General Theory of Relativity

After 1905, Einstein continued working in all three of his works in the 1905 papers. He made important contributions to the quantum theory, but increasingly he sought to extend the special theory of relativity to phenomena involving acceleration. The key to an elaboration emerged in 1907 with the principle of equivalence, in which gravitational acceleration was held a priori indistinguishable from acceleration caused by mechanical forces; gravitational mass was therefore identical with inertial mass. Einstein elevated this identity, which is implicit in the work of Isaac Newton, to a guiding principle in his attempts to explain both electromagnetic and gravitational acceleration according to one set of physical laws. In 1907 he proposed that if mass were equivalent to energy, then the principle of equivalence required that gravitational mass would interact with the apparent mass of electromagnetic radiation, which includes light. By 1911, Einstein was able to make preliminary predictions about how a ray of light from a distant star, passing near the Sun, would appear to be attracted, or bent slightly, in the direction of the Sun’s mass. At the same time, light radiated from the Sun would interact with the Sun’s mass, resulting in a slight change toward the infrared end of the Sun’s optical spectrum. At this juncture Einstein also knew that any new theory of gravitation would have to account for a small but persistent anomaly in the perihelion motion of the planet Mercury.About 1912, Einstein began a new phase of his gravitational research, with the help of his mathematician friend Marcel Grossmann, by phrasing his work in terms of the tensor calculus of Tullio Levi-Civita and Gregorio Ricci-Curbastro. The tensor calculus greatly facilitated calculations in four-dimensional space-time, a notion that Einstein had obtained from Hermann Minkowski’s 1907 mathematical elaboration of Einstein’s own special theory of relativity. Einstein called his new work the general theory of relativity. After a number of false starts, he published the definitive form of the general theory in late 1915. In it the gravitational field equations were covariant; that is, similar to Maxwell’s equations, the field equations took the same form in all equivalent frames of reference. To their advantage from the beginning, the covariant field equations gave the observed perihelion motion of the planet Mercury. In its original form, Einstein’s general relativity has been verified numerous times in the past 60 years, especially during solar-eclipse expeditions when Einstein’s light-deflection prediction could be tested.

The 1905 Papers

In the first of three seminal papers that were published in 1905, Einstein examined the phenomenon discovered by Max Planck, according to which electromagnetic energy seemed to be emitted from radiating objects in quantities that were ultimately discrete. The energy of these emitted quantities, the so-called light-quanta, was directly proportional to the frequency of the radiation. This circumstance was perplexing because classical electromagnetic theory, based on Maxwell’s equations and the laws of thermodynamics, had assumed that electromagnetic energy consisted of waves propagating in a hypothetical, all-pervasive medium called the luminiferous ether, and that the waves could contain any amount of energy no matter how small. Einstein used Planck’s quantum hypothesis to describe visible electromagnetic radiation, or light. According to Einstein’s heuristic viewpoint, light could be imagined to consist of discrete bundles of radiation. Einstein used this interpretation to explain the photoelectric effect, by which certain metals emit electrons when illuminated by light with a given frequency. Einstein’s theory, and his subsequent elaboration of it, formed the basis for much of quantum mechanics.

The second of Einstein’s 1905 papers proposed what is today called the special theory of relativity. At the time Einstein knew that, according to Hendrik Antoon Lorentz’s theory of electrons, the mass of an electron increased as the velocity of the electron approached the velocity of light. Einstein also knew that the electron theory, based on Maxwell’s equations, carried along with it the assumption of a luminiferous ether, but that attempts to detect the physical properties of the ether had not succeeded. Einstein realized that the equations describing the motion of an electron in fact could describe the nonaccelerated motion of any particle or any suitably defined rigid body. He based his new kinematics on a reinterpretation of the classical principle of relativity, that the laws of physics had to have the same form in any frame of reference. As a second fundamental hypothesis, Einstein assumed that the speed of light remained constant in all frames of reference, as required by classical Maxwellian theory. Einstein abandoned the hypothesis of the ether, for it played no role in his kinematics or in his reinterpretation of Lorentz’s theory of electrons. As a consequence of his theory Einstein recovered the phenomenon of time dilatation, wherein time, analogous to length and mass, is a function of the velocity of a frame of reference. Later in 1905, Einstein elaborated how, in a certain manner of speaking, mass and energy were equivalent. Einstein was not the first to propose all the elements that went into the special theory of relativity; his contribution lies in having unified important parts of classical mechanics and Maxwellian electrodynamics.

The third of Einstein’s seminal papers of 1905 concerned statistical mechanics, a field of study that had been elaborated by, among others, Ludwig Boltzmann and Josiah Willard Gibbs. Unaware of Gibbs’ contributions, Einstein extended Boltzmann’s work and calculated the average trajectory of a microscopic particle buffeted by random collisions with molecules in a fluid or in a gas. Einstein observed that his calculations could account for brownian motion, the apparently erratic movement of pollen in fluids, which had been noted by the British botanist Robert Brown. Einstein’s paper provided convincing evidence for the physical existence of atom-sized molecules, which had already received much theoretical discussion. His results were independently discovered by the Polish physicist Marian von Smoluchowski and later elaborated by the French physicist Jean Perrin.

Albert Einstein

lived from 1879 to 1955

Einstein contributed more than any other scientist to the modern vision of physical reality. His special and general theories of relativity are still regarded as the most satisfactory model of the large-scale universe that we have.

Albert Einstein

Physicist

Thanks to his theory of relativity, Albert Einstein became the most famous scientist of the 20th century. In 1905, while working in a Swiss patent office, Einstein published a paper proposing a “special theory of relativity,” a groundbreaking notion which laid the foundation for much of modern physics theory. (The theory included his famous equation e=mc².) Einstein’s work had a profound impact on everything from quantum theory to nuclear power and the atom bomb. He continued to develop and refine his early ideas, and in 1915 published what is known as his general theory of relativity. By 1920 Einstein was internationally renowned; he won the Nobel Prize in 1921, not for relativity but for his 1905 work on the photoelectric effect. In 1933 Einstein moved to Princeton, New Jersey, where he worked at the Institute for Advanced Studies until the end of his life. Einstein’s genius is often compared with that of Sir Isaac Newton; in 2000 Time magazine named him the leading figure of the 20th century.Extra credit: Einstein was famously rumpled and frizzy-haired, and over time his image has become synonymous with absent-minded genius… He sent a famous letter to Franklin Roosevelt in 1939, warning that Germany was developing an atomic bomb and urging Allied research toward the same goal… Einstein married Mileva Maric in 1903. They had two sons: Hans Albert (b. 1904) and Eduard (b. 1910). They also had a daughter born before their marriage, Leiserl (b. 1902). She apparently was given for adoption or died in infancy. Mileva and Albert were divorced in 1914… He married his cousin Elsa Löwenthal in 1919, and they remained married until her death in 1936… The Institute for Advanced Studies has no formal link to Princeton University; however, according the IAS website, the two institutions “have many historic ties and ongoing relationships”… The Albert Einstein College of Medicine opened in New York City in 1955. It is part of Yeshiva University. Einstein did not create the school, but gave his permission to have his name used.

Quotations by Albert Einstein

(During a lecture)
This has been done elegantly by Minkowski; but chalk is cheaper than grey matter, and we will do it as it comes.
[Attributed by Pólya.]
Quoted in J E Littlewood, A Mathematician’s Miscellany, 1953. Everything should be made as simple as possible, but not simpler.
Reader’s Digest. Oct. 1977.

I don’t believe in mathematics.
Quoted in Carl Seelig. Albert Einstein.

Imagination is more important than knowledge.
On Science.

The most beautiful thing we can experience is the mysterious. It is the source of all true art and science.
What I Believe.

The bitter and the sweet come from the outside, the hard from within, from one’s own efforts.
Out of My Later Years.

Gott würfelt nicht.

Common sense is the collection of prejudices acquired by age eighteen.
Quoted in E T Bell Mathematics, Queen and Servant of the Sciences. 1952.

God does not care about our mathematical difficulties. He integrates empirically.
Quoted in L Infeld Quest, 1942.

How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?

(About Newton)
Nature to him was an open book, whose letters he could read without effort.
Quoted in G Simmons Calculus Gems (New York 1992).

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
Quoted in J R Newman, The World of Mathematics (New York 1956).

What is this frog and mouse battle among the mathematicians?
[i.e. Brouwer vs. Hilbert]
Quoted in H Eves Mathematical Circles Squared (Boston 1972).

Raffiniert ist der Herr Gott, aber boshaft ist er nicht.
God is subtle, but he is not malicious.
Inscribed in Fine Hall, Princeton University.

Nature hides her secrets because of her essential loftiness, but not by means of ruse.

The human mind has first to construct forms, independently, before we can find them in things.

Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.
Quoted in P A Schilpp, Albert Einstein, Philosopher-Scientist (Evanston 1949).

Do not worry about your difficulties in mathematics, I assure you that mine are greater.

The truth of a theory is in your mind, not in your eyes.
Quoted in H Eves Mathematical Circles Squared (Boston 1972).

These thoughts did not come in any verbal formulation. I rarely think in words at all. A thought comes, and I may try to express it in words afterward.
Quoted in H Eves Mathematical Circles Adieu (Boston 1977).

A human being is a part of the whole, called by us “Universe,” a part limited in time and space. He experiences himself, his thoughts and feelings as something separated from the resta kind of optical delusion of his consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us. Our task must be to free ourselves from this prison by widening our circle of compassion to embrace all living creatures and the whole of nature in its beauty. Nobody is able to achieve this completely, but the striving for such achievement is in itself a part of the liberation and a foundation for inner security.
Quoted in H Eves Mathematical Circles Adieu (Boston 1977).

The world needs heroes and it’s better they be harmless men like me than villains like Hitler.
Quoted in H Eves Return to Mathematical Circles (Boston 1988).

It is nothing short of a miracle that modern methods of instruction have not yet entirely strangled the holy curiousity of inquiry.
Quoted in H Eves Return to Mathematical Circles (Boston 1988).

Everything that is really great and inspiring is created by the individual who can labor in freedom.
Quoted in H Eves Return to Mathematical Circles (Boston 1988).

The search for truth is more precious than its possession.
The American Mathematical Monthly 100 (3).

If my theory of relativity is proven successful, Germany will claim me as a German and France will declare that I am a citizen of the world. Should my theory prove untrue, France will say that I am a German and Germany will declare that I am a Jew.
Address at the Sorbonne, Paris.

We come now to the question: what is a priori certain or necessary, respectively in geometry (doctrine of space) or its foundations? Formerly we thought everything; nowadays we think nothing. Already the distance-concept is logically arbitrary; there need be no things that correspond to it, even approximately.
“Space-Time.” Encyclopaedia Britannica, 14th ed.

Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone.
The Evolution of Physics.

Science without religion is lame; religion without science is blind.
Reader’s Digest, Nov. 1973.

(To a student)
Dear Miss —
I have read about sixteen pages of your manuscript … I suffered exactly the same treatment at the hands of my teachers who disliked me for my independence and passed over me when they wanted assistants … keep your manuscript for your sons and daughters, in order that they may derive consolation from it and not give a damn for what their teachers tell them or think of them. … There is too much education altogether.
The World as I See It, (New York, 1949), 21-22.

(Written in old age)
I have never belonged wholeheartedly to a country, a state, nor to a circle of friends, nor even to my own family.
When I was still a rather precocious young man, I already realized most vividly the futility of the hopes and aspirations that most men pursue throughout their lives.
Well-being and happiness never appeared to me as an absolute aim. I am even inclined to compare such moral aims to the ambitions of a pig.
Quoted in C P Snow, Variety of Men, (Harmondsworth 1969) 77.

The relativity principle in connection with the basic Maxwellian equations demands that the mass should be a direct measure of the energy contained in a body; light transfers mass. With radium there should be a noticeable diminution of mass. The idea is amusing and enticing; but whether the Almighty is laughing at it and is leading me up the garden path — that I cannot know.

When I am judging a theory, I ask myself whether, if I were God, I would have arranged the world in such a way.

Great spirits have always encountered violent opposition from mediocre minds.

.. common sense is nothing more than a deposit of prejudices laid down in the mind before you reach eighteen.
Quoted in E T Bell, Mathematics: Queen and Servant of Science

Thus the partial differential equation entered theoretical physics as a handmaid, but has gradually become mistress.
The World as I See It

But the creative principle resides in mathematics. In a certain sense, therefore, I hold true that pure thought can grasp reality, as the ancients dreamed.
Quoted in H R Pagels, The Cosmic Code

But there is another reason for the high repute of mathematics: it is mathematics that offers the exact natural sciences a certain measure of security which, withut mathematics, they could not attain.
Quoted in E T Bell Men of Mathematics

One reason why mathematics enjoys special esteem, above all other sciences, is that its laws are absolutely certain and indisputable, while those of other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts.
Sidelights on Relativity

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
Sidelights on Relativity

How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things?
Sidelights on Relativity

Mathematics are well and good but nature keeps dragging us around by the nose.
Quoted in A P French, Einstein: a Centenary Volume

Education is that which remains when one has forgotten everything learned in school.
Ideas and opinions (New York, 1954).

A hundred times every day I remind myself that my inner and outer life depend on the labours of other men, living and dead, and that I must exert myself in order to give in the same measure as I have received.
Quoted in Des MacHale, Wisdom (London, 2002).

Before God we are all equally wise – equally foolish.
Quoted in Des MacHale, Wisdom (London, 2002).

Each of us visits that Earth involuntarily and without an invitation. For me, it is enough to wonder at its secrets.
Quoted in Des MacHale, Wisdom (London, 2002).

If at first the idea is not absurd, then there is no hope for it.
Quoted in Des MacHale, Wisdom (London, 2002).

It is my contention that killing under the cloak of war is nothing but an act of murder.
Quoted in Des MacHale, Wisdom (London, 2002).

My religion consists of a humble admiration of the illimitable superior spirit who reveals himself in the slightest details we are able to perceive with our frail and feeble minds.
Quoted in Des MacHale, Wisdom (London, 2002).

Reading after a certain time diverts the mind too much from its creative pursuits. Any man who reads too much and uses his own brain too little falls into lazy habits of thinking.
Quoted in Des MacHale, Wisdom (London, 2002).
[EFR: For a modern view replace reading by watching television.]

Sometimes one pays most for things one gets for nothing.
Quoted in Des MacHale, Wisdom (London, 2002).

The most incomprehensible fact about the universe is that it is comprehensible.
Quoted in Des MacHale, Wisdom (London, 2002).

The world we have made, as a result of the level of thinking we have done thus far, creates problems we cannot solve at the same level of thinking at which we created them.
Quoted in Des MacHale, Wisdom (London, 2002).

There are two ways to live your life. One is as though nothing is a miracle. The other is as though everything is a miracle.
Quoted in Des MacHale, Wisdom (London, 2002).