The “Thought Experiment” with Two Observers
June 2004
The excerpt that follows is a “Thought Experiment” taken from a college textbook to prove Time Dilation.
In this classic experiment, One Observer is on the sidewalk and the other is in a Train or Truck. This analysis will use the following exercise1:
As a thought experiment, let us imagine two boys playing a game. One boy, A, is on the sidewalk (frame of reference FR), while the other boy, B, is on a truck (frame of reference FR’) that is moving past A eastward at a uniform velocity v. The situation as seen by A is shown in Fig. 28-8a. The distance between the boys (at closest approach) is a, and they are equipped with identical clocks with which they can measure time intervals. The object of the game is for a short time pulse of light from A’s flashlight to be reflected from B’s mirror back to A. We have two events – when A sends the pulse, and when A receives the reflected pulse. What is the duration of the time interval between these two events?
Fig. 28-8. (a) Two events as seen in A’s frame of reference. (b) The same two events as seen in B’s frame of reference.
Suppose that A turns on his flashlight at just the right time to illuminate B when B is opposite A. The velocity of light relative to either A or B is c, independent of the motion of the source or the observer (Einstein’s second postulate). In order to hit his target, the boy A must turn on his light just before B is opposite him (B at the noted position in Fig. 28-8a); the light travels a distance 2a at velocity c, so the time required for the round trip, t0, is given by
(28-1)
This is A’s measurement of the interval between the two events. We call t0 the proper time between the two events. This technical term, proper time, refers to the duration between two events that take place at the same place in some frame of reference (A’s frame, in this case).
On the other hand, B describes the same experiment somewhat differently. As B sees it (Fig. 28-8b), he is stationary and A is moving backward (to the west) with velocity v. As before, A points his flashlight northward, and, as before he turns it on just before he is opposite B. The resultant velocity of the light relative to B is found by B to be equal to c, as it must be according to Einstein’s second postulate. According to B, his friend A has moved a distance vt during the interval between the two events, where t is the duration of the interval measured in B’s frame. B now uses the Pythagorean Theorem to find the distance the light has traveled during this time; he obtains 2
. Since the speed of light in B’s frame is c, B concludes that the time interval is given by
(28-2)
Now a little elementary algebra is needed to solve Eq. 28-2 for t. We obtain
We see that t is greater than t0. A and B find different answers for the duration of the interval between the “sending” and the “receiving” events, even though their clocks are identical. There is a difference between their frames, however, because the two events do not take place at the same place in B’s frame (see Fig. 28-8b). Thus only t0, measured at the same place in A’s frame, is a proper time; t is an improper time. A considers that B’s clock is “wrong” measuring an interval that is “too long.” In an exact similar experiment, B could hold the flashlight and A the mirror. Now the interval measured in B’s frame would be the proper time, and B would claim that A’s clocks are given too long an interval. Fortunately we do not have to decide the argument, for it is impossible to tell which of the observers is “really” at rest (Einstein’s first postulate). We are sure of one thing – there is not absolute time, and A and B are both correct, each according to his own standards.
In general terms, we conclude that if t0 is the time interval between two events that occur at the same position in a frame of reference, then the time interval between the same two events has a longer duration t as measured by an observer in a frame of reference that is in uniform motion relative to the first frame.
In our example, the ratio of the two time intervals representing the same experiment is
(28-4)
This same ratio, which is always greater than 1, holds true for any experiment viewed by two observers who are moving relative to each other. This is what is meant by time dilation, the stretching out of time.
Analysis
It is so touching and pitiful, to see how Einstein’s Mythology deceived four generations of humans, especially the self-proclaimed “Scientific Community.”
An example such as this and its conclusion is so depressing to the human intellect that the future of science itself is under consideration.
Of course, the Ptolemaic Mythology deceived humanity for two millennia but this can be understood, because we have not, in that long time, had any Scientific Community. The simple historical conclusion is terrifying: no Scientific Community existed in the last century and it still doesn’t exist today.
The experiment’s conclusion, when it is viewed from Special Relativity’s point of view, only reveals the naïveté of human beings and the necessity to believe in magic as it is in a Primitive Society with a Primitive Mentality.
Generally any “thought experiment” is developed to yield preconceived results. The problem is to know if such an experiment could occur in Nature. For example: SR’s Sum Velocity equation works within the world of Kinematics (motion without expending energy) but not within the world of Dynamics, as has been explained elsewhere.
Telling the truth and only the truth, this example is ONLY a BOOBY TRAP.
In case (a) if boy B doesn’t emit any signal, boy A doesn’t know his position and consequently he doesn’t know when to flash the flashlight. This is not a problem, of course, because it is possible to make the flashlight flash periodically and to encoding each pulse. But as Relativity must be universal the same will either happen to B when A is in motion with respect to him, or when both are in motion relative to each other.
As in many “thought experiments” the Booby Trap2 is in case (b) purporting that what happens to A happens to B. Or better, what A measures is supposed to be measured by B.
It is not true that B, in case (b), sees the flashlight send the flashlight pulse in the ARROW direction. The light will not reach boy B.
To reach boy B, boy A needs to send the flashlight in the B direction, that is, with an angular deviation. In this case B will only see the light coming from an angle, which is the only thing that he can measure. The same explanation serves for time and position detection.
The author clearly says in case (a) that what boy A observes is the “proper time”, t0. But also in case (b) the only Observer is boy A because he starts the flashlight and he receives the signal reflected from boy B. Under the same rule, consequently, the “proper time” is also t0, even though t0 will have a different value.
When Observer A is an AD scientist, he is not surprised by the larger, new t0 value because he knows, after being given an angular deviation to his flashlight, that the distance traveled by the light beam must be longer, as a simple trigonometric calculation will show.
This has nothing to do with “TIME DILATION”. It has only to do with LONGER DISTANCE.
The Booby Trap is not complete without analysis of the “duration” between the “sending” event and the “receiving” event.
The thought experiment’s author says that the “improper time of B” is bigger than the “proper time of A” because “proper time” is defined when the events happen in the same position and B observes that the event happens at two different positions.
This conclusion is untrue. B doesn’t observe any event. The only thing that he observes is a light coming and going. It is the only time measured by him, but this time cannot be compared with any other time value. To get results he needs to receive information from A and, of course, B will arrive at the same value measured by A. What is artificially divisive is the defining of “proper time” and “improper time.” These “definitions” prepare the way for the expected results when the “thought experiment” was developed.
To work, Relativity needs to be Universal, that is, each time value measured by an Observer is the “Proper Time.” The “Proper Time” measured will vary according to the traveling velocity or the traveled distance but the difference has nothing to do with “Time Dilation”: It has only to do with “Time Duration”. If the velocity is the same, only the distance will change when the time is greater as really happens in the present “Thought Experiment.”
This naïve and simple-minded Booby Trap is a degradation of the Human Intellect, and certainly not a viable teaching tool.
Footnotes
- “College Physics,” Franklin Miller, Harcourt Brace Jovanovich, Inc, New York, London, etc., Fifth Edition.1982, page 667.
- “The Billiard Ball Game According to AD,” in the book “Storm in Physics,” on Chapter 18. Also in the web at http://www.autodynamics.org or http://www.autodynamicsuk.org.