A trip to the end of the universe and the twin "paradox"

American Journal of Physics

Volumn 76, Issue 4-5, 2008, Pages 360-373

  Müller, Thomas ^a   King, Andreas ^a   Adis, Daria ^a  

^a UNIVERSITY OF TÜBINGEN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 41849088855     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.2830528     Document Type: Article

References (37)

1. R. D'Inverno, Introducing Einstein's Relativity (Clarendon Press, Oxford, 1992).
2. C. E. Dolby and S. F. Gull, "On radar time and the twin 'paradox,'" Am. J. Phys. 69, 1257-1261 (2001).
3. S. P. Boughn, "The case of the identically accelerated twins," Am. J. Phys. 57, 791-793 (1988).
4. R. P. Gruber and Richard H. Price, "Zero time dilation in an accelerating rocket," Am. J. Phys. 65, 979-980 (1997).
5. W. Rindler, Relativity: Special, General, and Cosmology (Oxford University Press, New York, 2001).
6. R. U. Sexl, and H. K. Urbantke, Relativity, Groups, Particles: Special Relativity and Relativistic Symmetry in Field and Particle Physics (Springer, Wien/New York, 2000).
7. R. Perrin, "Twin paradox: A complete treatment from the point of view of each twin," Am. J. Phys. 47, 317-319 (1979).
8. A. P. French, Special Relativity (W. W. Nelson & Co, London, 1968).
9. R. H. Good, "Uniformly accelerated reference frame and twin paradox," Am. J. Phys. 50, 232-238 (1982).
10. H. und M. Ruder, Die Spezielle Relativitätstheorie (Vieweg Studium, Braunschweig/Wiesbaden, 1993).
11. See EPAPS Document No. E-AJPIAS-76-002803 for the Java applets, RelSkyApplet and TwinApplet, and accompanying help manuals. This document can be reached through a direct link in the online article's HTML reference section or via the EPAPS homepage (http://www.aip.org/pubservs/epaps.html).
12. One light year (1 ly) equals the distance which is covered by light with velocity c=299, 792, 458 m/s in one year.
13. The negative acceleration of Tina away from the Sun equals the situation where Tina has positive acceleration but looks contrary to her direction of motion.
14. The limit can be calculated using l'Hospital's rule.
15. G. D. Scott and H. J. van Driel, "Geometrical appearances at relativistic speeds," Am. J. Phys. 38, 971-977 (1970).
16. The Hipparcos star catalogue consists of about 118,000 stars most of which are at a distance of roughly 100 pc. The data we are interested in are right ascension (H3), declination (H4), magnitude in Johnson V (H5), trigonometric parallax (H11), Johnson B-V color (H37), and Henry Draper (HD) catalogue number (H71). The digital catalogue 1/239 can be found at 〈cdsarc.u-strasbg.fr/viz-bin/Cat?I/239〉 1997HIP...C. . . . . .0EEuropean Space Agency SP-1200.
17. B. C. Reed, "The composite observational-theoretical HR diagram," J. R. Astron. Soc. Can. 92, 36-37 (1998).
18. The bolometric magnitude of a star is defined by the flux density integrated over all wavelengths. However, a real telescope can only measure the flux density within a small region of wavelengths. The bolometric correction compensates the difference between the bolometric and the visual magnitude (Ref. 19).
19. Fundamental Astronomy, edited by Hannu Karttunen, Pekka Kröger, Heikki Oja, Markku Poutanen, and Karl J. Donner (Springer-Verlag, Heidelberg, 2003).
20. We take the physical constants from the National Institute of Standards and Technology, 〈physics.nist.gov/cuu/Constants〉.
21. R. W. Lindquist, "Relativistic transport theory," Ann. Phys. 37, 487-518 (1966).
22. ⊙ ≈ 5778 K, see also 〈nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html〉.
23. U. Kraus, "Brightness and color of rapidly moving objects: The visual appearance of a large sphere revisited," Am. J. Phys. 68, 56-60 (2000).
24. The constellations were obtained from the free planetarium software Stellarium, 〈www.stellarium.org〉.
25. J. C. Mather et al., "Calibrator design for the COBE far-infrared absolute spectrophotometer (FIRAS)," Astrophys. J. 512, 511-520 (1999).
26. For more information on x rays and γ rays we refer the reader to the following missions. X-ray: ROSAT, 〈wave.xray.mpe.mpg.de/ rosat〉, Chandra, 〈chandra.harvard.edu〉, XMM-Newton, 〈sci.esa.int〉; γ-ray: INTEGRAL 〈www.esa.int/esaMI/ Integral/〉.
27. The rotational velocity of a star in a galaxy does not follow Newton's law, but can be read from the rotation curve of the galaxy. See Ref. 28 for the outer rotation curve of the Milky Way.
28. J. Brand and L. Blitz, "The velocity field of the outer galaxy," Astron. Astrophys. 275, 67-90 (1993).
29. The goal of the GAIA mission is to collect high-precision astrometric data for the brightest one billion objects. A detailed description of the ESA Science missions is available at 〈www.esa.int/esaSC/〉.
30. P. Anderson, Tau Zero (Orion Publishing Group, London, 2006).
31. Data are taken from 〈map.gsfc.nasa.gov〉.
32. Single-precision floating-point numbers are stored in a 32 bit word, whereas double-precision ones are stored in a 64 bit word. The word itself is composed of a 23(52) bit mantissa, a 8(11) bit exponent, and one bit for the sign. More information can be found in the IEEE standard for binary floating-point arithmetic for microprocessor systems (ANSI/IEEE Std 754-1985).
33. This fact is known as the tristimulus theory. For more information see, for example, Foley (Ref. 35).
34. Color matching functions can be found at the Institute of Ophthalmology, 〈cvrl.ioo.ucl.ac.uk/basicindex.htm〉.
35. J. D. Foley, A. Van Dam, S. K. Feiner, and J. F. Hughes, Computer Graphics: Principles and Practice (Addison-Wesley, 1991).
36. w) that serve to define the color white.
37. An excellent science fiction book on this topic is given in Ref. 30 which might be a useful basis for discussion.