1 Introduction
In flat space-time, the constant speed wave equation
(∇2 − ∂2t )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
Z
Ωe
Z
2
d~x ∑ φ j (∇ N j ) Ni − φ̈ j N j Ni = −
j
Ωe
ρ Ni d~x ⇔
(2)
Mφ̈ + Kφ =b
R
R
(3)
R
with Mi j = Ωe d~xNi N j , Ki j = Ωe d~x(∇ Ni ) · (∇ N j ) and bi = Ωe d~xρ Ni .
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 φ̇) 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
The analytical equation with a simple damping term becomes
∂2t φ =ρ + ∇2φ − βφ̇
(4)
which leads to the discrete equation
Mφ̈ + Bφ̇ + Kφ =b
with Bi j =
R
Ωe
(5)
d~xβ Ni N j . We can now use finite difference in time, and get
∆t ∆t2 b = M φl +1 − 2φl + φl −1 +
B φl +1 − φl −1 + ∆t2 Kφl ⇔
2
(6)
−1


∆t 

φl +1 =2φl − φl −1 + 
 M + 2 B
| {z }
− [ B + ∆tK ] φl + Bφl −1 + ∆tb ∆t
A
(7)
Using lumped M and B, A becomes diagonal, and so this should be efficient
to solve.
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 0 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 = 0, generating a wave that, ideally, would be sharp and circular,
with 0 both outside and inside the circle, propagating at constant speed
from the source.
I generated a homogenous circular grid t1.grid with 10000 elements
and 100 border nodes using the command
makegrid --batch -m PreproStdGeom -g ’DISK r=15 degrees=360’ \
-p ’e=ElmT3n2D nel=10000 resol=100’ --casename nreftest"
solving it for 400 time steps of size 0.1 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 βmax = 1 to be ideal, with a reflected
1
wave of amplitude about 10
of the original wave. Figure 1 to 4 illustrate
this. Be ware that the scale on these figures is not the same.
2
u at time=1
15
0.019
0.0171
10
0.0152
0.0133
0.0114
0
0.0095
0.0076
0.0057
0.0038
−10
0.0019
−15
−15
−4.17e−17
−10
0
10
15
Figur 1: Flat-space wave equation with damping outside r = 10. t = 1.
u at time=11
15
0.00486
0.00437
10
0.00389
0.0034
0.00291
0
0.00243
0.00194
0.00146
0.000971
−10
0.000486
−15
−15
0
−10
0
10
15
Figur 2: Flat-space wave equation with damping outside r = 10. t = 11.
3
u at time=21
15
0.000169
3.49e−05
10
−9.94e−05
−0.000234
−0.000368
0
−0.000503
−0.000637
−0.000771
−0.000906
−10
−0.00104
−15
−15
−0.00117
−10
0
10
15
Figur 3: Flat-space wave equation with damping outside r = 10. t = 1.
u at time=31
15
1.69e−05
−0.000304
10
−0.000624
−0.000945
−0.00127
0
−0.00159
−0.00191
−0.00223
−0.00255
−10
−0.00287
−15
−15
−0.00319
−10
0
10
15
Figur 4: Flat-space wave equation with damping outside r = 10. t = 1.
4
u at time=40
100
0.00188
0.00151
0.00114
0.000774
0.000406
0
3.79e−05
−0.00033
−0.000699
−0.00107
−0.00144
−100
−100
−0.0018
0
100
Figur 5: Pulse after 40 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
1000 elements with radius 100, refined at radii 25, 20 and 15 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 = 0 looks like at t = 40. 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
The metric (for 2D space with rectangular coordinates) is given by

0
0
−µ 2 r2
q
1 
R
R
2
2

g= 2
xy r
0
x + µy
, r = x2 + y2 , µ = 1 −
µr
r
0
xy Rr
µ x2 + y2
(8)

This metric can be seperated into a time and space part,
R
gtt = − 1 −
r
2
1
xy Rr
x + µ y2
γ= 2
xy Rr
µ x2 + y2
µr
2
1 µ x + y2
− xy Rr
−1
γ = 2
− xy Rr
x2 + µ y2
r
(9)
(10)
(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
j
i
Λ− 1 i′ Λ− 1 j′ γi j
δi′ j′
z}|{
= γi′ j′ , where marked
with
=
transformation
coordinates belong the locally flat system. The last equation is, in matrix
form, Λ2 = 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
p
determinant of the transformation matrix, so d~x′ → |Λ|d~x = |γ |d~x, and
′
dxi
′
Λii dxi ,
i′
every derivate must be modified according to ∇i = Λ−1 i ∇i′ . In our case,
∇’s only occur in the combination
k′
kl ′
(∇ Ni ) · ∇ N j =Λ−1 k ∇k′ Ni Λ−1 ∇l ′ N j
=Λ − 1
=γ −1
T
l′ k
l ′ k′
k′
Λ − 1 k ∇ k ′ Ni ∇ l ′ N j
∇ k ′ Ni ∇ l ′ N j
(12)
(13)
(14)
Finally, the time speed must be taken into consideration. This is a simple as
√
letting dt = µ dt′ . Since B occurs everywhere with a factor ∆t and K and b
√
with ∆t2 , time speed is accounted for by including respectively µ and µ
in their integrals. This is all taken care of in the function integrands.
6
u at time=1
15
0.0294
0.0265
10
0.0235
0.0206
0.0176
0
0.0147
0.0117
0.00878
0.00583
−10
0.00289
−15
−15
−5.69e−05
−10
0
10
15
Figur 6: R = 3, particle at 5, damping from 10, 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 = 0. 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 0 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 0. 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
u at time=6
15
0.0169
0.015
10
0.0131
0.0112
0.00929
0
0.0074
0.0055
0.00361
0.00172
−10
−0.000171
−15
−15
−0.00206
−10
0
10
15
Figur 7: R = 3, particle at 5, damping from 10, t = 6.
u at time=11
15
0.0196
0.0175
10
0.0154
0.0133
0.0112
0
0.00913
0.00703
0.00493
0.00283
−10
0.000727
−15
−15
−0.00137
−10
0
10
15
Figur 8: R = 3, particle at 5, damping from 10, t = 11.
8
u at time=16
15
0.0216
0.0183
10
0.0149
0.0116
0.00824
0
0.00489
0.00154
−0.00181
−0.00516
−10
−0.00852
−15
−15
−0.0119
−10
0
10
15
Figur 9: R = 3, particle at 5, damping from 10, t = 16.
u at time=21
15
0.0129
0.0101
10
0.00742
0.00469
0.00197
0
−0.000755
−0.00348
−0.0062
−0.00893
−10
−0.0117
−15
−15
−0.0144
−10
0
10
15
Figur 10: R = 3, particle at 5, damping from 10, t = 21.
9
u at time=31
15
0.00717
0.00441
10
0.00164
−0.00112
−0.00388
0
−0.00665
−0.00941
−0.0122
−0.0149
−10
−0.0177
−15
−15
−0.0205
−10
0
10
15
Figur 11: R = 3, particle at 5, damping from 10, 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 0 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 0 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.
10
u at time=6
15
0.0173
0.0155
10
0.0137
0.0118
0.00998
0
0.00813
0.00629
0.00445
0.00261
−10
0.000763
−15
−15
−0.00108
−10
0
10
15
Figur 12: R = 3, extra dense within 3.2, forced zero inside hole, damping
from 10, t = 6.
u at time=11
15
0.015
0.0122
10
0.00939
0.00659
0.00379
0
0.000988
−0.00181
−0.00461
−0.00741
−10
−0.0102
−15
−15
−0.013
−10
0
10
15
Figur 13: R = 3, extra dense within 3.2, forced zero inside hole, damping
from 10, t = 11.
11
u at time=21
15
0.0102
0.00792
10
0.00561
0.00329
0.000981
0
−0.00133
−0.00365
−0.00596
−0.00827
−10
−0.0106
−15
−15
−0.0129
−10
0
10
15
Figur 14: R = 3, extra dense within 3.2, forced zero inside hole, damping
from 10, t = 21.
5.1 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 x̄ = a cos θ,
ȳ = a sin θ. 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 100 timeunits to stabilize and figure 15 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
u at time=200
15
0.673
0.606
10
0.539
0.471
0.404
0
0.337
0.269
0.202
0.135
−10
0.0673
−15
−15
0
−10
0
10
15
Figur 15: 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
u at time=200
15
1
0.9
10
0.8
0.7
0.6
0
0.5
0.4
0.3
0.2
−10
0.1
−15
−15
0
−10
0
10
15
Figur 16: The field of a stationary particle at r = 5 outside a black hole with
R = 3 without forcing inside the horizon.
14
u at time=190
15
0.851
0.764
10
0.676
0.589
0.501
0
0.414
0.326
0.239
0.152
−10
0.0642
−15
−15
−0.0233
−10
0
10
15
Figur 17: The field of an orbiting particle at r = 5 outside a black hole with
R = 3 with forcing inside the horizon.
15