Outline of this series:
I. The Impetus for This Series
II. Special Relativity: The Standard Explanation
IV. A Further Look at Simultaneity
VII. Further Thoughts on the Meaning of Spacetime and the Validity of STR
VIII. Mettenheim on Einstein’s Relativity
IX. Spacetime Isn’t Time
Bibliography (appended to each part of the series)
There’s some sloppy nomenclature in the special theory of relativity (STR). (See, for example, “Part VIII. Mettenheim on Einstein’s Relativity.) One significant error is the conflation of time and spacetime. As a result, STR claims wrongly that an object’s “time” depends on the inertial velocity of the object. (See especially the thought experiment described in the footnote to the linked essay.) In fact, time is invariant between objects, though clocks may run at various rates for reasons other than their inertial velocity. The purpose of this essay is to trace the source of the confusion.
Consider a rigid object of length L (e.g., Einstein’s railway embankment), which (arbitrarily) is situated horizontally. Call x0 the space coordinate of the left end of L, and x1 the space coordinate of the right end of L. The spatial distance between x0 and x1 can be expressed as L = ct , where t is the time it takes a particle moving at constant velocity to traverse L. Expressing c as light-years/year and t as years we get L in light-years (the “year” dimensions cancel). Thus if a particle traverses L at a constant velocity of 0.5c (1/2 light-year/year), it takes 2 years for the traversal, and L = 1 light-year in length.
Now, visualize a graph with a vertical axis representing time (t) in years and a horizontal axis representing distance (x) in light-years. From its initial position at t0 = 0 , x0= 0 , and x1 = 1 L moves “vertically” through time while the particle traverses it. The clocks at x0 and x1 are synchronized; both will show that 2 years elapse during the particle’s traversal of L. At the end of 2 years, these are the coordinates at each end of L: t1 = 2 , x0= 0 , and t1 = 2 , x1 = 1 .
In terms of the graph, everything that “moves” only in a vertical direction (e.g., L) moves only in time. (If nothing can move instantly through space, nothing can move horizontally along or parallel to the x axis.) Spacetime (s) is represented by diagonal movements through time and space, such as the movement of the particle from t0 = 0 , x0= 0 to t1 = 2 , x1 = 1 . The amount of spacetime “used” by the particle can be determined by applying the Pythagorean formula:
s = √[22 + 12] = √5 = 2.24 .
The first term in the square root is the “vertical” leg of the right triangle formed by the “ascent” of x0 through time, namely, the 2 years that elapse at x0 while the particle moves across L. The second term represents the “horizontal” distance between x0 and x1 traversed by the particle, namely, 1 light-year. The hypotenuse that connects t0 = 0 , x0= 0 and t1 = 2 , x1 = 1 represents s, where s is neither t nor x, but a compound of the two.
I must underscore the point: spacetime (s) is not time (t). Which brings me to the Lorentz transformation. It converts t into t’, the time “belonging” to an object that is moving relative to a “stationary” frame of reference. But t’ isn’t time — it’s another compound of space and time. Given the values that I chose for my example, the Lorentz transformation yields t’ = 2.31 . This is remarkably close to s = 2.24 , but the resemblance is purely coincidental because the underlying mathematical formulae are different. Moreover, by the standard interpretation of the Lorentz transformation, the elapsed time for the particle would be the reciprocal of 2.31: 1/2.31 = 0.43t . That is, the particle would somehow have “aged” less than the object (L) it was traversing.
But that’s a meaningless interpretation because t and t’ — like t and s — are different things.
BIBLIOGRAPHY
Online courses in special relativity
Lecture 1 of “Special Relativity”, Stanford University
All lectures of “Special Relativity”, Khan Academy
All lectures of “Understanding Einstein: The Special Theory of Relativity”, Standford University
Selected books, articles, and posts about special relativity
Barnett, Lincoln. The Universe and Dr. Einstein. New York: Time Incorporated, 1962.
Bondi, Hermann. Relativity and Common Sense: A New Approach to Einstein. New York: Doubleday & Company, 1946.
Buenker, Robert J. “Commentary on the Work of Thomas E. Phipps, Jr. (1925-2016)”. 2016.
Einstein, Albert. “On the Electrodynamics of Moving Bodies”. Annalen der Physik, 322 (10), 891–921 (1905).
———. Relativity: The Special and General Theory. New York: Henry Holt, 1920.
Epstein, Lewis Carroll. Relativity Visualized. San Francisco: Insight Press, 2000.
Hall, A.D. “Lensing by Refraction…Not Gravity?“. The Daily Plasma, December 23, 2015.
Marrett, Doug. “The Sagnac Effect: Does It Contradict Relativity?“. Conspiracy of Light, 2012.
———. “Did the Hafele and Keating Experiment Prove Einstein Wrong?“. Conspiracy of Light, 2013.
von Mettenheim, Christoph. Popper versus Einstein. Heidelberg: Mohr Siebeck, 1998.
———. Einstein, Popper and the Crisis of Theoretical Physics (Introduction: The Issue at Stake). Hamburg: Tredition GmhH, 2015.
Noyes, H. Pierre. “Preface to Heretical Verities [by Thomas E. Phipps Jr.]”. Stanford: Stanford Linear Accelerator Center, Stanford University, June 1986.
Phipps, Thomas E. Jr. “On Hertz’s Invariant Form of Maxwell’s Equations”. Physics Essays, Vol. 6, No. 2 (1993).
———. Old Physics for New: A Worldview Alternative to Einstein’s Relativity Theory. Montreal: Apeiron, first edition, 2006.
———. Old Physics for New: A Worldview Alternative to Einstein’s Relativity Theory. Montreal: Apeiron, second edition, 2012 (The late Dr. Phipps — Ph.D. in nuclear physics, Harvard University, 1950 — styled himself a dissident from STR, for reasons that he spells out carefully and exhaustively in the book.)
Rudolf v. B. Rucker. Geometry, Relativity, and the Fourth Dimension. New York: Dover Publications, 1977.