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The Physics of Time Travel: Part I

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Time Machine Tales

Part of the book series: Science and Fiction ((SCIFICT))


Before we start talking about the physics of time travel, let me say a few more words on time itself, in a way slightly less metaphysical that was the discussion in the previous chapter (which is why I’m writing this here, in a chapter with an increased emphasis on the analytical). When we speak of journeying to either the future or the past, we are implicitly making a distinction in the direction of the time traveler’s trip. But does time actually have a direction? Is there an arrow that points the way? The answer seems obvious: of course time has a direction. After all, everybody ‘knows’ it flows from past to future. There is a curious language problem here, however, because we also like to say the present recedes into the past, which implies a ‘flow’ in the opposite direction, from future to past. Well, despite this snarled syntax, can we at least distinguish past from future, whichever way time flows?

“… within forty-eight hours we had invented, designed, and assembled a chronomobile. I won’t weary you with the details, save to remark that it operated by transposing the seventh and eleventh dimensions in a hole in space, thus creating an inverse ether-vortex and standing the space-time continuum on its head.”

—almost certainly not the way to build a time machine

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  1. 1.

    L. Sprague de Camp, “Some Curious Effects of Time Travel,” in Analog Readers’ Choice, Dial 1981.

  2. 2.

    See note 54 in Chapter 2.

  3. 3.

    R. Gale, “Some Metaphysical Statements About Time,” Journal of Philosophy, April 25, 1963, pp. 225-237. For many, this analogy may well bring to mind a pile of baloney rather than one of salami (and I think this was Gale’s intention).

  4. 4.

    J. J. C. Smart, “The Temporal Asymmetry of the World,” Analysis, March 1954, pp. 79-83.

  5. 5.

    Two events A and B are non-causally related if their separation in spacetime is such that a particle would have to travel at a superluminal speed (faster than light) to go from A to B. We’ll discuss the physics of causally related events later in this chapter.

  6. 6.

    Gödel clearly states this in his 1949 philosophical essay (note 15 in the Introduction) concerning his 1949 technical paper (note 11 in the Introduction).

  7. 7.

    This figure is based on the interpretation of Gödel’s reasoning as presented by the philosopher Palle Yourgrau’s 1991 book The Disappearance of Time: Kurt Gödel and the idealistic tradition in philosophy (Cambridge), which was expanded and reprinted a few years later under the new title Gödel Meets Einstein: time travel in the Gödel universe, Open Court 1999. Yourgrau later wrote a less technical version: A World Without Time: the forgotten legacy of Gödel and Einstein, Basic Books 2005.

  8. 8.

    See, for example, S. Savitt, “Time Travel and Becoming,” The Monist, July 2005, pp. 413-422.

  9. 9.

    P. L. Csonka, “Advanced Effects in Particle Physics,” Physical Review, April 1969, pp. 1266-1281.

  10. 10.

    J. J. C. Smart, “The River of Time,” Mind, October 1949, pp. 483-494.

  11. 11.

    And how about this image of time: Time is a snowball, with the center marking the beginning of the past, with ever new ‘presents’ accreting on the ever increasing surface as the snowball rolls down the hill of history!

  12. 12.

    E. A. Manley and W. Thode, “The Time Annihilator,” Wonder Stories, November 1930. This is the same magazine that, months later, finally published Bell.

  13. 13.

    J. Wyndham, “Wanderers of Time,” Wonder Stories, March 1933. Notice again, that we have the same magazine (whose editor must have had a particular fancy for such tales).

  14. 14.

    M. Black, “The ‘Direction’ of Time,” Analysis, January 1959, pp. 54-63.

  15. 15.

    D. Zeilicovici, “Temporal Becoming Minus the Moving-Now,” Nous, September 1989, pp. 505-524.

  16. 16.

    C. W. Webb, “Could Time Flow? If So, How Fast?” Journal of Philosophy, May 1960, pp. 357-365.

  17. 17.

    I’ve taken this quotation from A. Grϋnbaum, “Is There a ‘Flow’ of Time or Temporal Becoming?” in Philosophical Problems of Space and Time, Knopf 1963.

  18. 18.

    L. R. Baker, “Temporal Becoming: The Argument from Physics,” Philosophical Forum, Spring 1975, pp. 218-236.

  19. 19.

    R. Ray, “Today’s Yesterday,” Wonder Stories, January 1934.

  20. 20.

    A. and P. Eisentein, “The Trouble With the Past,” in New Dimensions 1 (R. Silverberg, editor), Doubleday 1971.

  21. 21.

    Quoted from The Philosophy of Rudolp Carnap (P. A. Schlipp, editor), The Library of Living Philosophers, Open Court 1963, pp. 37-38. For a view contrary to Einstein’s, from another physicist, see K. B. M. Nor, “A Topological Explanation for Three Properties of Time,” Il Nuovo Cimento B, January 1992, pp. 65-70, which claims to develop a geometrical explanation for the flow of time, and so (says Nor) there is an objective, mathematical reality to the ‘moving now.’

  22. 22.

    A science fiction character pretty accurately sums-up what a modern physicist would tell you today, in L. Eisenberg’s story “The Time of His Life,” The Magazine of Fantasy & Science Fiction, April 1968.

  23. 23.

    J. P. Cullerne, “Free Will and the Resolution of Time Travel Paradoxes,” Contemporary Physics, July-August 2001, pp. 243-245.

  24. 24.

    J. J. Sylvester, “A Plea for the Mathematician,” Nature, December 30, 1869, pp. 237-239.

  25. 25.

    H. Margenau, “Can Time Flow Backwards?” Philosophy of Science, April 1954, pp. 79-92.

  26. 26.

    S. S. Schweber, “Feynman and the Visualization of Space-Time Processes,” Reviews of Modern Physics, April 1986, pp. 449-508.

  27. 27.

    G. N. Lewis, “The Symmetry of Time in Physics,” Science, June 6, 1930, pp. 569-577.

  28. 28.

    O. Penrose and I. C. Percival, “The Direction of Time,” Proceedings of the Physical Society (London), March 1962, pp. 605-616.

  29. 29.

    H. Mehlberg, “Philosophical Aspects of Physical Time,” in Basic Issues in the Philosophy of Time (E. Freeman and W. Sellars, editors), Open Court 1971.

  30. 30.

    P. A. M. Dirac, “Forms of Relativistic Dynamics,” Reviews of Modern Physics, July 1949, pp. 392-399.

  31. 31.

    A. Boucher, “The Chronokinesis of Jonathan Hull,” Astounding Science Fiction, June 1946.

  32. 32.

    A. S. Eddington, The Nature of the Physical World, Macmillan 1929.

  33. 33.

    The H-theorem was a direct continuation of the work by the Scottish physicist James Clerk Maxwell (1831-1879) on the statistical properties of gas molecules (determining the probability density function of the molecules’ speeds). In 1866 Maxwell found this function for the particular case of thermodynamic equilibrium. In 1872 Boltzmann found the differential-integral equation the function satisfies in general, even if the condition of thermodynamic equilibrium doesn’t hold. From this Boltzmann was able to define a quantity H that he showed evolves in time such that solution to his differential-integral equation approaches Maxwell’s equilibrium solution. The H-theorem says that H always decreases in systems not in equilibrium and is at a minimum in systems in equilibrium.

  34. 34.

    J. A. Wheeler, “Frontiers of Time,” in Problems in the Foundations of Physics (G. T. diFrancia, editor), Proceedings of the International School of Physics (Course 72), North-Holland 1979. See also W. J. Cocke, “Statistical Time Symmetry and Two-Time Boundary Conditions in Physics and Cosmology,” Physical Review, August 25, 1967, pp. 1165-1170.

  35. 35.

    I used MATLAB, and you can find the code — gasclock.m — in Appendix C, written in such a low-level way as to be virtually 100% transferable to just about any of the popular scientific programming languages, and easily executed on an inexpensive laptop. Note that there are no K-mesons in the code (!) and so, as stated in Chapter 1, they aren’t responsible for the uni-directional time behavior depicted in the figure.

  36. 36.

    The Austrian-British philosopher Karl Popper (1902-1994) called Boltzmann’s willingness to consider the possibility that different regions of the universe could have different directions of time “staggering in its boldness and beauty,” but when on to say that Boltzmann must be wrong because “it brands unidirectional change an illusion [which] makes the catastrophe of Hiroshima an illusion.” That is an emotional argument, of course, and although one of great power, I fail to see how it is related to physics. See Volume 1 of The Philosophy of Karl Popper (P. A. Schlipp, editor), Open Court 1974, pp. 127-128.

  37. 37.

    For more on Boltzmann’s views on entropy, see the end of his letter “On Certain Questions of the Theory of Gases,” Nature, February 28, 1895, pp. 413-415.

  38. 38.

    C. F. Hall, “The Man Who Lived Backwards,” Tales of Wonder, Summer 1938. The modern classic of a time-reversed world is Philip K. Dick’s 1967 novel Counter-Clock World. We’ll encounter another time-reversed world again in Chapter 4.

  39. 39.

    M. J. Breuer, “The Time Valve,” Wonder Stories, July 1930.

  40. 40.

    F. B. Long, “Temporary Warp,” Astounding Stories, August 1937.

  41. 41.

    R. M. Farley, “Time for Sale,” Amazing Stories, August 1938.

  42. 42.

    Also citing Eddington was a tale by D. W. O’Brien, “The Man Who Lived Next Week,” Amazing Stories, March 1941, which uses entropy to explain time travel. This curious story has the traveler arriving in the future with his clothing aged, which later ‘de-ages’ when the return trip is made!

  43. 43.

    P. Cross, “Prisoner of Time,” Super Science Stories, May 1942.

  44. 44.

    P. Anderson, “Time Heals,” Astounding Science Fiction, October 1949.

  45. 45.

    In, for example, his story of the tragic end of a geologist fifty million years in the past: “Time’s Arrow,” Science Fantasy, Summer 1950.

  46. 46.

    “What We Learned from This Morning’s Newspaper,” Infinity 4, November 1972.

  47. 47.

    J. M. Blatt, “Time Reversal,” Scientific American, August 1956.

  48. 48.

    The googol is a gigantic number, far greater than the number of raindrops that have fallen on the Earth during its entire history. And the googolplex is light years beyond that.

  49. 49.

    R. B. Braithwaite, “Professor Eddington’s Gifford Lectures,” Mind, October 1929, pp. 409-435.

  50. 50.

    Pulp science fiction writers, of course, were not discouraged by such calculations, as they depended on the certainty of recurrence over infinite time. See, for example, S. G. Weinbaum, “The Circle of Zero,” Thrilling Wonder Stories, August 1936, and L. D. Gunn, “The Time Twin,” Thrilling Wonder Stories, August 1939.

  51. 51.

    R. Small, “Incommensurability and Recurrence: From Oresme to Simmel,” Journal of the History of Ideas, January-March 1991, pp. 121-137.

  52. 52.

    J. Krueger, “Nietzschean Recurrence as a Cosmological Hypothesis,” Journal of the History of Philosophy, October 1978, pp. 435-444.

  53. 53.

    See “Human Repetends” by Marcus Clarke (1846-1881), a story originally published in 1872 and reprinted Australian Science Fiction (V. Ikin, editor), Academy Chicago 1984.

  54. 54.

    For more on this point, see D. W. Theobald, “On the Recurrence of Things Past,” Mind, January 1976, pp. 107-111.

  55. 55.

    C. F. Ksanda, “Forever Is Today,” Thrilling Wonder Stories, Summer 1946.

  56. 56.

    A. S. Eddington, “The End of the World,” in New Pathways in Science, Macmillan 1935.

  57. 57.

    F. J. Tipler, “General Relativity and the Eternal Return,” in Essays in General Relativity (F. J. Tipler, editor), Academic Press 1980.

  58. 58.

    It has been estimated that over the next 10116 years the entropy of the universe will increase by a factor in excess of 1014. See S. Frautschi, “Entropy in an Expanding Universe,” Science, August 13, 1982, pp. 593-599.

  59. 59.

    See note 54 in Chapter 2.

  60. 60.

    P. W. Bridgeman, Reflections of a Physicist, Philosophical Library 1955, p. 251.

  61. 61.

    Taken from K. G. Denbigh, “The Many Faces of Irreversibility,” British Journal for the Philosophy of Science, December 1989, pp. 501-518.

  62. 62.

    T. Gold, “The Arrow of Time,” American Journal of Physics, June 1962, pp. 403-410.

  63. 63.

    P. C. W. Davies and J. Twamley, “Time-Symmetric Cosmology and the Opacity of the Future Light Cone,” Classical and Quantum Gravity, May 1993, pp. 931-945.

  64. 64.

    S. Hawking, “Arrow of Time in Cosmology,” Physical Review D, November 15, 1985, pp. 2489-2495. See also the next paper in the same journal, D. N. Page, “Will Entropy Decrease if the Universe Recollapses?,” pp. 2496-2499.

  65. 65.

    For why he abandoned that claim, see S. Hawking et al., “Origin of Time Asymmetry,” Physical Review D, June 15, 1993, pp. 5342-5356.

  66. 66.

    S. W. Hawking, “The No Boundary Condition and the Arrow of Time,’ in Physical Origins of Time Asymmetry (J. J. Halliwell et al., editors), Cambridge University Press 1994.

  67. 67.

    F. J. Bridge, “Via the Time Accelerator,” Amazing Stories, January 1931.

  68. 68.

    For v > c the time dilation formula says that time becomes imaginary, and this is one reason for claiming that v > c is not possible. The time dilation formula has been experimentally verified: see H. E. Ives and G. R. Stilwell, “An Experimental Study of the Rate of a Moving Atomic Clock,” Journal of the Optical Society of America, July 1938, pp. 215-226.

  69. 69.

    Named after the Dutch physicist H. A. Lorentz (1853-1928) and the Irish physicist G. F. FitzGerald (1851-1901).

  70. 70.

    J. H. Haggard, “Faster Than Light,” Wonder Stories, October 1930.

  71. 71.

    J. H. Haggard, “Relativity to the Rescue,” Amazing Stories, April 1935.

  72. 72.

    D. Wandrei, “A Race Through Time,” Astounding Stories, October 1933.

  73. 73.

    N. Schachner, “Reverse Universe,” Astounding Stories, June 1936.

  74. 74.

    L. R. Hubbard, “To the Stars,” Astounding Science Fiction, February and March 1950.

  75. 75.

    Resistance to this conclusion persisted for years. See, for example, the letter “Relativity and Radio-activity,” Nature, January 8, 1920, p. 468. The author of that letter wondered whether a clock based on radioactive decay might not somehow beat the ‘conspiracy’ of moving clocks running slow compared to stationary ones. And in a letter to Science (December 7, 1962, p. 1180), a reader objected to applying the laws of physics to biological systems, first asserting (incorrectly) that time dilation “has never been proved or disproved experimentally,” and then “there is no known causal means by which greatly increased velocity could alter, without destroying the very biochemical basis of the life process, the metabolic changes which are responsible for the aging process.”

  76. 76.

    Gravitational time dilation was experimentally observed in 1960, more than half a century after Einstein predicted it.

  77. 77.

    A science fiction use of both the red and the blue gravitational shifts appears in the novel by J. P. Hogan, Out of Time, Bantam 1993.

  78. 78.

    Poul Anderson, “Kyrie,” in The Road to Science Fiction (J. Gunn, editor), volume 3, New American Library 1979.

  79. 79.

    A mathematical discussion of how signals take forever (even though they are emitted in a finite time interval) to travel from the event horizon of a black hole to a distant receiver can be found in James B. Hartle, Gravity: an introduction to Einstein’s general relativity, Addison Wesley 2003, pp. 264-268.

  80. 80.

    D. Knight, “Extempore,” in Far Out, Simon and Schuster 1961. Similar words (“If only I had more mathematics”) were spoken by Einstein the day before he died — see Walter Isaacson, Einstein: his life and universe, Simon & Schuster 2007, p. 542.

  81. 81.

    This variation of mass with speed was experimentally observed in 1901.

  82. 82.

    This result was found by the French physicist Henri Poincaré (1854-1912) in June 1905, three months before the publication of Einstein’s special theory of relativity which also contains the result. And it was Poincaré who first stated (in 1904) that “no velocity can surpass that of light, any more than any temperature could fall below the zero absolute.”

  83. 83.

    Einstein’s postulate was experimentally confirmed in 1932.

  84. 84.

    The editor who bought “To the Stars” was the same editor editor of Astounding when Asimov’s letter appeared, and so he was certainly aware of it.

  85. 85.

    Poul Anderson, “The Little Monster,” in Science Fiction Adventure from WAY OUT (R. Elwood, editor), Whitman 1973.

  86. 86.

    R. Feynman, “The Theory of Positrons,” Physical Review, September 15, 1949, pp. 749-759. In a paper published the year before (“A Relativistic Cut-Off for Classical Electrodynamics,” Physical Review, October 1948, pp. 939-946), he wrote “This idea that positrons might be electrons with the proper time reversed was suggested to me by Professor J. A. Wheeler.” The identification of anti-matter with backward time travel occurred in science fiction (see note 124 in Chapter 1) almost simultaneously with Wheeler’s speculation.

  87. 87.

    See R. Weingard in note 114 in Chapter 1.

  88. 88.

    H. Price, “The Asymmetry of Radiation: Reinterpreting the Wheeler-Feynman Argument,” Foundations of Physics, August 1991, pp. 959-975.

  89. 89.

    J. Earman, “On Going Backward in Time,” Philosophy of Science, September 1967, pp. 211-222.

  90. 90.

    Feynman declares the view of a positron as a time traveling electron to be of value, for example, in his famous book Quantum Electrodynamics, W. A. Benjamin 1961, p. 68.

  91. 91.

    The “others” Feynman had in mind included, in particular, the eminent Swiss physicist Ernest C. G. Stϋkelberg (1905-1984), who in a 1942 article in the journal Helvetica Physica Acta also wrote of waves scattering backward in time.

  92. 92.

    M. Hogarth, “Predicting the Future in Relativistic Spacetimes,” Studies in the History and Philosophy of Science, December 1993, pp. 721-739.

  93. 93.

    What is meant by a spacetime being flat will be formalized when we get to spacetime metrics later in this section. The Minkowski spacetime of special relativity is a flat spacetime, has no tilted light cones, and as such does not support time travel to the past.

  94. 94.

    N. Schachner, “When the Future Dies,” Astounding Science Fiction, June 1939.

  95. 95.

    Mathematicians have defined the general properties of a distance function as follows: if A and B are any two points, and if d(A, B) is the distance between A and B , then (1) d(A, B) = d(B, A); (2) d(A, B) = 0 if and only if A = B; and (3) if C is any third point, then d(A, B) ≤ d(A, C) + d(C, B). The Pythagorean distance function possesses all three of these properties, but so do many other functions (for example, ds = |dx| + |dy|).

  96. 96.

    The \( \frac{\partial }{\partial x} \) and \( \frac{\partial }{\partial t} \) symbols denote the partial derivatives with respect to x and t (see any good calculus book to brush-up on this). The rest of this chapter will have some more math in it, involving derivatives and even an integral or two, but nothing beyond freshman calculus. I’ve included it mostly for those who would feel cheated without some math!

  97. 97.

    Writing in 1972, one famous physicist said of his first encounter with the metric of special relativity (Minkowski spacetime), “Now, when I saw that minus sign [in −(dt)2], it produced a tremendous effect on me. I immediately saw that here was something new.” See P. A. M. Dirac, “Recollections of an Exciting Era,” in History of Twentieth Century Physics (C. Weiner, editor), Academic Press 1977.

  98. 98.

    V. Rousseau, “The Atom Smasher,” Astounding Stories, May 1930.

  99. 99.

    The fifth dimension was introduced in the 1920s by the German physicist Theodor Kaluza (1885-1954), but just what the nature of this fifth dimension might be remains a mystery. A few years after Kaluza, the Swedish physicist Oscar Klein (1894-1977) speculated that it might be a spatial dimension curled-up in a tiny circular path, so tiny that we don’t notice it; the issue remains open. The idea of a fifth dimension appeared early in pulp science fiction, as in the January 1931 tale “The Fifth-Dimension Catapult,” (Astounding) by Murray Leinster.

  100. 100.

    An elementary, quite interesting discussion of the enormous computational complexity of the field equations is presented in Richard Pavelle and Paul S. Wang, “MACSYMA from F to G,” Journal of Symbolic Computation, March 1985, pp. 69-100.

  101. 101.

    Clifford almost surely found inspiration in this part of his work from the even earlier efforts of the German mathematician Bernhard Riemann (1826-1866). See, for example, Clifford’s translation of Riemann’s famous 1854 lecture “On the Hypotheses Which Lie at the Bases of Geometry,” Nature, May 1, 1873, pp. 14-17, and continued in the next issue (May 8, 1873, pp. 36-37).

  102. 102.

    Quoted in P. Kerszberg, “The Relativity of Rotation in the Early Foundations of General Relativity,” Studies in History and Philosophy of Science, March 1987, pp. 53-79.

  103. 103.

    R. W. Brehme and W. E. Moore, “Gravitational and Two-Dimensional Curved Surfaces,” American Journal of Physics, July 1969, pp. 683-692.

  104. 104.

    For wormhole spacetimes that are not asymptotically flat, see (for example) K. Narahara, et al., “Traversable Wormhole in the Expanding Universe,” Physics Letters B, September 29, 1994, pp. 319-323.

  105. 105.

    Note carefully that this is a purely spatial problem, with no time, and we are taking all of the metric coefficients as positive (unlike in the case of a spacetime metric).

  106. 106.

    A science fiction scientist babbles incoherent nonsense, not special relativity, about how to travel into the future, in a story by J. H. Haggard, “He Who Masters Time,” Thrilling Wonder Stories, February 1937.

  107. 107.

    R. S. Shankland, “Conversations with Albert Einstein,” American Journal of Physics, January 1963, pp. 47-57.

  108. 108.

    H. J. Hay, et al., “Measurement of the Red Shift in an Accelerated System Using the Mössbauer Effect in Fe57,” Physical Review Letters, February 15, 1960, pp. 165-166.

  109. 109.

    J. Bailey, et al., “Measurements of Relativistic Time Dilation for Positive and Negative Muons in a Circular Orbit,” Nature, July 29, 1977, pp. 301-305.

  110. 110.

    The twin paradox is hinted at in Einstein’s 1905 paper, but it is in a 1911 address to the International Congress of Philosophy in Bologna, by the French physicist Paul Langevin (1872-1946), that a human space traveler is first introduced (in a cannonball moving at near light-speed, an idea motivated by Langevin’s reading of Jules Verne’s 1872 novel From the Earth to the Moon). The writer Pierre Boulle proudly mentioned this contribution by his fellow Frenchman in the time travel story “Time Out of Mind” (you can find it in Boulle’s collection Time Out of Mind, Vanguard Press 1966).

  111. 111.

    W. P. Montague, “The Einstein Theory and a Possible Alternative,” The Philosophical Review, March 1924, pp. 143-170.

  112. 112.

    See C. H. Brans and D. R. Stewart, “Unaccelerated-Returning Twin Paradox in Flat Space-Time,” Physical Review D, September 15, 1973, pp. 1662-1666. For a similar treatment, this time by a mathematician, see Jeffrey R. Weeks, “The Twin Paradox in a Closed Universe,” American Mathematical Monthly, August-September 2001, pp. 585-589.

  113. 113.

    In Chapter 6 we’ll discuss the idea of traveling into the past by moving faster than light (superluminal motion). A treatment of such travel, in Bob’s cylindrical spacetime, is by S. K. Blau, “Would a Topology Change Allow Ms. Bright to Travel Backward in Time?” American Journal of Physics, March 1998, pp. 179-185, which answers that question in its last line: “Ms. Bright cannot [return] ‘the previous night’ and alter history,” a conclusion that no doubt met with Hawking’s approval. The ‘Ms. Bright’ in the title is the heroine of a 1923 limerick that you can find quoted in the first For Further Discussion of Chapter 6.

  114. 114.

    Originally appearing in the journal Science, under the title of “The General Limits of Space Travel,” von Hoerner’s analysis was reprinted in the classic anthology Interstellar Communication (A. G. W. Cameron, editor), W. A. Benjamin 1963. The arithmetic was, alas, just a bit sloppy (the final formulas, fortunately, are correct), and many of von Hoerner’s numerical evaluations are incorrect. Later, the British mathematician Leslie Marder cleaned-up the analysis in his beautiful little book on the twin paradox, Time and the Space-Traveler, George Allen & Unwin, Ltd. 1971.

  115. 115.

    Consider, for example, this remark by 1965 Nobel physics laureate Julian Schwinger (1918-1994) about the twin paradox in his 1986 book Einstein’s Legacy: “The observer on the spaceship … is not in uniform, unaccelerated motion … The special theory of relativity does not apply to such an accelerated observer.” Schwinger was wrong in this conclusion (see the next note).

  116. 116.

    As a physicist wrote on this point after Einstein’s death, “A good many physicists believe that [the twin] paradox can only be resolved by the general theory of relativity. … However, they are quite wrong. The twin effect … is one of special relativity.” See A. Schild, “The Clock Paradox in Relativity Theory,” American Mathematical Monthly, January 1959, pp. 1-18. Alfred Schild (1921-1977) was professor of physics at the University of Texas, and a recognized expert in the general theory.

  117. 117.

    In using the T, T formula, one has to be careful to use MKS units, that is, length and time measured in meters and seconds, respectively. Thus, a = 1 gee = 9.81 meters/second2 and c = 186 , 210  miles/second = 2.997 × 108 meters/second.

  118. 118.

    For a science fiction use of such a trip, see Robert Heinlein’s 1956 novel Time for the Stars.

  119. 119.

    A. Schild, “Time,” Texas Quarterly, Autumn 1960, pp. 42-62.

  120. 120.

    In his 1956 novel The Door Into Summer, the always ingenious Robert Heinlein used both ideas, with the cold-storage method of reaching the future combined with a true time machine to allow his hero to return to his ‘present’ (the future’s past).

  121. 121.

    A. Gallois, “Asymmetry in Attitudes About the Nature of Time,” Philosophical Studies, October 1994, pp. 51-59.

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Nahin, P.J. (2017). The Physics of Time Travel: Part I. In: Time Machine Tales. Science and Fiction. Springer, Cham.

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