The last equation is, in matrix form, Λ 2 = γ, so it seems like one needs to find a sqare root of the matrix γ (can be done as an eigenvalue problem). This, however, will turn out to be unneccesary. The translation between these global and local coordinates is entierly similar to that already present in the finite element formalism, and we can reuse results from here. Every volume element must be inflated by the determinant of the transformation matrix, so d x Λ d x = γ d x, and every derivate must be modified according to i = Λ 1i i i. In our case, s only occur in the combination ( N i ) ( N j ) =Λ 1 k k k N i Λ 1kl l N j (12) =Λ 1Tl k Λ 1 k k k N i l N j (13) =γ 1l k k N i l N j (14) Finally, the time speed must be taken into consideration. This is a simple as letting dt = µdt. Since B occurs everywhere with a factor t and K and b with t 2, time speed is accounted for by including respectively µ and µ in their integrals. This is all taken care of in the functionintegrands. δ i j 6

7 u at time= e 5 Figur 6: R = 3, particle at 5, damping from 1, t = 1. 5 Results To see how waves propagate in the Schwarzschild geometry, I made a new run run with almost the same settings as the last run, but now with a Schwarzschild radius of 3 and a particle at a radius of 5, which is still, for the time being, only present at t =. The input file for this run is hole1.i, and figure 6 to 11 display the result. This simulation displays many of the aspects of what would really happen: The wave propagates only slowly in the direction of the black hole, since time runs slower there, and space is compressed there. Instead, the wave front winds itself around the horizon, meets itself on the other side of the hole, and continues winding up while diluting. In the end, it seems like a relic field is trapped at the horizen, while the field is practically elsewhere, which also seems reasonable. Closer inspection, however, reveals a problem. The relic field turns out to be inside the horizon, where the field isn t supposed to be able to go in finite time, and where the time speed in this simulation is set to be. This must be due to elements that cross the horizon. Some of the integration points of an element lie outside the horizon, then the basis functions in that element will get non-zero contributions. In that case, this problem should be avoidable by ensuring that element boundaries line up with the horizon. If the DISK_WITH_HOLE grid generator supported full circles, this would have been easy to achieve, while at the same time avoiding wasting time 7

8 u at time= Figur 7: R = 3, particle at 5, damping from 1, t = 6. u at time= Figur 8: R = 3, particle at 5, damping from 1, t = 11. 8

9 u at time= Figur 9: R = 3, particle at 5, damping from 1, t = 16. u at time= Figur 1: R = 3, particle at 5, damping from 1, t = 21. 9

10 u at time= Figur 11: R = 3, particle at 5, damping from 1, t = 31. on elements inside the horizon. Lacking the means to line up the element boundaries with the black hole horizon, an alternative could be to look at the horizon as a boundary, and force the field to be there, even though it is in the middle of the grid, and not a real boundary. One could also increase the grid resolution around the horizon, so there is a buffer of elements with very low time speed between the wave and the horizon, which could prevent the wave from reaching the horizon during the simulation. This would most easily be achieved with an adaptive grid with refinements based on time speed, which is what I originally implemented, only to discover that grid refinement is an extension of diffpack, which didn t seem to be available. Instead, I will use makegrid to make the grid extra dense in this area. I ran new simulation with doubly refined grid within r = 3.2 and forced value of inside the field, but otherwise identical settings (hole2.i). Figure 12 to 14 show the results. Increasing the density of the grid around the horizon helped. It is now more obvious that the field is compressed as it enters an area with slower time speed (much like the density of cars on a road increases when the speed limit decreases) and more compressed space. Figure 12 illustrates this nicely. The next figures, however, reveal that forcing the field to be zero inside the horizon leads to reflection. I will still use this forcing in the following section, to avoid the accumulation of the field inside horizon. 1

11 u at time= Figur 12: R = 3, extra dense within 3.2, forced zero inside hole, damping from 1, t = 6. u at time= Figur 13: R = 3, extra dense within 3.2, forced zero inside hole, damping from 1, t =

12 u at time= Figur 14: R = 3, extra dense within 3.2, forced zero inside hole, damping from 1, t = Other ways to avoid the horizon problem One could also try to avoid getting nonzero values inside the black hole by using a coordinate system where only the outside of the hole exist, for example by letting r a = r R, and introducing coordinates x = a cosθ, ȳ = a sinθ. The program includes an option for doing this, but the results are unphysical, probably due to horizon being replaced by a single point, and I will not discuss this further here. 5.2 The field of a stationary particle outside a black hole By letting the source stay inside the simulation region during the whole simulation, instead of just initially, I can calculate the field of a stationary particle outside a black hole. The input for this simulation is in hole3.i. The field took about 1 timeunits to stabilize and figure illustrates the final, stationary field. The same situation without zero inside the horizon can be seen in figure 16. It is unclear which of these is most correct. The first stabilizes relatively quickly, but does not display a strong field near the horizon. The latter has this, but also has artifacts inside the horizon, which prevent the solution from stabilizing completely. 12

13 u at time= Figur : The field of a stationary particle at r = 5 outside a black hole with R = 3 with forcing inside the horizon. 5.3 The field of an orbiting source outside a black hole Using the same parameters as in the last simulation, but letting the source orbit, results in a field that is surprisingly similar, even for high rotation speeds, as figure 17 illustrates. I am not sure if this is physical or not (a moving particle needs the vector wave equation in the case of electromagnetism in any case). 6 Conclusion These simulations have demonstrated that it is possible to get reasonablelooking results for the scalar wave equation in curved space-time by using this simple method of solving the flat-space equation locally, and applying a transformation to get the global solution. Any involvement of the horrible and numerous Christoffel symbols and their derivatives can thus be avoided. There were some difficulties with this method, particularily with the boundary conditions, both the ones at big and small radius, and it is not which of the approaches used for handling the horizon is closest to an exact solution. 13

14 u at time= Figur 16: The field of a stationary particle at r = 5 outside a black hole with R = 3 without forcing inside the horizon. 14

15 u at time= Figur 17: The field of an orbiting particle at r = 5 outside a black hole with R = 3 with forcing inside the horizon.