Happy 3.14!

Albert Einstein's birthday is easy for mathematicians to remember, March 14 or 3.14! The Einstein-de Sitter Universe has a "critical" density of (6 $\pi$G t^2)^(-1), a density that keeps it from collapsing or expanding without bounds. Scientists long wondered why the density is exactly this, invoking strange ideas like "inflation" to explain it.

A cosmology where GM=tc^3 actually predicts that the "critical" density is in fact the stable density. If the Universe were less than this density, matter would be created via pair production until this density were reached. For a 4-dimensional spherical Universe of mass M, initial density is just (2 $\pi$^2 G t^2)^(-1). Difference between initial and final density is the difference between 3 and $\pi$ or 4.507034%. The density of baryonic matter that has been measured by the WMAP spacecraft may be precisely predicted from pure math.

Ancient Greek mathematicians first thought that the ratio between a circle's circumference and diameter was exactly 3, then later wondered why it should be an irrational number like 3.14. Fortunately $\pi$ does not equal 3, or the matter that we are made of would not exist!

Higher Authority

Here is someone who has revolutionized science, was selfless in pursuit of the truth, overcame skepticism and outright prejudice, while maintaining a child's wonder about the universe. Rather than just learn equations from a book, this scientist thought up new equations! The statue of Albert Einstein sits in front of the National Academy of Sciences in Washington.

The sculptor, Robert Berks, gave Einstein a notebook with three of his famous equations.

E=mc^2

The most famous Einstein equation, equating energy with mass. It was published as a quick addendum to one of his 1905 papers. Einstein published 4 revolutionary papers in a short time. The world took years to realize their significance. Today E=mc^2 is ubiquitous, even to those who don't understand what it means.

eV = h $\nu$ - A

The photoelectric effect, which Einstein explained using quantum mechanics. This probably got the attention of Annalen der Physik editor Max Planck, one of the founders of quantum mechanics. There was no system of "peer review" in 1905; publication of papers was solely the decision of editors like Planck. If not for Planck, Einstein might have waited indefinitely to be published. Einstein's solution to the photoelectric effect was the official reason for his 1921 Nobel Prize, for even in 1921 Relativity was controversial.

Ruv-½guvR=-κTuv

The Einstein equation for gravitation, part of his General Theory of Relativity published in 1915. General Relativity predicted that gravity is actually a curvature of Space/Time. Einstein tried to apply this curvature to the entire universe, imagining it as a sphere of four dimensions rather than 3. We move in the 3-D surface of the sphere. Photons travel like satellites in orbit around the sphere with velocity c.

Einstein realized that the gravity which causes Space to be curved into a sphere would also cause it to collapse, unless it were expanding. Einstein could have predicted an expanding universe, which would have been one of history's great scientific predictions. Certainly the prediction would have been ridiculed at first. 15 years later Edwin Hubble's observations would show that the universe was indeed expanding.

To support his spherical Space/Time, Einstein added a repulsive "cosmological constant" to the equation. When Hubble showed that the universe was expanding, Einstein removed the constant. Later Einstein would call the CC his greatest blunder. Recently it has been fashionable to add the repulsive constant again. So far, Einstein has still not been convinced. His notebook shows that there is no cosmological constant.

We can learn a lot from Einstein. The very formulation of original equations is proof of his unusual mind. Finding equations that made testable predictions made Einstein even more special. We learn that even Nobel Prize winners make blunders. The inclusion of a cosmological constant also hides another prediction, that the speed of light c is not always constant.

NEXT: We visit another Nobel Prize Winner.

Fire

Astronomer Michael Brown is known for his observations of Kuiper Belt objects like Sedna. This weekend, from his 9th floor window at Caltech, he made this photo of the fire raging in the California hills. JPL has already been evacuated, but appears to be out of danger. Presently Mount Wilson and it's famous telescopes are in the line of fire.

From Mount Wilson's 100 inch telescope Edwin Hubble found evidence that the Universe was expanding. Hubble's initial graph of redshift vs. magnitude was so scattered that it is amazing that anyone could see a trend. Despite the fuzzy data, Hubble's chart convinced even Einstein that the Universe expanded. In a well-publicized 1932 visit to Mount Wilson, Einstein conferred with Hubble and peered through the telescope. To the assembled reporters Einstein happily admitted that his Cosmological Constant was an error.

Old ideas eventually give way to better theories. Out of the ashes of the cosmological constant came the Big Bang theory. A simple expression like R = ct can predict an expanding Universe. Out of today's speculation about cosmological constants and "dark energies" will come a better theory of the Universe.

Earth Stands Still

A hat-tip to the wondrous Kea, who will soon be working at Oxford. In the new version of DAY THE EARTH STOOD STILL, Klaatu is deeply concerned about he mess humans have made on the planet. Again he comes across the scientist's blackboard filled with our primitive math. He begins by crossing out the "lambda" term in Einstein's field equation, which is a useless contrivance. Einstein himself called this "cosmic constant" his greatest blunder. Without it, Einstein might have predicted expansion of the Universe.

Now the equation looks like:

Ruv-½guvR=-κTuv

Klaatu then writes in the solutions of his more advanced species. Today, as was done 2 years ago, we can guess at what he wrote:




Today's physicists, trained for years in our primitive science, may raise petty objections when someone alters their equations. On the question of units, the Friedmann equations are valid in units of mass density or energy density.

The stress-energy tensor T_uv can have units of mass density ρ or energy density ρ(c^2). Removing for a moment the c^2 from Friedmann we would have:
8πGρ/3 = ⅓κρ (c^2)
4πGρ/3 = ⅙κρ (c^2)

To normalise the left-hand and right-hand sides, some physicists chose κ=8πG/(c^2). This led to decades of misconception that Relativity requires a fixed c. Some "geometrized" unit systems give κ=8πG/(c^4), which is too convoluted to describe.

If T_uv has units of energy density, we must use the same for Friedmann:
8πGρ(c^2)/3 = ⅓κρ(c^2)
4πGρ(c^2)/3 = ⅙κρ(c^2)
κ=8πG
This is very trivial. The (c^2) on both sides simply cancels out. Einstein called constant κ "related to the gravitational constant" without mentioning c.

Now the Einstein equation becomes Ruv-½guvR=8πGTuv. The Bianchi identities become:
▽u(Ruv-½guvR)=0
8πG▽u(Tuv)=0
The world is much simpler without that pesky (c^2) factor.

Finally, the Einstein-Hilbert action becomes:
S=∫(16πGR+Lm)d^4(x)
Thus we can do everything General Relativity can without a fixed c. Some problems, like the deflection of bodies by the Sun, work even better with a varying c.

In reality, a human professor would throw Klaatu out for messing with his blackboard. A Universe that can be described in a few equations just might be beyond human understanding. Humans tend to complicate their Universe with epicycles, luminiferous ether, or cosmic constants. Are humans ready for new physics? Perhaps we should ask Klaatu.

Wilbur and Orville's 105th

December 17 was the 105th anniversary of the Wright Brothers flight. Though a local reporter wrote about the story, it took some months for governments to show interest. In September 1908, while Wilberr was in Europe, Orville demonstrated his plane for the US Army. Despite a crash which killed an observer and severely injured Orville, the Army would buy its first plane in June 1909. In 1911 the US Navy purchased its first aircraft, signalling the start of Naval Aviation. In the Great War of 1914-1918 aeroplanes played an important role. Today flight in aeroplanes is so commonplace as to be a nuisance.

In 1905 a patent clerk named Einstein had performed the amazing feat of publishing 4 groundbreaking papers in a very short time. If a sympathetic Max Planck had not been editor, Einstein might not have published. The first reaction was a deafening silence--by 1908 few had heard of Einstein. (During 1908 and 1909 Einstein had a fascinating disagreement with Walter Ritz on the time arrows of electrodynamics and entropy.) In 1911 Einstein would finally get an academic appointment. Not until 1919 and Eddington's eclipse expedition would Einstein become a celebrity.

We hope that in 100 years the achievements of today, like aircraft and Space flight, become commonplace.

Einstein and Gamow Take a Walk

George Gamow, another giant of 20th century physics, made mistakes of his own. He had a tempestuous marriage, drank heavily, was fond of gossip and practical jokes. His famous "Alpha, Beta, Gamma" paper was co-authored by Ralph Alpher with Hans Bethe added to complete the title. This 1948 paper describes the formation of elements after the Big Bang. A follow-up paper predicted a cosmic microwave background with a temperature of 5K. In his book "Creation of the Universe" Gamow mistakenly calculates the temperature at 50 degrees K!

Prediction of the cosmic microwave background was ignored and forgotten for nearly two decades. Robert Dicke and James Peebles of Princeton, unaware of Gamow's work, independently calculated CMB temperature at 10K. Nearby at Bell Labs, physicists Arthur Penzias and Joseph Wilson were tuning a large microwave antenna to communicatewith satellites. Finding a strange background signal, Penzias and Wilson accidentally discovered the CMB. It's temperature would eventually be measured as 2.7K, barely half Gamow's original prediction.

One day in the 1940's Einstein and Gamow were walking through Princeton. Gamow mentioned that one of his students had calculated that it was possible to make a star from nothing; its gravitational energy is equal and opposite to rest energy. Einstein, realising that this could apply to the whole Universe, stopped in the middle of the street. Unfotunately history has not recorded the math behind this calculation or who Gamow's student was.

The gravitational potential energy U of two particles is: U = -GMm/R. The total mechanical energy of a star works out to:

E ~ -(3/10)(GM^2)/R

Where M and R are the star's Mass and Radius. If our Sun's luminosity came from gravitational collapse, this energy would be used up in only 10 million years. Early in the 20th century, this led astronomers to conclude that something beyond gravity powered the stars.

What had Gamow's student found? If we equate an object's total potential energy with rest energy, we get:

GMm/R = mc^2

R = GM/c^2

Where M and R relate to a distant centre of Mass. If G, M and c are fixed this is the spherical Einstein Space. Such a Space would collapse unless it were expanding or supported by some repulsive force.

As nige and others have found, we can insert R = ct. Applied to the Universe, Scale R is distance to the Big Bang, age t multiplied by conversion factor c.

ct = GM/c^2

GM = tc^3

This can also be derived from Relativity, but doing so raises petty mathematical objections. There are multiple ways to derive GM = tc^3, but this is among the simplest and the most difficult to contest.

Thanks to George Gamow, we learned many things about physics and cosmology. His prediction of the CMB, though inaccurate in value, was eventually proved by Penzias and Wilson. We may never know exactly what Einstein and Gamow had hit upon while crossing the street. Perhaps they suspected that total energy of the Universe is zero. Our Universe may be the ultimate free lunch, which has allowed it to grow from a tiny point to the immensity we enjoy today.

Einstein's Mistakes

Fuld Hall at the Institute for Advanced Studies, a beautiful but chilly day in November 007.

EINSTEIN'S MISTAKES: THE HUMAN FAILING OF GENIUS by Hans Ohanian goes on sale today. The book does not try to diminish the man's extraordinary achievements. Physicist Ohanian shows that Einstein was human like the rest of us. Like a real scientist trying different approaches, Einstein made many errors. Part of his greatness was the willingness to make mistakes

The first derivation of m=E/c^2 is in a 2-page addendum to "The Electrodynamics of Moving Bodies." There is an error in this hastily written proof, and more mistakes in the second, third and fourth proofs of 1906-07. Only in 1911 did Max Laue produce a full proof. Einstein's fifth proof in 1914, his sixth proof in 1934 and his seventh proof in 1946 also contained errors. By this time nuclear reactions and the atomic bomb had proved to the world that E=mc^2.

EINSTEIN'S MISTAKES finds errors in 4 of the 5 seminal papers, including the photoelectric effect and the size of molecules from Brownian motion. Einstein's PhD thesis was full of mistakes. The road to General Relativity contained many errors and dead ends. In 1916 Einstein made a mistake interpreting Mach's Principle. The last decades of Einstein's life were spent in a fruitless search for Unified Field Theory. Attempting to unite electromagnetism with gravity, he tried many approaches which all failed. The book lists dozens of mistakes stretching across an entire career.

Einstein's greatest blunder was the cosmological constant. In a 1917 paper Einstein dared to imagine the entire Universe. According to General Relativity, mass causes Space/Time to be curved. Einstein realised that enough mass would cause the Universe to be curved into a sphere of 4 dimensions. Travel in any spatial direction would be confined to the sphere. Such a sphere would collapse under its own gravity, unless it were already expanding. To support the sphere Einstein invoked the fudge factor of a cosmological constant.

If Einstein had proposed an expanding Universe it would have been one of history's great predictions. No doubt he would have been ridiculed for having no supporting evidence. That's why it is called a prediction, boys! If Einstein had waited long enough, Edwin Hubble would eventually have proven his amazing prediction. When faced with Hubble's evidence Einstein had to admit that the cosmological constant was a blunder.

Fortunately Einstein had a great patron in Max Planck. Planck was an editor of Annalen Der Physik, otherwise Einstein's great papers may not have been published in 1905. In later years Planck was first to hail Einstein as a new Copernicus. Planck's blackbody formula was an experimental result; many years passed before someone found a mathematical derivation.

Einstein had another regret: Submitting to refereed journals. Around 1936 he submitted a paper to Physical Review, the leading American journal. Despite Einstein being by then the world's most renowned scientist, his paper was returned with anonymous comments from a referee. Einstein withdrew the paper and henceforth avoided PRL and any journal with anonymous referees. This was a blessing in disguise, for the paper contained an error.

A child could figure out that R=ct and GM=tc^3. Proposing this gets the most tangled mathematical objections: the units don't add up (they do), the metric is (1, 1, 1, 1) rather than (1, -1, -1, -1),...and so on. We can see why these people, despite their expensive education, have never come up with anything original. By their own admission, such people referee papers. If they were reviewing Copernicus, they would insist that everything be expressed in terms of a fixed Earth.

History gives us many lessons for today. Even an Einstein can make mistakes. The cosmological constant was a blunder and still is. Predicting an expanding Universe or a shrinking speed of light is worth the wait for confirmation. Don't feel bad if refereed journals reject your stuff. A great physicist is prepared to make mistakes, lots of them. The book admires Einstein's "mystical, intuitive" approach. He used mistakes as "stepping stones and shortcuts" to success. We can still learn a lot from Einstein.

TOMORROW: Einstein and Gamow take a walk

A to Z with the Carnival of Space!

3.14

DR. WHO David Tennant as Sir Arthur Eddington in the upcoming production EINSTEIN AND EDDINGTON.

For today's mathematicians, it is easy to remember that 3.14 is Einstein's birthday. Today we are used to overnight fame, and from this perspective it seems that Einstein's ideas were accepted quickly. It was years before Einstein and his papers became public knowledge. A century later we can review how long that took.

In the 5th year of the century Einstein's four major papers are published. He is still working in the patent office and has not received his PhD. Fortunately Max Planck is an editor and sees the value in Einstein's photoelectric effect. If not for Planck, Einstein's papers might not have been published for years.

After Einstein's publication, the response is a deafening silence. Not until the 9th year of the century does Einstein get an academic job. Not until the 11th year does SCIENTIFIC AMERICAN magazine make any mention of Einstein's work. In the 15th year Einstein finally completes his General Theory, but few people notice. The 14th to 18th years are occupied by a long and costly war.

In the 19th year of the century Arthur Eddington makes his famous eclipse expedition. Though a respected British scientist and Director of the Cambridge Observatory, Eddington at 36 is younger than Einstein. He would have been college age upon first reading Einstein's papers. Some jokingly claim that Eddington is one of only three people including Einstein who understood Relativity. As one of the younger generation, Eddington is open to testing new ideas.

On November 6 of the 19th year Eddington's results are announced at the Royal Society. Today Wendy Freedman and others suspect that Eddington "cooked the books" to support Einstein. The next morning the Times of London announces: "Revolution in Science--New Theory of the Universe--Newton's Ideas Overthrown." Three days later the New York Times picks up the story and Einstein's fame spreads like wildfire. (This is about the time a JDEM would begin returning data.)

Copernicus published his "Revolutionibus Oblure Coelestium" in 1543, but not until the late 17th century was the Sun-centred model taught in universities. Along the way Giordano Bruno was burned at the stake and Galileo sentenced to house arrest. As late as the 1660's Isaac Newton heard lectures on the Ptolemaic system in Cambridge. Before Einstein came a communications revolution more profound than today's internet, telephones and telegraphs transmitting at the speed of light. Even with this quantum leap in technology, Einstein's ideas took more than a decade to spread around the world.

The number 3.14159 may have still more significance. As the wondrous Kea has noted, the difference between $\pi$ and 3 is 4.507034 percent, exactly the proportion of baryonic matter in the Universe. This is a check on the shape of Space/Time. If the Universe were a shape other than a sphere, that proportion would be different. This may be part of something our century will discover.

Griffith Park Above the Fire

Griffith Observatory was first opened in 1935 and reopened after a long renovation in November 2006. It has been backdrop for movies from REBEL WITHOUT A CAUSE to THE ROCKETEER. The site has a wonderful view of Los Angeles lights, which makes it terrible for observing but a great asset to the community.

Visitors: You can no longer drive here. You must purchase your tickets online and take one of the shuttle buses up the hill. Thus place is popular as a Hollywood premiere, and is expected to be very busy this Summer. It is nice to see so many people interested in astronomy!

Nearby atop Mount Wilson stands the 100-inch Hooker telescope, once the largest in the world. When this telescope saw first light in 1917, astronomers disagreed whether our Milky Way represented the entire Universe. Andromeda was called a nebula, and no one agreed whether it was part of our galaxy or outside. Using the Hooker telescope, Edwin Hubble determined that "spiral nebulae" were in fact distant galaxies.

Albert Einstein theorised that the Universe was spherical in four dimensions rather than three. Every bit would resemble every other bit. He realised that gravity would cause the Universe to collapse, UNLESS it were already expanding. Einstein could have predicted an expanding Universe, but instead introduced a repulsive "cosmological constant" opposing gravity.

Hubble and his colleague Milton Humason relied upon a class of stars called Cepheid Variables. These stars vary periodically in brightness, with a period related to their luminosity. Observing a Cepheid’s period would tell them how much light the star gave off. By measuring how much of that light reached Earth, they cold determine distance to the star and galaxy. Cepheid Variables were standard candles measuring the distance to their galaxies.

Redshift is approximately v/c, an object’s velocity divided by the speed of light. When the redshifts of many galaxies were plotted against their distances, those redshifts increased linearly with distance. This did not indicate that our galaxy was unpopular. If the Universe was spherical like a balloon, the galaxies were like spots on its surface. As a balloon expanded, the spots would increase their distance uniformly. A galaxy twice as distant would recede twice as fast. The distance-redshift relation was convincing evidence that our Universe was expanding.

In a well-publicised 1931 visit to Mount Wilson, Einstein conferred with Hubble and peered through Hubble’s telescope. The world’s most famous scientist happily accepted the expanding Universe. Hubble’s data convinced Einstein to drop the cosmological constant, later calling the CC his “greatest blunder.” If Einstein called something a blunder he was probably right.

UPDATE: As this was posted, a huge fire has erupted in Griffith Park, visible for miles. Fortunately it is a big park and the fire is on the opposite side. The observatory has been closed as a precaution. Southern California has been very hot and dry.

UPDATE: As of midnight PST, the fire has not been contained and is inching closer to the observatory. Residents of Los Feliz have been ordered out. The view is eerily similiar to the Oakland Hills fire of 1991. This could get bad.


UPDATE: This morning the observatory appears out of danger. The fire had been moving North toward the zoo, and was expected to die down by sunset. Instead the wind shifted South toward the observatory and the Los Feliz neighbourhood. The fire is still burning and large parts of Griffith Park are black.