NOVA: Einstein’s Big Idea – Special and General Relativity

In the middle of the 17th century, Sir Isaac Newton unified the Celestial and the Terrestrial with his description of gravity as an influence between two objects, which can best be considered as a force of attraction between objects as a result of the product of their masses and the square of the distance between them. The effect of this influence is to cause the velocity of objects to change, in accordance with Newton’s Laws of Motion. While his work in this area alone would have been sufficient secure his place as one of the greatest scientists in history, he is also credited with a number of achievements (see http://www.pbs.org/wgbh/nova/newton/legacy.html):  Inventing the reflecting telescope  Proposed a new theory of light and color  Discovered calculus  Developed three laws of motion  Devised the law of universal gravitation  Advanced early modern chemisty  Became the father of modern science

Independent of Newton’s work, Michael Faraday and James Clerk Maxwell collaborated to unify electricity and magnetism, into the electro-magnetic force. In my interpretation of the historical accounts, Michael Faraday was responsible for the majority of the science while James Clerk Maxwell was responsible for the mathematics. One of the most important results of their efforts was that electromagnetic energy can be thought of as a wave that travels at a single speed, regardless of its frequency … the speed of light. Below are excerpts from the NOVA Program entitled “Einstein’s Big Idea” that follow this body of work (all from the Phistory Channel on youtube):

Part 1 (6:20 to end): https://www.youtube.com/watch?v=ROC8zSiw1x4 Part 2 (beginning to 7:45): https://www.youtube.com/watch?v=XUL9D0aHLmE Part 4 (6:10 to end): https://www.youtube.com/watch?v=BeOKwEUOTt0 Part 5 (Beginning to 0:42): https://www.youtube.com/watch?v=gT3BAWOpQ7Q Part 6 (Beginning to 7:58): https://www.youtube.com/watch?v=lPGWMUdaPYE

In his efforts to unify gravity and electro-magnetism, Albert Einstein made some of the most incredible scientific advances in history. Through a variety of *thought experiments*, he developed the theories of special and general relativity, each of which have been supported by a variety of observations.

Complete the following after reading the attached materials and conducting additional research:  Special Relativity: o Write a paragraph that explains Special Relativity in your words o State the Two Postulates of Special Relativity o Submit and explain an illustration that you have found helpful in improving your understanding of time dilation, length contraction, and/or the twin paradox (include the source)  General Relativity: o Write a paragraph that explains General Relativity in your own words o State the Equivalence Principle o Submit and explain an illustration that you have found helpful in improving your understanding of gravitational bending of light, and/or gravitational time dilation (include the source)

Name: ______Total Grade: ____ /30

Topic Advanced (5 pts) Proficient (4pts) Working toward Submitted (2pts) proficiency (3pts)

[some good stuff] [shows insight] [follows prompt] Special Relativity: Paragraph

Special Relativity:

Postulates

Special Relativity: Time dilation

Illustration and Length Explanation contraction

Twin Paradox General Relativity:

Paragraph

General Relativity:

Equivalence Principle General Bending of light Relativity:

Illustration and Explanation Time dilation http://www.pbs.org/wgbh/nova/einstein

The Theory Behind the Equation by Michio Kaku

Imagine a police officer chasing after a speeding motorist. If he drives fast enough, the officer knows that he can catch the motorist. Anyone who has ever gotten a ticket for speeding knows that. But if we now replace the speeding motorist with a light beam, and an observer witnesses the whole thing, then the observer concludes that the officer is speeding just behind the light beam, traveling almost as fast as light. We are confident that the officer knows he is traveling neck and neck with the light beam.

But later, when we interview him, we hear a strange tale. He claims that instead of riding alongside the light beam as we just witnessed, it sped away from him, leaving him in the dust. He says that no matter how much he gunned his engines, the light beam sped away at precisely the same velocity. In fact, he swears that he could not even make a dent in catching up to the light beam. No matter how fast he traveled, the light beam traveled away from him at the speed of light, as if he were stationary instead of speeding in a police car.

But when you insist that you saw the police officer speeding neck and neck with the light beam, Einstein realized that the world within a hairsbreadth of catching up to it, he says you are crazy; he never even got close. To described by Isaac Newton Einstein, this was the central, nagging mystery: How was it possible for two people to see the (left), in which one could add same event in such totally different ways? If the speed of light was really a constant of nature, and subtract velocities, and that then how could a witness claim that the officer was neck and neck with the light beam, yet the described by James Clerk officer swears that he never even got close? Maxwell, in which the speed of light is constant, could not both be right. He decided to solve the Einstein had realized earlier that the Newtonian picture (where velocities can be added and problem—and special relativity subtracted) and the Maxwellian picture (where the speed of light was constant) were in total was the result. contradiction. Newtonian theory was a self-contained system, resting on a few assumptions. If only one of these assumptions were changed, it would unravel the entire theory in the same way that a loose thread can unravel a sweater. That thread would be Einstein's daydream of

racing a light beam.

SPECIAL RELATIVITY IS BORN

One day around May of 1905, Einstein went to visit his good friend Michele Besso, who also worked at the patent office, and laid out the dimensions of the problem that had puzzled him for a decade. Using Besso as his favorite sounding board for ideas, Einstein presented the issue: Newtonian mechanics and Maxwell's equations, the two pillars of physics, were incompatible. One or the other was wrong. Whichever theory proved to be correct, the final resolution would require a vast reorganization of all of physics. He went over and over the paradox of racing a light beam. Einstein would later recall, "The germ of the special relativity theory was already present in that paradox." They talked for hours, discussing every aspect of the problem, including Newton's concept of absolute space and time, which seemed to violate Maxwell's constancy of the speed of light. Eventually, totally exhausted, Einstein announced that he was defeated and would give up the entire quest. It was no use; he had failed.

Although Einstein was depressed, his thoughts were still churning in his mind when he returned home that night. In particular, he remembered riding in a streetcar in Bern and looking back at the famous clock tower that dominated the city. He then imagined what would happen if his streetcar raced away from the clock tower at the speed of light. He quickly realized that the clock would appear stopped, since light could not catch up to the streetcar, but his own clock in the streetcar would beat normally.

Then it suddenly hit him, the key to the entire problem. Einstein recalled, "A storm broke loose in my mind." The answer was simple and elegant: time can beat at different rates throughout the universe, depending on how fast you moved. Imagine clocks scattered at different points in Einstein in the Bern patent office space, each one announcing a different time, each one ticking at a different rate. One second on in 1904, just months away from Earth was not the same length as one second on the moon or one second on Jupiter. In fact, the the brilliant insight that led to faster you moved, the more time slowed down. (Einstein once joked that in relativity theory, he his theory of special relativity— placed a clock at every point in the universe, each one running at a different rate, but in real life and, a few weeks later, to E = 2 he didn't have enough money to buy even one.) This meant that events that were simultaneous mc in one frame were not necessarily simultaneous in another frame, as Newton thought. He had finally tapped into "God's thoughts." He would recall excitedly, "The solution came to me suddenly with the thought that our concepts and laws of space and time can only claim validity insofar as they stand in a clear relation to our experiences.... By a revision of the concept of simultaneity into a more malleable form, I thus arrived at the theory of relativity."

“Thank you, I’ve completely solved the problem.”

For example, remember that in the paradox of the speeding motorist, the police officer was traveling neck and neck with the speeding light beam, while the officer himself claimed that the light beam was speeding away from him at precisely the speed of light, no matter how much he gunned his engines. The only way to reconcile these two pictures is to have the brain of the officer slow down. Time slows down for the policeman. If we could have seen the officer's wristwatch from the roadside, we would have seen that it nearly stopped and that his facial expressions were frozen in time. Thus, from our point of view, we saw him speeding neck and neck with the light beam, but his clocks (and his brain) were nearly stopped. When we interviewed the officer later, we found that he perceived the light beam to be speeding away, only because his brain and clocks were running much slower.

THE PAPER THAT CHANGED EVERYTHING

The day after this revelation, Einstein went back to Besso's home and, without even saying hello, he blurted out, "Thank you, I've completely solved the problem." He would proudly recall, "An analysis of the concept of time was my solution. Time cannot be absolutely defined, and there is an inseparable relation between time and signal velocity." For the next six weeks, he A streetcar trundles below the furiously worked out every mathematical detail of his brilliant insight, leading to a paper that is clock tower in Bern that Einstein arguably one of the most important scientific papers of all time. According to his son, he then made famous with his thought went straight to bed for two weeks after giving the paper to his wife Mileva to check for any experiment about racing a light mathematical errors. The final paper, "On the Electrodynamics of Moving Bodies," was scribbled beam. on 31 handwritten pages, but it changed world history.

In the paper, he does not acknowledge any other physicist; he only gives thanks to Michele

Besso. It was finally published in Annalen der Physik in September 1905, in volume 17. In fact, Einstein would publish three of his pathbreaking papers in that famous volume 17. His colleague

Max Born has written, volume 17 is "one of the most remarkable volumes in the whole scientific literature. It contains three papers by Einstein, each dealing with a different subject and each today acknowledged to be a masterpiece." (Copies of that famous volume sold for $15,000 at an auction in 1994.)

With almost breathtaking sweep, Einstein began his paper by proclaiming that his theories worked not just for light, but were truths about the universe itself. Remarkably, he derived all his work from two simple postulates applying to inertial frames (i.e., objects that move with constant velocity with respect to each other):

1. The laws of physics are the same in all inertial frames. 2. The speed of light is a constant in all inertial frames.

These two deceptively simple principles mark the most profound insights into the nature of the universe since Newton's work. From them, one can derive an entirely new picture of space and time.

LENGTH, LIKE TIME, IS RELATIVE

First, in one masterful stroke, Einstein elegantly proved that if the speed of light was indeed a constant of nature, then the most general solution was the Lorentz transformation*. He then showed that Maxwell's equations did indeed respect that principle. Last, he showed that velocities add in a peculiar way. Although Newton, observing the motion of sailing ships, concluded that velocities could add without limit, Einstein concluded that the speed of light was the ultimate velocity in the universe. Imagine, for a moment, that you are in a rocket speeding at 90 percent the speed of light away from Earth. Now fire a bullet inside the rocket that is also going at 90 percent the speed of light. According to Newtonian physics, the bullet should be going at 180 percent the speed of light, thus exceeding light velocity. But Einstein showed that because meter sticks are shortening and time is slowing down, the sum of these velocities is actually close to 99 percent the speed of light. In fact, Einstein could show that no matter how hard you tried, you could never boost yourself beyond the speed of light. Light velocity was the ultimate speed limit in the universe.

We never see these bizarre distortions in our experience because we never travel near the speed of light. For everyday velocities, Newton's laws are perfectly fine. This is the fundamental reason why it took over 200 years to discover the first correction to Newton's laws. But now imagine that the speed of light is only 20 miles per hour. If a car were to go down the street, it Volume 17 of the German might look compressed in the direction of motion, being squeezed like an accordion down to physics journal Annalen der perhaps one inch in length, for example, although its height would remain the same. Because Physik, in which Einstein the passengers in the car are compressed down to one inch, we might expect them to yell and published no fewer than three scream as their bones are crushed. In fact, the passengers see nothing wrong, since everything groundbreaking papers at age inside the car, including the atoms in their bodies, is squeezed as well. 26.

As the car slows down to a stop, it would slowly expand from one inch to about 10 feet, and the passengers would walk out as if nothing happened. Who is really compressed? You or the car?

According to relativity, you cannot tell, since the concept of length has no absolute meaning.

THE GREATEST AFTERTHOUGHT IN HISTORY

Einstein then pushed further and made the next fateful leap. He wrote a small paper, almost a footnote, late in 1905 that would change world history. If meter sticks and clocks became distorted the faster you moved, then everything you can measure with meter sticks and clocks must also change, including matter and energy. In fact, matter and energy could change into each other. For example, Einstein could show that the mass of an object increased the faster it moved. (Its mass would in fact become infinite if you hit the speed of light—which is impossible, which proves the unattainability of the speed of light.) This meant that the energy of motion was somehow being transformed into increasing the mass of the object. Thus, matter and energy are interchangeable. If you calculated precisely how much energy was being converted into mass, in a few simple lines you could show that E = mc2, the most celebrated equation of all time. Since the speed of light was a fantastically large number, and its square was even larger, this meant that even a tiny amount of matter could release a fabulous amount of energy. A few teaspoons of matter, for example, has the energy of several hydrogen bombs. In fact, a piece of matter the size of a house might be enough to crack the Earth in half.

“Imagine the audacity of such a step ... every speck of dust becoming a prodigious reservoir of untapped energy.” Other scientists came close to stumbling upon relativity before Einstein, including the Dutch physicist Hendrik Lorentz Einstein's formula was not simply an academic exercise, because he believed that it might (seated fourth from left) and the explain the curious fact discovered by Marie Curie, that just an ounce of radium emitted 4,000 French mathematician Henri calories of heat per hour indefinitely, seemingly violating the first law of thermodynamics (which Poincaré (seated far right, next states that the total amount of energy is always constant or conserved). He concluded that to Marie Curie). Einstein is there should be a slight decrease in its mass as radium radiated away energy (an amount too standing second from right in small to be measured using the equipment of 1905). "The idea is amusing and enticing; but this photo from a 1911 whether the Almighty is laughing at it and is leading me up the garden path—that I cannot conference. know," he wrote. He concluded that a direct verification of his conjecture "for the time being probably lies beyond the realm of possible experience."

Why hadn't this untapped energy been noticed before? He compared this to a fabulously rich man who kept his wealth secret by never spending a cent.

Banesh Hoffman, a former student, wrote, "Imagine the audacity of such a step.... Every clod of earth, every feather, every speck of dust becoming a prodigious reservoir of untapped energy. There was no way of verifying this at the time. Yet in presenting his equation in 1907 Einstein spoke of it as the most important consequence of his theory of relativity. His extraordinary ability to see far ahead is shown by the fact that his equation was not verified ... until some twenty-five years later."

Once again, the relativity principle forced a major revision in classical physics. Before, physicists believed in the conservation of energy, the first law of thermodynamics, which states that the total amount of energy can never be created or destroyed. Now physicists considered the total combined amount of matter and energy as being conserved.

The world's most famous *Named for the Dutch physicist Hendrik Lorentz, who calculated them, the Lorentz equation, as it appears in transformations are the distortions of space and time inherent in the equations for light, i.e., modified form in a manuscript Maxwell's equations. These transformations state that the faster you move, the slower time on special relativity theory that beats for you and the more compressed you become. (At the speed of light, hypothetically time Einstein wrote in 1912 would stop and distances would shrink to nothing, both of which are impossible.) These transformations are necessary to keep the speed of light a constant in all inertial frames.

Michio Kaku, a theoretical physicist at the City University of New York, is the author of Einstein's Cosmos: How Albert Einstein's Vision Transformed Our Understanding of Space and Time (Norton, 2004), from which this article was adapted with kind permission of the author and publisher

http://www.pbs.org/wgbh/nova/einstein

Relativity and the Cosmos

by Alan Lightman

In November of 1919, at the age of 40, Albert Einstein became an overnight celebrity, thanks to a solar eclipse. An experiment had confirmed that light rays from distant stars were deflected by the gravity of the sun in just the amount he had predicted in his theory of gravity, general relativity. General relativity was the first major new theory of gravity since Isaac Newton's more than 250 years earlier.

Einstein became a hero, and the myth-building began. Headlines appeared in newspapers all over the world. On November 8, 1919, for example, the London Times had an article headlined: "The Revolution In Science/Einstein Versus Newton." Two days later, The New York Times' headlines read: "Lights All Askew In The Heavens/Men Of Science More Or Less Agog Over Results Of Eclipse Observations/Einstein Theory Triumphs." The planet was exhausted from World War I, eager for some sign of humankind's nobility, and suddenly here was a modest scientific genius, seemingly interested only in pure intellectual pursuits.

THE ESSENCE OF GRAVITY

What was general relativity? Einstein's earlier theory of time and space, special relativity, If it were not for Einstein, proposed that distance and time are not absolute. The ticking rate of a clock depends on several decades might have the motion of the observer of that clock; likewise for the length of a "yardstick." Published passed before another physicist in 1915, general relativity proposed that gravity, as well as motion, can affect the intervals worked out the concepts and of time and of space. The key idea of general relativity, called the equivalence principle, is mathematics of general that gravity pulling in one direction is completely equivalent to an acceleration in the relativity, Lightman says. opposite direction. A car accelerating forwards feels just like sideways gravity pushing you back against your seat. An elevator accelerating upwards feels just like gravity pushing you into the floor.

If gravity is equivalent to acceleration, and if motion affects measurements of time and space (as shown in special relativity), then it follows that gravity does so as well. In particular, the gravity of any mass, such as our sun, has the effect of warping the space and time around it. For example, the angles of a triangle no longer add up to 180 degrees, and clocks tick more slowly the closer they are to a gravitational mass like the sun.

Many of the predictions of general relativity, such as the bending of starlight by gravity and a tiny shift in the orbit of the planet Mercury, have been quantitatively confirmed by experiment. Two of the strangest predictions, impossible ever to completely confirm, are the existence of black holes and the effect of gravity on the universe as a whole (cosmology). Visit: http://www.pbs.org/wgbh/nova/ COLLAPSED STARS einstein/rela-i.html to see:

A black hole is a region of space whose attractive gravitational force is so intense that no  Einstein racing a light beam, a matter, light, or communication of any kind can escape. A black hole would thus appear thought experiment that led him black from the outside. (However, gas around a black hole can be very bright.) It is to special relativity; believed that black holes form from the collapse of stars. As long as they are emitting heat and light into space, stars are able to support themselves against their own inward gravity  Einstein in an elevator, which with the outward pressure generated by heat from nuclear reactions in their deep interiors. shows how gravity and acceleration are the same;  and the sun warping space-time, Every star, however, must eventually exhaust its nuclear fuel. When it does so, its a visualization of general unbalanced self-gravitational attraction causes it to collapse. According to theory, if a relativity burned-out star has a mass larger than about three times the mass of our sun, no amount of additional pressure can stave off total gravitational collapse. The star collapses to form a black hole. For a nonrotating collapsed star, the size of the resulting black hole is proportional to the mass of the parent star; a black hole with a mass three times that of our sun would have a diameter of about 10 miles.

General relativity may be the biggest leap of the scientific imagination in history.

The possibility that stars could collapse to form black holes was first theoretically "discovered" in 1939 by J. Robert Oppenheimer and Hartland Snyder, who were manipulating the equations of Einstein's general relativity. The first black hole believed to be discovered in the physical world, as opposed to the mathematical world of pencil and paper, was Cygnus X-1, about 7,000 light-years from Earth. (A light-year, the distance light travels in a year, is about six trillion miles.) Cygnus X-1 was found in 1970. Since then, a dozen excellent black hole candidates have been identified. Many astronomers and astrophysicists believe that massive black holes, with sizes up to 10 million times that of our sun, inhabit the centers of energetic galaxies and quasars and are responsible for their enormous energy release. Ironically, Einstein himself did not believe in the existence of black holes, even though they were predicted by his theory. A swirling gas disk around a probable black hole in M87 THE START OF EVERYTHING Galaxy

Beginning in 1917, Einstein and others applied general relativity to the structure and evolution of the universe as a whole. The leading cosmological theory, called the big bang theory, was formulated in 1922 by the Russian mathematician and meteorologist Alexander Friedmann. Friedmann began with Einstein's equations of general relativity and found a solution to those equations in which the universe began in a state of extremely high density and temperature (the so-called big bang) and then expanded in time, thinning out and cooling as it did so. One of the most stunning successes of the big bang theory is the prediction that the universe is approximately 10 billion years old, a result obtained from the rate at which distant galaxies are flying away from each other. This prediction accords with the age of the universe as obtained from very local methods, such as the dating of radioactive rocks on Earth.

According to the big bang theory, the universe may keep expanding forever, if its inward gravity is not sufficiently strong to counterbalance the outward motion of galaxies, or it may reach a maximum point of expansion and then start collapsing, growing denser and denser, gradually disrupting galaxies, stars, planets, people, and eventually even individual atoms. Which of these two fates awaits our universe can be determined by measuring the density of matter versus the rate of expansion. Much of modern cosmology, including the construction of giant new telescopes such as the new Keck telescope in Hawaii, has been an attempt to measure these two numbers with better and better accuracy. With the present accuracy of measurement, the numbers suggest that our An image of distant galaxies universe will keep expanding forever, growing colder and colder, thinner and thinner. taken by Hubble Deep Field

General relativity may be the biggest leap of the scientific imagination in history. Unlike many previous scientific breakthroughs, such as the principle of natural selection, or the discovery of the physical existence of atoms, general relativity had little foundation upon the theories or experiments of the time. No one except Einstein was thinking of gravity as equivalent to acceleration, as a geometrical phenomenon, as a bending of time and space. Although it is impossible to know, many physicists believe that without Einstein, it could have been another few decades or more before another physicist worked out the concepts and mathematics of general relativity.

Note: This feature originally appeared on NOVA's "Einstein Revealed" Web site, which has been subsumed into the "Einstein's Big Idea" Web site. Alan Lightman, a physicist and novelist, is currently Adjunct Professor of Humanities at MIT. Some of his recent books are Einstein's Dreams, The Diagnosis, Reunion, A Sense of the Mysterious, and The Discoveries.

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