Sciences

One reason why Einstein's Special Theory of Relativity was needed was because of the special problems that cropped up when scientists noticed that the

The velocity addition problem is illustrated in the figures above and below. In the top figure we see the red driver's frame of reference or

In the figure below the point of view of the blue driver is illustrated. To the driver of the blue car, the purple object is flying forwards out of her window at

The velocity addition problem asks the question:

If we use Newton's model for time as being experienced exactly the same for all observers, we wind up with the answer:

Suppose, as Newton believed, the red and blue drivers measure time and distance precisely the same. According to the blue driver, in time T the distance of the purple ball from the blue car is

But

Sounds reasonable, right?

Okay, now suppose that

This is contrary to observed behavior of Nature. Therefore the Newtonian model of time as being experienced equally by all observers must not be a good model for Nature.

Let's take the red and blue cars from the previous example, and let's mount a laser on top of the blue car. We'll put a mirror on the blue car as shown in the figures, and aim the laser at the mirror perpendicular to the direction the blue car is travelling.

## Here is the laser pulse viewed in the frame of reference of the blue car. |

Meanwhile, the driver of the red car sees the blue car go whizzing by. According to the red driver, the laser took a total time

## Here is the laser pulse viewed in the frame of reference of the red car. |

Now how can we sort this out and find out how their measurements of time should differ in order to account for the observed constancy of the speed of light?

What both drivers agree on is the distance

This has profound implications for the mathematical modelling of space and time as observed in Nature. This means we have to expand the idea of Euclidean analytic geometry to include the observer-dependent

What we've learned here is called Relativistic
time dilation. The process that occurred in the blue driver's
rest frame with in time T' was perceived by the red
driver to have occurred in time T = T' / (1 - (U/c),
which can be much much greater than ^{2})^{1/2}T' if U
is close to the speed of light c. |

Now we will figure out whether these observers measure space differently as well.

Let's suppose the driver of the blue car below has measured the length of her car with a ruler to be length

L' is the blue car length, and T' is the time
for the red car to pass, as measured by the blue driver. |

So we have

L is the blue car length, and T is the time
for the blue car to pass, as measured by the red driver. |

The red driver saw the laser pulse and return happening in

Constraining the speed of light to be constant gave us this relationship between the time

Therefore, the relationship

But this means that the relationsip between the lengths

What we've learned here is called Relativistic
length contraction. The blue car measured to have length L'
in the blue driver's rest frame was measured by the red driver to have
have the length L = L' (1 - (U/c),
which can be much much smaller than ^{2})^{1/2}L' if U
is close to the speed of light c. |

Is there anything that is the same for all observers? That is what we'll look at next.