Simultaneity: An Improved Definition of Simultaneous Events

by David Bryan Wallace
Cape Coral, Florida, USA
Copyright © 2013-04-29
Edited 2014-03-07, 2014-09-01, 2018-01-13

The definition of “simultaneous events” Albert Einstein included in his special theory of relativity (1905)[1] is no longer the best possible definition. After you read this essay, you too will see that technological advances since 1905, make a better operational definition feasible.When Einstein published his special theory of relativity, the standard for time was rotation of the earth. With such a unique standard of time, all that is needed to define time elsewhere is a method of synchronization at a distance. Accordingly, Einstein offered a definition of simultaneous events by which to synchronize clocks at a distance.

Intuitively, our notion of synchronization has to do with equivalence of time, and equivalence is characterized by three properties: symmetry (if the time of event X equals the time of event Y then the time of event Y equals the time of event X), transitivity (if the time of event X equals the time of event Y and the time of event Y equals the time of event Z then the time of event X equals the time of event Z), and reflexivity (the time of event X equals itself). After stating his definition of simultaneous events, Einstein affirmed the intention of equivalence by stating the first two properties, symmetry and transitivity, as follows in translation.

… we establish by definition that the “time” required by light to travel from A to B equals the “time” it requires to travel from B to A. Let a ray of light start at the “A time" tA from A towards B, let it at the “B time” tB be reflected at B in the direction of A, and arrive again at A at the “A time” t'A. In accordance with definition the two clocks synchronize if tB − tA = t'A − tB. We assume that this definition of synchronism is free from contradictions, and possible for any number of points; and that the following relations are universally valid:--
(1)[symmetry] If the clock at B synchronizes with the clock at A, the clock at A synchronizes with the clock at B.
(2)[transitivity] If the clock at A synchronizes with the clock at B and also with the clock at C, the clocks at B and C also synchronize with each other.
I suppose that Einstein considered reflexivity, “the clock at A is synchronized with itself,” too obvious to be worth stating. It should be noted that the one way speed of light relative to the observer at A is rendered, by this definition, equal in every direction.

In the next few paragraphs, however, Einstein demonstrates that when the definition is applied in different frames of reference this assumption is false, in particular, for clocks in motion relative to one another. He derives this discrepancy, not from properties of clocks (about which he adduces no evidence), but from the application of his definition in multiple frames of reference. He concludes that strict equivalence of time is an illusory notion and that synchronization is necessarily frame of reference dependent. Indiscriminate use of his definition creates that appearance; but that is a reason to reject application of his definition in multiple frames of reference, not a reason to reject equivalence of time.

A consistent equivalence relation for time is not an illusory notion. Einstein's definition applied in a single frame of reference succeeds in establishing an equivalence relation for time that could be shared by all frames of reference, but if the definition is applied in different frames of reference it gives us a different equivalence relation for each inertial frame of reference. The choice we make is between, on the one hand, a universally valid equivalence relation based on a single frame of reference and, on the other hand, infinitely many conflicting equivalence relations based respectively on infinitely many frames of reference. The essential problem is to find a universal standard frame of reference on which to base a unique universally acceptable equivalence relation for time.

An interval of time is marked by an event at the beginning and an event at the end. Consider two billiard balls that tap together twice, each tap being an event. These two events can be a considerable time apart. They mark the beginning and end of a time interval. If with each billiard ball we associate a clock we gain the possibility that the two clocks will not agree as to the measure of that time interval. Yet, it is clear that there is only one time interval. A disturbance of the time rate of a clock is not a disturbance of time itself.

Since Einstein published his special relativity theory the standard for time has changed. No longer relying on Earth rotation, the current SI definition of time is based on oscillations characteristic of a cesium atom at rest. Now we must be concerned with what we mean by “at rest.” If the astronomical proof for constancy of light speed proposed by Daniel Frost Comstock (1910)[2] and executed by Willem de Sitter (1913)[3] is sound, then the basis for a universal standard absolute rest frame of reference exists. The proof in question holds that light from a distant binary star, one of a binary pair, takes the same time to reach Earth regardless of whether the star in its orbit is approaching Earth or receding from Earth. The speed of light is constant, not in the frame of reference of its production (nor, by analogous arguments, in the frame of the observer), but it is constant and thereby reveals the existence of a frame of reference relative to which it is constant. This is a refutation of Ritz theory and of ballistic theories of light in general and also of the constancy of relative light speed declared by Einstein. If we apply, as we are accustomed to do, the principle of symmetry of space, then a unique frame of absolute rest is defined by the property that light propagates at the same speed in every direction. Only in the unique frame of absolute rest does the relationship tB − tA = t'A − tB assure the synchronization that defines our unique standard of time equivalence.

Measuring one-way speed of light was impossible in Einstein's day. Even now it is a daunting task, but possible. Like the longitude determination problem[4] that existed before the late eighteenth century because clocks did not keep accurate time on a rolling ship, the solution lies not in avoiding dependence on clocks, but in improvement of clocks so that they keep sufficiently accurate time.

Measuring one-way speed of light requires stable calibration of atomic clocks. Atomic clocks must be designed to compensate for all disturbances from the “at rest” condition. An analytical approach considers how the workings of an atomic clock may be affected. We can predict some influences on the time-keeping process. For example, if the design of an atomic clock involves one way light transmission from emitting atoms to a detector, the effect of movement may be dependent on clock orientation. An empirical approach uses experiments with clocks to observe the effects. For example, juxtaposed clocks can be synchronized before moving one or both in a controlled way that ultimately brings them back together for comparison and measurement of any effect. Using results of such studies, clocks compensated for movement might be devised. Then one way speed of light might be measured between two clocks which, after being synchronized while juxtaposed, have been moved some distance apart.

Because the speed of light appears to be a universal constant, it seems appropriate that speed of light has been adopted as the international standard for speed. Also, the international atomic standard for time may remain the base standard of time. It should be noted that the current time standard is the oscillation rate from atoms at rest. Length and distance thus become derived units. The Michelson-Morley experiment[5] demonstrated that length of a solid object is influenced by the object's speed, so we must doubt that a solid material length standard is appropriate to measure spatial distance. Also, the Michelson-Morley experiment and experience with interferometric calibration[6] of gage blocks demonstrate that round trip light time (interferometric) measurement seems to be equivalent to measurement with a solid material standard. From multiple measurements of one way light transit time, in opposing directions, the calculation of spatial distance and absolute velocity is possible. In another essay I present an algorithm to accomplish this calculation.

The equivalence of interferometric and solid length standards is the only remarkable discovery due to the Michelson-Morley experiment. The experiment did not compare clocks or even involve clocks; there was only one light source. The coordinate transformations derived by Einstein (1905)[1] may be correct (almost) but differ from the Lorentz[7] transformations; alas, after deriving his set of transformations, Einstein mistook them (apparently) to be the same as the Lorentz transformations to which he turns after the end of the derivation with the words “Substituting for x' its value, we obtain …”, [note: the substitution fails to achieve the claimed effect.] Einstein's coordinate transformations may be correct, except that he saw the velocity variable as relative velocity and considered the reference frames interchangable. I favor the transformations below. I would appreciate hearing from anyone who recognizes my transformations, including the identity transformation for time: τ = t, as having been described by anyone else than me prior to my publishing them; see Wikipedia: History of Lorentz transformations.

Wallace coordinate transformations (2013)
τ = txi = `/`(`*`(`+`(x, `-`(`*`(beta, `*`(c, `*`(t)))))), `*`(`+`(1, `-`(`*`(`^`(beta, 2)))))),  eta = `/`(`*`(y), `*`(sqrt(`+`(1, `-`(`*`(`^`(beta, 2))))))),  Zeta = `/`(`*`(z), `*`(sqrt(`+`(1, `-`(`*`(`^`(beta, 2)))))))
where β is absolute velocity as a fraction of light speed.

[1] Einstein, Albert, “Zur Elektrodynamik bewegter Körper,” Annalen der Physik 17:891, 1905, “On the Electrodynamics of Moving Bodies,” online in English

[2] Comstock, D.F., “A Neglected Type of Relativity,” Physical Review, February 1910, 30 (2): 267 (1910), online

[3] de Sitter, Willem, “Ein astronomischer Beweis für die Konstanz der Lichgeshwindigkeit," Physik. Zeitschr, 14, 429 (1913). “A proof of the constancy of the velocity of light,” Proceedings of the Royal Netherlands Academy of Arts and Sciences 15 (2): 1297-1298, online in English

[4] Sobel, Dava, Longitude: The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time, Walker Publishing (1995) ISBN 0-8027-1312-2.

[5] Michelson, Albert Abraham & Morley, Edward Williams (1887). “On the Relative Motion of the Earth and the Luminiferous Ether.” American Journal of Science 34: 333-345. online

[6] Doiron, Ted and Beers, John, The Gauge Block Handbook, NIST (1995), online

[7] Lorentz, Hendrik Antoon, “De relatieve beweging van de aarde en den aether,” Zittingsverlag Akad. V. Wet. 1: 74-79, (1892), online in English

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