Sunday, May 15, 2011

Einstein and Newton- The Dark Connection!

The Newton Story
In 1687 Newton proposed the universal law of gravitation relating the force between two masses. The force was attractive in nature and varied as inverse square of the distance between the masses. Newton did not understand why the force varies as inverse-square the understanding of this came later with relation to space being three dimensional.
In Newton's theroy there was no mechanism of interaction between the particles and it was essentilly an action at a distance theory. Newton was surprised when he found out that the force of gravity on a particle outside a spherical object was independent of distribution of the mass. The force behaved as if the whole mass is concentrated at the center (known as
Birkhoff's theorem). We now know that there are only two forces that follow this property namely: the inverse square law and when the force is proportional to the distance. Incidently these two are the only two central forces that are stable against radial perturbations (known as Bertrand's theorem). The trajectories of particles in this general force field are elliptical for bound orbits. We know from observations that bodies moving in the solar system follow bound
and stable orbits. Thus the most general force law consistant with this observation can be written as a combination of these two forces

Since we observe gravity always as an attractive force, the coefficient of the second term if positive (meaning a repulsive force) has to be very small. If we divide the equation (1) by the mass of the test particle we get the following equation

At this stage we leap some 240 years into the picture where our friend Einstein awaits!

The Einstein story
Newton's laws were heralded as s great success to man's intellect. It explained with great precission the motion of all the heavenly bodies. One of the instances where the observations didn't match with the observations was the motion of the planet Mercury. It was observed that the orbit of Mercury was precessing at a rate which was slightly more than what was predicted from Newtonian mechanics. The difference between theory and observations was 43 arc seconds/century. It was first thought that this was the effect of some hitherto unknown object which was christined as Vulcan (a similar thing has happened for the orbit of Uranus which led to the discovery of Neptune). In this case however no such object was found. Einstein proposed a new theory of gravity which accounted for this discrepancy. This theory required a revision of our understanding of gravity. In Einstein's theory of general relativity space and time were coupled and played a dynamic role in motion of objects. When Einsteins equations were applied to study the evolution of universe they gave unstable solutions for a static universe (which is expected because with only attractive force either everything is coming
closer or moving apart). In order to get a static and stable solution Einstein introduced a repulsive term in his equations known as the cosmological constant L. With this term the
equations for an homogeneous and isotroopic universe take the following form

If we substitute

we find that equation(1) and equation (2) take the same form! So, its interesting that two of our greatest minds were so close in this regards even though they were centuries apart from each other.


Tuesday, May 3, 2011

Godel and Goodstein theorems

I have started reading Penrose's book the Emperor's new mind and came across these interesting theorems in the preface about Godel's and Goodstein's theorem. Here I briefly mention the two following the treatment from the book:

Godel's theorem:
Suppose that we are given a computational procedure P for establishing mathematical assertions of a particularly well defined type such as the Fermat's last theorem. Then if we are prepared to accept the successful derivation of some assertion by use of rules of P provides us with an unassailable demonstration of the truth of that assertion- then we must also accept as unassailably true some other assertion G(P) which is beyond the scope of the rules of P.

Goodstein's theorem:
Consider any positive number, let us say 3. First, we express this as a sum of distinct powers of 2:
3 = 2^1 + 1.
We now apply a succession of simple operations to this expression, these alternating between:
  1. increasing the 'base' by 1.
  2. subtract 1.
It may seem that the numbers would be ever increasing, but the theorem tells us that no matter what positive number we start with, we always end up with zero!
The reader is encouraged to verify this by taking some number. An example starting with 3 can be seen here: click

What is rather more extraordinary is that Goodstein's theroem is actually a Godel theorem for the procedure of mathematical induction. Recall that mathematical induction provides a way of proving that some mathematical statement S(n) holds for all n = 1,2,3 ... The procedure is to show that if it holds for n =1, then it also holds for n+1. If P stands for the procedure of mathematical induction, then we take G(P) to be Goodstein's theorem.

Wednesday, March 23, 2011

The Grand Design

The mystery of being:

This book is an effort to understand why the universe the way it is, rather than the traditional question- how does the universe behave? It tries to answer the following three profound questions:
  1. Why is there something rather than nothing?
  2. Why do we exist?
  3. Why this particular set of laws and not some other?
These questions are traditionally dealt in philosophy. With the recent development in theoretical physics we are, as the author's belief, in a better position to answer these questions than before. Its time that we consider these questions again. As they say
Most of us do not spend most of our time worrying about these questions, but almost all of us worry about them some of the time.

The rule of law:
This chapter deals with the nature of physical laws. If nature is governed by physical laws three questions arise:
  1. What is the origin of the laws?
  2. Are there any exceptions to the laws, i.e., miracles?
  3. Is there only one set of possible laws?
The first two questions are taken up in the spirit of scientific determinism. The universe is a collection of large number of object which interact with each other following a specific set of laws and there are no exceptions to these set of laws. The first and the third question answered in later chapters . As to scientific determinism: given the state of universe at one time, a complete set of laws fully determines both the future and the past, the question of free will is discussed.
Do people have free will?

Do blue-green algae or bacteria have free will?
Descartes believed that human mind is different from the physical body and the laws of physics do not apply to the mind. We consist of two entities a body (which obeys physical laws) and a mind (or soul which has a free will). The authors take a different stand and assert that human behavi0ur is indeed determined by the physical laws. It is just that it not practical to define an initial state which takes into account all the trillions of molecules and compute their evolution with time in accordance with laws. So we adopt an effective theory, which is a framework created to model certain observed phenomena without describing in detail all the underlying process. Since we cannot solve the equations that determine our behaviour, we use an effective theory that people have free will.

What is reality?

The basic idea of this chapter is: there is no picture- or theory independent concept of reality. It adapts the viewpoint of model dependent realism: the idea that a physical theory or world picture is a model ( generally of a mathematical nature) and a set of rules that connect the elements of the model to observations.
Model dependent realism removes the idea of "reality" as more than one model may be consistent with the given observations. In that case both are equally valid models are reality and none is preferred. A model can have elements which may not be observables like quarks int the standard model. We don't ask the question whether quarks really exist. All we say is that the quarks exist in a model which so far agrees with the observations. As to characteristics of a good model is that it:
  1. is elegant
  2. contains few arbitrary of adjustable elements
  3. agrees with and explains all existing observations
  4. makes predictions about future observations that can disprove of falsify the model if they are not borne out.
Alternative Histories

Saturday, March 19, 2011

Tsunami and Earthquakes: the dynamic connection

Here I would like to briefly discuss the physics connected with the Tsunami. We heard the magnitude of the earthquake was 8.9 in Richter scale. Richter scales are logarithmic as most of the other things in physics, like magnitude of stars, the sound intensities (decibels) and the wind speeds (Beaufort scale). Richter introduced the logarithmic scale to quantify the earthquakes. So suppose you know there was an earthquake of Richter 8.9, how do we estimate the energy released? The formula that is normally used is given as
log E=2.5 M-1.2.
Here E is the energy released in Joules and M is the magnitude of the earthquake.
It is a coincidence that a similar formula is also used for the luminosity scale of stars. So now we can calculate that for 8.9 it works out to be about 10^21 J. For a typical 1 Megaton nuclear weapon the energy released is about 10^17 J . So the intensity of the earthquake was equivalent to that of 10 000 large nuclear bombs.And this earthquake also triggered a tsunami. To understand this we need a little more information. The earth is made up of tectonic plates. During an underground earthquake a tectonic plate goes over another tectonic plate and the material in between is pushed up. This doesn’t happen in any other planet. Venus and Mars don’t have tectonics. There is an essential condition for the formation of plates that there should be lot of water in the planet. This water goes deep inside and mixes with the rock forming a molten mixture which is a highly viscous and elastic fluid called Magma.This phenomenon affects only the crust and the upper part of the mantle. The upper mantle goes to about 20 km. So most of the earthquakes have their origin about 10 km deep. So essentially as far as the diameter of the earth is concerned these are all surface phenomena.There is an increase of pressure and temperature with depth, So if we go a few kilometres down, the rock are molten. Originally these rocks became molten because of the radioactivity in the earth's crust coming from isotopes like 40 K, Rh, U etc. which initially provided much of the heat to keep the earth hot. The continental plates are floating on this magma. Every continent has got its plate, even other landmasses have got their plate. India has got its own plate called the Indian plate. Alfred Wegener was the first to notice that like in a jigsaw puzzle, the west coast of Africa fits nicely to the east coast of south America. Similarly the other landmasses could also be fitted together like a jigsaw puzzle . He then proposed that all the continents were part of only one landmass and that the continents are actually drifting. Initially there was only one super continent Pangaea, then two huge continents Laurentian and Gondwanaland. Apparently about 200 Myrs back India was near the Atlantic in the southern hemisphere. Then the Indian plate broke up and joined the Asian plate. Now with GPS (Global Positioning System), we know this is the reality because we are now actually able to measure the small drift in continents . The Indian plate is still going northwards and pushing against China’s plate. The Himalayas are the result of Indian plate colliding with the Asian plate about 40 Myrs ago. So Himalayas are quite young mountains. As an evidence for this we find fossils of marine animals on top of mount Everest! This shows that once upon a time it was all part of the sea. This also explains why the Himalayas are growing taller. All these mountain building and movements of continents are all connected. The earth surface is in a state of continuous flux. All the continents are moving around. Even now the Mediterranean is closing because the African plate is pushing northwards again and Europe is trying to merge with America and thereby shrinking the Atlantic! In another 100 Myrs the face of the continents will be very different. The pacific is the biggest ocean right now , but it may not be so after 100 Myrs. There is a ring of fire around Japan and all these islands in the pacific. These are highly earthquake prone regions because their pleates collide with each other every now and then and the intensity of the collision depends on how much matter is thrust up. In 2004 there was an underground earthquake in Indonesia that led to a big tsunami and death of 2 lakh people including several in the Sri-Lankan coast and Chennai. Last year in Chile 2010 there was a big earthquake and one in Haiti which also caused a tsunami. In the past year there have been three or four such events.
Tsunami literally means harbor waves in Japanese because when they break on the coast they rise to great heights. So how do we estimate the height of the tsunami wave? Waves can propagate by two means namely : gravity waves and capillary waves. For any body of water if there is a small perturbation, like throwing a pebble on water, the ripples on the surface can be treated as capillary waves. These are generated because of surface tension in the water. Now suppose we have a bigger perturbation like an explosion under water, gravity waves will be created. There is a column of water formed which is denser than the surrounding and they will be pushed up because of buoyancy. There is a formula for the frequency of these waves which depends on the wave number k (inverse of the wavelength) and g is the acceleration due to gravity. For gravity waves the formula is given as
omega^2=g k .
For capillary waves this formula depends on the surface tension gamma , the density of the liquid rho and k^3 given by:
omega^2=k^3 gamma/rho.
So the general expression for a combination of gravity waves and capillary waves.
is given by
omega2=g k +k^3 gamma/rho.
There is a transition between one form of wave to another depending on the acceleration due to gravity, the density and the surface tension. For larger values of k capillary waves will dominate and for longer wavelengths, gravity waves will dominate. Tsunami waves are nothing but the gravity waves. The velocity of the waves depends on the depth of the wave, d
The waves which are moving with a higher velocity are submerged to a greater depth. That's why tsunami waves could be quite high and they move at high velocities. We have heard that the velocities of the tsunami waves were about 1000 km/hr, comparable to a jet liner. We can roughly estimate these velocities in the following way: a substantial part of the energy released in the earthquake (10^21 J) would be transmitted to the water causing the wave motion.
We saw that the area involved was something like 1000 square km. The depth of the wave would be about 100 m which can be deduced from the epicentre where the earthquake took place (how much of the plate was broken off at the fault line). So the volume covered is roughly about 10^14 cubic meters. So we know the mass of the water was about 10^17 kg. We can now work out the velocity. The kinetic energy of the water mv^2 , is roughly of the order of 10^21 J. So, if you do this calculation so you get a v roughly about close to what they have said, about 1000 km/hr. Off course for high values of wave number k, there would be a surface wave and the transition is interesting to work out. For example, the surface tension of water is about 70 dynes/cm and density 1g/cc. So for water the transition frequency is something like 10 Hertz. For a frequency smaller than this gravity wave dominates and for a larger frequency capillary waves dominates. So we can see that for the case of tsunami surface waves are not important and the deeper gravity waves are important. So we are justified in estimating the velocity of the waves in the above fashion.
If we have different planet or satellite say like Titan which is supposed to have oceans of liquid methane (one of the findings of Cassini spacecraft was that Titan has got weather similar to earth but running on methane's cycle). Now suppose a similar thing happens there, the surface tension for alcohol is much lower (compared to water) say around 20 whereas the densities of the order of comparable about 0.8 g/cc . So again the crossover frequency can be worked out. The surface gravity on Titan is quite small compared to the earth . The frequency of waves would get affected. For a given wave of particular frequency the velocity will be different now. Its velocity will be smaller and wavelength probably 10 times longer. If you take another example another example of Europa ,one of the satellites around the Jupiter, they think there might be a lot of water below the surface (which is why some space craft are being planned to exclusively monitor the satellite Europa). This huge liquid ocean underneath the surface might be subjected to all these kind of wave phenomena. For the Europa, the surface gravity is only one seventh of the earth. So you see any velocity of any of these waves will be much less . It is easy to deduce many things from this kind of analysis . So next time you read about some earthquakes and all these things about Richter scales, you can plug in the formulae and see how much energy is released and what would be the velocities of the waves.

Thursday, August 12, 2010

Giant concerns

Considering the second voyage of Gulliver, i.e. to the land of giants, Brobdingnag, the giants are supposedly 12 times taller than humans. In the case of Lilliputians, they were 12 times shorter. So did Swift get his biology right?
If linear dimension is scaled 12 times, the volume would be 12 cubed, so the mass of such a giant could be easily a thousand times more, whereas the bone area would increase only by a factor of hundred. So each unit area of bone would have to bear a load ten times more than that of in humans. But bones would break at such loads. So as Haldane writes in his essay "On being the right size", each time such giants took a step they would break their bones.
Similarly many other quantities scale with linear size. Even the frequency of the vocal chord will be much lower, so Gulliver would not have been able to communicate with them!

What about trees in Brobdingnag? Would they be 12 times taller than our trees? The reader is encouraged to try it as a problem. Hint: The height of a tree is governed by a balance between gravity and surface tension (which is responsible for capillary action for transport of water and minerals).

What about birds? What would be the size of the largest birds in the land of Brobdingnag. Hint: The flight of a bird is a balance between air drag and gravity. The velocity of take off scales as square root of the wingspan.
Similarly the heat production in the brain would also be ten times more. So the Giants are literally hot-headed!

For details click here
Further read:
1. On being the right size - J B S Haldane.
The pair of numbers (5,4),(11,5),(71,7) are called Brown numbers. They satisfy the condition that
for a pair (m,n). Erdos had given a conjecture that these are the only three numbers that satisfy this condition. Can you find other pairs?
Image courtesy:

Thursday, August 5, 2010

Remembering Hiroshima and Nagasaki

World War II still remains fresh in our minds, both for the destruction
that was caused worldwide and for all the new innovations it helped
spawning in terms of science and technology. Though these were initially
aimed at giving a strategic military advantage to those countries who
reaped its benefits, in the years that followed after the war, these
innovative ideas have been put to use for more peaceful means. An object
of fascination, dread and ridicule the world over took birth during these
troubled times during which it was also put to use, the atomic bomb. It
brought the end of the war closer when its effects were experienced by the
speechless masses of people in the cities of Hiroshima and Nagasaki in
Japan. The much debated events occurred on 6th August and 9th August, 1945
respectively. By then, Germany had surrendered and Japan was still
battling against the Allied forces. America, which had earlier in 1941
suffered heavy casualties at Pearl Harbor owing to attacks by the Japanese
were prompted to join into the war efforts on a full time basis. They had
been developing the atomic bomb for some time under the Manhattan project
(1942 - 1946) where the research was directed by Robert Oppenheimer.

The “Little Boy” was dropped on 6th August, 1945 on Hiroshima, Japan
making it the first atomic bomb to be used in war. The sustained fission
of Uranium – 235 was used to create a massive explosion at a height of
around 580 m above the city. The mass of the potentially fissile uranium
used was 68 kg of which only 1 kg effectively took part in the fission
reaction. It was estimated that the energy released in the explosion was
roughly 54 TJ, the equivalent of exploding 13 kT (kilo tons) of TNT.
Working backwards using Einstein's mass-energy relation, the amount of
mass that was actually converted into energy turns out to be only 0.6 g
out of the 1 kg of active fissile material.

There were various components that contributed to the destructive power of
the bomb. The initial destruction was in regions close to the explosion
where people present were witness to the high flux of gamma radiation,
neutrons and extreme heat (~ 4000 K) which led to them being instantly
vapourized. This was followed by a blast wave of very high pressure which
coupled with a sufficient time duration of sustenance and a high wind
velocity resulted in the loss of many more lives and destruction on a
massive scale to buildings within a radius of 1.6 km from the blast site.
At a distance of 1.6 km, it was estimated that the pressure would have
been roughly 34 kPa, still enough to cause structural damage to buildings.
Also, the radiation emanating from fissile material that got scattered
would have also contributed to some long term effects in people such as
radiation sickness, cancer, genetic mutations, etc. which have been
studied over the years.

1. Calculate the amount (critical mass) of Uranium-235 that would be required to sustain a chain reaction.
Hint: For the reaction to continue the number of neutrons produced should be more than the number of neutrons captured in the process. Now fission of each Uranium-235 nucleus creates 3 neutrons, a fraction of which is absorbed or lost through the surface. At equilibrium there is a balance between the amount of neutrons produced in the volume and lost through the surface. The actual value of critical mass depends on the geometry and density of the fissile material. For a pure Uranium-235 sample of spherical geometry, the critical mass is about 30 kg. For details of the calculation for critical mass
click here.

2. Calculate the pressure and the radius of shock wave 10 s after the explosion.

3. What would be temperature at the epicentre few milliseconds after the explosion?


10662526601 is the only known palindromic number whose cube root (2201) is not a palindrome.
Image courtesy : wikipedia.

Sunday, August 1, 2010

Some uplifting thoughts !

Jonathan Swift describes the flying island of Laputa in Gulliver's travels in the following way- " The flying or floating island is exactly circular, its diameter 7837 yards, or about four miles and a half, and consequently contains ten thousand acres. It is three hundred yards thick...But the greatest curiosity, upon which the fate of the island depends, is a loadstone of a prodigious size, in shape resembling a weaver's shuttle. It is in length six yards, and in the thickest part at least three yards over...By means of this loadstone, the island is made to rise and fall, and move from one place to another...But it must be observed, that this island cannot move beyond the extent of the dominions below, nor can it rise above the height of four miles."

Was Swift foreseeing a day when objects could be levitated by the use of technology? One can draw some parallels to our current technology that uses a very advanced version of the crude loadstone. Magnetic levitation has become a reality in the recent past, being put to use for transport at very high speeds. Below, we offer a few hints based on the information provided by Swift to see if it is possible today to levitate a island like Laputa based on our current technology.

1) Calculate the magnetic field strength necessary to keep the island afloat?
When magnetic force balances the island against gravity we can equate the total magnetic energy to the gravitational potential energy .

where B represents the magnetic field strength in a volume V, m is the mass of the island ( assume the density of Laputa to be same as of Earth), g is the acceleration due to gravity and h is the height to which the island is levitated.
In comaparision the magnetic field strength of the Earth is typically 0.5 Gauss.

Suppose we want to levitate a house with a giant balloon (like in the movie Up), what would be the size of the balloon required? We can use the following expression

where rho is the density of gas, V is the volume of the balloon and m is the mass of the house. Compare with the volume of the largest gas balloon launched which is about 60 million cubic feet.


The Euler-Mascheroni constant is the limiting difference between the harmonic series and natural logarithm. It is denoted by the greek letter gamma.

There is a prime number version of the Euler-Mascheroni constant known as the Meissel Mertens constant which is the difference between harmonic series and natural logarithm of natural logarithm.

where p is a prime number less than or equal to n.
Image courtesy: