SPECIAL RELATIVITY

After the Michelson-Morley experiment, the scientists were left with two equally unpleasant alternatives:

• either Maxwell's equations were incorrect (but they explained all the electromagnetic phenomena so well!...)
• or Galilean principle was wrong (yet Newtonian mechanics works so well everywhere, from orbits of planets to everyday experience on Earth!...)

Various explanations were immediately proposed.

• George FitzGerald suggested that all objects moving through the ether were physically contracted depending on their speed: the faster they were moving, the larger was this contraction. Objects moving with the speed of light were contracted to zero length (!).
• Others proposed ``ether drag'': all moving bodies dragged ether around them along with them.

Ether drag was partially supported by the fact that the speed of light in a medium is always smaller than the speed of light in vacuum. However, in that case shouldn't the Earth experience some friction, and as the result spiral down and fall to the Sun?

Both explanations were not satisfactory. Austrian physicist and philosopher Ernst Mach offered a different explanation:

The Michelson-Morley experiment was designed to detect ether. No ether was detected, therefore, there was no ether at all.

This explanation fully follow the rules of science. However, it was difficult to accept since there was no alternative theory!

Lorentz transformation

Based on the FitzGerald hypothesis, Hendrik Lorentz discovered a coordinate transformation, i.e. a way of relating two different reference frames, which kept the speed of light, and thus Maxwell's equations invariant. This transformation has been called Lorentz transformation ever since.

• Maxwell's equations are invariant under the Lorentz transformation.
• Newton's equations are invariant under the Galilean transformation.

Important note: for speeds much less than the speed of light, both transformations are identical.

However, for speeds close to the speed of light, Lorentz transformation predicted weird things: lengths should contract and time intervals should increase (time dilation). This seemed so radical, that few people were ready to accept this.

Then there came a patent examiner from Bern, named A.E.

He realized, that these weird things like length contraction and time dilation were not absolute, but relative. In other words, they only appeared.

Einstein based his theory on the relativity principle:

• The laws of nature are the same in all inertial frames of reference.

Einstein also believed in Maxwell's equations, and since Maxwell's equations require that the speed of light is the same in all reference frames, he simply stated that:

• The speed of light in the vacuum is the same in all inertial frames of reference.

Thus, he accepted Lorentz transformation and discarded Galilean transformation.

In doing so, he created the Special Theory of Relativity, or SR.

You book says:

Einstein had the audacity and courage to abandon Galilean relativity completely, and with it Newtonian mechanics...

A

agree

B

disagree

Important conclusions of SR:

• the property of simultaneity becomes relative: events that are simultaneous in one reference frame become not simultaneous in the other;
• no material object can move faster than the speed of light; only massless particles (like photon) can move with the speed of light.

$\Gamma$ factor

Both, the length contraction, and the time dilation, are described by one quantity, called the boost factor, or simply $\Gamma$ factor, because it is traditionally denoted by a capital Greek letter $\Gamma$.

If an object is moving with respect to a specific reference frame, it appears that all lengths along the direction of motion are contracted $\Gamma$ times, and all times are slowed down $\Gamma$times on this object.

If the speed of an object is much smaller than the speed of light, the $\Gamma$ factor is almost exactly 1, and Newtonian mechanics with Galilean relativity rules. When the speed of an object approaches the speed of light, the $\Gamma$ factor becomes infinitely large, and then deviations from the Galilean relativity become very large.

$$\begin{array}{ccc} v = 0.9c & \phantom{AAAAA} & \Gamma = 2.3... ...1 \\ v = 0.999c & \phantom{AAAAA} & \Gamma = 22.4 \end{array}$$

Proper time and proper length

Since the time and the length appear differently to different observers, i.e. they become relative, it is important to have some invariant quantities as well.

Proper time is the time that is measured in the reference frame that is at rest with respect to an object.

Proper length is the length that is measured in the reference frame that is at rest with respect to an object.

Thus, if you want to measure a proper length of an object, or a proper time interval between two events, you need to be in the reference frame that is at rest with respect to this object or events.

• Proper time interval is always the smallest as measured by various observers.
• Proper length is always the largest.

The meaning of the Lorentz transformation

Modern physics has shown that all fundamental forces result from the exchange of particles. Two objects feel an electromagnetic force between them because they exchange a massless particle - photon. The gravitational force appears when two objects exchange another kind of massless particles, called graviton.

They both move with the same speed - the speed of light. Hence, the speed with which the gravity force propagates is the speed of light, not infinite as predicted by the Newton's law.

The Lorentz transformation appears if there is a finite speed for any force propagation. Galilean transformation will hold in a universe where forces propagate with the infinite speed.