Outline of this series:
I. The Impetus for This Series
II. Special Relativity: The Standard Explanation
IV. A Further Look at Simultaneity
V. The Velocity Conundrum
VII. Further Thoughts on the Meaning of Spacetime and the Validity of STR
VIII. Mettenheim on Einstein’s Relativity
Bibliography (appended to each part of the series)
THE VELOCITY CONUNDRUM
Einstein’s special theory of relativity (STR) depends on two postulates:
The laws of physics are invariant (i.e. identical) in all inertial systems (non-accelerating frames of reference).
The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source.
In sum, the speed of light is the same for all observers, regardless of their relative motion and regardless of the motion of the source of light.
The postulates of STR purportedly negate Galilei–Newton relativity, where time is the same in all frames of reference (i.e., there is absolute time) and spatial differences between frames of reference depend simply on their relative velocity (i.e., there is absolute space). In mathematical terms:
t’ = t ,
x’ = x – vt ,
y’ = y ,
z’ = z ,
where t’, x’, y’, and z’ denote the time and position of a frame of reference S’ that is in motion along the x-axis of frame of reference S, the coordinates of which are denoted by t, x, y, and z, and where v is the velocity of S’ relative to S (in the x-direction); and where x’ = x at x = 0 .
Graphically:
Source: Susskind Lectures, Special Relativity, Lecture 1 (Galilean relativity).
(It would seem that x’ = x + vt , but x’ = x – vt because the x’ axis is represented by the vector τ .)
The “kinematical” part of STR is summarized in the equations of the Lorentz transformation:
[T]he Lorentz transformation … relates the coordinates used by one observer to coordinates used by another in uniform relative motion with respect to the first.
Assume that the first observer uses coordinates labeled t, x, y, and z, while the second observer uses coordinates labeled t’, x’, y’, and z’. Now suppose that the first observer sees the second moving in the x-direction at a velocity v. And suppose that the observers’ coordinate axes are parallel and that they have the same origin. Then the Lorentz transformation expresses how the coordinates are related:
where c is the speed of light.
Here is a graph of the relationship between the velocity of a body (frame of reference) and the distance it travels in, say, a year of proper time (the time that would elapse in a “stationary” body):
This graph is based on Epstein (2000), specifically, figures 5-6 through 5-10 of and the accompanying discussion on pages 79-85. The graph implies that an object moving at the speed of light would use no time; it would move only along the spaced-used axis, at a constant time value of zero. Here is Epstein’s explanation:
The object does not age at all. The object has the maximum speed through space, the speed of light. Its speed through time is zero. It is stationary in time. “Right now is forever….”
This seems inconsistent with the fact that light has a finite velocity. Even an object that moves at the speed of light would take some amount of time to go any distance, which is what the graph means.
The “stationary” body (an abstract point) would use no space; it would move only along the time-used axis, at a constant space value of zero. The designation of a stationary body is problematic, as discussed below.
In any event, the curve depicts the intermediate relationships between velocity (and space used) and proper time used. For example, a body that moves at 0.7c would travel 0.7 light-year in a year, and would age only a bit more than 0.7 year.
The relationship between v and t’ can be computed as follows:
t’ = (1 – v2)1/2 ,
where t’ is the elapsed time (proper time), as perceived by S, when S’ is traveling at v (expressed as a fraction of c).
The faster a body moves the slower it ages relative to an observer in a “stationary” reference frame; that is, a clock in the moving body is seen by the “stationary” observer to advance at a slower rate than his own, identically constructed clock. (This is a reciprocal relationship, which I address elsewhere in this page.)
The aging rate is determined by the first of the four Lorentz equations given above, where t’ refers to the length of a time interval in S’ relative to the length of a time interval in S . For t > 0 and x = 0 (i.e.. a “stationary” body), t’ is always greater than t ; that is, a clock that ticks every second in S would tick every 1+ seconds in S’ . The greater the interval, the slower the clock runs, which is why, according to STR, time slows as velocity increases.
This is so — according to STR — because of the inextricable relationship between space and time, which is really a unitary four-dimensional “thing” called spacetime. Reverting to the usual conceptions of space and time as separate entities, we are all moving through time, whether or not we are moving through space. (STR is limited to inertial movement on a hypothetical plane, and does not address the gravitational movement of a body or of Earth, the Sun’s solar system, and the Milky Way galaxy.) Some of the time involved in moving through space would have been used anyway, just by standing still. So the time required to move through space is reduced by some amount. The amount depends on how fast a body is moving; the faster it is moving, the greater the reduction in the amount of time required to travel a given distance.
Here is a rough analogy: I can go downstream without paddling my canoe, but I can only go as fast as the current will take me. If I paddle, It will take me less time to travel a given distance. The faster I paddle, the less time it will take. A plot of the time required to go a given distance at various paddling rates would resemble the graph above, though it is impossible to reduce the travel time to zero, and the mathematical relationship between travel time and paddling rate isn’t the same as that represented by the graph.
Putting aside time, which I will address below, STR hinges on the meaning and measurement of velocity. In fact, v is assumed to have an absolute and determinate value for a “moving” frame of reference. Otherwise, the equations of STR are meaningless; that is, if v simply represented the relative velocity of two bodies (both in motion), the solutions for a given body would vary with the body with which it is being compared.
Accordingly, STR assumes that all comparisons are made with a hypothetical stationary body with no velocity — a “fixed point” in the universe, if you will. But the “fixed point” would have to be an object around which the universe revolves, and there is no such body. It is therefore impossible to compute the absolute velocity of a body, which STR requires.
And if one can’t determine the velocity of a body, the Lorentz transformation is really meaningless. The values of t’ and x’ are indeterminate — unless one reverts to Galilei-Newton relativity, which assumes, in effect, that every part of the universe is comprised in a unitary frame of reference, with absolute space and absolute time (and, therefore, absolute measures of velocity).
I will end here with two thoughts. The first is that Phipps (2012) may well have resolved the internal contradictions of STR. Near the end of the book (page 306) Phipps summarizes with this:
By means of CT [collective time, which is not the same as Einsteinian time], the science of mechanics simplifies formally to its nineteenth-century canonical forms … ; … three-space geometry reverts to Euclidean … and … absoluteness of distant simultaneity … [is] restored.
(Regarding the absoluteness of simultaneity, see Part III.)
More speculatively, it seems to me that there might be a physical phenomenon which serves as the unitary frame of reference implicit in Galilei-Newton relativity: the Higgs field, the lattice of Higgs bosons which is believed to permeate the universe. But this is only a preliminary thought.
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.