The theory of special relativity is a scientific theory that focuses on the interaction between time and space and how the laws of physics are the same in all inertial frames. In this article, we will explore Einstein's theory of special relativity, the Michelson-Morley experiment, simultaneity and time dilation, length contraction, and some examples of special relativity.

Einstein's Theory of Special Relativity

Albert Einstein's theory of special relativity is one of the most important developments in physics. It explains how speed affects mass, time, and space and says that:

When we consider movement across enormous distances, this cosmic speed limit stimulates new areas of physics and science fiction. Einstein created the theory of special relativity with two simple postulates and careful measurements.

Although special relativity includes observations and experiments, it is founded on logically connected postulates, much like trigonometry.

First Postulate of Special Relativity

Einstein's first postulate concerns reference frames. All velocities are measured in relation to a reference frame. Here are some basic examples:

• If a man is running, his motion is measured relative to his starting point or the ground he is running on.
• If you throw a ball in the air, the ball's motion is measured relative to your standing position.

According to the first postulate of special relativity, the laws of physics are the same and can be stated much more simply in all inertial frames of reference than in non-inertial ones.

In an inertial frame of reference, a body at rest remains at rest, or a body in motion continues at a constant speed in a straight line unless impacted by an outside force. Also, the laws of physics seem to be much simpler in inertial frames. See the following example:

When you are on a plane flying at a constant speed and altitude, physics seems to work the same way as when you are standing on the surface of the earth. But if the plane is taking off, things are a little trickier.

In such a case, ie, when a plane is taking off, F, which is the net force of an object, does not equal the multiplication of mass and acceleration (ma). Instead, it is equal to ma plus a postulated force. Here is an example:

Let's say the speed of the plane is V0. When you throw a ball inside the plane at a velocity of v, you will see the ball moving at a velocity of v, but to a person who is standing on the earth, it will appear to be moving at a velocity of v + V0.

Not only are the laws of physics much simpler in inertial frames; they also are the same for all inertial frames since there is no preferred frame or absolute motion.

One of the most important outcomes of this first postulate is the famous mass-energy equivalence equation \(E = mc ^ 2\) that applies to a force in a near light moving frame.

Second Postulate of Special Relativity

The second postulate deals with the speed of light. The laws of electricity and magnetism say that in a vacuum, light travels at approximately \(3.00 \cdot 10 ^ 8 m / s\). However, they make no mention of the frame of reference in which light travels at this speed. The question is whether c (the speed of light) is constant or whether it is relative, in which case, for instance, an observer travelling at the speed of light might see the light waves as stationary.

Einstein concluded that an object with mass could not travel at the speed c. He also stated that light in a vacuum must travel at the speed c, which is \(3.00 \cdot 10 ^ 8 m / s\), relative to any observer. It follows that:

Special Relativity: Michelson-Morley Experiment

The Michelson - Morley experiment, conducted in 1887, was designed to determine the presence of the luminiferous aether, a postulated medium pervading space that is assumed to carry light waves. If the aether were to carry the light waves, the flow of the aether would change the light velocity by carrying the photons and adding more speed.

Can time intervals be different from one observer to another? Intuitively, we think of time as a process that is the same for everyone. However, in some cases, time seems to be going faster or slower. These different perceptions are related to the accuracy of the measuring of time. If you consider how time is measured, you will see that this is determined by an observer's relative motion with regard to the process being measured.

A foot race's elapsed time is the same for all observers, except that it is affected by the observer's relative velocity.

Simultaneity

Simultaneity describes the relationship between two occurrences supposedly happening at the same moment in a frame of reference. See the following example:

A fireworks show in Paris and another one in New York appear to be happening at the same moment. However, these two events will appear to be observed at different times by an observer on earth and another one moving from New York to Paris at near the speed of light. The second observer will see the fireworks in Paris earlier than the ones in New York, while the observer on earth will see the two events happening simultaneously.

Time Dilation

Time dilation is the concept that time is measured differently for moving objects than for stationary objects as they travel through space.

Time dilation occurs when one observer moves relative to another observer, causing time to flow more slowly. For example, time moves slowly in the International Space Station, with 0.01 seconds elapsed for every 12 earth months.

\[\Delta t = \frac{\Delta t_0}{\sqrt{1 - \frac{v^2}{c^2}}} = \gamma \Delta t_0\]

where

\[\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\]

Special Relativity: Length Contraction

Imagine you are travelling with a friend and discussing how many kilometres you have left to go. You and your friend may give different answers, but if you measured the road, you would come to an agreement because, travelling at everyday speeds, you would arrive at the same measurement.

Let's say that, to an observer on earth, a muon is travelling at a velocity of 0.950c for \(7.05 \cdot 10 ^ {-6} s\) from the time it has been seen until it disappears. It travels a distance of:

\[L_0 = v \Delta t= (0.950) \cdot (3.00 \cdot 10^8 m/s) \cdot (7.05 \cdot 10^{-6} s) = 2.01 [km]\]

This is relative to earth. In muon's frame of reference, its lifetime is Δt₀:

\[\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} = \frac {1}{\sqrt{1 - \frac{(0.950c)^2}{c^2}}} = 3.203\]

Our example envisages an observer on earth, so \(7.05 \cdot 10 ^ -6 s\) is Δt, and you need Δt₀ to find the length from muon's reference. As we saw before:

\[\Delta t = \gamma \Delta t_0\]

So if you put in the known parameters, you get:

\[\Delta t_0 = 2.20 \mu s\]

Now you can determine the length relative to the observer (L):

\[L = v \Delta t_0 = (0.950) \cdot (3.00 \cdot 10^8 m/s) \cdot (2.20 \cdot 10^{-6} s) = 0.627 [km]\]

In conclusion, the distance between the muon appearing and it disappearing depends on who measures it and how the observer is moving relative to it.

People may describe distances differently; in relativistic speeds, they really are different.

Special Theory of Relativity Examples

Examples of us being able to observe special relativity in our daily lives include:

• Gold's yellow colour. White light is a combination of all the colours of the rainbow. For gold, wavelengths tend to be longer when light is absorbed and re-emitted. As a result, the spectrum we perceive has a lower concentration of blue and violet waves. Because yellow, orange and red light have a longer wavelength than blue light, they make gold look yellowish.

• Mercury in liquid form. Similar to gold, electrons are kept close to the nucleus because of their speed and increased mass in mercury. Because mercury's atoms are only bound together weakly, it melts at lower temperatures and appears as a liquid.