Let’s get Relative: Theory of Relativity in Brief

Albert Einstein is probably the most scientific figure in the modern world for his messy hair and the theory of relativity. As the title suggests, in this article, we are going to discuss Einstein’s Theory of Relativity.

Actually, classical Physics is only applicable to the objects which are far slower than light and are far larger than an atom. This created a need for a new type of Physics which can explain any kind of motion and of every sized object including very large scale structures like galaxies, black holes etc. and also the small-sized objects like electrons, quarks, photons etc. 

Read the full article on ‘Classical Physics’ by clicking here.

This Physics is called the ‘Modern Physics’ which consists of the two major theories of Physics — ‘Theory of Relativity’ or ‘Relativity’ and ‘Quantum Mechanics’. 

Relativity was mainly developed Albert Einstein, but, some contributions of Hermann Minkowski, Hendrik Lorentz, Max Planck is also there.  Theory of Relativity comes in two different ‘flavours’ — The ‘Special Theory of Relativity’ and the ‘General Theory of Relativity’ or simply special relativity and general relativity.

Image taken from PicturesCafe.com

In this article we shall discuss various concepts of special and general relativity.


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The Difference

By name, special relativity seems to be more complex than general relativity, but actually, special relativity is much simpler than general relativity.

Special relativity is called ‘special’ because it is applicable only on objects moving with a constant speed. This is a very rare or ‘special’ case because in our universe hardly anything moves with a constant speed, almost everything accelerates in our universe. 

General relativity is applicable to objects moving with a constant speed as well as accelerating objects. Acceleration is a common or ‘general’ case in our universe. The most common reason for acceleration is ‘gravity’, so, the general theory of relativity explains gravity in terms of acceleration and vice versa.

Most of the part of general relativity can be understood without its complex mathematics and most of the part of special relativity can be understood only with its (simple) mathematics. 


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 The Meaning of ‘Relativity’

Now, we shall know what is actually the meaning of relativity. Relativity means ‘to be relative’. 

It can be understood by a simple example. Suppose you are inside a train and let the train is moving with 40 mph (miles per hour) of speed. When you see a person sitting in the train in front of you, the speed of the person will seem to be zero for you but for an observer outside the train, the person will be moving with a speed of 40 mph. 

So, this can be elaborated as, with respect to you, the person is at rest, but, with respect to an observer outside the train, the person is in motion. This is the basic principle on which the theory of relativity is based on. There is no state of absolute motion or rest, everything is relative. So, relativity, in a single sentence can be defined as –

Nothing is absolute in this universe, but everything is relative to each other” 

Only the speed of light is absolute, it doesn’t depend on the velocity of the observer.

Illustration of Relativity of Motion

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The Special Theory of Relativity

As described above, all the special and general relativity is based on the basic idea that nothing is absolute in this universe, but everything is relative to each other.

The special theory of relativity is based on two simple postulates:

  1. The fundamental laws of nature, that is, the laws of Physics (which govern the whole universe), don’t change for an object (or observer) at rest or moving with a constant speed.
  2. The speed (or velocity) of light remains constant for every object (or observer) and is independent of the velocity of the object or observer.
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The Idea of Four Dimensional Space-time

In this section, we shall discuss the ‘Minkowski Space’ or simply ‘Space-time’. To understand what is space-time, we should know what is actually ‘Space’ and what is ‘Time’.

Space is a three-dimensional structure, that is, it is a structure made up of three dimensions and time is a single dimensional structure.

This means that time is a single dimension like an ‘arrow’ going only in one direction (forward). That is why it is different from the three dimensions of space known as ‘Spatial Dimensions‘. For, example, a cube has some length, breadth and height. These length breadth and height are the three dimensions of the cube, and these are spatial dimensions.

Firstly we should discuss something about ‘absolute quantity’ and ‘non-absolute quantity’. An absolute quantity is a quantity, which remains the same for all observers, (for example, the speed of light) and a non-absolute quantity can differ from observer to observer, depending on the observer’s (or object’s) velocity. 

As described above, the basic principle of relativity is that nothing is absolute, but is relative to each other, so everything other than the speed of light should be considered as a non-absolute quantity. 

Classical Physics ends the idea of absolute space (or distance), that is, the distance between two points can differ for two observers. For example, when you look at the night sky filled with stars, the distance between them seems to be very small, but they are actually many light years away from each other! (One ‘Light Year’ is the distance travelled by light in a year. One light year is equal to  9,461,000,000,000 kilometers). So, this proves that distance (or space) is not absolute. 

According to the second postulate of the special theory of relativity, velocity or speed of light is absolute, that is, every observer will measure the same speed of light. We all are familiar with the equation, speed = distance / time, so, speed of light = distance travelled by light / time taken, and, we know that the speed of light is absolute. 

This situation can be possible in two cases, whether both distance and time are ‘absolutes’ or both are ‘non-absolutes’. If they are non-absolutes, their values should be dependent on each other in such a way that the speed of light ‘automatically’ becomes absolute, and, because the distance is not absolute, so only the second situation is possible. 

So, we can conclude that time measured by two observers can differ. Every observer has its own measure of time. Space (or distance) and time are now non-absolute quantities and both are dependent on each other. 

This predicts another possibility which was put forward by Hermann Minkowski in 1908. Because space and time are dependent on each other, so both space (three-dimensional structure) and time (one-dimensional structure) are always fused with each other to form a single ‘four-dimensional structure’ called ‘Space-time’ or ‘Minkowski Space’. Actually, Hermann Minkowski was one of the mathematics teachers of Albert Einstein, who refused to believe that the theory of relativity was developed by Einstein!


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Length Contraction – Squeezed Space

Special Relativity predicts that the measure of distance or length for a moving observer and a stationary observer will differ.

For example, consider a train moving with any velocity, a person is standing inside a train and a person is standing outside the train which is at rest. 

Suppose the person in the train has a torch and he switches on it when he is at position ‘A’. There is a distant screen ‘O’ in front of the person. Suppose light strikes screen ‘O’ when the person in the train comes at position ‘B’. 

Now, for the person outside the train, light would have covered a distance from ‘A’ to ‘O’, that is, ‘AO’ and, for the person inside the train light would have covered a distance from ‘B’ to ‘O’ only, that is, ‘BO’. This example is illustrated in the figure below.

The dotted ‘line’ represents the distance travelled by light with respect to the two observers and the ‘arrow’ represents the emitted light ray.

From the above figure, it is clear that the distance travelled by light is smaller for the moving observer as compared to the stationary observer. 

But, the difference in these distances will be almost zero at ‘daily life’ speeds or ‘slow’ speeds. ‘Slow’ here refers to far slower than the speed of light’. Only at speeds near the speed of light, this difference in length measured by a stationary and a moving observer can be observed. 

There is another case for the ‘contraction of size’. Consider an object moving at a speed near the speed of light (with respect to a stationary observer), when it will move with such a high speed, there will be a noticeable contraction in the object’s dimension (size or length) which is parallel to the direction of its motion. 

The size will be contracted even at slow speeds or ‘daily life speeds’, but that contraction will not be observable. 

There is a proper mathematical relation between ‘original length or size’ and ‘contracted length or size’ which is given below.

In this equation, ‘L0‘ is the ‘Original Size‘ or ‘Size measured by stationary observer‘.

L’ is the ‘Contracted Size‘ or ‘Size measured by Moving Observer‘.

v‘ is the velocity of the moving object or the moving observer and ‘c‘ is the speed of light.


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Time Dilation – Stretched Time

According to relativity, time is relative, every observer has its own measure of time. Time interval, like size, is also affected by velocity or speed. Time runs slow for a moving observer. This can be understood with the same example given in the previous section. For the moving observer, the distance travelled by light is shorter than that for the stationary observer. And, we know that:

Speed of Light = Distance Travelled by Light / Time Taken

So:

Time Taken = Distance Travelled by Light / Speed of Light 

As the speed of light is constant for all observers, so, the time will differ. For the moving observer (in the figure in the previous section), light took less time to reach the screen whereas for the stationary observer light took more time to reach the screen.

We can also say that less time has passed for the moving observer as compared to the stationary observer. 

So, in simple words, time runs slowly for a moving observer (as less time has passed for the moving observer in the example) as compared to a stationary observer.

This effect of ‘Time Dilation’ has also been confirmed experimentally with ‘Muons’ (a muon is a kind of particle). A muon is a type of particle which has a ‘lifetime’ of about 0.000002 seconds. ‘Lifetime of a Particle’ means the time for which it remains in its ‘original form’ and after completing its lifetime, it decays into another type of particle.

Muons have a lifetime of  about 0.000002 seconds or 2 micro-seconds, but it was found that muons travelling at ninety-eight per cent of the speed of light (very close to light’s speed) have their lifetime extended to five times, that is, a muon travelling at 98 % of the speed of light lives five times longer than a stationary muon. 

There is a well defined mathematical relation between ‘original time’ (at zero velocity) and ‘dilated time’ or ‘stretched time’ which is given below.

In th equation, Δt is the ‘Proper Time‘ or ‘Time measured by a stationary observer‘.

Δt’ is the ‘Dilated Time‘ or ‘Time measured by a moving observer‘.

v‘ is the velocity of the object or observer and ‘c‘ is the speed of light.

Suppose an object is moving with some velocity. When the velocity will increase with time, time becomes further slower for the object and when it reaches the speed of light, the time will suddenly stop for the object, that is, time will not pass for the given object. 

However, anything having some ‘rest mass’ (mass of an object when its velocity is zero) can’t travel at the speed of light or greater than the speed of light. The reason for this will be discussed soon in this article.

Now, a question may arise in your mind that if time doesn’t pass for anything travelling at speed of light, so does light itself experience time? The answer is no, light is made up of a type of particle called ‘Photon’, and, a photon actually doesn’t experience time. Time doesn’t pass for a photon. This is the reason why it doesn’t decay into other particles normally.


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Variation of Mass and Energy-Mass Equivalence

Mass of an object is also relative. It will increase when the velocity of the object will increase. The relation between ‘Rest Mass‘ (at velocity equals zero) and increased mass is given below.

In this equation, ‘m0‘ is the rest mass, ‘m‘ is the increased mass, ‘v‘ is the velocity of the object and ‘c‘ is the speed of light.

But, according to the law of conservation of mass, mass can neither be created nor be destroyed, in other words, mass can’t be changed. But in this case, actually, the total mass remains the same because the mass is not ‘created’, actually energy ‘converts’ into the mass. 

According to special relativity, energy and mass are the same things and can be converted into each other. Mass is only a very dense form of energy. The mathematical relationship between mass and energy is given by Einstein’s famous equation:

In this equation, ‘E‘ is the total energy of an object with a mass ‘m‘ (note that this is the relativistic mass).

c‘ is the speed of light.

Any moving object possesses some ‘Kinetic Energy’, and, this energy increases with an increase in velocity. This energy converts into mass and hence, the mass increases. On a daily basis, this effect of increasing mass is not observable because velocity is far smaller than light. This effect is observable when objects travel with velocities near the speed of light.

Now, we can discuss the reason for the speed of light being the ‘speed limit of the universe’. 

Suppose an object is travelling very close to the speed of light, because its velocity is very high, so its kinetic energy will also be very large, so now, much of the kinetic energy will be converted into mass, that is, most of the kinetic energy will be ‘used up’ to increase the mass and not velocity, so, it will become difficult to increase the velocity and it will never reach the speed of light.

It can also be understood in another way. When an object will travel at the speed of light, its velocity, ‘v’ will be equal to the speed of light, ‘c’ and hence, ‘v2 will be equal to ‘c2, and after solving the value of ‘Relativistic Mass‘ (by putting the value of ‘v‘ in the formula of relativistic mass), we get:

m = Infinite

Also because E = mc2, so energy will also become infinite.
This simply means that to accelerate an object to the speed of light, an infinite amount of energy will be required, which can’t happen. This is the reason why nothing can travel with the speed of light.


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The General Theory of Relativity

The equation for the strength (or force) of gravity, which Newton has given us is so accurate, that we still use it for practical purposes like flying a rocket from Earth and landing it on the moon or any other planet. But, there are two problems with Newton’s theory –  It fails for strong gravitational fields like a neutron star; The way Newton described gravity was not correct, in fact, Newton himself wasn’t satisfied with his theory. He didn’t know how actually gravity works. So, he ‘invented’ something called ‘Gravitational Pull’. But, he didn’t actually explain the reason of this gravitational pull. He didn’t explain how gravity works.

We still use this equation of Newton because it is much simpler than equations of general relativity. As Sun’s gravitational field beyond planet Mercury is weak, so we can calculate gravity between any planet (except Mercury) and Sun using Newton’s equation.

The orbits of all planets (except Mercury), predicted by general relativity are the same as those predicted by Newton’s theory of gravity.

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The Equivalence Principle

The general theory of relativity is the currently accepted model of gravity which describes gravity in another way. 

You may have searched for general relativity on the web and found this type of description – “Gravity is a consequence of distortion of space-time”. 

But what Einstein meant by distortion of space-time and how he developed this idea?

Einstein used his equivalence principle to develop this idea. The principle of equivalence is the foundation of the general theory of relativity. 

The equivalence principle states that — The effects in a gravitational field are the same as the effects in a uniformly accelerated body. In other words, you can’t be a hundred per cent sure whether you are in a gravitational field or you are being accelerated uniformly.

We can understand it with the help of the following examples.:

  1. A man is in a rocket at rest on Earth, he drops a ball in the rocket.
  2. The man is now in the same rocket without any contact with a gravitational field and the rocket is accelerating at an acceleration of 9.8 m/s2 (m/s2 is unit in which acceleration is measured and 9.8 m/s2 is the acceleration of a falling object due to the gravity of  Earth.) Now, again he drops a ball.

He will see the ball fall in the same way in both cases. So, we can conclude that the effects in the accelerated rocket are the same as the effects in a gravitational field. This is actually because of inertia. A  ‘pseudo-force’ is experienced by a person in a bus, when it accelerates. The same concept is here. The rocket is accelerating upwards, so, the person in the rocket will experience the pseudo-force in downwards direction.

Image taken from Wikipedia.

This is what we call the ‘Principle of Equivalence’. Einstein developed the most revolutionary idea with this principle of equivalence, that is, the idea of ‘Curved Space-time’.


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Curved Space-time and Geodesics

Einstein used the equivalence principle to predict that space-time curves. We can take an example to know how he did it.

Consider a rocket (which doesn’t have any contact with a gravitational field) which is 300,000 kilometres long, this means that light takes one second to reach from top to bottom (and vice versa) of the rocket and according to special relativity, light’s speed will remain constant no matter what the speed of the rocket is.

Consider, currently it is at rest or moving with a constant speed. Now consider two people, ‘A’ and ‘B’ are standing at the top and the bottom of the rocket respectively. 

A’ sends two pulses or waves (or rays) of light to the person ‘B’, at the bottom of the rocket at a time gap of one second, that is, he sends the second pulse of light after one second of sending the first pulse of light. The person ‘B’ with no doubt, will receive these two pulses at the time gap of one second

The blue-coloured arrow (pointing downwards) represents the light ray coming towards the person ‘B’. The green-coloured arrow (pointing upwards) represents the direction in which the rocket is moving/accelerating (with respect to a stationary observer).

Now, suppose that the rocket starts to accelerate and ‘A’ gets out of the rocket. Consider that ‘A’ is at rest. Now, if ‘A’ sends two pulses of light at a time gap of one second, the person, ‘B’ standing at the bottom will receive the second pulse in less than one second

Why does it happen? The answer is because the light has to travel less distance in this case as ‘B’ is moving towards the light rays.

The person ‘B’ is moving towards the light ray and the light ray is moving towards the person, and, because the speed of light is absolute, the distance (or space) which the light ray has to travel is decreased. So, the person at the bottom will receive the two pulses in a time interval less than one second, that is, time is lengthened (or dilated or slowed down) for ‘B’ because less time has passed for him.

This means that space has contracted (or shortened) for the person ‘B’ at the bottom of the rocket and time has dilated (or lengthened) for him. But, what can we conclude from this? 

According to the equivalence principle, the effects in accelerating bodies are the same as the effects in gravitational fields. So, we can conclude that in a gravitational field, space contracts and time dilates, in other words, near a source of mass or energy, space is contracted and time is dilated and this effect goes on decreasing as we move away from the source of mass (or energy). 

Gravitational Time Dilation’ has been experimentally measured using atomic clocks on airplanes. The clocks aboard the airplanes were slightly faster than clocks on the ground. This effect is so significant that the GPS satellites need to have their clocks corrected!

It should be noted that time dilates (and space contracts) for both moving and accelerating objects (with respect to an observer at rest).

But, how this concept explains gravity? Now, consider an apple falling on Earth. Why does it fall? The answer is because the earth has mass, so, near the earth, space is contracted and time is dilated, and because space is contracted, it pushes the apple to the ground. This is how gravity works. This ‘Gravitational Space Contraction’ and ‘Gravitational Time Dilation’ is known as ‘Curving of Space-time’. 

Space pushes objects only in some definite paths, the straight path between any two points in four-dimensional space-time. This straight path between two points is called ‘Geodesics’. 

All the objects in four-dimensional space-time follow straight lines in four-dimensional space-time or they follow the shortest path between any two points (because space pushes them to move in these straight paths) in space-time. This shortest or straight path is called geodesics. 

Now, the question arises, if space pushes an object to travel in straight paths, why Earth and other planets are orbiting the Sun in elliptical orbits? We can take an example to understand this. 

The surface of Earth is two-dimensional and is curved because it is a sphere. Now, on a globe of Earth, mark two points far enough and join them, do you get a straight line after joining them? The answer is no. The line or shortest path between these two points is curved. Now, if a rocket flies over Earth in a straight path, its shadow on Earth, that is, its path on a two-dimensional surface, will be curved. This means that the rocket is moving straight in three-dimensional space but its path seems to be curved on the two-dimensional surface of Earth.

This can be understood the figure below.

The rocket is moving straight in three dimensional space but its path seems to be curved on the two dimensional surface of earth.

This effect is the same for four-dimensional space-time and three-dimensional space. For example, space is pushing Earth and other planets to follow the geodesics, that is, the straight path in four-dimensional space-time but they seem to move in curved paths in three-dimensional space. 

But when an apple falls, it moves in a straight path in three-dimensional space also, that is, the apple is not moving in curved paths in three-dimensional space when it falls. Why? When you will mark two very nearby points on a very large globe of Earth, and join them, you will find that this line or shortest path is straight. This means that the distance between the two points in space (which the apple covers) is too short that its path in three-dimensional space is also straight.

Now, how can gravity be defined in brief? We can define gravity as follows:

Gravity is not a force like others but is a consequence of the ‘curvature of space-time’. The presence of mass and energy curves space-time. 

‘Curving’ of space-time means that near a mass, space contracts and time dilates and this effect goes on decreasing as we move away from the source of mass (or energy). 

All the objects follow the shortest path, that is, a straight path in four-dimensional space-time. These paths are known as ‘Geodesics’ and space pushes the objects to follow these geodesics (straight paths). However, the objects may seem to move in curved paths in three-dimensional space.


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Gravity as Waves!

As we can’t imagine a four dimensional structure as our brain creates a three dimensional model of everything, so we can visually represent space-time as a two dimensional ‘fabric’ (below).

The distortion in space-time ‘fabric’ can be represented like this (below).

Space-time can also be represented three dimensionally and distortion in it can be represented like this (below).

Space-time can be ‘disturbed’ and these disturbances in it are called ‘Gravitational Waves’. A wave can be defined as a disturbance in anything (like a medium or field). 

The waves which disturb a medium [like water waves are disturbances in the water, sound waves disturb all solid, liquid and gas] are called ‘Mechanical Waves’. 

The waves which disturb a field, like light waves disturb the ‘Electromagnetic Field’, are called ‘Electromagnetic Waves’. 

Similarly, the gravitational waves disturb space-time. Gravitational waves can be represented like this (below).

Gravitational waves also travel at the speed of light. This means that Earth will take eight minutes to leave its orbit if the Sun suddenly dissapears. Newton’s theory predicts that if Sun disappears, all the planets will instantly leave their orbit, but this would mean gravity travels faster than light, which violates the special theory of relativity. But, as gravitational waves travel at a finite speed, so, it will take some time for the planets to leave their orbit. This is illustrated in the image below.


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References

  1. Einstein, Albert, & Renn, Jurgen (2015). Relativity – The Special and General Theory. Princeton: Princeton University Press.
  2. Hawking, Stephen (2000). A Brief History of Time. New York: Bantam Books.
  3. Hawking, Stephen (2005). A Briefer History of Time. London: Bantam Press.
  4. Theory of Relativity. (n.d.). Retrieved from Wikipedia: https://en.wikipedia.org/wiki/Theory_of_relativity
  5. Chen, Susan (2017, January 1). The Basics of Special Relativity. Retrieved from Passion for STEM: https://passionforstem.wordpress.com/2017/01/02/the-basics-of-special-relativity/

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Author: Madhur Sorout

Madhur Sorout is currently a fifteen-year-old eleventh-grader from India. His main fascination lies with the subject of physics, mainly in the field of the general theory of relativity and topics related to it like the Big Bang, black holes and the evolution of the universe. He likes to make sense of what he sees in this universe. He has founded the (popular) science website – Maddyz Physics (maddyzphysics.com). He is also a physics and astrophysics editor for the Young Scientists Journal. He loves to read books by Stephen Hawking, Neil deGrasse Tyson and other authors (and physicists). Inspired by their work, Madhur wrote Astrophysics Simplified: A Simple Guide to the Universe. He started to write this book when he was 14. A diehard fan of fiction, Madhur also likes to play cricket and wants to continue down the route of research in theoretical astrophysics.

4 thoughts

    1. There is not a fixed limit of speed where classical physics fails but the speeds are near the speed of light where the “Relativistic Effects” can be observed.

      Btw, classical physics also fails at objects of sizes 10^-9 metres and smaller than this value.

      Liked by 1 person

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