TIME DILATION due to Velocity

Imagine two mirrors kept parallel to each other with a photon bouncing back and forth between them. Consider this apparatus as a photon clock where one strike of the photon with each mirror counts as one tik.

In the first case, the apparatus is observed on the ground by a man who is stationary relative to it. In the second case, the apparatus is carried on a plane moving at a certain velocity relative to the man.

The photon clock as observed by the person on the plane.

Now, consider the man standing on the ground and observing the photon clock on the moving plane. As the plane is moving forward, it would appear to the man that the photon is not just bouncing up and down but also moving along with the plane.

The photon appears to move in this manner for the man on the ground observing the photon clock on the moving plane.

So, let during each tik of the photon clock the plane moves forward by ‘x’ m (or the velocity of the plane is x m/tik). so, the man on the ground would observe that after each strike or after each tik of the photon clock, the photon has to move ‘x’ m forward to reach and strike the other mirror. This means the distance that light has to travel has increased in the inertial frame of reference of the man on the ground.

Now, as special relativity had already proved the speed of light is constant for all observers and as in this case, the distance travelled by light increases, this could only mean that the time elapsed between two consecutive tiks decreases according to the man on the ground. (As velocity=distance/time, if velocity is constant and distance increases, the time is bound to decrease) It would appear to him that the photon clock is moving slower than normal. In fact everything on the plane would appear to be moving slowly according to him because the effect observed by him in the photon clock would also apply to everything else on the plane (as the whole plane itself is moving at a constant velocity).  Time would appear to have slowed down inside the plane!

So, what he would be observing is the result of time dilation due to relative velocity. It states that time slows down for an object moving with a relative velocity (as in this case the plane is moving relative to the man). 

It’s spooky but is tested and true.

Click Here for ‘Time dilation due to gravity’ 

TIME DILATION due to Gravity

Time dilation is an interesting phenomenon that was introduced by special relativity. Time dilation, as the name suggests, is the effect of slowing down of time. Time dilation introduces possibility of actual time travel into the future. It may sound science fiction for some, but it’s a well tested and practical phenomenon.

Time dilation has two independent causes- gravity and relative velocity. In this article we will see what's time dilation due to gravity and how it is caused.

When light moves against a gravitational field caused by any massive body, its energy decreases. We know that energy of an electromagnetic wave of frequency 'µ' is given by E=hµ where ‘h’ is the plank's constant. So, from this equation we get that energy of an electromagnetic wave is directly proportional to its frequency. This means as the light moves up in a gravitational field, it energy goes down causing its frequency to go down as well.

You must be familiar with the equation v=λf. It gives us the relation between the velocity of a wave ‘v’, its frequency ’f’ and wavelength ‘λ’. Now here comes the role of special relativity. Special relativity states that the speed of light just like any other electromagnetic wave is constant or absolute for all observers no matter where they are or in what speed they are moving. So, in the equation stated above, the velocity 'v' becomes constant. As a result we get that the frequency 'f' is inversely proportional to the wavelength 'λ'. So, as the frequency of the light moving against gravity decreases, it wavelength increases.

We know that f=1/t, where 't' is the time period of oscillation of the wave. So the above equation can be rewritten as v=λ/t. Now, as 'v' is constant (as proved earlier), we get that wavelength is directly proportional to the time period. So, as the wavelength of the light wave moving against gravity decreases, the time period of oscillation of the light wave increases.

So, when a light wave (coming from an event near the source of gravity) with lower time period reaches an observer who is far away from the source of gravity, it would appear to him that the event is taking more time to happen. In other words, it will appear to him that the time has slowed down at the place and everything is taking longer to happen.

This strange effect he would be observing is nothing but time dilation which, in this case, is caused by difference in gravitational field intensity for the observer and the event.

Relativity of SIMULTANEITY- A logical proof for special relativity

One of the special things about special relativity is that it has many interesting consequences. 

One of the direct consequences of special relativity is the relativity of simultaneity. It means that one cannot be sure that two events takes place simultaneously or at different times. This can be demonstrated by a famous thought experiment by Einstein. Einstein imagined a train buggy moving with constant velocity and inside it are three people. One person is standing exactly at the middle of the buggy and the other two are standing at the opposite ends of the buggy so that they are equidistant from the person standing at the middle. Now the person at the middle lights up his lighter which sends out two beams of light towards the two other men standing at the ends of the buggy. Now as the two are standing equidistant from the middle and as the speed of light does not depend upon the speed of the source (here the speed of the source i.e the person at the middle is same as the speed of the train), the person at the middle should observe that the light from his lighter should reach the other two persons at the exact same time.

Now let’s view the situation from perspective of a person outside the train. Suppose the train passes through a platform and a person standing on the platform happens to observe the experiment. So, what would he observe?

He would observe that both the people at the ends of the buggy would be moving along with the train but the 1st person would be moving towards the source of light (the man at the middle) while the 2nd person would be moving away from the source of light. So, it is obvious that the man on the platform would conclude that the 1st person receives the light earlier than the 2nd person. So, now who is correct, the person in the middle of the buggy or the person on the platform? This may sound strange, but according to special relativity, both are correct because in special relativity time is relative to the observer. 

Now you see how awkward it is that two events occurring simultaneously for one observer may not be simultaneous for another observer. So, this is relativity of simultaneity. It can be considered as a logical and simple proof for special relativity as it clearly demonstrates how time can be relative.

Special Theory of Relativity

Einstein’s theory of special relativity came up in 1905 in his paper named “On the Electrodynamics of Moving Bodies”. It was one of Einstein’s Annus mirabilis papers published that year.

Special Relativity for the rescue!

(Read this first) Until special relativity, Galileo, Newton and the others believed that only space was absolute. But time was believed to be a completely absolute quantity irrespective of observer or condition. But special relativity quite surprisingly eradicated this logic and stated that time just like space was relative to observer. It means that different observers might also disagree upon the time that an event takes place just as they might disagree upon the position where the event takes place.

Now let’s implement special relativity into the previous analogy(two observers observing a certain event in space) . So, due to special relativity, time becomes relative for the two observers. As a result they will get different values of time just as they got different values of distance for the same event. But ultimately, when they calculate using their dissimilar values of distance and time, they get the same value for velocity of light. So, anyone no matter anywhere he or she is will observe the same velocity of light. Also speed of light will not depend upon the speed of the observer. So, this is how special relativity effectively solved the inconsistency between principle of relativity and Maxwell’s theory.

So, the main thing that special relativity did was that it extended and improvised the notion of relativity by including time in it and thus stated that space and time are not two different things but are combined into a single interwoven continuum known as the spacetime.

Einstein’s Postulates of Special Relativity

So, the following are the postulates of the improvised theory of relativity given by Einstein-

1. The principle of relativity applies to all physical objects
2. Maxwell’s equations are valid in all inertial frames
3. Light travels without a medium and thus no ether is required
4. The speed of light is constant in all inertial frames

Consequences of Special Relativity

Some amazing consequences of special relativity are the follows-

• Relativity of simultaneity (two simultaneous events may seem to be occurring at different times) 
•  Mass- Energy Equivalence (E=mc² -the most famous equation on the earth)
• Time Dilation due to Gravity (time slows down for a body in a strong gravitational field)
• Time dilation due to relative velocity (time slows down for a body moving with a relative velocity)
•  Length contraction (the apparent length of a body moving with a relative velocity increases)
• Relativistic mass (Mass of a body moving with a relative velocity increases)

The time, length and clock rate vary by the same factor i.e. γ(v) . It is known as the Lorentz Factor. Its value is given by-

Suppose a spaceship is observed in two cases – when it is at stationary relative to observer and when it is moving with velocity ‘v’ relative to the observer. Then the following results will be obtained-

	SPACE SHIP WHEN STOPPED	SPACESHIP WHEN MOVING WITH VELOCITY ‘v’
SPACESHIP’S LENGTH	L	L / γ(v)
SPACESHIP’S MASS	M	M x γ(v)
SPACESHIP’S CLOCK RATE	T	T / γ(v)

Maxwell Vs. Relativity

People already knew about the principle of relativity well before Einstein. (Click here for 'Relativity before Einstein). Galileo introduced relativity and Newton’s equations confirmed it. Newton’s equations of motion had shown that every event in space is relative to the observer and that every motion in space or velocity of anything is not absolute, but are relative to something else. In other words Newton’s equations of motion eradicated the notion of an absolute space which was till then the accepted norm. The lack of an absolute space seemed bizarre (even Newton found it difficult to accept it because he believed that it would indicate towards lack of an absolute god), but had to be accepted.

Ether

The principle of relativity originally predicted that everything moved in space relative to something else. But soon it was realized that this statement applied to everything except one that is light. Light was believed to be moving in vacuum, or simply, it was moving relative to nothing. This was no doubt contradictory for the principle of relativity. So, to save the theory, scientists assumed that the whole universe was filled with a substance known as ether and that everything including light travelled relative to this universal ether. Ether was just assumed to be there, it was never detected.

Now, the possibility of ether introduced a new hypothesis. Imagine the universe filled with an infinite sea of ether and all celestial bodies including earth moved through it. So this meant the ether moved relative to the motion of the celestial bodies. And as light also moved relative to the ether, the velocity of light would seem to be different in different parts of the universe. Different observers moving relative to ether would see light coming towards them at different. speeds, but the speed of light relative to ether would remain fixed. So, this meant that the speed of light was also relative to the observer. 

A SIMPLE ANALOGY-

This can be understood by a simple analogy. Imagine two observers observing a certain event at a certain place in space. So, now as space is relative, both the observers would get different values for the distance at which the event occurred, but as time was believed to be absolute, both of them will get the same value for the time taken for the light coming from the event to reach them. Now, as speed simply means distance divided by time, the observers would get different values for the speed of light. Well this fact about light was readily accepted (though not proved) but only until 1861 when Maxwell's theory of electromagnetism came up.

Maxwell’s theory & Michelson-Morley Experiment

According to Maxwell’s theory of electromagnetism, light was an electromagnetic wave which moved at a constant speed irrespective to the observer. This theory also got its proof when the famous Michelson-Morley experiment was conducted. In 1887, two scientists named Albert Michelson and Edward Morley conducted the first experiment to test whether light's speed did vary. It was expected that as earth was moving through the ether, there would be ether wind blowing all over the earth in a direction opposite to the direction of motion of earth through it. This meant that light would travel faster along the direction of the ether wind and it would travel slower against the direction of the ether wind. But, in which ever direction they measured, the speed of light appeared to be exactly the same! This result was quite unexpected and shocking for the scientists themselves. Michelson was later awarded the Nobel Prize in physics for his astounding experimental work.

The experiment was conducted by a device known as the interferometer invented by Albert Michelson. It could precisely measure changes in the speed of light along two arms set perpendicularly with mirrors at ends.

So the Michelson-Morley experiment proved Maxwell’s
electromagnetic theory but at the same time, posed a threat against the theory of relativity. This incompatibility of principle of relativity and Maxwell’s theory was quite a great problem. 
But in 1905 Einstein’s special relativity came for the rescue. (Click Here for SPECIAL RELATIVITY)

Albert Abraham Michelson

Edward Williams Morley

Principle of Relativity (before Einstein)

Many think relativity is a concept introduced by the genius Einstein. It is wrong. Relativity actually dates back to the times of Galileo. But it was Einstein who extended and improvised the idea and showed that it was not a paradox but a reality of nature by his special and general theory of relativity.

So, let’s first go to the origins of relativity. The principle of relativity is considered to be first proposed by Galileo in 1632 in his 'Dialogue Concerning the Two Chief World Systems'. Galileo gave the first law of relativity. And then, when Newton proposed his three laws of motion in his infamous 'Principia Mathematica', the second law of relativity was derived and the first law was confirmed.

The principle of relativity before Einstein originally had the following postulates-

1st- Laws of nature appear to be same for all observers moving with constant velocity.

2nd- Absolute velocity is has no physical meaning, only relative velocities are meaningful.

Consider a completely sealed chamber whose interior is completely free from any outside interference. This chamber is occupied by an intelligent scientist who has access to a lot of scientific instruments and is free to conduct any kind of experiment, but the condition is that he can conduct his experiments within the premises of the chamber not outside it (which means he even cannot send or receive any sort of signals from the outside). Now, without knowledge of the scientist, the chamber is strapped on to a rocket flying at a high but constant speed heading straight towards an active volcano! Now, he is given a challenge, that if he could anyhow find out the speed of the rocket (while within the chamber), then only he will be released.

So, can he do it?

The answer is NO! Because the first postulate of relativity forbids him to do so. According to the first postulate, laws of nature appear to be same for all observers moving with constant velocity. So, any experiment would appear to be the same for objects stationary on the ground and objects moving with a uniform velocity. So, any experiment conducted by the scientist within the chamber on the rocket will give him normal results (just as he was stationary on the ground). If he hadn't even been told that he was on a rocket he wouldn't even have known it and would have certainly met his death.

Therefore, there is no way he could have found out the speed of the rocket, no matter how intelligent he is.

Let’s come to the 2nd rule of relativity. The 2nd rule states that no one can calculate ones absolute velocity. You can only calculate your velocity relative to something else. So, if the scientist could have carved out a window out of the wall of the chamber, he could have got an idea of his speed relative to the wind blowing backwards. (But that would be cheating of course).

Now, consider a train moving at 100 mph. Now, this speed is not its absolute speed because, as earth spins around its axis we are being carried at 1000 mph, as it orbits the sun we are being carried at 70,000 mph,  as the sun orbits the milky way galaxy, we are moving at 500,00 mph. Also our galaxy is moving at some 800,000 mph through the cosmic microwave background. So, what is the actual or absolute speed of the train? Theory of relativity states that one’s absolute velocity is unobtainable. We can only state that the train is moving at 100 mph relative to the earth.

So, due the rules of relativity, we could never know what our actual position is or what our actual state of motion is. We can only state that we are here relative to something else or we are moving relative to something else. So, the original theory of relativity completely eradicated the concept of an absolute space, which was believed in until then.

Relativity proved that space is not absolute. The position of events occurring in space depends on the one who observes it. Different observers might disagree on the actual position of an event. But, the all are correct. Because it is what they observe according to their frame of reference, (provided that the frame is inertial, as relativity does not apply to non inertial frames just as Newton’s laws of motion does not apply in non inertial frames.) For example, a person on the train might argue that the train is stationary and the earth is moving relative to it and there is no reason to disagree with him.

So, basically relativity eradicated the notion of an absolute reality which was unacceptable by many. Although Newton’s laws of motion indicated absence of an absolute space, he denied to accept it because it did not agree with the notion of an absolute god.

Why can't we see light?

A peculiar question indeed! You might think it needs complex physics and maths involved to be answered. But no it is not so the answer is quite simple and can be explained by means of simple logic.

Suppose a genius inventor somehow builds a high tech camera capable of shooting at some trillion frames per second and sets it up for an experiment to observe light or supposedly the particles which compose light, the photons.

So, he uses a simple torch as the source of light which he intends to fire up on a screen in front of it and will record the path of light on his camera expecting to see the photons and their interactions in slow motion. He also ensures absolute vacuum conditions between the torch and the screen so that he can get a clear observation of the photons. Finally he turns the room completely dark and fires his torch and records this action on the camera.

Now,he runs towards his camera to watch the less than 1 second footage that was recorded. He runs the footage at the slowest possible rate that the camera is capable of. But, he finds nothing. Nothing but the illuminated screen and the burning torch.

The reason is quite clear. When the torch is turned on, photons are emitted and are scattered to all directions and some of them reach our eye and some of them reach the screen. Our intention is to view the photons that are moving towards the screen. We know that we are only able to see things when the photons reach our retina after bouncing off the thing (actually this is not exactly that happens. This is just an over simplification). So, to view them, they need to reach our eye first of all (or the camera lens in this case). Also as photons are massless they cannot interact with each other. So, one photon cannot also bounce off another photon to enable us to see it.

Thus, we only see the objects that light illuminates (in this case- the screen) or the sources from which it is produced (in this case- the torch) but not light itself. Simply speaking we cannot observe something that that is itself required for observation.

Reasons and Consequences of THEORY of EVERYTHING

Reasons to obtain Theory of Everything

I think that currently there are no formal reasons or necessity for obtaining a Theory of everything. Well, then why should the scientists go through all this labor and pain to obtain it? Won’t it be simply sheer waste of time? Well, the answer is that science is not just to comfort lives of people by its inventions and discoveries. Unlocking the secrets of the universe, the search for reality is also part of the game even if it may not have any possible benefit to mankind. Some might even argue that such absurd fairytale theories are not required in this utilitarian world and the world is happy with what technology has to offer. But they forget that technology has itself originated from science. Pursuit for reality never goes vain and even if reality is absurd, it has to be accepted. Scientists are characterized as people in pursuit of science and ToE is just the pursuit for the ultimate science. Also, applications of a scientific theory cannot be well predicted unless it is itself discovered. Who knows, discovery of a Theory of Everything might have the power to completely change the way of mankind.

Consequences of Theory of Everything

Well, the obvious consequences of discovery of a ToE is that we will be able to completely understand the universe we live in, various scientific debates relating to different topics will finally resolve once and for all.

Apart from these, I think there is another powerful consequence of a ToE. It would be the 'END OF PHYSICS' (which was postulated long before, during the time of Einstein). Because after we understand EVERYTHING, what would be left to discover and understand would be NOTHING. So, it is possible that ToE would be the long-predicted end of physics. 

But in this 21^st century end of physics seems to be nowhere near. So, it is certain that discovery of a Theory of Everything has quite a long way to go.

Happy Republic Day !

I wish a Happy Republic Day to all my Indian friends !

GöDEL's THEOREMS- "the argument against Theory of Everything"

Gödel's incompleteness theorems are two theorems of mathematical logic that establish limitations against completeness or consistency of any physical or arithmetic theory. They were devised by Kurt Gödel in 1931.

Kurt Gödel 

1st INCOMPLETENESS THEOREM-

“For any consistent axiomatic system in arithmetic there always exists a set of arithmetic propositions which can’t be proved by it.”

It means that, a consistent (free from any sort of contradiction) mathematical theory-X that completely covers up certain mathematical topic-x cannot be stated as a complete theory because it is always possible to formulate a statement belonging to topic-X whose validity or invalidity cannot be proved by theory-X.

So, basically, Gödel’s first theorem states that a theory can’t be consistent as well as complete at the same time and therefore there would always exist unprovable statements.

2nd INCOMPLETENESS THEOREM-

“Any consistent axiomatic system cannot prove its own consistency and if it does so, it is inconsistent.”

So, the 2nd theorem clearly states that any consistent mathematical theory cannot be used to prove itself.

Now, what has Gödel’s theorems got to do with Theory of Everything?

First we must remember that a physical theory is always in form of a mathematical model. So, the Gödel’s theorems hold as good for physics as they do for mathematics. Now, suppose we get a theory that could be claimed as a theory of everything (call it ToE v1.0). For it to be accepted as valid, it is bound to be consistent. So, as it is consistent, by Gödel’s 1st theorem, it will be incomplete or there will always be something that couldn't be explained by it (thus contradicting it from a theory of “EVERYTHING”). Now, even if we construct a even higher theory (call it ToE v2.0) which could explain the thing that was unexplainable by ToE v1.0, Gödel’s 1st theorem would still apply, due to which something else will get formulated which would be unexplainable by ToE v2.0. So, this process of refinement of theories will go on forever but still none of them will succeed to be claimed as a true Theory of Everything.

Suppose we are able to create a satisfying theory of everything disregarding Gödel’s first theorem. Now, this theory by definition would be able to explain everything. And everything includes itself. This is where Gödel’s 2nd theorem will come to play. By Gödel’s 2nd theorem, we know that no theory can prove itself. So, if this theory of everything could explain itself, it will be inconsistent and eventually be invalid.

Therefore Gödel’s incompleteness theorems will deviate any proposed theory of everything from its very definition of being “of EVERYTHING” and will disprove its validity. So, this is how Gödel’s theorems prove to be strong arguments against a theory of everything.

A number of scholars claim that Gödel's incompleteness theorem suggests that any attempt to construct a ToE is bound to fail. Stephen Hawking was originally a believer in the Theory of Everything but, after considering Gödel's Theorem, concluded that one was not obtainable. 

He says In his lecture ‘Gödel and the End of Physics’ -

“Some people will be very disappointed if there is not an ultimate theory that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind. I'm now glad that our search for understanding will never come to an end, and that we will always have the challenge of new discovery. Without it, we would stagnate. Godel’s theorem ensured there would always be a job for mathematicians. I think M theory will do the same for physicists.”

Still many physicists believe that a ToE is possible to formulate. Most physicists believe that Gödel's Theorem does not mean that a ToE cannot exist. They argue that the theory of everything may not refer to a set of underlying rules but to the understanding of how and why the universe exists and how it works. Surely they will leave no stone unturned to get to it.

(To tell the truth I as a lover of physics am quite scared and startled of the consequences of these theorems)

Click Here for Part-3 (Pillars for the Toe)

Click Here for Part-5 (Reasons and Consequences of the ToE)

Pillars for the THEORY OF EVERYTHING

THE FOLLOWING ARE SOME THEORIES SIMILAR OR CLOSE TO THE ToE.

Kaluza–Klein theory-

Kaluza–Klein theory is a model that seeks to unify the two fundamental forces of gravitation and electromagnetism. It says that the world is 5 dimensional and not 4 dimensional (which is the accepted norm), but we can't see the 5th dimension because it is incredibly small. This extra dimension is packed up tightly into a tiny circle of size in order of Planck length and it would contain just one force and just one type of field, the gravitational force or the gravitational field. The ripples or vibrations produced in this gravitational field would seem to the observers like vibrations in an electromagnetic field and a scalar field in the remaining three spatial and one time dimension. So, it states that all other forces are actually derived from the force of gravity which is itself higher dimensional.

David Hilbert

Hilbert's program-

In mathematics, Hilbert’s program, formulated by German mathematician David Hilbert, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies. Hilbert proposed to ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. Basically, it is a theory of everything for mathematics.

The Standard Model-

It is the theory of particle physics concerning the electromagnetic, weak, and strong nuclear interactions which govern the motion and interactions of the sub atomic or elementary particles. The standard model is also referred as the set of all elementary particles known till date (can be considered as a 'periodic table' for particle physics) It includes the force carriers- the gluons, photons, W bosons, Z bosons which are underneath any force or interaction between big and small bodies. The standard model also has the mass carrier- the Higgs boson which are believed to be the fundamental particles that provide mass to any object. The standard model is sometimes referred as "theory of almost everything"due to its inability to incorporate gravity and its foundations being based on simplifying assumptions. Still, it is a quite successful and well tested theory in physics.

Grand Unified theory-

A Grand Unified Theory is a model in particle physics which attempts to merge the electromagnetic, weak, and strong interactions into one single interaction.

Loop Quantum gravity-

Loop quantum gravity is a theory that attempts to describe the 

quantum properties of gravity. It proposes that the structure of space is granular and has a certain critical density of matter that can be contained within it, thus contradicting general relativity (which proposes that space can be stretched infinitely).  Area and volume of any portion of space are quantized.

Granular Structure of Space as proposed by Loop Quantum Gravity

These theories are not ToE but are close to it and will certainly form the FOUNDATIONAL PILLARS of ToE if such a theory would ever be possible to construct.

Click Here for Part-2 (String Theory & E8 Theory)

Click Here for Part-4 (Godel's Theorems)