Special Relativity: A Simple Matter of Perspective?

Popular belief has it that Eintein's Theory of Special Relativity is something difficult. It is not. Special Relativity is nothing more than what we usually call "Perspective", but applied to space and time, and not only to space.

On this page, you will see how easy Relativity paradoxs appear once you have accepted this idea.

The Special Relativity "Paradoxs"

Special relativity introduced a number of well known "paradoxs", such as: Explanations for these paradoxs usually involve lengthy calculations. See for instance the chapter on the twins paradox in the Relativity FAQ. The goal of this article is to show how these effects can be explained in a much simpler way, without any computation, as a simple effect of perspective.

The "magic recipe"

However, all special relativity paradoxs can be explained without any computation with the help of the following "magic recipe":
  1. Plot all events in the experiment in a plane, where time is an axis, and space is the other axis, as measured by any given observer.
  2. For any mobile in the diagram, the trajectory on the diagram is called the time line.
  3. Any time measurement made by this mobile will be made along its own trajectory.
  4. Any space measurement made by the mobile will be made perpendicular to this trajectory.
  5. Any effect of perspective needs to be inverted. Where contractions happen on the diagram, special relativity will predict a dilatation. Where dilatations happen on the diagram, special relativity will predict a contraction.
  6. Time and space don't mix. A rule of the thumb is to look suspiciously at any angle above 45° on your diagram.
Examples of how to apply the magic recipe are shown below. You can also read the mathematical explanation of why this magic recipe works.

The twins paradox

This paradox is the most well known and explained paradox, at least as far as the web is concerned. Several other very good sites on the Web also explain this same paradox. See for example the Relativity FAQ, the explanation of the University of Virginia, or the nicely presented one of Howard C. McAllister, as well as some more puzzling ones...

Our magic recipe, however, makes this paradox look quite simple. If we apply step 1 of the magic recipe, the twins paradox can be plotted as follows:

The red curve indicates the trajectory of the travelling twin. The blue line indicates the trajectory of the twin remaining on earth.

Step 2 indicates that the age of the travelling twin is represented by the length of the red curve, and the age of the twin remaining on earth by the length of the blue line.

It is a well known fact that the shortest path between two points is the straight line. So any trajectory the twin follows on our diagram is always longer. Applying step 3 of our recipe, the travelling twin will always age less than the twin remaining on earth.

The two rockets paradox

The two rockets paradox is solved equaly well. Applying step 1, the two rockets paradox can be plotted as follows (from the point of view of the red rocket):

The big dots represent the events in space-time corresponding to each pulse of light, for each of the two rockets. Since the blue rocket is moving with respect to the red rocket, its trajectory is slanted (there is a displacement along space).

Applying step 2, and considering that the blue rocket emits a dot every second of its own time, we see that the plots are separated by the same distance along its trajectory, not along our time. However, if the red rockets tries to measure the interval between two pulses emitted by the blue rocket, it does so using the red dotted lines, which are perpendicular to its own red time. Conversely, the blue dotted lines show how the blue rockets measures the interval between two red pulses.

On the diagram, a contraction appears in both cases. Applying step 3, we discover that rocket red will see pulses emitted by rocket blue separated by a larger time interval, but that rocket blue will also see rocket red pulses separated by a larger time interval !

And that's it...

Red shift

The red shift is exactly the same phenomenon as the rockets paradox, applied to the frequency of the light emitted by moving distant stars. For a given element, the possible frequencies of emitted light are known and supposedly constant accross the universe. The set of possible frequencies is known as the spectrum of the element.

Compared to the spectrum of the same element measured in a lab on earth, the spectrum of elements in distant stars appears shifted towards low frequencies. Since red is the lowest frequency in the visible spectrum, this is called the red shift.

See also the General Relativity FAQ on this topic.

The train paradox

The train paradox involves space measurements. To plot it correctly, we need to plot both the front and the rear of the train. In the figure below, we plot it from the observer on the track. The lightning bolts strike simultraneously for the observer on the track (at the same time), so they are on the red dotted line. We then make the space measurement on the track, so following the green dotted lines on the diagram.

Applying step 2, we can understand why the lightning bolts are not simultaneous for an observer in the train. The lightning at the front of the train will occur after the one at the rear of the train.

The distance d between the two bolts, measured in by an observer on the track (along the green dotted lines), is longer on our diagram than the length of the train measured by an observer in the train. Applying step 3, we can deduce that the distance between the bolts in special relativity is shorter than the length of the train.

The barn and the pole paradox

The barn and the pole paradox is similar to the paradox of the train. It can be thought of as follows: in "normal" space, it is more difficult to enter a garage in diagonal than straight. In special relativity, it becomes easier.

See also the detailed explanation in the Relativity FAQ.

The complete explanation
Redemonstrating Lorentz equations
The Relativity FAQ
© 1999, Christophe de Dinechin