Interstellar is the first Hollywood movie to attempt depicting a black hole as it would actually be seen by somebody nearby. To achieve this, we developed a code called DNGR (Double Negative Gravitational Renderer) to solve the equations for light-beam propagation through the curved spacetime of a spinning (Kerr) black hole, and to render IMAX-quality, rapidly changing images. Our ray-bundle techniques were crucial for achieving IMAX-quality smoothness without flickering. This paper has four main goals: (i) to describe DNGR for physicists and CGI practitioners; (ii) to present the equations we use to map light sources to camera images via elliptical ray bundles when the camera is in arbitrary motion near a Kerr black hole; (iii) to describe new insights from DNGR into gravitational lensing when the camera is near the spinning black hole, rather than far away as in most previous studies; and (iv) to describe how the images of the black hole Gargantua and its accretion disk in the movie Interstellar were generated with DNGR. While the accretion-disk section of this paper doesn’t present new astrophysical insights, it may be pedagogically interesting for those unfamiliar with disks, and movie buffs may find its discussion of Interstellar interesting.
[1502.03809] Visualizing Interstellar’s Wormhole
This paper describes opportunities for students in elementary courses on general relativity to learn from Interstellar. These include: (i) understanding how elementary relativity concepts underlie the wormhole visualizations in the movie; (ii) performing instructive calculations with simple wormhole metrics, including constructing embedding diagrams; (iii) constructing a light-ray-tracing map from a camera’s local sky to a wormhole’s celestial spheres; (iv) implementing this map to generate images of what a camera sees near or inside a wormhole; (v) exploring how the wormhole’s parameters influence the camera’s view—as Christopher Nolan did to choose the parameters for the movie’s wormhole; and (vi) exploring the wormhole’s Einstein ring and the peculiar motions of star images near it, and what it looks like to travel through the wormhole.
I. Introduction
In 1988, in connection with Carl Sagan’s novel Contact, one of us published an article in this journal about wormholes as a tool for teaching general relativity. This paper is a follow-up, a quarter century later, in the context of Christopher Nolan’s movie Interstellar and Kip Thorne’s associated book The Science of Interstellar. Like Contact, Interstellar incorporates real science, thanks to a strong commitment from the director, screenwriters, producers, and visual effects team, and Thorne’s role as an executive producer. While wormholes were central to Contact and many subsequent movies and TV shows, Interstellar is the first to depict a wormhole as it would actually be seen by a nearby human. We, along with Christopher Nolan, were responsible for that depiction.
This paper has two purposes: (i) to explain how Interstellar’s wormhole images were constructed and the decisions made along the way, and (ii) to present this explanation in a way that may be useful to students and teachers in elementary courses on general relativity.
Before explaining our work, we briefly describe physicists’ current understanding of wormholes, based on research since 1988. There is no known mechanism for creating wormholes, either naturally or artificially, though there are speculations. Any creation of a wormhole would require a change in topology, potentially involving negative energy and closed timelike curves. It’s likely the laws of physics forbid this, but it’s not certain. A wormhole will likely pinch off quickly unless it has “exotic matter” at its throat—matter with negative energy density. While negative energy density is permitted by physics (e.g., the Casimir effect), quantum inequalities limit its amount and duration, potentially preventing macroscopic, traversable wormholes. Research suggests these inequalities may prevent wormholes like the one in Interstellar from staying open long enough for travel.
Despite these challenges, wormholes remain a staple of science fiction, and a useful pedagogical tool in relativity.
This work is a collaboration between Caltech physicist Kip Thorne and the computer graphics artists at Double Negative Visual Effects in London. We came together in May 2013 when Christopher Nolan asked us to build realistic images of a wormhole, a fast-spinning black hole, and its accretion disk, with ultra-high (IMAX) resolution and smoothness. This provided an opportunity to create images for both the movie and for research in relativity and astrophysics.
II. Spacetime Metrics for Wormholes, and Embedding Diagrams
We begin by reviewing a simple wormhole metric: the Ellis wormhole. Then we discuss the three-parameter “Dneg wormhole” metric we designed for Interstellar.
II.1 The Ellis Wormhole
In 1973, Homer Ellis introduced the “drainhole” metric:
[Equation 1]
where is a function of the coordinate given by:
[Equation 2]
This metric represents a wormhole with a throat radius connecting two asymptotically flat regions of space.
II.2 The Double Negative Three-Parameter Wormhole
The Ellis wormhole wasn’t suitable for Interstellar because Nolan wanted to explore how the wormhole’s visual appearance depends on its shape. So we designed a wormhole with three adjustable parameters.
II.2.1 Wormhole with Sharp Transitions
Our initial wormhole was a simple cylinder of length with spherical cross-sections of radius . This cylinder was joined to flat three-dimensional spaces with balls of radius removed.
II.2.2 Dneg Wormhole Without Gravity
To smooth the transitions and introduce gravitational lensing, we modified the metric. We chose a function for the radius that is similar to the Schwarzschild metric:
[Equation 5a, 5b, 5c]
This gives us a three-parameter wormhole with length , radius , and a lensing width parameter.
II.2.3 Embedding Diagrams for the Dneg Wormhole
We constructed embedding diagrams to visualize the wormhole’s geometry. These diagrams show the wormhole’s two-dimensional equatorial surface embedded in a three-dimensional space. The shape of the embedding diagram depends on the wormhole’s length-to-diameter ratio and its lensing width.
II.2.4 Dneg Wormhole with Gravity
Nolan wanted the wormhole to exert a gravitational pull. We added a Newtonian-type gravitational potential to the metric:
[Equation 8]
This potential is small enough to be negligible for the image generation process.
III. Mapping a Wormhole’s Two Celestial Spheres onto a Camera’s Sky
The key to generating images is mapping light rays from the wormhole’s two celestial spheres to the camera’s local sky. This is done by solving the geodesic equation and tracing the rays backward in time.
III.1 Foundations for the Map
The map is generated by light rays that travel backward in time from the camera to the celestial spheres. These rays are null geodesics, solutions of the geodesic equation:
[Equation 10]
We use a Hamiltonian formulation of the geodesic equation to simplify the calculations.
III.2 Procedure for Generating the Map
The procedure involves choosing a camera location, setting up local coordinates, and numerically integrating the ray equations to find the corresponding points on the celestial spheres.
IV. The Influence of the Wormhole’s Parameters on What the Camera Sees
We used our map to generate images of a distant galaxy as seen through the wormhole. By varying the wormhole’s parameters, we explored their influence on the camera’s view.
IV.1 Influence of the Wormhole’s Length
Increasing the wormhole’s length shrinks the primary image and creates additional, fainter images.
IV.2 Influence of the Wormhole’s Lensing Width
Increasing the lensing width distorts the external star field and creates more prominent secondary images.
V. Interstellar’s Wormhole
After reviewing these images, Nolan chose a short wormhole with a modest lensing width for Interstellar.
V.1 Interstellar’s Distant Galaxy
The distant galaxy was created by a team led by Paul Franklin and Eugénie von Tunzelmann, combining space photography with artistic imagination.
V.2 View Through Interstellar’s Wormhole
The final images in Interstellar show the wormhole with the chosen parameters, revealing a distorted view of the distant galaxy.
V.3 The Einstein Ring
The wormhole’s Einstein ring is a prominent feature in the images, showing peculiar motions of star images near the ring.
VI. Trip Through the Wormhole
We also explored what it would look like to travel through the wormhole. However, the resulting images were not visually compelling, as they lacked a sense of depth and motion. The visual effects team combined our simulations with additional effects to create the final sequence in the movie.
VII. Conclusion
Our work on Interstellar provides valuable educational opportunities for students of general relativity. The tools and techniques we used to create the wormhole images can be used to explore the concepts of gravitational lensing and spacetime geometry.
References are included in the original document.