The Special Theory of Relativity in pictures - the lance paradox

The special theory of relativity (STR), unlike quantum mechanics, can be explained in pictures, in space-time diagrams, with only highschool geometry. All diagrams were done with Mathematica 9.

All the math and physics for this comes from the References, in particular, [1].

A Glossary follows to keep things straight.

This version of space-time diagrams is really useful, because the reader will be able to see the length and time dilations of STR, in one diagram, with the speed of light, c, having central importance.

Also explained, is the paradox of the knight with a long lance trying to ride into a barn. In this gedanken experiment, the knight is traveling 1/2 the speed of light, c/2, and is riding into his barn, lance first. A person beside the barn says, "Come on in there's plenty of room," but the knight says, "There's not enough room for my lance!"

The diagrams are space-time (originally called Minkowski space) diagrams. The diagrams do not depict Euclidean space, so the notion of length is different: they show a space of (x, t). The distance berween 2 points is called the interval, with units of length.

Space-time diagrams can easily show Einstein's great realization about the relativity of simultaneity and the speed of light, from his 1905 paper [5].

A point (x,t) in space-time is called an event. An observer is an hypothetical someone who could use clocks and rods to mark out a coordinate system (CS), or frame, marking out an x-axis with rods and keeping clocks around to measure durations.

We will deal with inertial observers, that is, unaccelerated observers.

Figure 0 shows the idea. This is a diagram for one observer. The diagonal red lines are light rays, one unit of time, into the future, for one unit of space. The origin is called "O". For a unit, one light second is nearly to the moon from earth.

The future light cone contains all the possible events that can be reached from O. The past lightcone comprises all the events that could affect O. Intervals between O and points in these lightcones are called time-like.

The line marked "t" is the worldline of a stationary observer at O. A world line is the string of events that a body makes over time. We call the observer at rest with respect to O, the observer O, in the O CS. There of course may be other observers moving with respect to O. Our favorite one will be some one a space ship moving along the increasing O x-axis, at c/2. Call this observer O'.

Intervals between O and points outside of the future or past light cones are called space-like and can't be traveled because they require going faster than the speed of light.

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See Figure 1

Understanding and using space-time diagrams requires working with 3 very weird ideas:

1) "The velocity of light is independent of its source" - the constancy of the velocity of light.

What? This is crazy, what if a spaceship goes by at c/2 and sets off a light bomb on its nose. Then, both you and person riding on the spaceship will measure the same velocity of light. But wait, so does someone riding in a spaceship going in any other direction, at any other velocity, but that's crazy. There is no goop that light travels in. In the O CS light from the tip of the space ship starts traveling away at c, with the space ship traveling away at c/2, but the space ship still sees light at the speed c. What keeps all this straight? - that's the weirdness.

2) The interval between 2 points is measured the same (preserved or invariant) in both CSs. Experiment has shown that clocks run slower and rods look shorter in a moving CS. Rods and clocks are not fundamental, the space-time interval is fundamental. What does the CS of a moving observer, O', look like from the point of view of O's CS? In other words, what do its clocks and rods look like? The weird idea is that to calculate the (x', t') values , (in the O' CS), from the (x, t) values , (in the O CS), space and time are not independent. Such a mathematical transformation must preserve the constancy of the velocity of light and length of any interval (in space-time). The Lorentz transformation does this, mathematically transforming between the (x, t) and the (x', t') numbers, giving the weird values for moving clocks and rods as seen experimentally.

Because of the constancy of the velocity of light, the x' and t' axes have to measure the speed of light as c, so the Lorentz transform squeezes them together.

3) Using the Lorentz transform, the unit length on any new x'-axis, by varying the speed, traces out an hyperbola. This hyperbola is called a gauge curve. All the intervals (along a new x'-axis) from O to a unit gauge curve are measured to be the same, even though the length in the diagram is longer. It's not Euclidean space. In Figure 1, the interval from O to the gauge curve on the x-axis, OD, is the same as the interval from O to the gauge curve on the x'-axis, OC. The interval measured along a light ray is zero!


Now we can show the most basic features of STR, the contraction of length and the slowing down of clocks, in moving frames, in one diagram.

Warning - weird units, the interval is in units of length, so what's one second up the t-axis? Ha, ha, it's 3x10^8 meters. To avoid algebra and to keep things geometric, we'll call the units on the t-axis "seconds," or clock ticks, the only cheat in this note. When the reader encounters the space-time metric, all will be made clear.

By the Lorentz transform, the OAand OB intervals must have the same interval - their tips lie on the same gauge curve. Call OA one second on the x-axis, then OB must also measure one second in its frame. This is the famous time dilation.

But!

if the O CS measures, lets say, 1 second on the t-axis for OA, then the O' CSs clocks must also measure 1 second along OB. The tick mark on the t-axis shows the elapsed time by the O CSs clocks from O to the event OB and it's longer than 1 second. The clocks in the O' CS appear to run slower.

Similarly - By the Lorentz transform, the OC and OD intervals must have the same length - their tips lie on the same gauge curve. Call OD one light-second on the x-axis, then OC must also measure one light-second in its frame.

But!

if the O CS measures, lets say, 1 light-second on the x-axis for OD, then the O' CSs clocks must also measure 1 light-second along OC. The tick mark on the x-axis shows the length by the O CSs rods from O to the event OC and it's longer than 1 light-second. The rods in the O' CS appear to be shorter. This is the famous length contraction.

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Figure 2 shows how to use these ideas to solve the lance-in-the-barn paradox.

A knight is riding at c/2 with his lance up, into a barn. For our gedanken experiment, the rest length of the lance in the O CS is one light-second (hey it's a gedanken experiment). For the moving lance, in the O CS the handle of the lance goes through O at time 0, and the handle is at the origin of the O' CS. It travels up the t-axis. The end of the lance traces out the green line, which crosses the x-axis and osculates with the gauge curve. This is to show that lance tip in the O' CS goes from O to the gauge curve on the x'-axis, the interval OA.

Aha! In the O CS the length of the lance is shorter than when at rest.

Let the cyan line parallel to the t-axis be the end of the barn in the O CS, the barn is at rest.

Let the green line be the tip of the lance in the O' CS, parallel to the t'-axis. The handle of the lance is held by the knight, the observer O' , on the t'-axis.

Using Einstein's ideas about simultaneity from his 1905 paper, we see that in the O CS, at time 0, t = 0, the tip of the lance, and the end of the barn are simultaneuous, with the tip of the lance still in the barn. We also see, that at time 0, t' = 0, in the O' CS, the tip of the lance went through the barn and at t' = 0, is outside the barn. No paradox, just different definitions of simultaneity.

N.B., please refer to:

http://www.phys.vt.edu/~takeuchi/relativity/notes/section10.html

for a beautiful space-time diagram explanation of what happens when signals can go faster then the speed of light.

Glossary

c - speed of light = 3x10^8 m/sec.

Clock - just a regular clock that ticks off seconds.

Coordinate system - a means of labeling points (or events) in space-time.

Event - a point in space-time, such that an observer can sit there with a clock and rod, and measure things.

Gedanken experiment - a thought experiment, a term made famous by Einstein.

Interval - the distance in space-time between points, with units of length.

Observer - an hypothetical someone who can measure with clocks and rods.

Rod - just some meter stick.

Worldline - the set of events that a body makes in its history through space-time.

References

[1] The Special Theory of Relativity, Aharoni, Oxford, 1965.

[2] Spacetime Physics, Taylor and Wheeler, Freeman, 1966.

[3] Relativistic Kinematics, Hagedorn, Benjamin, 1963.

[5] "Zur Elektrodynamik bewegter Körper". Einstein, Annalen der Physik 17 (10): 891–921, 1905.

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