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
Bibliography (appended to each part of the series)
SPECIAL RELATIVITY: THE STANDARD EXPLANATION
The principle of relativity predates Einstein. It says that all bodies are in relative motion, and that none of them holds a privileged place in the reckoning of time and space. The translation of time and space between inertial bodies (frames of reference) is straightforward in 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 . (I will later explain why it is x’ = x – vt instead of x’ = x + vt .)
Einstein introduced special relativity to account for the finite speed of light and its effects on time and space. Because light moves at the same, constant speed for all observers, an observer in one inertial frame of reference (S) will not necessarily see an event occur at the same time as an observer in another inertial frame of reference (S’). This, according to Einstein, destroys the idea of absolute time and, therefore, the idea of absolute simultaneity.
“Special Relativity” at Wikipedia gives a good account of the alternative devised by Einstein:
Reference frames play a crucial role in relativity theory. The term reference frame as used here is an observational perspective in space which is not undergoing any change in motion (acceleration), from which a position can be measured along 3 spatial axes. In addition, a reference frame has the ability to determine measurements of the time of events using a ‘clock’ (any reference device with uniform periodicity).
An event is an occurrence that can be assigned a single unique time and location in space relative to a reference frame: it is a “point” in spacetime. Since the speed of light is constant in relativity in each and every reference frame, pulses of light can be used to unambiguously measure distances and refer back the times that events occurred to the clock, even though light takes time to reach the clock after the event has transpired.
For example, the explosion of a firecracker may be considered to be an “event”. We can completely specify an event by its four spacetime coordinates: The time of occurrence and its 3-dimensional spatial location define a reference point. Let’s call this reference frame S.
In relativity theory we often want to calculate the position of a point from a different reference point.
Suppose we have a second reference frame S′, whose spatial axes and clock exactly coincide with that of S at time zero, but it is moving at a constant velocity v with respect to S along the x-axis.
Since there is no absolute reference frame in relativity theory, a concept of ‘moving’ doesn’t strictly exist, as everything is always moving with respect to some other reference frame. Instead, any two frames that move at the same speed in the same direction are said to be comoving. Therefore, S and S′ are not comoving.
Define the event to have spacetime coordinates (t,x,y,z) in system S and (t′,x′,y′,z′) in S′. Then the Lorentz transformation specifies that these coordinates are related in the following way:
where
is the Lorentz factor and c is the speed of light in vacuum, and the velocity v of S′ is parallel to the x-axis. The y and z coordinates are unaffected; only the x and t coordinates are transformed….
These effects are not merely appearances; they are explicitly related to our way of measuring time intervals between events which occur at the same place in a given coordinate system (called “co-local” events). These time intervals will be different in another coordinate system moving with respect to the first, unless the events are also simultaneous. Similarly, these effects also relate to our measured distances between separated but simultaneous events in a given coordinate system of choice. If these events are not co-local, but are separated by distance (space), they will not occur at the same spatial distance from each other when seen from another moving coordinate system. However, the spacetime interval will be the same for all observers….
Two events happening in two different locations that occur simultaneously in the reference frame of one inertial observer, may occur non-simultaneously in the reference frame of another inertial observer (lack of absolute simultaneity).
From the first equation of the Lorentz transformation in terms of coordinate differences
it is clear that two events that are simultaneous in frame S (satisfying Δt = 0), are not necessarily simultaneous in another inertial frame S′ (satisfying Δt′ = 0). Only if these events are additionally co-local in frame S (satisfying Δx = 0), will they be simultaneous in another frame S′….
The time lapse between two events is not invariant from one observer to another, but is dependent on the relative speeds of the observers’ reference frames (e.g., the twin paradox which concerns a twin who flies off in a spaceship traveling near the speed of light and returns to discover that his or her twin sibling has aged much more).
Suppose a clock is at rest in the unprimed system S. The location of the clock on two different ticks is then characterized by Δx = 0. To find the relation between the times between these ticks as measured in both systems, the first equation can be used to find:
for events satisfying
This shows that the time (Δt′) between the two ticks as seen in the frame in which the clock is moving (S′), is longer than the time (Δt) between these ticks as measured in the rest frame of the clock (S)….
The dimensions (e.g., length) of an object as measured by one observer may be smaller than the results of measurements of the same object made by another observer (e.g., the ladder paradox involves a long ladder traveling near the speed of light and being contained within a smaller garage).
Similarly, suppose a measuring rod is at rest and aligned along the x-axis in the unprimed system S. In this system, the length of this rod is written as Δx. To measure the length of this rod in the system S′, in which the rod is moving, the distances x′ to the end points of the rod must be measured simultaneously in that system S′. In other words, the measurement is characterized by Δt′ = 0, which can be combined with the fourth equation to find the relation between the lengths Δx and Δx′:
for events satisfying
This shows that the length (Δx′) of the rod as measured in the frame in which it is moving (S′), is shorter than its length (Δx) in its own rest frame (S)….
The usual example given is that of a train (frame S′ above) traveling due east with a velocity v with respect to the tracks (frame S). A child inside the train throws a baseball due east with a velocity u′ with respect to the train. In nonrelativistic physics, an observer at rest on the tracks will measure the velocity of the baseball (due east) as u = u′ + v, while in special relativity this is no longer true; instead the velocity of the baseball (due east) is given by the … equation: u = (u′ + v)/(1 + u′v/c2).
All of these effects, even if “real,” are symmetrical. An observer in the “moving” frame (S’) would see the same things happening in the “rest” frame (S). (Technically, anything moving inside a frame of reference is its own frame of reference, inasmuch as it is moving relative to the spacetime coordinates of the frame in which it moves.)
Now for the crucial question: What does the finite speed of light have to do with all of this? The light-clock thought experiment is a good place to start.
Imagine an improbably tall tube with a mirror at each end. A flash of light bounces back and forth from top to bottom. From the standpoint of an observer (S’) standing next to the tube It takes 1 second for the light to go from end to end. Imagine further that the tube is moving uniformly from left to right, as seen by a second observer (S), who can be thought of as stationary, for the purpose of this thought experiment.
As the tube moves, S sees the flashes of light moving diagonally rather than vertically. That is, the distance between the top and bottom of the tube seems to be longer (from the viewpoint of S). S therefore reckons that the light in the moving tube takes more than 1 second to go from top to bottom. In other words, S perceives that the moving light clock is running slow. That is time dilation.
There is a less lucid explanation here, but it gives the mathematical derivation of the time-dilation effect. The resulting relationship is as stated earlier, in the long quotation from the Wikipedia article about special relativity:
for events satisfying
The clock time at S’, as perceived by S, can be expressed as
T’t = T0 + (Δt/γ) , where
T’0 = T0 when the clocks at S’ and S are adjacent in spacetime and are set to the same time
T’t is the time at S’ (according to S) after the elapse of Δt.
Take the case where S’ and S are adjacent at noon, and both of their clocks read 12:00. When it is 13:00 at S (Δt = 1:00) and γ = 2 (at v ≈ 0.87), the clock at S’ will read 12:30.
Length contraction is derived from time dilation by applying a relativistic version of the classical relationship between distance, velocity, and time, d = vt . Given a velocity, time slows as discussed above. As time slows, distance must necessarily become smaller. Distance, in this case, is the span between the endpoints of a rigid object. The mathematical derivation is more complex, but that’s the gist of it. The result is as given earlier:
for events satisfying
Finally, the finite speed of light is said to undermine the traditional concept of simultaneity. Einstein (1916), draws this conclusion from his train-and-embankment thought experiment (discussed at length, later):
Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity). Every reference-body (co-ordinate system) has its own particular time; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event.
We shall see.
One more thing before I proceed. What I have said thus far leaves a lot of questions unanswered. Some of theme are merely what I would call mechanical ones; for example:
Is the speed of light really constant?
What is the experimental and observational evidence for the constancy of the speed of light, time dilation, and length contraction?
Are the relationships in STR real or merely perceptual?
If the Lorentz transformation is valid, doesn’t it implicitly admit the absoluteness of time and space?
Even if the train-embankment thought experiment is invalid, does that negate STR?
Is the light-clock thought experiment a valid application of the constancy and finitenesss of the speed of light?
I will address such questions, either directly or by implication, in the rest of this series.
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.