Special relativity

Special relativity (or the special theory of relativity) was developed and explained by Albert Einstein in 1905 because he thought the older physics was not good enough. The Michelson–Morley experiment had shown this.

Einstein believed that these physics theories gave unspoken preference to one group of observers (i.e., viewers) over another group of observers. Galileo had established the principle of relativity according to which physics events must look the same to all observers, and no observer can be said to have the "right" way to look at the things studied by physics.

E=mc²

Basics of special relativity

Suppose you are moving toward something that is moving toward you. If you measure its speed, it will seem to be moving faster than if you were not moving. Now suppose you are moving away from something that is moving toward you. If you measure its speed again, it will seem to be moving more slowly. This is the idea of "relative speed."

Before Einstein, scientists were trying to measure the "relative speed" of light. They were doing this by measuring the speed of starlight reaching the Earth. They expected that if the Earth were moving toward a star, the light from that star should seem faster than if the Earth were moving away from that star.

They noticed that no matter who performed the experiments, where they were performed, or what starlight they used, the measured speed of light in a vacuum was always the same.

Einstein said this happens because there is something unexpected about distance and time. He thought that as the Earth moves through space, our clocks slow down (ever so slightly). Any clock used to measure the speed of light is off by exactly the right amount to make light seem to be moving at its regular speed. Mentally constructing a "light clock" allow us to see exactly how to explain this remarkable fact.

Also, Einstein said that as the Earth moves through space, our measuring devices change length (ever so slightly). So, any measuring device used to measure the speed of light is off by exactly the right amount to make the starlight seem to be moving at its regular speed.

Other scientists before Einstein had written about light seeming to go the same speed no matter how it was observed. The idea that made Einstein's relativity so revolutionary is that light does not just seem to go the same speed, it is always going the same speed no matter how an observer is moving.

The Lorentz transformations

The mathematical basis of special relativity are the Lorentz Transformations, which mathematically describe the views of space and time for two observers who are moving with respect to each other but are not experiencing acceleration.

To define the transformations we use a Cartesian coordinate system to mathematically describe the time and space of "events".

Each observer can describe something being somewhere at a certain time, using coordinates (x,y,z,t).

The location of the event is defined in three dimensional x, y and z such as (0,0,0) for the center, or (3,3,3) which is a diagonal going 3 units out in each direction, in some unit of distance (like meters or miles).

The Time of the event is described with the fourth variable t in some unit of time (like seconds or hours or years).

Let there be an observer K who describes when events occur with a time coordinate t, and who describes where events occur with spatial coordinates x, y, and z. This is mathematically defining the first observer whose "point of view" will be our first reference.

Let us specify that the time of an event is given: by the time that it is observed_(observed) (say today, at 12 o'clock) minus the time that it took for the observation to reach us:

which can be calculated as:

the distance from the observer to the event d_(observed)

(say the event is on a star which is 1 light year away, so it takes the light 1 year to reach the observer)

divided by c (which is the speed of light - a very large number in miles per hour).

This is correct because distance, divided by speed gives the time it takes to go that distance at that speed (e.g. 30 miles divided by 10 mph: give us 3 hours, because if you go at 10 mph for 3 hours, you reach 30 miles).

So we have:

$t = d/c$

This is mathematically defining what any "time" means for any observer.

Now with these definitions in place,

let there be another observer K' who is

• moving along the x axis of K' at a rate of v,
• has a spatial coordinate system of x' , y' , and z' ,

where x' axis is coincident with the x axis, and with the y' and z' axes - "always being parallel" to the y and z axes,

This means that when K', the second observer, gives a location like (3,1,2), the x (which is 3 in this example) is the same place that K, the first observer would be talking about, but the 1 on the y axis or the 2 on the z axis are only parallel to some location on the K' observer's coordinate system.

and

• where K and K' are coincident at t = t' = 0

This means that the coordinate (0,0,0,0) is the same event for both observers.

In other words, both observers have (at least) one time and location that both agree on, which is location and time zero.

The Lorentz Transformations then are

$t' = (t - vx/c^2)/ \sqrt{1 - v^2/c^2}$

$x' = (x - vt)/\sqrt{1 - v^2/c^2}$

$y' = y$, and

$z' = z$.

Mass, energy and momentum

In special relativity, the momentum p and the energy E of an object as a function of its rest mass m₀ are

$p = \frac {m_0 v}{\sqrt{1 - \frac {v^2}{c^2}}}$

and

$E = \frac {m_0 c^2} {\sqrt{1 - \frac {v^2}{c^2}}}$.

A frequently made error (also in some books) is to rewrite these equation using a "relativistic mass" (in the direction of motion) of $m=\frac {m_0} {\sqrt{1 - \frac {v^2}{c^2}}}$. The fact why this is incorrect is that the light, for example, has no mass, but has energy. If we use this formula, the photon (particule of light) has a mass, which is experimentally (by the experiments) incorrect.

In special relativity, energy and momentum are related by the equation

$E^2 = p^2c^2 + m^2 c^4$.

History

The need for special relativity arose from Maxwell's equations of electromagnetism, which were published in 1865. It was later found that they call for electromagnetic waves (such as light) to move at a constant speed (i.e., the speed of light).

To have James Clerk Maxwell's Equations be consistent with both astronomical observations,^[1] and Newtonian physics^[2] Maxwell proposed in 1877 that light travels through an ether which is everywhere in the universe.

In 1887, the famous Michelson-Morley experiment tried to detect the "ether wind" generated by the movement of the Earth.^[3] The persistent null results of this experiment puzzled physicists, and called the ether theory into question.

In 1895, Lorentz and Fitzgerald noted that the null result of the Michelson-Morley experiment could be explained by the ether wind contracting the experiment in the direction of motion of the ether. This effect is called the Lorentz contraction, and (without ether) is a consequence of special relativity.

In 1899, Lorentz first published the Lorentz Equations. Although this was not the first time they had been published, this was the first time that they were used as an explanation of the Michelson-Morley experiment's null result, since the Lorentz contraction is a result of them.

In 1900, Poincaré gave a famous speech in which he considered the possibility that some "new physics" was needed to explain the Michelson-Morley experiment.

In 1904, Lorentz showed that electrical and magnetic fields can be modified into each other through the Lorentz transformations.

In 1905, Einstein published his article introducing special relativity, "On the Electrodynamics of Moving Bodies", in Annalen der Physik. In this article, he presented the postulates of relativity, derived the Lorentz transformations from them, and (unaware of Lorentz's 1904 article) also showed how the Lorentz Transformations affect electric and magnetic fields.

Later in 1905, Einstein published another article presenting E = mc².

In 1908, Max Planck endorsed Einstein's theory and named it "relativity". In that same year, Minkowski gave a famous speech on Space and Time in which he showed that relativity is self-consistent and further developed the theory. These events forced the physics community to take relativity seriously. Relativity came to be more and more accepted after that.

In 1912 Einstein and Lorentz were nominated for the Nobel prize in physics due to their pioneering work on relativity. Unfortunately, relativity remained so controversial then, and for a long time after that, that a Nobel prize was never awarded for it.

Experimental confirmations

• The Michelson-Morley experiment, which failed to detect any difference in the speed of light based on the direction of the light's movement.
• Fizeau's experiment, in which the index of refraction for light in moving water cannot be made to be less than 1. The observed results are explained by the relativistic rule for adding velocities.
• The energy and momentum of light obey the equation E = pc. (In Newtonian physics, this is expected to be $E = \begin{matrix} \frac{1}{2} \end{matrix} pc$.)
• The transverse doppler effect, which is where the light emitted by a quickly moving object is red-shifted due to time dilation.
• The presence of muons created in the upper atmosphere at the surface of the Earth. The issue is that it takes much longer than the half-life of the muons to get down to the surface of the Earth even at nearly the speed of light. Their presence can be seen as either being due to time dilation (in our view) or length contraction of the distance to the surface of the Earth (in the muon's view).
• Particle accelerators cannot be made to perform properly unless relativistic physics is used.

Notes

• ^[1] Observations of binary stars show that light takes the same amount of time to reach the Earth over the same distance for both stars in such systems. If the speed of light was constant with respect to its source, the light from the approaching star would arrive sooner than the light from the receding star. This would cause binary stars to appear to move in ways that violate Keppler's Laws, but this is not seen.
• ^[2] The second postulate of special relativity (that the speed of light is a constant for the observer) contradicts Newtonian physics.
• ^[3] Since the Earth is constantly being accelerated as it orbits the Sun, the initial null result was not a concern. However, that did mean that a strong ether wind should have been present 6 months later, but none was observed.

References

• W. Rindler, Introduction to Special Relativity, 2^nd edition, Oxford Science Publications, 1991, ISBN 0-19-853952-5.
• Web article on the history of special relativity
• Relativity Calculator - Learn Special Relativity Mathematics The mathematics of special relativity presented in as simple and comprehensive manner possible within philosophical and historical contexts.

Other pages

• General theory of relativity
• English Wikipedia's article about Special relativity for beginners