The Special Theory of Relativity

The Special Theory of Relativity

By Michelle Ng

Image source: https://rhapsodyinbooks.wordpress.com/2017/05/29/review-of-now-the-physics-of-time-by-richard-a-muller/large-5/

INTRODUCTION

What is ‘relativity’? The theory of relativity is often explained using ‘the train’, a situation that illustrates the basic idea of this revolutionary theory. This ‘train’ thought experiment is as follows: 

If someone throws a ball vertically upwards in a moving train, the person in the train will see the ball move upwards and eventually fall back down as a projectile with zero horizontal velocity. But someone standing on the platform observing the ball will see it follow a parabolic path: the ball will appear to have the same horizontal velocity as the velocity of the train. This shows that different observers can disagree on the position of events in space; while the person inside the train thinks the ball ends up where it started, the person on the platform believes the ball ends up at a certain horizontal distance away. 

But this is not the full story. In fact, the idea of relativity described by this train had already been realised a long time ago by classical physicists such as Galileo. Classical physicists have already established that velocities and positions in space are not absolute, but did you know that different observers can also disagree on the time at which events occur? 

THE EXPERIMENT THAT ‘FAILED’

It all started when people realised a contradiction between existing theories: according to the classical ideas of relativity, light would also be travelling at different speeds for different observers (imagine if light was the ball in the train scenario). But Maxwell’s equations showed that light is a wave comprised of the oscillations of the electric and magnetic fields, and propagates at a fixed speed, c. So does the speed of light depend on the person, or is it a universal constant?

Because of this, people thought that there was an invisible material that permeates through space called the ‘ether’, and that everything, including light, is travelling through this ether. People thought the ‘fixed’ speed of light calculated by Maxwell is relative to this stationary ether, that is, if the ether was an observer, Maxwell’s speed of light would be what the ether observes. 

Contemporary scientists wanted to measure the speed of the Earth travelling through the ether, so they designed an experiment to do so. They used an instrument called an interferometer to measure how long it takes for light to travel horizontally compared to vertically. Essentially, 2 beams of light are emitted at the same time from a point, with one travelling upwards, perpendicular to the motion of the Earth through the ether (called the transverse arm), and one travelling to the right, parallel to the motion of the Earth (called the longitudinal arm). The light beams are then reflected by mirrors and both travel back to a screen where they can be seen. [1]

If the speed of light is really dependent on the observer, then calculations show that the light travelling parallel will take longer to return (Imagine running next to someone: they will appear to be moving slower if you are running alongside them compared to if you stood at the side and watched, or if you ran in a direction perpendicular to them. In this case, the longitudinal light is moving alongside the Earth, so the Earth will ‘see’ the longitudinal light moving slower than the transverse light.) Because one of the light rays takes longer to return, the 2 light waves will not be in phase (i.e. at the same position in their cycles) when they arrive at the screen. When the waves overlap, constructive interference occurs when the waves add together, creating bright fringes; destructive interference occurs when the waves cancel out, creating dark fringes. This pattern can be used to find the time difference between the 2 beams of light, or the difference in the speed of the 2 light beams. This can then be used to find the speed of the Earth in the ether. [1]

Figure 1: The interferometer setup. [2]

Figure 2: Wave Interference [3]

To everyone’s dismay, the experiment failed spectacularly—there was no interference found. The 2 beams of light took equal amounts of time to travel even though one is moving alongside the Earth: the speed of light was not affected by the motion of the Earth. How is this possible? 

This experiment, famously known as the Michelson-Morley experiment, was a major contribution towards the development of Einstein’s theory. The physicist Lorentz resolved the problem by saying that lengths are relative. If there is an object, someone that is moving relative to the object will measure a foreshortened length than someone who is stationary relative to the object. [1] The ‘proper length’ of an object is defined as the length of the object measured by someone who is not moving relative to the object. [4]

L = length measured by moving observer

L0 = proper length measured by observer that is stationary relative to the object

v = speed of the moving observer

c = the speed of light

The Earth is moving in this case, so according to a ‘stationary’ ether, the distance that the parallel light has to travel is shorter, less time is needed than expected.

However, people struggled to explain this length contraction without assuming that this was just an intrinsic property of the ether caused by the electromagnetic force. 

THE DILATION OF TIME

Now enter Einstein, who saw this differently: he completely abandoned the idea of an ether and instead of saying that the speed of light depends on the observer, he postulated that it is the time at which events occur that is different for the observers, while the speed of light remains the same. He thought that a (‘stationary’) observer will see a moving clock (moving relative to the observer) run slower than a stationary (relative to the observer) clock: the moving clock time is dilated. [1]

Here is a thought experiment: imagine a clock inside the (horizontally) moving train. The clock is modelled as a beam of light being repeatedly reflected between a pair of mirrors so the light travels up and down; the clock itself measures regular time intervals as the time it takes for the light to reflect off a mirror, reach the opposite mirror, and return. [1]

Now consider the time taken for each observer to see this happening. For a person inside the train, the light is moving vertically away from one mirror to the other, and back. The time taken for this to occur, if the distance between 2 mirrors is L, will be equal to 2L/c.

But for an observer outside the train, the light is travelling in a diagonal path because the observer will also see the train moving horizontally forwards, so the observer outside the train will see that the light has to travel a greater distance. Using Pythagoras’ theorem, this distance is now √(L2+(vt/2)2) (where vt is the distance travelled by the train in the same time it takes for the light to reach the opposite mirror). Therefore, the observer outside the train will think that the time taken for light to travel between the mirrors is longer, at 2√(L2+(vt/2)2)/c, because the distance is greater but the speed of light is the same. Hence the observer will believe that the moving clock is running slower. In general, if there are 2 events, someone who sees the 2 events happening at 2 different locations will measure a longer time than someone who sees them happening at the same place. The ‘proper time’ between 2 events is defined as the time interval measured by the person seeing them at the same place. [4]

Figure 3: A diagram showing the clock from 2 different frames of reference [5]

T = time between 2 events measured by observer that sees the events happen at different places

T0 = time between 2 events measured by observer that sees the events happen at the same place

v = speed of the moving observer

c = the speed of light

How does this explain length contraction? If we consider the distance between 2 points AB which are not moving relative to each other: [6]

• For someone who is stationary relative to AB, the distance AB will be the proper length of A, L0. If there is another observer moving at speed v from A to B, then the stationary person will say that AB is also equal to vT, where T is the time taken for the trip as measured by the stationary person, and

  
  
  

• For the moving observer however, the time taken for the trip will be the proper time for the trip, T0. The moving observer will also think that AB is equal to vT0.

Since T is longer than T0 due to time dilation, the length measured by the moving observer will be shorter than the one measured by the stationary one:

• 

THE GROUNDBREAKING THEORY

Using these ideas, Einstein went on to discover the special theory of relativity, which states that:

1. The laws of physics are the same for all observers that are not accelerating.

2. The speed of light is constant for all observers. [1]

In most everyday situations, relativistic effects are negligible because the speeds of objects we normally observe are very small compared to the speed of light, so the effects of length contraction and time dilation are minuscule. However, this theory explained many phenomena involving objects travelling much faster, with a famous one being the muon decay problem. On Earth’s surface, we can detect many muon particles coming from space. However, the muon has a very short half-life so in classical theory, the vast majority should not even be able to reach the ground as they will have decayed during the journey downwards. This seems to be a paradox, but special relativity can explain this.

The muons are travelling at very high speeds close to the speed of light, so relativistic effects are significant. For the muons, the distance between the muon and the Earth becomes foreshortened compared to the distance measured by someone on Earth and in the muons’ point of view, they do not actually spend that much time travelling (the time measured by muons is equal to the proper time)—this proper time is short enough such that many muons reach the ground before decaying. For someone on Earth, the distance measured will be the ‘proper length’, but the time measured by someone on Earth is much longer than that measured by the muon, which explains why we think they could not have reached the ground. [7]

Figure 4: Explaining the muon decay problem from different frames of reference [11]

This theory also paved the way for many new scientific ideas, with one of the most interesting ones being that the temporal structure of the universe is based on a fabric of space-time. Since both distances and times depend on the observer, they cannot be defined absolutely on their own. However, space and time are not actually separate entities because they are linked together by the speed of light, which is fixed for all observers. Because of this, we had to abandon the idea that 3-dimensional space runs independently in 1-dimensional time and instead consider the idea of 4-dimensional space-time. [8]

Moreover, the theory led to the consequence that nothing can move faster than the speed of light. As an object tends towards moving at the speed of light, the Lorentz factor (1-v^2/c^2) tends towards 0, so the measured length of that object (by a ‘stationary observer’) will tend towards 0 whereas the time of the object’s clock will appear to run infinitely slowly.

The importance of light also became a new way to explain casualty, which can be illustrated by space-time diagrams. When an event at a point P in space-time occurs, light spreads out from P in all directions at the speed of light and this light can then reach further points in space-time—a future light cone can be constructed for the event P, which is the set of points that light from this event can reach. (This means that light from P can travel along the surface of this cone to reach other points on the cone.) Since nothing can travel faster than the speed of light, any points that are outside of this cone cannot be reached by light from P itself, and hence cannot be reached by anything else that is coming from this event (eg. a car): this means that the points that are outside of this cone cannot be affected by the original event. On the other hand, any point that lies on the surface of the cone will ‘see’ the event’s light and hence can be affected by it. [9]

Similarly, a past light cone can be constructed for the event P. The surface of the past light cone is the set of events that can affect P because anything that lies on this surface can emit light that will be able to reach P. [8]

Figure 5: Diagram of a light cone [10]

In fact there is such an overwhelming number of consequences that stem from this theory: mass-energy equivalence, space-time intervals, the loss of a common ‘present’... the list goes on and cannot be summarised in a single article. Even general relativity, which describes how masses govern the curvature of space time and is one of the two most significant theories in physics, was developed to explain the unresolved issues that came from special relativity. Special relativity revolutionised the way we see space and time, earning it the title of one of the most paramount discoveries in history.

BIBLIOGRAPHY

[1] Feynman, Richard Phillips, et al. Six Not-so-Easy Pieces: Einsteins Relativity, Symmetry, and Space-Time. Perseus Books Group, 2011. 

[2] Rami Arieli: "The Laser Adventure", perg.phys.ksu.edu/vqm/laserweb/Ch-10/F10s0p4.html. 

[3] Encyclopædia Britannica, Encyclopædia Britannica, Inc., kids.britannica.com/students/assembly/view/53869. 

[4] Length Contraction, Time Dilation, and Lorentz Transformations, phas.ubc.ca/~mav/p200/lttips.html. 

[5] “Time Dilation.” Wikipedia, Wikimedia Foundation, 6 Feb. 2021, en.wikipedia.org/wiki/Time_dilation. 

[6] Foundation, CK-12. “12 Foundation.” CK, ck12.org/book/ck-12-physics---intermediate/section/22.2/

[7] “Muon Experiment.” Muon Experiment in Relativity, hyperphysics.phy-astr.gsu.edu/hbase/Relativ/muon.html#c2. 

[8] Rovelli, Carlo, et al. The Order of Time. Allen Lane, 2019. 

[9] Hawkings, Steven. A Brief History of Time. Bantam Books, 1988. 

[10] “Light Cone.” Wikipedia, Wikimedia Foundation, 25 Dec. 2020, en.wikipedia.org/wiki/Light_cone. 

[11] Libretexts. “5.4: Time Dilation.” Physics LibreTexts, Libretexts, 26 Nov. 2020, phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Map%3A_University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/05%3A__Relativity/5.04%3A_Time_Dilation.