This is a new tutorial which will be integrated into
The Light Cone, a tutorial on relativity.
This new tutorial features a visual explanation of the Twin Paradox
using "lightclock diagrams" (spacetime diagrams of lightclocks).
This page features animated movies in MP4 and AVI formats.
These animations were produced with the VPython visualization module. 
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The Principle of Relativity  
The Principle of Relativity (Galileo)

From his Dialogue Concerning the Two Chief World Systems, Galileo wrote:
...have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still.
Galileo's Principle of RelativityBy observing the outcome of mechanical experiments, one cannot distinguish a state of rest from a state of constant velocity. 

The Principle of Relativity (Einstein) 
From his Relativity
The Special and General Theory, Einstein wrote:
As long as one was convinced that all natural phenomena were capable of representation with the help of classical mechanics, there was no need to doubt the validity of this principle of relativity. But in view of the more recent development of electrodynamics and optics it became more and more evident that classical mechanics affords an insufficient foundation for the physical description of all natural phenomena. At this juncture the question of the validity of the principle of relativity became ripe for discussion, and it did not appear impossible that the answer to this question might be in the negative. Einstein's Principle of RelativityBy observing the outcome of any experiment (mechanical, electromagnetic, opticalor any physical law whatsoever), one cannot distinguish a state of rest from a state of constant velocity. 

Light travels at 3.00 x 10^{8} m/s  
A light ray travels with speed c=3.00 x 10^{8} m/s,
independently of the motion of its source.
On a spacetimediagram with the axes calibrated as above, a lightray [shown in yellow] always makes at angle of 45^{o} with the timeaxis [which is, by convention, oriented vertically upward]. When a flash of light is emitted by a source lightrays spread out isotropically in space, tracing out a cone on a spacetimediagram. This cone is called the "[future] lightcone of the emission event". The lightcone depends only on the emission event. It does not depend on the motion of the source of the light. The source [modeled as a pointparticle in space] traces out a curve (called its "worldline") on a spacetimediagram. The green source is at rest, and the red source is in motion (with v=0.5c).



 
The Light Clock [at Rest]  
With a pair of mirrors and a light source, we can construct a simple clockcalled
the "light clock".
When we place a mirror at a distance L away from the source, a light ray travels at the speed of light for a time t=(L/c) to get to that distant mirror. Then, upon reflection, a light ray travels at the speed of light for an additional time t=(L/c) to get back to the source. The duration of the roundtrip is T=(2L/c). This defines one "tick" of this clock.
In these diagrams, we have chosen



The Light Clock [in Uniform Motion]  
Suppose we have an identically constructed clock (the red clock ),
which has a uniform velocity of v=0.5c along the xdirection.
What is the duration of one tick of this moving clock? We measure the duration of that red tick to be longer than the duration of our our green tick. Why? Because [to us] the moving mirror is moving away from the source, the light ray (traveling at the speed of light) that reaches that moving mirror must travel a longer distance for a longer duration. This is timedilation.
Let us refer to this type of light clock as a "transverse lightclock". Transverse means that the light clock is oriented perpendicular to the direction of relativemotion. (In the next section, we will also use a light clock that is oriented parallel to the direction of relativemotion. We refer to this type of clock as a "longitudinal lightclock".) 


A MichelsonMorley apparatus [at Rest]  
When we arrange two identical light clocks perpendicular to each other,
we obtain [what we will call] a "simplified MichelsonMorley apparatus".
Observe: Light rays, emitted by the source at a common event then reflected back by the equidistant mirrors, are received by the source at the same reception event.[To us,] the orientation of the two light clocks does not matter here. [To us,] the reflections on our distant green mirrors occurred at the same value of t. Let us refer to this as our "halftick".



A MichelsonMorley apparatus [in Uniform Motion]
 
Suppose we have an identically constructed MichelsonMorley apparatus
(the red set of clocks), which has a uniform velocity of v=0.5c along the xdirection.
We already know that the light ray (traveling at the speed of light) must travel a longer distance for a longer duration to reach the "transverse mirror". In fact, the light ray (traveling at the speed of light) that reaches the "longitudinal mirror" moving away from the source must travel an even longer distance for a even longer duration. [It had a headstart along the direction of motion!] Thus, the outgoing light ray reaches the moving longitudinal mirror after the light ray reaches the moving transverse mirror. Upon reflection, the return trip to the moving source doesn't take as long since the moving source is moving toward the incoming light ray. Although these are interesting observations (which we will deal with later), a more serious observation is this: Note carefully that the reflections are not received at the same event. The longitudinal reflection is received first, followed by the transverse reflection.



It appears that the red clock in motion
can now distinguish itself from the green clock at rest.
This situation, however, violates the principle of relativity! The red clock in motion does not receive its reflections simultaneously. In fact, the timedifference between the two receptions can be used to calculate the velocity of the moving red clock. The experiment of Michelson and Morley effectively tried to measure this time difference... but they always measured zero. (experimental data) Apparently, there is something wrong with this proposed model of the moving clock.



A MichelsonMorley apparatus [in Uniform Motion]
 
To satisfy the requirement that the two receptions occur
at the same event for the red clock in motion, it can be shown
that the position of the moving longitudinal mirror
must be shorter than L.
This is lengthcontraction.



Here are the two cases sidebyside. 


Absolute Simultaneity Lost
 
Observe that we regard
the "reflection events at our distant green mirrors"
to be simultaneous (i.e., they occur at the same value of
t for us).
However, observe that we do not regard the "reflection events at his distant red mirrors" to be simultaneous (i.e., they do not occur at the same value of t for us).
In accordance with the Principle of Relativity, he regards the "reflection events at his distant red mirrors" to be simultaneous (i.e., they occur at the same value of [call it] t' for him). The next visualization will elaborate on this. 


Circular Light Clocks
 
Let us generalize this to a circular arrangement of light clocks.
This provides us with a visualization of one tick of proper time. Since each mirror traces out its worldline, the collection of mirrors traces out a "worldtube".
Consider the lightcone of the first lightflash
Observe that these reflection events actually trace out a circle [for the green lightclock]. Consider the unique plane that contains this circle. Realize that all events on this plane are simultaneous [for the green lightclock]. One could call this the "t=0.5plane [for the green lightclock]". Following the reflected rays leads to the "first tick". (By the way, these reflected rays trace out the "[past] lightcone of the reception event".) Then, the procedure can be repeated for the other ticks. Upon considering all of the ticks for this clock, one obtains a "family of parallel planes of simultaneity, marking the "halfticks" [for the green lightclock]". The same procedure applies to the red lightclock, which is in motion. The intersection of that first lightcone and the red worldtube represents a set of "reflection events for the red lightclock". These events are "simultaneous for the red lightclock but NOT for the green lightclock". After tracing out the paths of all of these lightrays, the measurement of the intervals of propertime is now reduced to the "counting of the ticks".


In order to work with simple arithmetic,
we will use a circular light clock with v=0.8c.
By counting ticks, it is evident that, for the green lightclock, "when 3 redticks have elapsed, 5 green ticks have elapsed". The ratio 5/3 is the timedilation factor



The Clock Effect / Twin Paradox  
Here is the Clock Effect/Twin Paradox.
Here's how the story runs (adapted from Spacetime Physics, ex. 27): The use of "twins" was just to make the story sound more dramatic. Here is a lessdramatic formulation:
(The issue of the [false] "paradox" arises when one attempts to assert the brothers are "equivalent", so that, by the supposed symmetry, the age difference must be zero [which is at odds with the notion of timedilation]. The paradox is resolved when it is realized that the brothers are, in fact, notequivalent. Paul has been an inertial observer (traveling with a constant velocityvector ["zero", in this case]) between the separation and reunion events. Peter, however, has not been an inertial observerhe has been for most of his trip. However, he changed his velocityvector when he turned around. That event disqualified him from being "inertial".) 


Let us draw a pretty picture (a Spacetime Diagram) of the Clock Effect/Twin Paradox using light cones. 


Visualizing Proper Time (the paper)Visualizing Proper Time in Special Relativity (Salgado) (PDF) AAPT Summer 2001 meeting (Rochester, NY  July 25, 2001)
A slightlyrevised version appears in
It is also posted at the
Slides from my
AAPT 2004 Winter Meeting talk.


Other sites of interest
The Light Cone  an illuminating introduction to Relativity The VRML Gallery of Electromagnetism VPython applications for Teaching Physics other selected presentations visualrelativity.com 
Copyright © 2001
Roberto B. Salgado
Department of Physics  Syracuse University
Syracuse NY 132441130
All Rights Reserved.
They were captured to AVI/RealVideo format with SnagIt.