The Postulates of Special Relativity (Part 1 of the series)

Want to know how Special Relativity works in all its glory ? Read on then.

Preface:

In this many part series, I aim to present a proper representation of the Special Theory of Relativity, that was proposed by Einstein in 1905 (yes the title does bear a resemblance to Sean Carroll’s article on General Relativity). Here, I plan to start from the ground up, first explaining the foundations of Special Relativity (SR), and then gradually introducing the math as and when required. I am writing this for people with a background in physics (one year atleast, with a little bit of calculus) and will thus include equations and derivations at that level. I hope to proviHead and shoulders shot of a young, moustached man with dark, curly hair wearing a plaid suit and vest, striped shirt, and a dark tie.de all the mathematical rigor that comes with a deeper understanding of SR, mainly the concept of four-vectors, which will aid a transition to the General Relativity. Hope you understand this beautiful and magical theory. Best of luck.

 

What exactly do we mean by relativity ?

Relativity, in physics, means exactly the same thing as it does in English (shocking!). Every person, animal, or actually anything that is capable of “looking” at the events taking place around him sees them from his own unique perspective.

A simple example is that of when you are moving in a car. In your perspective, you are at rest, and the whole world around you is in motion. A car going at the same speed as you will appear to be at rest. A person standing on the sidewalk appears to (magically) move past you without even moving a foot. The surrounding cars moving in the same direction as you appear to be moving slower/faster to you depending on whether you are overtaking them or they you.The cars that are approaching you from the opposite direction, however, appear to be very fast(even above the speed limit!!).

However, for the person standing on the sidewalk, the picture can be entirely different. He will, likewise, consider himself at rest, and the rest of the world moving around him. He will see you go past him at speed. He will see cars going in the opposite direction move with the same speed as you are (approximately). And everything not moving with respect to him he will consider to be at rest.

This everyday experience was captured brilliantly by Galileo in his theory of relativity as early as the year 1632. To see the equations that Galileo put forth, lets see a few very important definitions which will be needed throughout our discussion.

Event : It is a physical situation, or occurrence, at a given point in space and at a particular instant in time. For example, a traffic light changing from red to green, or Chelsea defeating Liverpool.

Observer: An observer is simply an entity that is capable of recording information regarding the event that we are observing (such as its position, velocity, duration for which it occurred, etc.). Note that an observer is not a person, or an entity as such, but is instead spread over the entire frame of reference(see next definition) attached to that observer. For example, from the above examples, the person standing on the pavement is an observer, as is the person traveling in the car. An observer is thus an entity that can assign co-ordinate positions to each event it observes, along with the corresponding time at which the event took place.

Frame of Reference : It is the co-ordinate system fixed to each observer. The observer can use this set of co-ordinates to express the position of an event he observes and also indicate the instant of time at which the event occurred. For example, in the above example, we can associate a co-ordinate system to the person on the pavement, as well as to the person in the car which is fixed with respect to them individually, using which they measure the positions of the cars they observe.

On to Galileo then !

Galilean Relativity:

Galileo put forth his theory by quoting an example of a ship moving at a constant velocity. We will also do the same, however, in a bit of an idealized situation. Consider two observers, one who is at rest (with respect to us) denoted by S, and the other, denoted by S', is moving away with a constant velocity v along the X-axis.

File:Frames of reference in relative motion.svg

Let the clocks of both the frames start ticking when the origin of both S and S’ coincide. Then, according to Galileo, the measurements made in the x-direction by the frame S will exceed those made in the frame S’ by and amount vt , which is the distance S’ has moved in the x-direction. That is,

x' = x - vt

y' = y

z' = z

And in the absence of any indication to the contrary, our normal experience also dictates

t' = t

To convert velocity components measured in the S frame to their corresponding values in the S’ frame, we just take the derivatives wrt time:

v'_x = \frac {dx'} {dt} = v_x - v

where v_x is the velocity with which any object in S is moving

and v'_x is the velocity of the same object measured in S’.

And similarly for the other co-ordinates v'_y = v_y and v'_z = v_z.

Here, I want to draw your attention to a few key points which emerge from Galileo’s relativity:

  • Note that the time remains invariant (i.e. it does not change) between the two frames. The time measured by both observers in S and S’ is the same. Thus, the clocks throughout the reference frames are synchronized and tick off at the same rate.
  • If we restrict ourselves to any one frame, there is no way to know whether our frame is in motion relative to any other frame of reference, without looking outside our frame of reference. This means, that if we are in the frame S’, which is actually a spaceship with no windows(design issues !), then no matter what experiment we perform inside our spaceship, provided there is no influence from the “outside”, it will not give a result suggesting that our reference frame(i.e. the spaceship) is moving (this is true for ANY experiment conducted from ANY branch of physics). This curious phenomena stems from the fact that the frame S’ is moving with uniform speed, and in such frames, Newton’s laws are valid, especially his first law which says that ” a body at rest or in uniform motion continues to maintain its state of motion unless acted upon by an external force”. This is the effect that helps us discern whether we are in motion or not in everyday life, which is violated whenever there is an acceleration (braking of a car, or turning on the road). This can understood, since if v_x is constant, then v_x - v is also constant.
  • This also leaves Newton’s second law unaffected:

F = ma = m(\frac {dv} {dt})

a' = (\frac {dv'} {dt}) = (\frac {d(v_x - v)} {dt}) = (\frac {dv_x} {dt}) = a

{as v \rightarrow constant }

Thus, the second law will be valid according to measurements of both observers, provided we add the conditions that F and m themselves are invariant.

  • Newton’s third law, is also unaffected, as we are assuming the forces to be invariant, as explained above.
  • Thus, we can conclude the last three points by saying that the laws of physics are the same in all reference frames.

Thus, the most important fact to come out of Galilean relativity is that the notion of absolute velocity is wrong. There is no absolute velocity in this universe; everything is moving relative to everything else, and there is no single “absolute” rest frame from which the velocities of all these frames can be deduced.

This fact is so important, that it was correspondingly included in Einstein’s SR, and is still held valid today.

Inertial Frames and Observers:

From the above discussion, it becomes clear that when a frame is moving with constant velocity, no “new” phenomena are generated due to its motion. However, this is not the case when the frame is accelerating, which results in the appearance of new forces which we call pseudo-forces (for example centrifugal force).

As an illustration of the above claim, consider yourself drinking soup in a train. If the train is moving with constant velocity, then there is no noticeable difference in the bowl of soup. However, if the train is accelerating, either speeding up or turning on a bend, then we can see the soup climb up one side of the bowl.

Thus, it is reasonable and useful to single out a class of preferred observers: those who are unaccelerated. These observers are called inertial observers and each has a constant velocity with respect to any other one. The associated frames of these inertial observers are called as inertial reference frames.

A reference frame (or co-ordinate system) can be called as inertial if it meets the following three criterion:

  1. The distance between point P_1 (coordinates x_1,y_1,z_1) and point P_2 (coordinates x_2,y_2,z_2 ) is independent of time.
  2. The clocks that sit at every point ticking off the time t are synchronized and all run at the same rate.
  3. The geometry of space at any constant time t is Euclidean.{that is, there is no curvature in space.}

From here on, unless mentioned, all frames mentioned will be inertial frames.

Why was there a need for a theory like Special Relativity ?

So all was fine and fun till a man called James Clerk Maxwell came onto the scene.

It all started in the late 19th century. Maxwell proposed his famous four equations of electromagnetism. Hertz, then, on the basis of these equations had proved the existence of electromagnetic waves which were found to propagate at the speed of light, i.e. approximately 3 \times 10 ^8 m/s .

Now, at that time, there was a popular theory of the luminiferous aether going around in physics, which proposed the “aether” as the medium through which the light propagates in space. To test this hypothesis, Albert A. Michelson and Edward Morley devised their famous Michelson-Morley experiment to detect the changes in speed of light due to the motion of the earth through the aether. Simply put, the light sent vertically upward from the earth and reflected back to the detector should arrive earlier than light emitted in the direction of earth’s revolution around the sun, as it is perpendicular to the direction of earth and should not have any effect on the speed of light (just like our analysis in Galilean relativity). Click on the following links for their respective wikipedia pages.

Luminiferous aether

Michelson-Morley experiment

However, the results of this experiment were negative, which gave first strong indication of the absence of the hypothesized “aether”, and proved that there was no such thing as “absolute motion” or “absolute rest frame”. But along with that, it also produced a startling result that the speed of light was the same for all observers.

This formed the second foundation of Einstein’s SR: the speed of light in free space has the same value in all inertial frames of reference.

Thus, we now have the two foundations on which Einstein based his Special Relativity upon.

The Fundamental Postulates of Special Relativity:

  1. Principle of relativity (Galileo) : No experiment can measure the absolute velocity of an observer; the results of any experiment performed by an observer do not depend on his speed relative to other observers who are not involved in the experiment. Thus, all the laws of physics have the same form for every inertial observer.
  2. Universality of the speed of light : The speed of light relative to any unaccelerated observer is c = 3 \times 10^8 , regardless of the motion of the light’s source relative to the observer.

Coming up next time : The consequences of Einstein’s two postulates, which include time dilation, length contraction, and Lorentz transformations.

Here is the link for the next part:

Part 2

 

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Why gravitational waves are such a big deal

Understanding the Gravity of the situation:

What are these gravitational waves that everyone is so excited about, how they were detected and why they are so exciting

The BICEP (Background Imaging of Cosmic Extragalactic Polarization) results were announced recently, in which they claimed to have detected the presence of the much sought after gravitational waves. What really are these waves made of? What makes these so special? How were they detected? These surely are the questions that must have faced many people.
  • The Golden Question: What are these waves?

This is the first question that will come to the mind of many people. What really are these damned waves? Why the hype?

Well, these waves are ripples in the very fabric of space-time. These waves do not propagate in a background dependent form like electromagnetic waves, or the more familiar waves we see when we drop a stone into a pond. These waves actually distort the background (that is, the space-time fabric) as they travel through it, creating crests and troughs in the very space-time fabric itself. The reason that these waves are background independent stems from the fact the theory that predicts them is background independent. Which is the theory? Why it’s Einstein’s General Relativity.

So, how does General Relativity (GR) explain the generation and propagation of these waves? It is easier to understand the existence of these waves in empty vacuum. The starting point in the derivation is the assumption of the weak field limit of the gravitational field, but with the gravitational field allowed to be non-static and no restrictions on the motions of the particles in the gravitational field. Solving the corresponding Einstein equations by taking the proper gauge conditions leads to a wave equation which is analogous to the wave equation obtained in electromagnetic theory, i.e. a solution that permits sinusoidal waves.

However, when we get to the sources of these waves, the math gets very complicated and not so easy to explain. In technical terms, the condition for emission of gravitational waves is that the third time derivative of the quadrupole moment (or the l-th derivative of the l-th multipole moment) of the stress energy tensor must be non-zero. In simpler terms, only those sources emit gravitational waves which involve some amount of acceleration. Also, this motion should not have spherical or cylindrical symmetry, otherwise it does not radiate. A simple example to motivate the idea is that of a spinning dumbbell. If the dumbbell spins like the wheels of a car, it will not radiate any gravity waves. However, if it tumbles end over, like two planets orbiting each other, then it will radiate gravitational waves.

The sources of these waves are believed to be binary systems, and exploding supernovae (non-symmetric explosions only). Another source could be primordial gravitational waves created during inflationary epoch.

What makes these waves special and different from the ‘normal’ electromagnetic waves is the fact that they can be detected only at very low frequencies (below approximately 500 Hz), along with the fact that their amplitudes are very very tiny. Another reason for their elevated status is that we need special equipment to detect these waves. This equipment should be able to detect the stress and strain caused by the passage of the gravity wave through it, which is made difficult due to their low amplitudes, and various noise interferences.

  • How were these waves detected?:

These waves were detected in the Cosmic Microwave Background Radiation (CMBR or simply CMB) by the BICEP (Background Imaging of Cosmic Extragalactic Polarization) team. Simply put, the BICEP expedition is just a bolometer designed to detect the polarization in the CMB for a small patch of the sky.

The Bicep

What has the polarization got to do with anything? Well, what the BICEP team was looking for was evidence of primordial gravitational waves, which are caused due to a process occurring in the very very early universe called Inflation. The signature of the gravitational radiation from this time is believed to be imprinted into the CMB in the form of polarization of the CMB photons. In the absence of the gravitational waves, the CMB only has the standard polarization that is present in the electromagnetic fields, i.e. the E-mode polarization, which for linear polarization means that the oscillations of the wave are in the directions 900 to direction of propagation.

However, if gravitational waves are present, then there is an additional mode of polarization called the B-mode polarization. In this type, the oscillations of the wave are 450 tilted with respect to the E-mode polarizations. These are the tell-tale signs of gravitational radiation, as no other process can produce B-mode polarizations in the CMB.

These E-modes are primarily caused due to density fluctuations, along with gravitational waves and are also known as “scalar modes”. The B-modes however, have very few possible sources, chief among them being gravitational waves, with other sources being those that cause distortions in the E-modes by ‘twisting’ them(in a manner of speaking), like for example gravitational lensing (oh yeah, they are also known as “tensor modes”).

Correspondingly, a map of CMB polarizations takes the form of little lb-modesine segments- the direction of the net oscillation in the electric field. The manner of curling (or handedness) of the pattern can be used to determine what type of polarization they correspond to. The B-modes have a net twist to them, as compared to the E-modes. This is best illustrated in the following figure to the right.

The BICEP team collected the data from a particular patch in the CMB, and probed it for the appearance of the B-mode polarizations. Using this information, they computed, what is called the “tensor ratio (r)”, which is known as the “tensor to scalar” ration, where the tensor is the B-mode polarization and scalar is the E-mode polarization.

This is what the BICEP team observed.

They displayed in their findings, a tensor ratio of r = 0.2 with some measure of deviation. This disfavored the previous prediction of r = 0 obtained from previous observations of the CMB, and thus, prove to be a strong evidence in support of gravitational waves and primordial inflation, which is the theory that predicts the presence of these waves in the CMB.

  • So what’s the big deal? :

The big deal is actually two very big deals. Both of them have huge implications for the field of physics and how future research is carried out in the fields of cosmology and astronomy.

  1. A new avenue for astronomy:

Gravitational waves are generated only due to the mass of the source body and its motion in space-time. They are also relatively unique in the sense that they are not affected as much by interstellar medium as conventional electromagnetic waves, and thus arrive at the earth un-impeded and without losing much of their original information.

This opens up a totally new field of astronomy, namely gravitational wave astronomy. It is a very promising development which will allow us to observe processes and phenomena occurring in the universe which do not involve the emission of electromagnetic radiations, such as the interactions between the hypothesized dark matter particles. It also allows us to probe the centre of supernovae, stellar nebulae, and even colliding galaxies in particular detail. They could also help us look at the merger of two black holes, or binary systems in which one companion is either a black hole or an accreting neutron star. It may even serve to test some models of string theory which predict phase transitions in cosmic strings from the early universe (about 10-25 seconds after the birth). Thus, this will truly be a new revolution in astronomy, allowing us to observe what no one thought would even be possible to observe and will surely increase our understanding of the universe by a great amount.

  1. Inflation HAPPENED!! (no not the economic one) :

Another important consequence of the discovery of these waves is that they provide evidence for the occurrence of the process of “inflation”, which was postulated to occur when the universe was merely 10-35 seconds old. This would be the first observed evidence for the inflationary cosmology model. The theory of inflation was proposed in the late 20th century by Alan Guth, Andre Linde, and Alexei Starobinski among others to explain various problems that were observed in the universe, most prominent of which are the Flatness problem (crudely, why is the universe flat), the Horizon problem (crudely, why is the universe homogeneous) and the problem of structure formation in the universe (how were the seeds of galaxies introduced in the universe).

These problems cannot be solved by the standard Big Bang model, and thus the concept of inflation had to be introduced. Inflation theory basically says that when the universe was very young (10-35 seconds approximately), it underwent a huge expansion, of the order of 1026 times its size in a very small fraction of time.

This hypothesis brilliantly solves the flatness problem. The flatness problem essentially asks why the universe is flat, or in more technical language, why the matter-energy density is so close to the critical density required to make the universe flat. The inflation theory introduces a so-called field which permeates space and drives the inflation for a certain period. The field contains a certain energy density, but unlike the density of the matter or radiation present in the late universe, which decrease over time, the density of the inflationary field remains roughly constant as space expands. This forces the density of matter-energy to the required value of one, as can be seen from the Friedmann equation.

It also solves the horizon problem. Since everything in the universe was “close up” with each other (the universe had an approximate radius of 10-35 cm), all the disturbances could propagate throughout the universe then, and create homogeneity in the universe. This homogeneity would be preserved when the universe would undergo the inflationary expansion.

To understand how this model solves the problem of creation of galaxies in the universe, consider the example of a crumpled ball of bedcovers. This crumpled ball of covers has many wrinkles in it. Now, when you open up the covers to their complete size, these wrinkles still remain in the bedcover, even though are spread over a larger area than apparent when they were just crumpled up in a ball. In a similar manner, the universe before inflation contained a few quantum fluctuations (whose existence is demanded by quantum physics). These quantum fluctuations in this primordial universe were “frozen” when inflation occurred, and became the seeds of structure formation in the universe by introducing gravitational instability.

Before the discovery of gravitational waves, there was only theoretical evidence available in support of this theory. But, with this discovery, it seems very probable that inflation might be THE candidate to explain the behaviour of our universe during its early life.

  1. A tool to probe the earliest times of our universe! :

The detection of these primordial gravitational waves also opens up a possibility of probing the conditions of our universe during the very early stages of its life. These waves can be used to fine tune the initial conditions with which our universe was set up. It can also be used to examine the decoupling of gravity from the other three forces of nature, thus giving us a technique to test our GUT (Grand Unified Theory) more thoroughly than ever. In fact, all processes after the decoupling between gravity and other forces that can leave a gravitational imprint can be detected and studied with the help of observing the CMB, as well as direct observation of primordial gravitational waves.

Even though all these avenues are speculative at best at the current time, they can surely be developed to gain information about processes that we will NEVER be able to recreate on earth.

This is how one of the most interesting and engaging discoveries in the history of science has panned out. The only thing holding back physicists from having a full scale party is the verification of these results from the Planck team, which operate the Planck satellite, also used for CMB observations. If verified (most probably the results from the Planck team will be out by November) then surely, Nobel Prizes will be in order for the people involved in this great expedition.

Sources:

http://bicepkeck.org/visuals.html

http://www.astro.umass.edu/~myun/teaching/a100_old/longlecture25.html

http://cosmology.berkeley.edu/~yuki/CMBpol/CMBpol.htm

http://www.wikipedia.org