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/content/aapt/journal/ajp/83/6/10.1119/1.4916949
1.
1. Carl Sagan , Contact ( Simon and Schuster, New York, 1985).
2.
2.Contact, The Movie, directed by Robert Zemeckis (© Warner Bros., 1997).
3.
3. Michael S. Morris and Kip S. Thorne , “ Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity,” Am. J. Phys. 56, 395412 (1988).
http://dx.doi.org/10.1119/1.15620
4.
4.Interstellar, directed by Christopher Nolan, screenplay by Jonathan Nolan and Christopher Nolan (© Warner Bros, 2014).
5.
5. Kip Thorne , The Science of Interstellar ( W.W. Norton and Company, New York, 2014).
6.
6. Allen Everett and Thomas Roman , Time Travel and Warp Drives ( University of Chicago Press, Chicago, 2012).
7.
7. John L. Friedman and Atsushi HiguchiTopological censorship and chronology protection,” Ann. Phys. 15, 109128 (2006).
http://dx.doi.org/10.1002/andp.200510172
8.
8. Francisco S. N. LoboExotic solutions in general relativity: Traversable wormholes and ‘warp drive’ spacetimes,” Classical and Quantum Gravity Research 5 Progress ( Nova Science Publishers, Hauppauge, NY, 2008), pp. 178.
9.
9. Michael S. Morris , Kip S. Thorne , and Ulvi Yurtsever , “ Wormholes, time machines, and the weak energy condition,” Phys. Rev. Lett. 61, 14461449 (1988).
http://dx.doi.org/10.1103/PhysRevLett.61.1446
10.
10. See, e.g., chapter 13 of The Science of Interstellar4.
11.
11. James B. Hartle , Gravity: An Introduction to Einstein's General Relativity ( Addison-Wesley, San Francisco, 2003).
12.
12. Oliver James , Eugénie von Tunzelmann , Paul Franklin , and Kip S. Thorne , “ Gravitational lensing by spinning black holes in astrophysics, and in the movie Interstellar,” Class. Quant. Grav. 32, 065001 (2015).
http://dx.doi.org/10.1088/0264-9381/32/6/065001
13.
13. Homer G. Ellis , “ Ether flow through a drainhole: A particle model in general relativity,” J. Math. Phys. 14, 104118 (1973).
http://dx.doi.org/10.1063/1.1666161
14.
14.Fifteen years later, Morris and Thorne3 wrote down this same metric, among others, and being unaware of Ellis's paper, failed to attribute it to him, for which they apologize. Regretably, it is sometimes called the Morris-Thorne wormhole metric.
15.
15. Thomas Müller , “ Visual appearance of a Morris-Thorne-Wormhole,” Am. J. Phys. 72, 10451050 (2004),
15.which is based in part on Daniel Weiskopf , “ Visualization of four-dimensional spacetimes,” Ph.D thesis at der Eberhard-Karls-Universität zu Tübingen, available at http://nbn-resolving.de/urn:nbn:de:bsz:21-opus-2400. Wormhole images based on Müller's paper are available at https://www.youtube.com/watch?v=FHmupoY4nZU&index=2&list=PLdcIglDT8_FEnv0MmGrb04azONGpgqbq7,
15.and also in the paper Hans Ruder et al., “ How computers can help us in creating an intuitive access to relativity,” New J. Phys. 10, 125014 (2008). For a movie by Covin Zahn of what it looks like to travel through an Ellis wormhole, see http://www.spacetimetravel.org/wurmlochflug/wurmlochflug.html.
http://dx.doi.org/10.1119/1.1758220
16.
16. Tommaso Treu , Philip J. Marshall , and Douglas Clowe , “ Resource Letter GL-1: Gravitational Lensing,” Am. J. Phys. 80, 753763 (2012); http://arxiv.org/pdf/1206.0791v1.pdf and https://groups.diigo.com/group/gravitational-lensing.
http://dx.doi.org/10.1119/1.4726204
17.
17.This is the same as Eq. (7.46b) of Hartle,11 where, however, our is denoted ρ.
18.
18.See the technical notes for chapter 15 of The Science of Interstellar,5 pp. 294–295.
19.
19. From the embedding equation (6) and [Eq. (5b)], it follows that .
20.
21.
21. J. F. Blinn and M. E. Newell , “ Texture and reflection in computer generated images,” Commun. ACM 19, 542547 (1976).
http://dx.doi.org/10.1145/360349.360353
22.
22. M. BartelmannGravitational lensing,” Class. Quant. Grav. 27, 233001 (2010).
http://dx.doi.org/10.1088/0264-9381/27/23/233001
23.
23. Richard H. Price and Kip S. Thorne , “ Superhamiltonian for geodesic motion and its power in numerical computations,” Am. J. Phys. (in preparation).
24.
24. Charles W. Misner , Kip S. Thorne , and John Archibald Wheeler , Gravitation ( W. H. Freeman, San Francisco, 1973).
25.
25. The polar axis is arbitrary because the wormhole's geometry is spherically symmetric.
http://aip.metastore.ingenta.com/content/aapt/journal/ajp/83/6/10.1119/1.4916949
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/content/aapt/journal/ajp/83/6/10.1119/1.4916949
2015-06-01
2016-05-31

Abstract

Christopher Nolan's science fiction movie offers a variety of opportunities for students in elementary courses on general relativity theory. This paper describes such opportunities, including: (i) At the motivational level, the manner in which elementary relativity concepts underlie the wormhole visualizations seen in the movie; (ii) At the briefest computational level, instructive calculations with simple but intriguing wormhole metrics, including, e.g., constructing embedding diagrams for the three-parameter wormhole that was used by our visual effects team and Christopher Nolan in scoping out possible wormhole geometries for the movie; (iii) Combining the proper reference frame of a camera with solutions of the geodesic equation, to construct a light-ray-tracing map backward in time from a camera's local sky to a wormhole's two celestial spheres; (iv) Implementing this map, for example, in Mathematica, Maple or Matlab, and using that implementation to construct images of what a camera sees when near or inside a wormhole; (v) With the student's implementation, exploring how the wormhole's three parameters influence what the camera sees—which is precisely how Christopher Nolan, using our implementation, chose the parameters for 's wormhole; (vi) Using the student's implementation, exploring the wormhole's Einstein ring and particularly the peculiar motions of star images near the ring, and exploring what it looks like to travel through a wormhole.

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