Symplectic Raytracing:
Raytracing with Hamiltonian Dynamics
in Blackhole Spacetime

Tetsu Satoh
Haruo Takemura
Naokazu Yokoya
Nara Institute of Science and Technology
ftetu-s,takemura,yokoyag@is.aist-nara.ac.jp
http://yokoya.aist-nara.ac.jp/

Introduction

This technical sketch presents symplectic raytracing, which is a
novel approach to extend raytracing in the curved spacetime withblackholes. In conventional study for visualizing blackhole spacetime, the path of light is computed by solving geodesic equations
numerically. However, raytracing based on the geodesic equation
suffers from some problems concerning the computational cost andthe accuracy of results. In order to overcome such problems, wehave developed symplectic raytracing based on Hamilton’s canonical equation instead of the geodesic equation. Hamilton’s canonicalequation can be numerically solved by a symplectic solver suited tolong-time computation in blackhole spacetime.

(a)

(b)

(c)

(d)

Symplectic Raytracing

Original raytracing technique [1] traces the light projected fromlight sources (or a viewpoint) assuming that the path of light draws
a straight line. An equation of straight line is given by:2 id x= 0;

ds2

(1)

where xi indicates the component of three dimensional coordinatesand s is a parameter of the straight line. In gravitational raytracing [2] extended from original raytracing, the following geodesic
equation is employed to compute a path of light:
2 id x
ds2+ 0ikl

dx

kds

dx

lds

= 0;

(2)

where 0ikl , called Christoffel’s symbol, is a function to calculate
curvature of space and other variables are the same as in Eq. (1).
By adding the term of 0ikl to Eq. (1), Eq. (2) have some difficulties
as follows. Firstly, Eq. (2) is a second order nonlinear differential
equation of parameter s. Secondly, any suitable numerical method
for solving Eq. (2) have been not proposed. Thirdly, it is not easy
to concretely derive the geodesic equation of blackhole spacetime.
From the above consideration, we propose symplectic raytracing, which is a new raytracing method using the following Hamilton’s canonical equation instead of the geodesic equation.

8><>:

dpi
dsdqi
ds==0Figure 1: Visualization of a spherically symmetric blackhole; (a)no blackhole, (b) blackhole in the center, (c) superimposing x-y-z
axes and x-y plane, (d) volume rendering of scalar curvature as wellas three axes.
Visualization of Blackhole Spacetime

We assume the universe can be modeled by sphere and a blackhole
is located at the center of the sphere. An image of galaxies1 ismapped on the inside surface of the sphere. An observer is in thecelestial sphere. Images perceived by the observer are generated
with the proposed method as shown in Fig.1.

@H
@ qi
@H;

(3)

@ piwhere qi indicates the component of four dimensional coordinates(the same as xi in Eqs. (1) and (2)), pi indicates the momentum ofqi , and H called Hamiltonian is a function of qi and pi . Note that
symplectic raytracing needs eight dimensional phase-space constructed of four coordinate components and four momentum components. Compared with Eq. (2), Eq. (3) is a first order linear differential equation so that the suitable numerical method called symplectic numerical analysis [3] is applicable and the concrete derivation of Hamilton’s canonical equation for the blackhole spacetime
is simple. Especially, it is important to introduce symplectic numerical analysis. Because non-symplectic numerical analyses suchas classical Runge-Kutta methods and Euler method are known to
break the energy conservation law, they are not suitable for longtime calculation to trace the light in the universe. The main difference between the proposed and conventional methods for raytracing in curved spacetime is whether symplectic numerical analysis
is applicable or not.

Figure 2: Immersive environment for observing a
blackhole spacetime.

Because symplectic raytracing isspecialized for Hamilton system,
precisely and fast computation ispossible more than conventional raytracing with the geodesic equation.
All images in Fig. 1 are rendered on
the SGI Onyx2 and Origin2000 withparallel computation by OpenMP.
Each rendering took thirty to sixty
minutes using 13 MIPS R10000
CPUs. We have also implemented
the method on the cylindrical immersive projection display CYLIN-DRA(See Fig. 2).
References[1] R. A. Goldstein and R. Nagel. 3-d visual simulation. Simulation, 23(6):25–31,
1971.
[2] Y. Yamashita. Computer graphics of black holes: The extension of raytracings
to 4-dimensional curved space-time. Trans. Information Processing Society ofJapan, 30(5):642–651, 1989. (in Japanese).
[3] H. Yoshida. Recent progress in the theory and application of symplectic integrators. Celestial Mechanics and Dynamical Astronomy, 56:27–43, 1993.
1 http://oposite.stsci.edu/pubinfo/pr/1999/30/index.html