Conference proceedings

13. Prof. Hari Dass on gravity

The discovery of gravitational waves was preceded by revolutionary developments and followed by unprecedented success. It was unprecedented considering such waves are very hard to detect. It took several decades to come up with the experimental setup to detect gravitational waves, and it was thought that once detected, it would take some time for people to come up with applications. The reason for such thinking was parallel, in some sense, in astronomy.

Visible light was known to man for a long time. But it was a very long time before it was used as a tool for astronomical studies. Surprisingly, applications of gravitational waves were discovered soon after they were discovered. The highlight of this talk is the concept and the discovery of gravitational waves.

This chapter was written by Raghunandan R. as a guest author and has been edited for stylistic consistency. It presents Prof. Dass's talk paraphrased based on a transcript of his lecture titled ‘From Galileo to gravitational waves’ given at Yuvaraja’s College, Mysore, in November 2017.

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Recently, a new method of detection was invented, wherein we see the source as not only the source of gravitational waves, but other electromagnetic radiations as well. This is called multi-messenger astronomy. From the data obtained, one can extrapolate and derive a lot of information. The detection of gravitational waves is a long story starting with Galileo.

Galileo invented the telescope. It was this invention that taught mankind that there was a vast world beyond the limits of Earth. Essentially, Galileo, with this invention, opened a window, not only to modern physics, but to the universe as we know it today. That apart, this truly revolutionary scientist formulated a principle of relativity that is as foundational as Einstein's Special and General theories. It was this idea that paved the way to a long journey that culminated in the discovery of gravitational waves.

The Galilean principle of relativity is supposed to be based on a Gedanken—or thought—experiment. The whole point of a thought experiment is that in the end, what matters are the conclusions we draw from it. Galileo's experiment was as follows: consider two people, one on dry land and another inside a closed ship sailing in smooth waters. Both do the same experiments—in Galileo's time, 'experiments' meant purely mechanical stuff, like throwing a ball. It was concluded that no matter what experiments they do, there was no difference in the results. This shows that there is no difference in the mathematical form of mechanical laws of nature, whether you are stationary, or moving with a uniform velocity.

Lay emphasis on the 'mathematical form', since the actual values, the numbers provided by the experiments, may vary. This is the essence of the Galilean principle of relativity. It can also be said that the principle of relativity determines the form of laws. To appreciate how revolutionary Galileo's work was, we digress a bit to the work of another great person.

When Newton formulated the laws of classical mechanics, the were in conformity with the Galilean principle. So, if Newton's laws held in one frame, they held in every frame moving with a uniform velocity with respect to the first. It was Galileo that introduced the notion of inertial frames. And this notion lies very deep in the conceptual framework of mechanics, even of Einstein's general theory of relativity.

Coming back to our gedankenexperiment, if the fellow in the ship drops a ball, he will see it dropping vertically downwards. For a person standing outside the ship and looking in, the ball drops like an arc in the direction of motion of the ship. Both these views are valid solutions to Newton's laws. The only difference between them is the existence of a horizontal component of velocity in the direction the ship is traveling in.

By suitably choosing the initial velocity of the object, the last shred of difference between both frames can be eliminated. Galileo was rumoured to have done another experiment, which is famous—the dropping of two object from the Tower of Pisa. Again, whether or not he actually did the experiment is immaterial. Only the conclusions drawn matter.

To summarise the experiment, if two objects are dropped from the same height, no matter what the size of shape, they will reach the ground at the same time. If the effects of air resistance are eliminated, this phenomenon can be experimentally seen, even with the objects in question being a feather and a cannonball (see demonstration below, courtesy of BBC Two).

It was these two results of Galileo's experiments that laid foundation to Einstein's theory of gravitation, the theory that predicted the existence of gravitational waves.

Though he formulated the theory, Einstein himself was in a dilemma as to the existence of gravitational waves, owing to the theoretical complications and mathematics the led to the predictions.

It is a proof of the tenacity of physics that now, a hundred years after the prediction was made, it was proved to be true, and that too to a great degree of precision.

The next chapter in the story of gravitational waves is Newton's. He formulated three laws (see chapter 1 and chapter 2). At first glance, it feels like the first law is a corollary to the second. As a consequence, the first seems redundant and unnecessary. The catch is this: the first law does not hold in all frames. It is true only in frames that are inertial.

This shows the importance of the way Newton formulated the laws. They had to be done just so. It is only in the frames in which the first law holds, that the second and third Laws can be talked about. The efficiency of formulation of these laws is what made them rule the scientific world for centuries.

Another law that Newton formulated is the Law of gravitation: $$F = G\frac{m_1 m_2}{r^2}$$The most important difference between the laws of motion and the law above is that the first three laws can be applied to any force. They give a framework to all of mechanics. The Law of gravitation, on the other hand, is a description of a very specific phenomenon, and has to be put into the framework of mechanics to derive conclusions. For instance, Kepler's laws of planetary motion are derivable from Newton's Law of Gravitation and Laws of motion1.

Kepler's laws say the planets go in elliptical orbits. The actual orbits have slight deviations from a perfect ellipse, but this does not mean Kepler's or Newton's laws are wrong, just that they are partly right. Newton's law of gravitation applies to a two-body system. Taking the contributions of other planets into calculation, these deviations can be accounted for.

The reverse works as well: unaccounted deviations may mean an unknown planet. In the past, the predicted orbit of Uranus, then the farthest discovered planet, did not agree with observations. To account for the errors, a new planet beyond Uranus was hypothesised. Later, a planet in close agreement as regards the hypothesised mass and distance was discovered. We know this today as Neptune.

Newton's calculations made much more subtle predictions too. Newton's laws assume that the body in question is a point mass. But the Earth is obviously not a point mass; it is a big, imperfect sphere. So he was hesitant to submit his theory of gravitation to the scientific community. Therefore, he waited till he developed Calculus and had a solid mathematical justification to make the assumption before gifting the world his law of gravitation.

Further calculations predicted the precession of the Earth's axis of rotation, which is a very slow process and has a time scale of many thousand years. The next big revolution was Maxwell's. Though each of the four (Maxwell) equations were discovered by others, it was Maxwell who compiled them to form one, complete theory. The theory of electromagnetism is so beautiful, so complete, that scientific revelations and discoveries that came after it have not been able to change it to this day.

Electromagnetic waves and gravitational waves have some similarities. When a distribution of charges or currents is perturbed, the perturbations relapse into electromagnetic waves. These waves are a part of a wide spectrum, from radio waves to gamma rays. An electromagnetic wave propagates as shown below, with electric and magnetic components perpendicular to the direction of propagation and to each other.

Electromagnetic wave propagation2.

Consider a coil of conducting wire and a magnet. A current is induced in the coil if there is a change in magnetic flux inside it. This induction does not depend on which part is moving, whether the coil or the magnet, so long as there is relative motion. This is the Galilean principle of relativity applied to the laws of electromagnetism. This means, like mechanical phenomena, electrical experiments cannot distinguish between inertial frames.

Now consider Newton's law of gravitational force and Coulomb's law of electrostatic force. Both look exactly the same in form, just that the constants vary, and the former equation uses mass while the latter uses charge. So, like any perturbation of a charge distribution produces electromagnetic waves, perturbations of mass distributions produce gravitational waves. But this is where the similarity starts to end.

Charges come in two varieties, while mass does not. Two charges may attract or repel, depending on their relative variety, but two masses must always attract. Also, when push comes to shove, between two charged bodies, an electrostatic force is $10^35$ times stronger than a gravitational force. But, any mass is made of atoms, which are individually neutral, which is why, over astronomical distances, gravitational force dominates.

Einstein once performed an interesting thought experiment. Consider a box with a dog sitting inside and a ball held mid-air. The box is in empty space; there is no force acting on it. Imagine the box being accelerated upwards. The dog in the box sees the distance between the ball and the floor decreasing. The ball is 'falling' and without the help of gravity too.

The principle of equivalence3.

The same would have happened if the box was on Earth, and the ball was dropped from a height. Hence there is no particular difference between gravity and acceleration. This is the essence of Einstein's principle of equivalence.

The theory that ensued from this principle is now called the General Theory of Relativity. Keeping in mind that upward acceleration can have the same effect as gravity, consider another thought experiment. Imagine a person in a stationary box shining light onto a wall. The beam hits the wall at some point. If the box is now accelerated upwards, the observer inside sees the light beam hitting the wall slightly closer to the floor. Light bends in an accelerated frame. So, if Einstein was right and there is no difference between acceleration and gravity, then gravity should be able to bend light.

Here is an equation formulated by Einstein:

$$ R_{\mu\nu} - \frac{1}{2} Rg_{\mu\nu} + \lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} $$

The equation is beautiful in the sense that it is compact, but gives a lot of information to draw conclusions from. In short, it tells us that the spacetime continuum around a massive body is curved and gives you a measure of the curvature. Which means, the planets moving around the sun are actually moving in straight lines, as opposed to ellipses. Just that these straight lines are in a space whose geometry itself is curved.

Image courtesy, NASA.

For a person outside, in another frame of reference, a straight line in curved space looks like an ellipse, as in Kepler's orbit. (See figure above.) The core ideal of science was arguably best stated by Feynman: 'No matter how good a theory is, or how brilliant the person who made it, if it does not agree with experiment, it is wrong.' It still remained for Einstein's idea of gravity bending light to be proved.

This was done by Eddington, who thought of using a solar eclipse. The principle was simple: the stars that one could not see during daytime would be visible during a complete solar eclipse. One could note the position of a star in the sky near the Sun at such an event. Later on, one could note the position of the same star at night. The difference is that, at night, the Sun is physically present in the sky during an eclipse, but not so at night. If the Sun's gravity did bend light from the star, the two readings should have a discrepancy. When the experiment was done, the discrepancy was found, and Einstein's theory was proved.

Kepler's laws say that all planets move around the Sun in a elliptical orbit, with the Sun at one of the ellipse's foci. As a result, there is a point in the orbit where a planet is closest to the Sun, called the perihelion. Another prediction of the General Theory of Relativity was that the perihelion also revolves around the Sun. In case of Mercury, this perihelion motion was observed, and found to be in complete agreement with predictions. Another score for General Relativity.

A star, once it has burned up its fuel, gives up to gravity and becomes an extremely compact body. At this stage, it is called a neutron star, since it is made entirely of neutrons. The radius of a neutron star is around one part in a million of its original radius. When the star was born, it must have had some rotation. This implies, if the radius of the star decreases, to conserve angular momentum of the body, the star will now rotate really fast.

Such a rotating star radiates radio waves from its poles. This radiation is so regular that earlier people considered it to be a sign of extra-terrestrial life. Another prediction of the General Theory was black holes. It is a solution to Einstein's equation, developed by Schwarzchild. Einstein himself was a bit lukewarm about the existence of black holes, but today, there is very little doubt about their existence.

Black holes are very compact masses. They are so compact, that the escape velocity -- the requirement to escape its gravitational pull -- exceeds that of the speed of light. Chandrasekhar, an Indian scientist was so drawn to the concept that he has wrote voluminous treatises on the mathematical theory of black holes.

Gravitational waves are called `ripples in spacetime,' and it is as accurate a description as possible. Two bodies going around each other can set of these ripples. These ripples, or gravitational waves, in turn take carry away energy. This means that the orbital parameters of the revolving bodies changes.

From this reasoning, if there is a change in the time of Earth's orbit—if it takes slightly less than a year to complete a revolution, let's say—it is an indirect proof for the existence of gravitational waves. We now have direct evidence of existence of gravitational waves.

An interferometer can detect distortions in space. The idea was that, if a gravitational waves passed through an interferometer, the arms would get distorted in length, and this would cause a distortion in the interference pattern.

To make this practical, two observatories were set up—the LIGO project. These were phenomenally large scale interferometers. Each arm was four kilometres long. The chances of errors occurring were equally as large. Anything from a slight tremor deep down in the Earth to a sudden change in local temperature could cause a fringe shift. All these were taken into account while building the observatories.

So, when gravitational waves from the merging of two black holes far far away reached the Earth in 2015, mankind was ready to detect them. In 2017, a third observatory was made, called VIRGO. With the third detection system in place, one can actually pinpoint the source of gravitational waves, as opposed to just detecting their presence.

The first detection of VIRGO was that of two neutron stars merging. The conclusions and consequences of this phenomenon are yet to discovered. We are living in very interesting times.

  1. Prof. Das made a valid point here, aside from his talk: If you go back and read Newton's Principia you will likely not recognise our modern equations. Newton wrote in an older style, with a lot of verbose description, lots of geometrical figures, very few equations. To see Newton's Law of Gravitation, in Principia is very difficult. When we want to look at something in ancient texts it is not often easy to recognise what we currently accept as 'formalism'. The same is true about Maxwell's equations. If you look at Maxwell's original writings, you cannot recognise what you learn in your electrodynamics classes. And if Maxwell himself were to come and write exams, he would flunk all of them.

  2.  Image credit:

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