Einstein’s Principle of Relativity

In his latest flesh of genius Christopher Nolan made many spectacular claims. Interstellar takes you onto an unexpected convergence, owing heavily onto the world of physics. But what is relativity really about and should economists get their hands on it?

In order to appreciate the contribution of the theory of relativity to the modern view of natural sciences and especially of physics and astronomy we should begin this journey (of the foundation of modern physics) by mentioning the scientific ideas and problems that this theory is related to. To start with, a wave is a disturbance that travels through a medium. For instance, water waves are disturbances traveling through water, while sound waves are disturbances traveling through air. After developing the idea that light consists of electromagnetic waves, James Clerk Maxwell supported the prevailing belief of many famous scientists of the nineteenth century. Space, empty of matter, is filled with a substance which is certainly the largest, and probably the most uniform body in the universe: the ether. As such, the ether is the medium through which light waves and stellar objects are traveling. At the same time, the ether was the well-known fifth element of Aristotle’s theory of world and nature. This theory describes the motions of objects of the universe while the rest of his four elements (Earth, Air, Fire and Water) describe the motions of object on Earth and its atmosphere. According to Aristotle, each object has the tendency to move to its natural positions and the observed movements of stellar objects were explained as movements of ether around the center of the universe, the immobile Earth.

Careful measurements have shown that light propagates with velocity c=2.99*meters/sec. Assuming the ether to be stationary, we may say that light propagates relative to the ether with velocity c. If the Earth moves through the ether with a velocity u without disturbing it, the velocity of light relative to the Earth should depend on the direction of light propagation. For example, the velocity of light relative to the Earth should be c-u (minimum velocity) for a ray of light propagating in the same direction that the Earth is moving  through ether, while, the velocity of light should be c+u (maximum velocity) for the propagation in the opposite direction. Of course, the velocity of light takes values between those two limit cases if the light is not propagating parallel to Earth’s movement through the ether. Therefore, if scientists were able to observe such differences in the velocities of light propagating in different directions then they will have been able to provide an indirect proof of the existence of the ether. The most famous experiment that aimed to test  the existence of the ether took place in 1887 by Albert Michelson (Nobel prize in Physics, 1907) and Edward Morley. In that experiment, apart from measuring the velocity of light that propagates in different directions, they also intended to determine the velocity of the Earth relative to the ether. Their experiment was repeated several times in different conditions. To their great astonishment, they found that within the high accuracy of their measurements, the velocity of light relative to the Earth was the same in all directions! This negative result leaded to extensive discussions among prominent scientists of that time. The main and crucial questions  were: Is the world we inherited from Galileo Galilei and Isaac Newton the real one? Does the ether exist? And if not, how can we explain the movements of stellar objects? How is it possible for light waves to propagate without the existence of such a medium? It was clear that the scientific community was puzzled; only few had anticipated this negative result. The strong supporters of the ether hypothesis tried to explain the result by assuming that the Earth drags the ether with it, as it drags the atmosphere. Therefore, close to the Earth’s surface the ether should be  rest relative to the Earth and thus the velocity of the ether drag would manifest itself in other phenomena connected with light propagation, such as the change in the direction of light coming from stars as the Earth moves along its orbit. Despite the great effort, such phenomena were never observed, and as such this solution was abandoned.

1905 was the year that the foundations of modern physics were established. After eight years of discussions and debates about the existence of the ether, the speed of light and the movements of stellar objects a 26 years old scientist working as a teaching assistant in the Swiss Patent Office, postulated the most revolutionary theory of the previous century rejecting the notion of the ether and generalizing the Newton’s theory of gravitation and mechanics.  Albert Einstein introduced the theory of relativity which is based on the following principle: Light travels through a vacuum at a constant speed c that is independent of the motion of the light source. So, regardless the direction of the propagation the velocity of light has a constant value and it does not depend on the relative movement of Earth. The laws of physics are the same for all inertial reference frames. In other words, the laws of physics are the same in all frames where the first Newton’s law of motion is applicable (frames with uniform linear motion under constant velocity and zero acceleration). After the postulation of his principle, Einstein extended the use of transformations developed by the Dutch astronomer Lorentz to study the relativistic effects. He introduced two reference frames, one with velocity equal to zero and one  moving with constant velocity ν with respect to the fixed frame at direction x. Hencerforth both satisfied the first law of Newton. Using the Cartesian system of coordinates he could write the transformation equations of the coordinates from the moving frame to the fixed one (the well-known Lorentz transformations): Writing these in mathematical terms one realizes the following: if the test particle or object moves with constant velocity v that approaches the velocity of light c, some effects (relativistic effects) become important that are not present to the Newtonian theory.

One important effect of relativity is the length contraction: Consider an object that moves with velocity v with a respect to an observer A that is immobile (zero velocity) in direction x and an observer B that is moving together with the object. Therefore, according to observer B the object is immobile. The length of the object as measured by observer A will appear foreshortened in the direction of motion with respect to the length measured by observer B (proper length). The amount of contraction can be calculated as following. If the length is measured by observer B and the length is measured by observer A, then the Lorenz transformations lead to: But since the two measurements are made simultaneously by observer A:

The ratio v/cor the Lorentz factor as it is called is indicative of the importance of relativistic effects. The closer the velocity ν to c is, the greater the ratio v/c is, and so the contraction is also greater.

  Another important effect that proves the famous phrase “The time is relative” is the time dilation: Assume that each of the observers A and B (who continues moving with velocity v together with the object) have a perfectly accurate clock, clock A and B, respectively. Clock A will be seen to be dilated (run slower) with respect to clock B. If the time interval is measured by observer A and time is measured by clock B, then using again the Lorentz transformations we find that:

But since observer A makes her time measurement at the same location (x_1=x_2), we conclude to: The third famous effect of the principle of relativity is that events that are simultaneous to one observer need not be simultaneous to another. Whether or not two events are simultaneous depends upon your frame of reference. The time order of events that are close together in time but distant in space can be different in different frames.

Apart from the answers the theory of special relativity provides, it triggers further questions, something necessary for the scientific development. Some of those questions lead to paradoxes and discussions in order to explain them.