Emmy Noether: Probably the best female Mathematician in the history

Consider that you are launching a ball into the air. The skill of juggling involves a dance between motion and rest, with the ball's speed decreasing as it rises and increasing as it descends. One of the fundamental ideas of physics is revealed in this dance: conservation laws. Earth, or juggling for that matter, is not the only place where conservation principles apply. These are general laws of physics that hold true in many branches of the subject. However, they aren't always clear-cut. Here's where the woman from Germany enters the scene. Her discovery revealed an incredibly straightforward relationship between these conservation rules and the symmetries of the world.

Emmy Noether, the woman who developed one of the most beautiful theorems in physics. Today (March 23) is her birthday.

Emmy Noether was a German mathematician who made significant contributions to abstract algebra and theoretical physics. She was born on March 23, 1882, in Erlangen, Germany, and died on April 14, 1935, in Bryn Mawr, Pennsylvania, United States.

She developed what is now known as Noether's theorem, which is fundamental in the study of symmetries in physics. This theorem relates symmetries and conservation laws, showing that for every continuous symmetry in a physical system, there is a corresponding conserved quantity.

Despite facing discrimination as a woman in academia during her time, Noether made significant contributions to mathematics and physics. She worked at the University of Göttingen, where she collaborated with prominent mathematicians and physicists such as David Hilbert and Albert Einstein. Her work laid the foundation for modern algebra and had a profound impact on theoretical physics, particularly in the development of quantum mechanics and general relativity.

Noether thought about the issue raised by Hilbert and Einstein, namely that the General Relativity Theory appeared to violate the law of conservation of energy. This led to the development of her well-known theorem, which fundamentally altered the path of physics.

It's likely true that Emmy Noether is the best female mathematician in history. Noether's theorem fundamentally changed our view of the cosmos. Her pioneering work in abstract algebra also revolutionized mathematics.

Contributions

Emmy Noether made several significant contributions to mathematics and theoretical physics during her career. Some of her most notable contributions include:

1. Noether's Theorem: This is arguably her most famous contribution. Noether's theorem states that for every continuous symmetry in a physical system, there is a corresponding conserved quantity. This theorem has profound implications in theoretical physics, particularly in the conservation laws of energy, momentum, and angular momentum.

2.  Abstract Algebra: Noether made significant contributions to abstract algebra, particularly in the development of ring theory and group theory. She introduced the concept of "ideals" in ring theory, which has become a fundamental concept in modern algebra. Her work provided important insights into the structure of algebraic systems.

3. Invariant Theory: Noether worked on invariant theory, a branch of mathematics concerned with properties that remain unchanged under a group of transformations. Her work in this area helped to clarify and extend the understanding of the relationships between symmetry and invariance.

4.  Algebraic Geometry: Noether made contributions to algebraic geometry, particularly in the study of algebraic curves and surfaces. She developed the theory of "Noetherian rings" and made significant advances in understanding the properties of algebraic varieties.

5.  Physics: While primarily a mathematician, Noether's work had a profound impact on theoretical physics. Her theorem has applications in various areas of physics, including classical mechanics, quantum mechanics, and general relativity. She collaborated with notable physicists such as Albert Einstein and David Hilbert, contributing to the development of their theories.

6.  Teaching and Mentorship: Noether was also known for her exceptional teaching skills and mentorship of students. Despite facing discrimination as a woman in academia, she mentored several prominent mathematicians and physicists, inspiring future generations of researchers.

Emmy Noether's contributions to mathematics and theoretical physics have had a lasting impact on these fields, and she is widely regarded as one of the most influential mathematicians of the 20th century. She has left a lasting legacy, with her work continuing to influence research in various fields of science and mathematics.

Prabir Rudra

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Abel Prize for 2024 goes to Michel Talagrand

 

The 2024 Abel Prize has been awarded to Michel Talagrand of the French National Center for Scientific Research (CNRS), Paris, France for his ground-breaking work in functional analysis and probability theory, which has excellent applications in statistics and mathematical physics.

Michel Talagrand is recognized for his contributions to stochastic processes and probability theory. Problems arising in the context of gambling or risk assessment served as the first impetus for the creation of probability theory. An overarching theme throughout Michel Talagrand's seminal discoveries is interacting with and comprehending the stochastic processes surrounding us. It is now evident that in the modern world, a deep comprehension of random events is crucial. For instance, our huge language models and weather forecasts are based on random methods.

Random events are constantly at work in the modern world and have far-reaching implications for everything from condensed matter physics to business logistics. Talagrand's work mostly focuses on comprehending and applying the "Gaussian distribution," which is often referred to as the "Normal distribution" or, because of its form, the "bell curve". The Gaussian distribution governs every aspect of our lives. Weight at birth, grades at school, and retirement ages of athletes are all seemingly random phenomena that nicely fit into the Gaussian distribution.

Specific areas of research

The prize has been awarded for the following three basic areas:

Suprema of stochastic processes: 

The greatest value that can be anticipated from a group of random values is known as the "supremum" of stochastic processes. A stochastic process generates a series of random values. Knowing the expected height of the biggest wave that will hit the beach the following year is helpful if the height of waves that smash onto a beach is determined by a stochastic process.

Concentration of measures: 

It may seem counterintuitive, but when a process relies on a variety of random sources, rather than becoming more complex, the various random components may be able to balance each other out and yield more predictable outcomes. For this, Talagrand has provided accurate quantitative estimations.

Spin glass: 

Leaving abstract probability theory behind, physicists were first surprised to learn that atoms may arrange themselves in a unique form of matter called a "spin glass". Talagrand finished the proof of Giorgio Parisi's work, which won him the Nobel Prize in 2021, by using his understanding of statistics and probability to establish bounds on the behavior of spin glass matter.

Talagrand is a strong problem solver in addition to being an outstanding mathematician. He has significantly advanced our knowledge of random processes, particularly Gaussian processes. His contributions have changed probability theory in several ways. Professor Helge Holden, chair of the Abel Prize Committee, adds, "His proof of the renowned Parisi formula for the free energy of spin glasses is an amazing accomplishment."

Life

Michel Talagrand was born in France in 1952 and graduated from the University of Paris VI with a Ph.D. in mathematics in 1977. He studied for a while at Ohio State University in the United States. He is married and has two sons. He is a member of the French Academy of Sciences and has won various honors and awards. One of the amazing things that he runs on his website is that he challenges mathematicians to solve riddles under the banner "Become rich with my prizes."

Prabir Rudra

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Pi Day 2024

Pi Day is an annual celebration observed on March 14th (3/14) around the world. It's a day dedicated to the mathematical constant π (pi), which is approximately equal to 3.14159. Here are some key aspects and traditions associated with Pi Day:

•  Origin: 

Pi Day was first celebrated in 1988 by Larry Shaw, a physicist at the Exploratorium, a museum of science, art, and human perception in San Francisco. Shaw organized various activities and celebrations to mark the day, including a circular parade and eating pie.

• Significance of Pi: 

Pi is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It's an irrational number, meaning its decimal representation never ends or repeats. Pi has fascinated mathematicians, scientists, and enthusiasts for centuries due to its unique properties.

• Activities and Events: 

Pi Day is often celebrated in schools, universities, museums, and math clubs through various activities and events. These may include pi recitation competitions, pi-themed quizzes, math art contests, and hands-on activities related to circles and geometry.

• Pie Eating: 

One of the most popular traditions associated with Pi Day is indulging in pies, both savory and sweet. Whether it's apple pie, pumpkin pie, pizza pie, or any other kind, enjoying a slice of pie on Pi Day has become a widespread tradition.

• Educational Outreach: 

Pi Day provides an opportunity for educators to engage students in learning about mathematics and its applications in a fun and interactive way. Teachers often use the day to introduce concepts related to circles, geometry, and the history of mathematics.

• Pi Memorization: 

Some enthusiasts challenge themselves to memorize and recite as many digits of pi as possible. While memorizing pi to hundreds or even thousands of digits is impressive, it's important to note that the practical applications of pi typically require only a few decimal places.

• Online Celebrations: 

With the rise of social media and online communities, Pi Day celebrations have expanded to the digital realm. People share pi-related memes, jokes, artwork, and trivia on platforms like Twitter, Instagram, and Reddit, fostering a sense of camaraderie among math enthusiasts worldwide.

Overall, Pi Day is a fun and inclusive celebration of mathematics that encourages people of all ages to appreciate the beauty and ubiquity of numbers in our lives. Whether you're a seasoned mathematician or just someone who enjoys a good slice of pie, there's something for everyone to enjoy on Pi Day. Wish everyone a very happy Pi Day 2024!!

Prabir Rudra

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Classical Mechanics: A journey from Newton to Lagrange and Hamilton

Classical mechanics, as understood today, is largely attributed to the foundational contributions of Sir Isaac Newton, building upon the work of earlier thinkers like Galileo Galilei. Here is a brief overview of the contributions of Newton and Galileo to classical mechanics:

 

Galileo Galilei (1564-1642):

• Experimental Approach:

Galileo is often considered the father of modern physics due to his emphasis on experimental methods. He conducted a series of experiments on inclined planes and falling bodies to study the fundamental principles of motion.

•  Law of Inertia:

Galileo formulated the principle of inertia, which states that an object at rest will remain at rest, and an object in motion will continue in motion with a constant velocity unless acted upon by an external force. This laid the groundwork for Newton's first law of motion.

• Uniform Acceleration:

Galileo made significant progress in understanding uniformly accelerated motion. He developed mathematical descriptions of falling bodies, showing that the distance traveled is proportional to the square of the time elapsed.

 

Isaac Newton (1642-1727):

• Three Laws of Motion:

Newton formulated three laws of motion, which are fundamental principles of classical mechanics.

First Law (Law of Inertia): An object at rest stays at rest, and an object in motion remains in motion with a constant velocity unless acted upon by a net external force.

Second Law: The force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma).

Third Law: For every action, there is an equal and opposite reaction.

• Law of Universal Gravitation:

Newton proposed the law of universal gravitation, which states that every mass attracts every other mass in the universe with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

• Mathematical Formulation:

Newton's work was not only conceptual but also highly mathematical. He introduced calculus to describe motion and developed differential equations to express the relationship between force, mass, and acceleration.

• Unified Theory:

 Newton's laws of motion and the law of universal gravitation provided a unified framework for understanding a wide range of phenomena, from the motion of celestial bodies to the behavior of objects on Earth. This marked a significant departure from earlier fragmented approaches.

Together, the contributions of Galileo and Newton laid the foundation for classical mechanics, providing a systematic and mathematical framework for understanding the motion of objects and the forces acting upon them. Their work is essential for understanding the physical world and continues to be a fundamental part of the study of physics.

Development into an advanced form of mechanics

There was a lull following Newton, and it took until the end of the eighteenth century for classical mechanics to advance further. The development of classical mechanics from Newton to Lagrange and Hamilton represents a significant evolution in the understanding of physical systems. Here is a more detailed overview of this progression:

 

• Newtonian Mechanics (17th century):

Isaac Newton (1642-1727):

Newton formulated the three laws of motion and the law of universal gravitation, establishing a comprehensive framework for classical mechanics. His work laid the groundwork for understanding the motion of objects and the force interactions between them. Newtonian mechanics was basically a force-based mechanics where all motions were considered to be directly associated with forces.

• Principle of Least Action (18th century):

Pierre Louis Maupertuis (1698-1759):

    Maupertuis introduced the principle of least action, suggesting that natural processes follow paths that minimize or maximize a certain quantity called action. The principle states that "the actual path taken by a physical system between two points in its configuration space is the one for which the action, defined as the integral of the Lagrangian over time, is stationary—either a minimum, maximum or a saddle point".

• Analytical Mechanics and Euler-Lagrange Equations (18th century):

Leonhard Euler (1707-1783):

 Euler made significant contributions to mechanics, formulating the Euler-Lagrange equations. These equations provided a mathematical framework for expressing the equations of motion using the principle of least action.

• Lagrangian Mechanics (18th century):

Joseph-Louis Lagrange (1736-1813):

Lagrange further developed the analytical approach to mechanics, introducing the Lagrangian formulation. In his work "Mécanique Analytique" (1788), Lagrange employed generalized coordinates and the principle of least action to derive the equations of motion.

 

• Hamiltonian Mechanics (19th century):

William Rowan Hamilton (1805-1865):

Hamilton built upon Lagrange's work and introduced Hamiltonian mechanics. He reformulated the principle of least action in terms of the Hamiltonian function (H= Kinetic Energy + Potential Energy) leading to Hamilton's principle. In 1833, he published "On a General Method in Dynamics," which introduced the Hamiltonian formulation of mechanics. This approach uses generalized coordinates and momenta to describe the dynamics of a system, and it is equivalent to the Lagrangian formulation. Hamiltonian mechanics provides an alternative perspective, particularly useful in certain physical problems and in the context of quantum mechanics.

In summary, the development of classical mechanics from Newton to Lagrange and Hamilton involved a transition from empirical laws to more abstract and general mathematical formulations. The most striking feature of this advanced form of classical mechanics is that it is no longer a force-based mechanics like its Newtonian counterpart, but it is now based on energy. Euler, Lagrange, and Hamilton each contributed significantly to the analytical and variational approaches that are foundational to classical mechanics today. The Lagrangian and Hamiltonian formulations provide powerful tools for solving complex problems and have applications in various branches of physics.

Prabir Rudra

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Mathematical Theory of Probability: A historical perspective from Pascal to Laplace

 

                                     3 dice rolling problem

Terminology

The term 'Probability' literally means chance or odds or expectation or likelihood. The term originates from the medieval Latin word 'probabilis' meaning plausible. Probability is always utilized to study the behavior of stochastic (random) processes like tossing a coin or rolling a dice. Historically probability has always been closely associated with the term chance and used synonymously with it until we had a proper mathematical perspective in the 18th century. The early form of the theory was called the 'Doctrine of Chances'.

Origin

Probability finds its place in the ancient and medieval laws of evidence where they had to deal with proofs, credibility, and uncertainties of evidence in a court. Games of chance are believed to have existed as early as the Egyptian civilization. In the excavations of different tombs of the Pharaohs, they found a game called 'Hounds and Jackals' which matches closely with our modern-day game of 'Snakes and Ladders.' This must have been the early stage of the creation of dice. Throwing around a set of dice and betting on its outcome has been an ancient habit of humans and this has been passed on from one civilization to another. The first dice game found in the literature of the Christian era was known as 'Hazard', played with 2-3 dice. The game was thought to have been brought to Europe by the knights returning from the crusades. The pottery of the Greek civilization showed the existence of some games which involved various degrees of uncertainty. Present-day casinos all other the world have successfully carried on the legend to modern times. 

During Renaissance, Europe was a center of gambling and other games of chance which involved a humongous amount of wealth put at stake. Maritime insurance plans were supposed to be estimated based on the risks involved. But during that time (17th century) there was no proper mathematical framework that could provide a logical basis to calculate or predict the degree of risks involved in these. People would always like to know an estimate of returns for the wealth they put at stake. These could only be provided by a sound mathematical theory that could take into account the randomness (unpredictability) of the systems and thoroughly perform a risk analysis. Under such situations, the stage was set for the mathematicians of that era to step forward and start developing a mathematical framework that would fulfill the needs of the prevailing society.

The early form of the theory

The early form of the mathematical theory of probability can be attributed to four mathematicians of that era, namely, the Italian polymath Gerolamo Cardano, the French mathematicians Pierre de Fermat and Blaise Pascal, and the Dutch mathematician Christiaan Huygens. Cardano began his investigations as early as 1560, but his work was unknown to society for 100 years. He basically put his thoughts into investigating the sum of the numbers obtained from the throws of three dice. The randomness involved amazed him and he tried to find a pattern in it. This was such a booming topic in those days that Galileo could not stay away from it. In the early 17th century he considered the problem of throwing 3 dice and declared that it is possible to throw some numbers more often than others because there are more ways to create that number.  From the middle of the 17th century, there began a correspondence between Fermat and Pascal aiming to find a solution for the games of chance. This triggered a serious attempt towards the development of a mathematical basis of probability. In 1657 Huygens gave a comprehensive treatment to the concept. 

Subsequent developments

In the 18th century, the subject was taken up by the Swiss mathematician Jacob Bernoulli and the French mathematician Abraham De Moivre. In his Arcs Conjectandi (1713), Bernoulli derived the first version of the Law of Large Numbers (LLN), which states that the average of the results obtained from a large number of trials of a random experiment should be close to the expected value and the gap gradually decreases as the number of trails are increased. De Moivre in his Doctrine of Chances (1718) showed the method of calculating a wide range of complex probabilities.

By the 19th century, it was almost evident that the mathematical theory of probability is a powerful mathematical tool having a wide range of real-life applications. The randomness or uncertainties in various activities of human life and natural phenomenon can be well addressed by a well-formulated theory of probability. To re-affirm this idea German mathematician and physicist Gauss used the theory in astronomical studies and the predictions were great. From a few observational data, he determined the orbit of Ceres (a dwarf planet in the asteroid belt between Mars and Jupiter). He used the method of least squares to perform an error analysis to correct the errors in the observations, which became a routine analysis in astronomy thereafter. To do this he used the normal distribution of errors in his calculations, which is a probabilistic tool. In 1812 French scholar and polymath Laplace further developed the theory by introducing fundamental concepts of mathematical expectations which include the moment generating function, the method of least squares, and the testing of hypothesis. From here the mathematical theory of probability took a turn and slowly started to develop a bonding with the mathematical theory of statistics. It was understood that the two concepts are related and one cannot do without the other. 

                                       Dwarf planet Ceres

Probability in Physics

By the end of the 19th century, physics was gaining ground and was the leading science of the era. Classical mechanics developed by Newton, Lagrange, Hamilton, etc were no longer valid for the sub-atomic worlds. Science needed new physical theories to describe the observations. Who knew that the mathematical theory of probability will form the cornerstone of the new theories to come? It was found that the properties of gases like the temperature could only be expressed in terms of the motions of a large number of particles. This could not be done without the help of statistics as the number of particles that we are talking about here is huge. Ludwig Boltzmann and J. Willard Gibbs developed the field of Statistical Mechanics to address the problem, which involved the concepts of probabilities and statistics. 

                                    Gas particles

The laws followed by the sub-atomic (micro) particles were queer and totally different from the classical world. To address this issue Quantum Mechanics was developed in the 20th century by people like Max Planck, Albert Einstein, Niels Bohr, Werner Heisenberg, Erwin Schrodinger, Paul Dirac, Wolfgang Pauli, Richard Feynman, etc.  The basis of the modern quantum theory is the Uncertainty principle proposed by Heisenberg and it is built on the concepts of probability.

Probability in today's World

The twentieth century saw the mathematical theory of probability develop leaps and bounds. One of the basic problems of probability is finding a formal unambiguous definition of the mathematical origin. The theory is so realistic, obvious, and application-based that it was really tough to find a theoretical definition of probability. The classical definition was initially formed which was far from being sufficient and made way for the frequency definition. Finally, the frequency definition was replaced by the axiomatic definition given by the Russian mathematician Andrey Kolmogorov in 1933. The axiomatic definition is based on three axioms which are logical and accepted worldwide. It settled the long-standing disputes between mathematicians regarding the definition of probability. 

Probability and Statistics found a link and came together through the concept of hypothesis testing introduced by the Polish mathematician Jerzy Neyman and the British polymath R.A. Fisher. In modern times the concept of hypothesis testing is applied in various fields like biological and psychological experiments. It is used in the clinical trial of drugs and also in economics. Nowadays the idea of probability is used in concepts like the Markov process, Brownian motion, and other places where we have to deal with an aggregate of entities. Random fluctuations of stock markets are studied using probabilistic mathematical models to provide predictions for investors. So Mathematical finance has emerged as a new area of mathematics.

The modern era is an era of computer simulations, artificial intelligence, quantum computing, data science, etc. In almost all these areas the mathematical theory of probability plays a significant role. It is understandable that there is still a lot of room for development. Mathematicians all over the world work on stochastic models with the aim of improving the theory and increasing its applicability. We hope that a theory that developed from within the human society out of utmost necessity, will continue to develop and help humanity reach new heights in science and technology.

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Prabir Rudra

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Relativity for All Contd.!! (The General theory of Relativity)

On the 140^th birth anniversary of Albert Einstein it is our great pleasure to revisit one of his greatest masterpiece, the General theory of Relativity (GR) (arguably the best theory of Einstein). Today we discuss GR in his memory and ponder over the extraordinary genius of the man who redefined Physics in the 20^th century.

Albert Einstein

Background

After his miraculous year in 1905, when he laid the foundation of 4 groundbreaking theories of nature (Photoelectric effect, Brownian motion, special theory of relativity and “the world’s most famous equation, E= mc2”), Einstein came to the limelight. 

World’s most famous equation

From a patent clerk in Bern, Switzerland Einstein became a lecturer in the University of Bern in 1908. The following year he was appointed as an associate professor of theoretical physics in the university of Zurich. Einstein became a full professor in the Charles-Ferdinand university in Prague in April, 1911. In 1912 he returned to Zurich as a professor of theoretical physics in ETH, Zurich. He decided to move to Germany and on 1^st April, 1914, he joined the Berlin University.

During all these switch overs from one institute to the other Einstein kept nourishing his original idea of generalizing the theory of relativity that he proposed back in 1905. It was as situation of crisis to humanity. First world war was about to break out. But all these seemed to have minimal effect on the genius. A sense of incompleteness kept haunting him through days and nights.

General theory of Relativity

                                       The Problem 

If we recall our last discussion of Special theory of Relativity we will note that the theory was valid only for inertial (non-accelerated) frames of references. But in reality almost all frames are accelerated . A moving car needs to accelerate or de-accelerate. Similarly a cyclist has to use brakes while cycling. A football rolling on the ground de-accelerates due to friction. All these are non-inertial (accelerated) frames of references where special theory of relativity is not applicable. 

So we need a general theory of relativity that will be applicable to all frames of references (both inertial and non-inertial frames).

Why a theory of Gravity?

In the attempt to generalize his theory of relativity, Einstein knew that he has to incorporate acceleration in his theory. Now acceleration comes from force. In a great insight Einstein understood two things which he stated as the Principle of Equivalence.

1)  The equivalence of inertial and gravitational mass.    

2)  The gravitational force experienced locally while standing on a massive body like Earth is the same as the pseudo force experienced by an observer in a non-inertial frame of reference.

He had the feeling that this can really be a theory of gravity. He started to consider the problem of gravity seriously. He consulted Max Planck (who laid the foundations of Quantum mechanics in 1900) and told him about his intentions to work out the gravity problem. 

The legend says that Planck told him “There are two possibilities: (1) The gravity problem is so difficult that probably you will never succeed.  (2) Even if you are able to find a solution to the problem nobody will believe you.”

Max Planck

The weak link of the prevailing Newton’s theory of Gravitation

The first comprehensive theory of gravitation was stated by Isaac Newton in the 17^th century. It stated that:

Any two masses separated by a certain distance will attract each other by a force directly proportional to the product of the masses and inversely proportional to the square of the distance between them.

The theory seemed to be fine at that time with no flaws and was universally accepted. Newton became the God and very few people had the guts to question his intellect. Moreover the problem of gravity was very difficult to address and so only a few showed any further interest and believed in Newton.

Isaac Newton

But some natural questions arise:

What is the source or the origin of the force?

What is creating it?

Why is it present at all?

Newton was silent to these questions. He did not have any logical answer to it. 

The Theory that answered the above questions

Einstein’s theory of General Relativity is a geometric theory of gravitation that answered the prevailing questions of the critics.

Einstein argued that space-time (space and time together) is like a stretched fabric.  Now when we put a mass on such a fabric naturally it will create a curvature around itself. This is warping (curving, bending or twisting) of space-time that gives rise to a gravity well around the mass. Now when a second mass comes within the curvature quite naturally it moves towards the first body due to the gravity well produced. 

Curvature in space-time

Einstein argued this is what gravitation is!! It is not a force at all as Newton had proposed almost 250 years back. It is a phenomenon that is created by the geometry or curvature of the space-time.

Greater the mass greater the curvature on the space-time fabric

What a masterstroke indeed!! What a beautiful insight!! The common mind can only bewilder and get lost in its aura!! Such was the extraordinary genius of a curious mind.   

The Race to Glory

Einstein had developed the physics of the theory back in 1912, but he did not have any clue about the Mathematics. He needed sufficient knowledge about the geometry of curved space-time (Non-Euclidean geometry) before he could proceed towards developing any meaningful mathematics of the theory. 

Einstein was far from being an expert in such a geometry. But one man by the name of David Hilbert was. He was a mathematical giant of that era. Hilbert took up the problem and started developing the mathematics, which came as a terror to Einstein. His theory was in danger of being credited to somebody else. He had to act and act fast.

David Hilbert

Einstein’s friend Marcel Grossmann (a great mathematician) came to his rescue. With his help Einstein learnt the concepts of Riemannian (Non-Euclidean) geometry and started developing the mathematics of his theory. Hilbert on the other hand continued his effort silently but swiftly. 

As the race to glory became more and more intense the scientific community watched the battle of the two scientific giants (one physicist, the other mathematician) and waited eagerly for a result.

By the grace of God, it so happened that both these great minds reached their destination independently and almost simultaneously. By 1915 both had their own versions of the mathematics of GR ready. On careful scrutiny it was found that both were correct.

Hilbert accepted that the contributions to the theory was far more loaded in favour of Einstein and the theory of general relativity must go by the name of Einstein. This ended all confusions if at all there was any. Finally in 1916, Einstein published his theory of general relativity in the Annals of Physics.

It is difficult to contemplate the feelings of the great man. During his quest, at some point he must have felt that he was losing the battle. After all Hilbert was a great mathematician and matching him was a herculean task. Therefore after the publication, it could have felt like winning a battle once thought to be lost. But it should be remembered that the physical basis of the theory was solely laid by Einstein and probably that’s enough to retain the credit. 

The Equations of General Relativity

This is a relation between mass and spacetime geometry.

Equation of GR

The Reaction from the scientific community

Accepting this theory was not at all easy for the scientific community. The theory will shake the very foundations of physics. Shock-waves were abundant and far reaching. Scientists had many years of scientific career jeopardized. Isaac Newton’s theory of gravitation was in danger. So it happened as Max Planck had predicted some years back. Everybody demanded a proof for the theory.

The radical nature of the theory was criticized by many great minds. Some even thought that GR is not a theory of physics, but the whims and fantasies of Einstein, who is living in a fantasy world.

Experimental physicists like Philipp Lenard (Nobel Laureate, 1905) went into direct confrontations with Einstein and mocked him publicly. He said that physics is a subject of experiments and observations. We believe what we see. There is no place for dreamers like Einstein in the scientific community. Science got really ugly!!

Einstein retaliated but he understood that he had to give a proof to save his theory from being demolished by the critics (specially nazi).

The Proof of General Relativity

The paper was published. The theory was complete. But it didn’t matter to anybody because nobody believed it. Now the genius of Einstein faced a bigger challenge. How will he provide a proof for the theory for which he fought for over a decade. Moreover he was not an expert in experimentation at all. It was an uphill task for him.

But again his extraordinary genius came out with an extraordinary idea. He proposed the bending of starlight around the solar limb (gravitational lensing) as a possible proof for general relativity. 

He argued that the light from the distant stars that pass by the sun before reaching us on Earth will bend while their journey near the sun. The sun being a massive mass will create a huge curvature in the space-time around it. The starlight while passing by the sun will follow this curvature and get bent. As a result multiple images of the star will be formed in the sky. This is in complete compliance with general relativity.

So if Einstein’s theory is correct starlight will bend around the sun and if it is not then the light will pass unaffected. Einstein requested the astronomers all over the world to go and check it.

Many astronomers including Arthur Eddington, a renowned British astronomer took up the challenge. Watching the faint starlight in the presence of powerful sunlight was always going to be difficult. So they chose total solar eclipse for their observations.

In 1919, Eddington’s observations of a total solar eclipse provided concrete proofs in favour of Einstein’s general theory of relativity. 

A British proof to a German theory provided a thrust towards world peace in the post first world war era. Einstein dreamt of a peaceful world without any wars. Even his greatest scientific theory advocated his humane feelings.

In 1921, Einstein was awarded the Nobel prize, not for his general theory of relativity, but for his discovery of the photoelectric effect (may be to his disappointment).

General theory of relativity has been termed as ‘the most beautiful theory of physics’ ever. Till date it remains arguably the best theory of nature not to win a Nobel prize.

I am not a genius. I am just curious...I ask many questions and when the answer is simple, then God is answering.     

         --Albert Einstein.

By Prabir Rudra

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