The March of Time

Time is a mystery to physicists.  The Newtonian notion of absolute time, that is, all clocks run at the same rate, was demolished by Einstein’s relativity theory.  Despite the fact that we know clocks in different reference frames can run at different rates, we don’t know what exactly time is.  Does time run continuously or in discrete packets?  Can one travel backwards in time?  Why don’t the laws of physics tell us what time is?  An intriguing book, Now by Richard Muller, gives us some paths to seek these answers.

During the Victorian age, time was considered a constant for everyone.  A conductor’s watch on the way to London would run at the same rate as the station master’s clock.  Perhaps this is why Big Ben is the perfect symbol for that era.  Einstein proved this notion wrong.  The conductor’s pocket watch will run slower than the motionless station master’s clock.  Given the speeds we travel, the difference is too small to discern.  Einstein’s genius was to follow a theory to its natural conclusion and not allowing false intuition to lead him astray.  Clocks run slower the faster you go.  Gravity also slows the rate of time.  Once you hit the speed of light, or enter the infinitely deep gravity well of a black hole, time stands still.

The question remained, why does time always flow forward?  The most common response to that has been quoting the 2nd law of thermodynamics.  This is the only law in physics that suggests a flow of time.  It states that entropy of the universe always increases with time.  More specifically, a closed system, one which energy cannot enter or leave, must become more disordered over time.  An open system, like the Earth which receives a constant stream of energy from the Sun, can experience a decrease in entropy and an increase in ordered states.  The universe, as far as we know, is a closed system.  This argument was first advanced by Arthur Eddington, but Richard Muller proceeds to poke some holes in it.

Eddington was one of the most accomplished astronomers of the early 20th Century.  One of the first to grasp Einstein’s relativity theory, Eddington led an expedition to the island of Príncipe off the west coast of Africa to observe the solar eclipse of 1919.  Eddington was able to measure the bending of starlight by the Sun as predicted by Einstein.  Once this result was reported by the media, Einstein became the most famous scientist in the world.  Eddington was also what we would call today a populizer of science.  His hypothesis that entropy mandates the flow of time was published in his book The Nature of the Physical World, written in a manner for the general public to enjoy.

However, as Muller notes, clocks do not run slower in local regimes where entropy is decreasing.  A recent experiment verified that, at least on a quantum level, heat can run from a cold to warmer object.  This is a violation of the 2nd law of thermodynamics where energy runs from hot to cold objects.  Some of the news articles on this experiment also claim this has reversed the arrow of time.  Muller considers entropy and time to be two separate concepts.  Rather than rely on entropy, Muller’s hypothesis on time is tied into one of the most asked questions I receive from my students.

When going over the Big Bang, I am often asked what existed before then?  The answer is…nothing.  Space did not exist before the Big Bang.  And neither did time.  Muller speculates that just as space is being created with the expansion of the universe, so is time.  And it is this expansion that gives us the flow of time and a sense of now.  Unlike Eddington’s entropy argument, Muller provides a means of falsifying his theory.

The idea of falsifying a theory might seem odd as we are taught in grade school that experiments prove a theory right.  That’s not quite correct.  As Richard Feynman would say, we don’t prove a theory correct, only that it is not wrong.  Newton’s law of gravity was not wrong for a couple of centuries.  It predicts the motion of most celestial objects quite well.  By the late 1800’s, observations came in that Newton’s laws could not predict, specifically the precession of Mercury’s orbit.  Einstein’s relativity theory provided accurate predictions in two areas Newton could not, when an object is near a large gravity well like the Sun and when an object moves at velocities near the speed of light.

When Einstein realized his theory predicted Mercury’s orbit correctly, he was so excited he suffered from heart palpitations.  For a hundred years, Einstein’s theory has been proven not wrong.  It may take a unification of quantum mechanics and relativity to change that.

Muller speculates that as dark energy is accelerating the expansion of the universe, it must also accelerate time.  That is, time runs faster now than in the past.  To detect this, we must look at galaxies at least 8 billion light years away and make highly precise measurements of their red shifts.  Any excess in the red shift not predicted by space expansion would be caused by time expansion.  At this time, we do not have instruments to make this precise a measurement.  It’s not unusual for theory to race ahead of experimental ability.  After all, it took one hundred years to prove gravitational waves predicted by Einstein actually exist.

Is there any way to falsify relativity or quantum mechanics? To date, both have held up to rigorous testing.  One possibility is the simultaneous collapse of the quantum probability curve upon observation.  With the Copenhagen interpretation of quantum mechanics, atomic particles exist in all possible states along a probability curve.  Once observed, the probability curve collapses instantaneously to its exact state.  As Muller notes, this is in direct odds with relativity where nothing, not even information, can exceed the speed of light.  Perhaps, this can provide a crack in the theory that can lead to a unification of the physics of atoms and of large-scale objects.

Muller’s exploration of time delves into other topics, often Star Trek related. In the case of the transporter, Muller questions if the person assembled at the other end is the same person or a duplicate with the original destroyed.  I thought this was interesting as it follows the plot of James Blish’s one off Star Trek novel Spock Must Die.  Published in 1970, the opening chapter involves a rec room conversation between Scotty and Dr. McCoy, where McCoy frets over the possibility he is no longer his original self.  That is, the transporter destroyed the original McCoy the first time he used it and constructed a replicate each time afterward.  Scotty is nonplussed – “a difference that makes no difference is no difference.”

Would it make a difference?

As Muller notes, our bodies are mostly made of different atoms and cells than it was years ago, yet we maintain our sense of self.  The only thing that does change is the sense of now.  So, when I bought Spock Must Die in the mid-70’s, the body of atoms that searched through department and stationery store bookshelves, is markedly different than the body of atoms that purchased Now online.  In that sense, I am a replicate of my childhood self.  Yet, throughout that whole time, my mind has maintained a continuous state of consciousness.

That brings me back to an argument made in a college philosophy class.  If you take a boat, and replace each plank of wood over time, is it still the original boat?  Boats do not experience the sensation of time, it takes a mind to do that.  The brain, in some regions, does replace neurons throughout life and this may lead to memory loss.  For other regions, it appears not to replicate.  This may explain our continuity of consciousness, but as many a journal article has ended, more research is required in this area.

In the same philosophy class, our professor discussed how we lack access to each other’s state of consciousness.  Unless we could perform a mind meld a la Mr. Spock, our life experience and sense of time is locked up within each individual.  So, is time a matter that can be solved by physics alone?  Or does in require an interdisciplinary approach?  My instinct is that time is a problem for physics to solve.  We require eyes to see light and the mind to interpret it, but the electromagnetic waves that create light was solved by physics.  Until some evidence based results come in, we’ll have to keep an open mind. Many a time instincts have led a scientist astray.  How will this story end?

I honestly don’t know.  Only time will tell.

*Image atop post is a Munich clock store.  Credit:  Gregory Pijanowski

Black Holes – Past, Present, and Future

On November 27 1783, two days after the last of the British troops evacuated New York City to conclude the Revolutionary War, the rector of St. Michael’s Church near Leeds postulated the existence of stars so massive light could not escape its gravitational field.  The rector, John Michell, was also a scientist and the first to conceptualize what we now call a black hole.  Michell was using Newton’s theory of light consisting of corpuscles that had mass and were affected by gravity in the same manner any other body of mass would be.  This didn’t quite turn out the case and would take Einstein’s new theory of gravity described by relativity in 1915 to formalize the concept of a black hole.

George Washington enters New York City on November 27, 1783. Across the pond, John Michell had just published his theory of dark stars. Evacuation Day was last celebrated in New York City in 1916, a year after Einstein published his theory of general relativity paving the way for the modern understanding of black holes. Credit: Library of Congress.

Before we get into all that, we need to familiarize ourselves with the concept of escape velocity.  This is the velocity required to escape the gravity of a body of mass and is defined as follows:

Vescape = √(2GM/r) where:

G is the gravitation constant = 6.67408 × 10-11 m3 kg-1 s-2

M = mass of the body

r = radius of the body

To calculate the escape velocity of Earth:

         Vescape = √[2(6.67408 × 10-11 m3 kg-1 s-2)(5.972 x 1024 kg)/6,371 km]

= 11.2 km/s or 7 miles/second

What would have to happen for Earth to become a black hole?  Earth’s radius would have to be reduced to the point where the escape velocity is equal to the speed of light at 3.0 x 108 m/s or 186,282 miles per second.  For this to occur, Earth’s radius has to be reduced to 9 mm or about a third of an inch.  For the Sun to become a black hole, its radius would have to be reduced to 3 km or 1.9 miles.  As you probably now have surmised, black holes have to be very dense and/or very small.  This is where Einstein comes in.

By the time the 20th Century rolled around, it was thought that light consisted only of electromagnetic waves.  As such, gravity would not affect light and thus, Michell’s idea of a dark star had been forgotten.  In 1905, Einstein discovered the photoelectric effect.  Light striking a metal ejected electrons from the surface meaning light had to consist of particles as well as waves.  In 1915, Einstein’s general relativity theory viewed gravity as a bending of space-time rather than a force between two objects.  Light would be affected by gravity as it would travel along the bend on space-time around a body of mass.  The next step in formalizing a theory of black holes would come from the Eastern Front in Russia during World War I raging at the time general relativity theory was published.

The more mass an object has, the more it warps space-time as predicted by general relativity. Light and mass move along the curved space-time fabric or gravity well. Credit: ESA/C.Carreau

Karl Schwarzschild was a German astrophysicist who had volunteered for military duty in World War I.  While calculating artillery trajectories, Schwarzschild somehow found the time to solve Einstein’s field equation for a gravitational field around a non-rotating object.  If a mass was smaller than a certain radius, space-time would curve into itself in a manner that would not allow light to escape.  This is in some sense, the Michell solution but using Einstein’s relativity theory to describe gravity instead of Newton’s theory.  This radius, now called the Schwarzschild radius, is defined as:

rs = 2GM/c2

c = speed of light

Using the Sun as an example:

rs = 2(6.67408 × 10-11 m3 kg-1 s-2)(1.989 x 1030 kg)/(3.0 x 108 m/s)2

rs = 2944 m or 2.94 km

This is to say any mass the size of the Sun with a radius less than 2.94 km will form a black hole.  Any light or matter within the 2.94 km radius will not be able to escape the gravitational field of the black hole.  The radius defined by this equation is the event horizon surrounding a black hole.  The more mass in a black hole, the larger the event horizon.  Once an object or light passes the horizon, it can never get back out.  However, outside the radius, the effect of gravity is the same.  If the Sun’s radius was reduced to the point of being a black hole, Earth’s orbit would remain the same as the Sun’s mass is the same.

Unlike Michell’s concept, rather than a dark star smaller than this radius, a singularity would form.  A singularity is an object of only one dimension and of infinite density and is infinitely small (volume = 0).  Admittedly, this is a mathematically abstract concept that is difficult to imagine.  Think of a gravity well in the image above that has an infinite depth, the proverbial bottomless pit. Time also stands still in a black hole from the perspective of an outside observer.  The nature of a singularity seemed so bizarre that Einstein himself doubted there was a physical process that could create such an object.  It would be the father of the atomic bomb, Robert Oppenheimer, who would confirm that theoretically black holes could exist.

Gravity well of black hole is infinitely deep. Credit: NASA

As a star the mass of the Sun nears the end of its life, it runs out of hydrogen to fuse into helium atoms.  What’s left are helium atoms to fuse into carbon, and this type of nuclear fusion burns hotter.  This pushes the outer layers outward to form a red giant, a star so large it will swallow up the Earth.  Eventually, the helium runs out and the outward expansion ceases.  The red giant sheds its outer layers and what’s left over is a planetary nebula surrounding a shrinking core.  The core is shrinking as the inner force of gravity is now greater than the outward force of heat produced by fusion.  The remaining core is compressed to a white dwarf the size of Earth.  That’s pretty dense, in fact one teaspoon weighs 15 tons, but not quite small enough to be a black hole.  The radius of a white dwarf is on the scale of a few thousand kilometers and the Sun, as noted above, would have to collapse to smaller than 3 kilometers to be a black hole.

The Hourglass Nebula – a planetary nebula with a white dwarf embedded in the center. If you are a Pearl Jam fan you’ll recognize this from the Binaural CD cover. Credit: NASA, Raghvendra Sahai, John Trauger (JPL), and the WFPC2 Science Team

What keeps a sun-like star from collapsing into a black hole are the nuclear forces that bind atoms together.  This force is strong enough to keep atoms intact and prevents a gravitational collapse beyond the white dwarf stage.  When a star is 8-20 times the mass of the Sun, it ends its life in a supernova explosion.  These stars fuse elements up to iron at which point fusion can no longer occur.  The resultant supernova leaves an iron core that becomes a neutron star.  Here, the gravitational force is strong enough to compress electrons and protons to form neutrons.  The density becomes higher than in a white dwarf, one teaspoon of a neutron star weights about 10 million tons.  The gravity here is pretty intense but still not quite enough to form a black hole.  More mass is required, and this is where Robert Oppenheimer comes in.

Embedded in the Crab Nebula is a pulsar which is a rotating neutron star. The Crab Pulsar rotates 30 times a second. Credit: NASA, ESA, J. Hester, A. Loll (ASU)

In 1939, Oppenheimer, along with his student, George Volkoff, published a paper demonstrating that a collapsing star, with sufficient mass, could overcome nuclear forces and form a singularity.  As World War II was about to commence, Oppenheimer found himself busy with the Manhattan Project and the paper generally went forgotten.  At the time, general relativity and singularities were considered fallow ground for experimental research.  Black holes were still considered an odd offshoot of relativity theory.  The problem is, how to observe an object that by definition, does not emit light.  The solution could be found in Michell’s 1783 paper and some 20th Century technological advancements.

Albert Einstein and Robert Oppenheimer, 1950. Credit: Wiki Commons/Department of Defense.

Michell noted in his paper that a black hole would have to be detected by observing the impact on the mass around it.  By the 1960’s, interest had been revived in the topic, especially by John Wheeler.  It was Wheeler, in fact, who popularized the term black hole in 1967 (The Star Trek episode Tomorrow is Yesterday, aired in January 1967, refers to a black star).  Just three years prior, the first black hole candidate was detected as an x-ray source dubbed Cygnus X-1.  Why would the presence of x-ray emissions possibly be a sign of a black hole?  The answer lies in the surrounding accretion disk of matter falling into the black hole.

Matter falling in the surrounding accretion disk can be heated up to several million degrees.  At this temperature, matter will begin to emit high energy, short wavelength x-rays.  We are not able to observe x-rays from the Earth’s surface as they are absorbed in the upper atmosphere.  That’s a good thing as x-rays are harmful to life, but it does require observations above the surface.  The first observations of Cygnus X-1 in the 1960’s were made by sounding rockets and high-altitude aircraft.  The launch of the Chandra X-Ray Observatory in 1999 gave astronomers an opportunity to take a good look at Cygnus X-1 from space.

Hot gas surrounding Cygnus X-1. Credit: NASA/CXC

The hot gas is siphoned off from an orbiting blue giant that is visible.  This star orbits Cygnus X-1 every 5.6 days and from that, it can be deduced that Cygnus X-1 is 15 solar masses.  There is nothing known that can be that large and not be visible besides a black hole.  In 1975, Stephen Hawking bet Kip Thorne that Cygnus X-1 was not a black hole.  Since then, Hawking has conceded that he lost the gamble.  The age of orbiting telescopes would also reveal a different kind of black hole, one much larger in mass than the remnants of supernova explosions.

M87 is a very large elliptical galaxy containing several trillion stars and is 54 million light years from Earth.  Back in the 1950’s, there were hints of something unusual in M87 when a large radio source was detected.  When charged particles are accelerated, they emit radio waves.  This is the principle behind radio towers as electrons are accelerated up and down the tower producing a radio broadcast.  Sounding rockets during the 1960’s detected x-ray sources from the galaxy as with Cygnus X-1.  In 1998, the Hubble Space Telescope imaged a jet of electrons and sub-atomic particles protruding from M87.  Originally discovered in 1919 at the Lick Observatory, Hubble’s high resolution capabilities determined this 5,000 light year jet was caused by a black hole with a mass 2 billion times that of the Sun.

Jet emanating from M87. Credit: The Hubble Heritage Team (STScI/AURA) and NASA/ESA

How does a jet of matter become ejected from a region with a black hole?  Astronomers are not quite sure but it appears so much mass is trying to enter the black hole that it results in a traffic jam of sorts.  Think of it as shooting a fire hose into a bathtub drain.  The rejected material gets shot out along the intense magnetic field surrounding the black hole as charged particles will travel along the path of magnetic field lines.  M87 is not the only galaxy with a central black hole, in fact, most galaxies have been discovered to have these including the Milky Way.

Composite image of the Milky Way center. The blue and white at right center are x-ray observations of jets emanating from the galactic black hole. Yellow is near-infrared observations from Hubble and are areas of star formation. Red represents dust clouds detected by the Spitzer infrared telescope. Credit: NASA, ESA, SSC, CXC and STScI

During the summer months, the constellation Sagittarius is visible.  Located in this constellation is the center of the Milky Way.  We cannot see the center as it is shrouded by dust.  However, infrared observations allow us to peer behind the dust.  The UCLA Galactic Center Group has been using the 10-m Keck Telescope to observe the galactic center since 1995 to track the motions of stars in the region.  Just like using the orbit of the blue giant around Cygnus X-1 to determine the properties of the black hole, the UCLA team has been able to determine that the Milky Way’s central black hole is 4 million times the mass of the Sun and has a Schwarzschild radius 17 times the Sun’s radius.  Below are the observations from the UCLA team.

The star marks where the Milky Way central black hole is located. Credit: Keck/UCLA Galactic Center Group.

What does the future hold for black hole research?  One intriguing prospect is the possible existence and detection of atomic sized black holes.  Speculation is these would have formed during the Big Bang and pass routinely through our bodies.  The CERN supercollider may be able to produce such black holes.  No need to worry, it would not present a danger to Earth.  Most importantly, black holes represent where quantum mechanics and general relativity theory intersect.  Quantum mechanics provides the physics for atomic sized particles, relativity provides the physics for gravity and large objects.  Relativity breaks down once you reach the singularity.  As the universe was a singularity at the beginning of time, understanding the physics of gravity at this scale is required to understand the universe when it originated.  Black holes, once considered an abstract oddity of relativity theory, may be able to provide the key to the answer of how the universe came to exist.

* Image atop post is a computer simulation of a galactic black hole.  The edge of the black region is the Schwarzschild radius.  The light from stars passing near, but not inside, the Schwarzschild radius is smeared by the curvature in space-time caused by the black hole.  Credit:  NASA, ESA, and D. Coe, J. Anderson, and R. van der Marel (STScI)

The Education of Albert Einstein

Most historic figures have myths attached to them and certainly Albert Einstein is no exception.  Among them, Einstein failed math in high school and did his famous work on relativity in “splendid isolation”.  After reading Walter Isaacson’s biography on Einstein, one can see the social influences that shaped Einstein in his early years and how it enabled him to make advances in physics that others could not.  And much of that is rooted in modern educational theory.

Jean Piaget’s research on child development concluded there are four stages of development.  The final transition usually occurs around age eleven when a child moves from a concrete understanding of the world to an ability to solve abstract and hypothetical problems.  The age this transition occurs can vary with each individual and also with the subject matter.  Contrary to the struggling student myth, Einstein began thinking in abstract terms at a very early age.  A compass given to Einstein at age five demonstrates this.  Rather than thinking of the compass in concrete terms, that is, a mechanical device that points north, Einstein conjectured on the invisible magnetic field that caused the compass to always point north.  And this trend continued in Einstein’s early life.

During the 1930’s, a Ripley’s Believe It or Not! column stated Einstein failed math in high school and has remained part of the Einstein lore.  Truth is, Einstein had learned calculus by age 15.  And physics?  Einstein was at a college level by age 11.  How did this myth begin?  More than likely from Einstein’s days as a student in Germany’s authoritarian educational system.  Einstein thought little of rote learning, and was not afraid to make his teachers aware of that.  In today’s parlance, that bit of acting out probably gave the impression of a troubled student.  So what was it in Einstein’s background that allowed him to advance so quickly in his studies?

The second pillar of modern educational theory is Lev Vygotsky’s theory of learning by social interaction.  Part of that theory is the concept of the zone of proximal development.  Here, a student is placed in contact with a more skilled partner to help master a subject.  In Einstein’s case, his parents provided the first zone of proximal development.  Hermann Einstein, Albert’s father, partnered with his brother Jakob building electric generators and lighting.  This surrounded Albert with a technical/scientific background from the get-go not unlike, say, Bill Belichick growing up in a household with a football coach as a father.  Pauline, Albert’s mother, was a pianist and Albert would play the violin most of his life to catch a break from physics.

Einstein plays the violin during the charity concert in the New Synagogue, Berlin, January 29, 1930. Credit: Institute of Czech Literature, Czech Academy of Science.

At age 10, Einstein was introduced into another zone of proximal development in the person of Max Talmud, a 21-year-old medical student who had dinner with the Einstein’s weekly.  Talmud introduced Einstein to many subjects including geometry and Kant’s Critique of Pure ReasonTalmud’s greatest gift to Einstein may have been Aaron Bernstein’s 21 volume People’s Book on Natural ScienceBernstein encouraged constructive learning techniques, in particular, thought experiments such as what it would be like to ride along a light beam.  These thought experiments played a crucial role in Einstein’s relativity breakthroughs and his attempt to describe the theory to the public in his book, Relativity:  The Special and General Theory.

As one might imagine, Einstein raced out of Talmud’s zone of proximal development in short order.  Not unlike the first time a student realizes they have raced ahead intellectually of their teacher.  Nonetheless, Talmud served as a rich pipeline of learning resources for Einstein.  In some sense, Talmud was Einstein’s version of the internet without all the negative distractions.  This resource enabled Einstein to think in ways that provided insights to solve problems other physicists were not able to.  Young Albert Einstein also possessed a fierce streak of individuality.

Self-identity is typically formed during high school years, but can be delayed beyond college.  By all indications, Einstein’s self-identity was molded by his family and his ethnicity.  Of the four general parenting characteristics, the Einsteins would fall into authoritative (not to be confused with authoritarian).  This engaged parenting style typically endows a child with high self-esteem and confidence, which certainly Albert Einstein possessed.  As a Jew in Germany, Einstein was an outsider in German society (as Isaacson notes, only 2% of Munich’s population was Jewish) and this reinforced Einstein’s contempt for the German authoritative educational system.  The Swiss educational system was another story.

Aarau, Switzerland. Credit: Roland Zumbuhl/Wiki Commons

Fed up with Germany, Einstein moved to Switzerland at age 16 and spent a year at the Aarau Cantonal School.  This school favored a constructionist educational philosophy where students build their own knowledge rather than simply accepting what was told to them by an authority figure.  Part of the instructional technique at Aarau included an emphasis on visualization of mathematical concepts based on the ideas of Johann Heinrich Pestalozzi who also valued student individuality.  Einstein thrived at Aarau and its visualization techniques played a significant role in Einstein’s breakthroughs in relativity.

Einstein’s Aarau transcript. Grade scale is 1-6 with 6 being best grade. Credit: Wiki Commons. Translation can be found at:

However, Einstein’s professional academic career did get off to a slow start.  In fact, he was working at a Swiss patent office in 1905 when he published four landmark papers on special relativity, mass-energy equivalence (E = mc2) the photoelectric effect (proving light acts as particles as well as waves) and Brownian motion (which established the existence of atoms).  Einstein’s anti-authoritarianism during his college years at Zurich Polytechnic rubbed some of his professors the wrong way and he had difficulty obtaining good references.  This has led to the myth of Einstein working in “splendid isolation” during this time.  And in a sense, Einstein was isolated from the heavy hitters in physics.  However, this may have been a godsend as those heavy hitters made discoveries that pointed towards relativity, but lacked the creativity Einstein possessed to put all the pieces together.  In pursuit of this, Einstein found one more learning social component in Zurich.

The Olympia Academy founders Conrad Habicht, Maurice Solovine, and Albert Einstein. Credit: Wiki Commons/Emil Vollenweider und Sohn

Had Einstein been discussing the current problems of physics in academia after the turn of the century, he would have been hamstrung by the Newtonian concept of absolute time.  That is, clocks run at the same pace for every observer in the universe.  Einstein and a group of friends formed what they jokingly dubbed the Olympia Academy.  Of the many topics discussed during these weekly sessions were David Hume’s and Ernst Mach’s rejection of absolute time.  This skepticism of Newtonian absolute time is the linchpin of special relativity, which states the speed of light is constant to all observers in the universe and time is variable as a function of velocity (times moves more slowly the faster you go, reaching a standstill at the speed of light).  Special relativity also put the universal speed limit at light speed leading to general relativity, which redefined gravity as curvatures in space-time which ripple throughout the universe at the speed of light and not instantaneously via Newton’s gravitational fields.

So is there anything we can apply from Einstein’s education?

To begin, don’t expect your students to become Einstein – the human race is lucky to experience such a genius once a century.  Great disasters are usually the result of many little things going wrong, great successes require many little things going right.  Replicating Einstein’s education will not likely produce another Einstein anymore than putting a hockey stick in a child’s hand will make him a Wayne Gretzky.  But to continue the sport’s analogy, Red Auerbach expressed a coaching philosophy that his job was to help his players reach their differing levels of maximum potential.  To illustrate, I am the same height as Larry Bird and Magic Johnson, but my maximum potential as a basketball player is significantly lower.  Rather than concern myself with that, with proper instruction, I should focus on reaching my personal potential level.

For example, if a student is struggling putting the ball in the hoop, rather than give a wedgie George Costanza style, have the player perform a thought experiment Albert Einstein style.  Instead of traveling with a light beam, imagine moving along with a basketball headed for the rim.  Take two scenarios, a shot with a low arc and one with a high arc.  How does the hoop appear as you are headed with the ball towards it?  The ball with the high arc “sees” more area in the hoop to enter, increasing the odds of making two points.  It  might not make the child into Larry Bird, but will move forward into reaching their full basketball potential wherever that may fall.

Techniques such as this allows a student to internally construct knowledge and not simply take a teacher’s word for it.  And student’s can apply these techniques in other subjects.  Also, the social component of learning cannot be ignored.  Ridiculing, instead of providing instruction, for a poor performing student causes social isolation not only in that class, but can cascade throughout the educational experience.  All the educational resources in the world cannot help a student who is socially isolated.  And likewise, lack of community resources in the educational system can thwart good instruction.  Teaching someone to fish may keep them well fed, but it only works if they actually have a fishing rod to use.

To maximize a student’s potential a rich social experience is required where ideas are passed back and forth as well as contact with more experienced learners.  This does not stop after childhood.  As the great economist Alfred Marshall noted, inexperienced workers are more productive when teamed with more experienced workers.  This is also why industries tend to form geographic clusters such as Silicon Valley.  In fact, despite his disdain for Germany, Einstein moved to Berlin in 1914 as that was the center of physics on the continent.  The diaspora of Jewish scientists, including Einstein, in the 1930’s had the opposite effect of diminishing Germany’s physics research.  Also, adequate resources must be available to apply what is learned.  Can a student without computer resources expect to function well in today’s society?  Finally, do not burden the student with unrealistic expectations.  Focus on what the student can do, not what they cannot do, and use that as a base to build upon to reach their own level of maximum potential.

*Image on top of post is Einstein presenting a lecture at American Association for the Advancement of Science in Pittsburgh on December 28, 1934.  Credit:  AP/Public Domain.

Beware of Outliers

As we currently digest the run-up to the 2016 presidential election, it can be expected that the candidates will present exaggerated claims to promote their agenda.  Often, these claims are abetted by less than objective press outlets.  Now, that’s not supposed to be the press corps job obviously, but it is what it is.  How do we discern fact from exaggeration?  One way to do that is to be on the lookout for the use of outliers to promote falsities.  So what exactly is an outlier?  Merriam-Webster defines it as follows:

A statistical observation that is markedly different in value from the others of the sample.

The Wolfram MathWorld website adds:

Usually, the presence of an outlier indicates some sort of problem. This can be a case which does not fit the model under study, or an error in measurement.

The most simple case of an outlier is a single data point that strays greatly from an overall trend.  An example of this is the United States jobs report from September 1983.

Credit: Bureau of Labor Statistics

In September 1983, the Bureau of Labor Statistics announced a net gain of 1.1 million new jobs.  As you can tell from the graph above, it is the only month since 1980 that has gained 1 million jobs.  And why would we care about a jobs report from three decades ago?  It is often used to promote the stimulus of the Reagan tax cuts.  When you see an outlier such as this being used to support an argument, you should be wary.  As it turned out, there is a simpler explanation for this that has nothing to do, pro or con, with Reagan’s economic policy.  See the job loss immediately preceding September 1983?  In August 1983, there was a net loss of 308,000 jobs.  This was caused by the strike of 650,000 AT&T workers who returned to work the following month.

If you eliminate the statistical noise of the striking workers from both months, you have a gain of over 300,000 jobs in August 1983, and 400,000 jobs in September 1983.  Those are still impressive numbers and require no need for the use of an outlier to exaggerate.  However, it has to be noted, it was the monetary policy of the Fed Chair Paul Volcker, rather than the fiscal policy of the Reagan administration that was the main driver of the economy then.  Volcker pushed the Fed Funds rate as high as 19% in 1981 to choke off inflation causing the recession.  When the Fed eased up on interest rates, the economy rebounded quickly as is the normal response as predicted by standard economic models.  So we really can’t credit Reagan for the recovery, or blame him for the 1981-82 recession, either.  It’s highly suspect to use an outlier to support an argument, it’s even more suspect to assume a correlation.

To present a proper argument, your data has to fit a model consistently.  In this case, the argument is tax cuts alone are the dominant driver determining job creation in the economy.  That argument is clearly falsified in the data above as the 1993 tax increases were followed by a sustained period of job creation in the mid-late 1990’s.  And that is precisely why supporters of the tax cuts equals job creation argument have to rely on an outlier to make their case.  It’s a false argument intended to rely on the fact that, unless one is a trained economist, you are not likely to be aware of what occurred in a monthly jobs report over three decades ago.  Clearly, a more sophisticated model with multiple inputs are required to predict an economy’s ability to create jobs.

When dealing with an outlier, you have to explore whether it is a measurement error, and if not, can it be accounted for with existing models.  If it cannot, you’ll need to determine what type of modification is required to make your model explain it.  In science, the classic case is the orbit of Mercury.  Newton’s Laws do not accurately predict this orbit.  Mercury’s perihelion precesses at a rate of 43 arc seconds per century greater than predicted by Newton’s Laws.  Precession of planetary orbits are caused by the gravitational influence of the other planets.  The orbital precession of the planets besides Mercury are correctly predicted by Newton’s laws.  Explaining this outlier was a key problem for astronomers in the late 1800’s.

At first, astronomers attempted to analyze this outlier within the confines of the Newtonian model.  The most prominent of these solutions was the proposal that a planet, whose orbit resided inside of Mercury’s, perturbed the orbit of Mercury in a manner that explained the extra precession.  This proposed planet was dubbed Vulcan, after the Roman god of fire.  Several attempts were made to observe this planet during solar eclipses and predicted transits of the Sun with no success.  In 1909, William W. Campbell of the Lick Observatory stated no such planet existed and declared the matter closed.  At the same time, Albert Einstein was working on a new model of gravity that would accurately predict the orbit of Mercury.

Vulcan’s Forge by Diego Velázquez, 1630. Apollo pays Vulcan a visit. Instead of having a real planet named after him, Vulcan settled for one of the most famous planets in science fiction.  Credit: Museo del Prado, Madrid.

The general theory of relativity describes the motion of matter in two areas that Newton could not.  That is, when located near a large gravity well such as the Sun or moving at a velocity close to the speed of light.  In all other cases, the solutions of Newton and Einstein match.  Einstein understood that if his new theory could predict the orbit of Mercury, this would pass a key test for his work.  On November 18, 1915, Einstein presented his successful calculation of Mercury’s orbit to the Prussian Academy of Sciences.  This outlier was finally understood and a new theory of gravity was required to do it.  Nearly 100 years later, another outlier was discovered that could have challenged Einstein’s theory.

Relativity puts a velocity limit in the universe at the speed of light.  A measurement of a particle traveling faster than this would, as the orbit of Mercury did to Newton, require a modification to Einstein’s work.  In 2011, a team of physicists announced they had recorded a neutrino with a velocity faster than the speed of light.  The OPERA (Oscillation Project with Emulsion-tRacking Apparatus) team could not find any evidence for a measurement error.  Understanding the ramifications of this conclusion, OPERA asked for outside help in verifying this result.  As it turned out, a loose fiber optic cable caused a delay in firing the neutrinos.  This delay resulted in the measurement error.  Once the cable was repaired, OPERA measured the neutrinos at its proper velocity in accordance with Einstein’s theory.

While the OPERA situation was concluding, another outlier was beginning to gain headlines.  This being the increase in the annual sea ice in Antarctica, seemingly contradicting the claim by climate scientists that global temperatures are on the rise.  Is it possible to reconcile this observation within the confines of a model of global warming?  What has to understood is this measurement is an outlier that cannot be extrapolated globally.  It only pertains to sea ice surrounding the Antarctica continent.

Glaciers on the land mass of Antarctica continue to recede, along with mountain ranges across the globe and in the Arctic as well.  Clearly something interesting is happening in Antarctica, but it is regional in nature and does not overturn current climate change models.  At least, none of the arguments I’ve seen using this phenomenon to rebut global warming models have provided an alternative model that also explains why glaciers are receding on a global scale.

Outliers are found in business as well.  Most notably, carelessly taking an outlier and incorporating it as a statistical average in a forecasting model is dangerous.  Lets take a look at the history of housing prices.

Credit: St. Louis Federal Reserve.
Credit: St. Louis Federal Reserve.

In the period from 2004-06, housing prices climbed over 25% per year.  This was clearly a historic outlier and yet, many assumed this was the new normal and underwrote mortgages and derivative products as such.  An example of this would be balloon mortgages, where it was assumed the homeowner could refinance the large balloon payment at the end of the note with newly acquired equity in the property as a result of rapid appreciation.  Instead, the crash in property values left these homeowners owing more than the property was worth causing high rates of defaults.  Often, the use of outliers for business purposes are justified with slogans such as this is a new era, or the new prosperity.  It turns out to be just another bubble.  Slogans are never enough to justify using an outlier as an average in a model and never be swayed by any outside noise demanding you accept an outlier as the new normal.  Intimidation in the workplace played no small role in the real estate bubble, and if you are a business major, you’ll need to prepare yourself against such a scenario.

If you are a student and have an outlier in your data set, what should you do?  Ask your teachers to start with.  Often outliers have a very simple explanation, such as the 1983 jobs report, that will not interfere with the overall data set.  Look at the long range history of your data.  In the case of economic bubbles, you will note a similar pattern, the “this time is different” syndrome.  Only to eventually find out this time was not different.  More often than not, an outlier can be explained as an anomaly within a current working model.  And if that is not the case, you’ll need to build a new model to explain the data in a manner that predicts the outlier, but also replicates the accurate predictions of the previous model.  It’s a tall order, but that is how science progresses.

*Image on top of post is record Antarctic sea ice from 2014.  This is an outlier as ice levels around the globe recede as temperatures warm.  Credit:  NASA’s Scientific Visualization Studio/Cindy Starr.

Gravitational Waves – A New Window to the Universe

Some 1.3 billion years ago, as plant life was making its first appearance on Earth, two black holes 29 and 36 times the mass of our Sun, collided.  The result of this collision was a single black hole 62 times the mass of the Sun.  The remaining mass, equal to three Suns, was expelled as energy.  This energy created a ripple in the space-time fabric referred to as gravitational waves.  These waves, which emanated from the colliding black holes like pond waves formed by a rock tossed into it, were detected by the LIGO team on September 14, 2015.  The announcement made today, culminates a 100 year effort by physicists to confirm Albert Einstein’s prediction of gravitational waves.

What are gravitational waves?

Issac Newton’s theory postulates that gravity acts as an instantaneous force throughout the universe.  That is, the gravitational force from the Sun, Earth, even your body, is felt immediately on every other body everywhere.  As Einstein worked up his theory of relativity, he knew there was a problem with this.  According to relativity, there is a firm speed limit in the universe, this limit being the speed of light.  As nothing, whether it is matter or energy, could travel faster than this, it would not be possible for the effect of gravity to travel faster than light as well.  Clearly, a new way of explaining gravity was required.

Einstein found this explanation in the form of gravitational waves.  If there was to be some sort of perturbation in the Sun’s gravitational field, we would not sense it right away on Earth.  Instead, the disturbance would radiate from the Sun at the speed of light in the form of gravitational waves.  It takes light eight minutes to reach Earth.  Thus, a time lag of eight minutes would occur before we would feel the gravitational disturbance on Earth.  In the same manner, there was a 1.3 billion year lag to detect the gravitational waves from colliding black holes located 1.3 billion light years away.  Had Newton’s theory of gravity been correct, the gravitational effect of the colliding black holes, however faint, would have reached Earth instantly 1.3 billion years ago rather than last September.

I want to emphasize that Newton’s theory of gravity works in most situations.  Newton’s predictions deviate from Einstein’s predictions in two key situations.  One is when a body is located very close to a large mass, such as Mercury is to the Sun.  The other is when a body is traveling near the speed of light.  In other situations, Newton’s and Einstein’s equations yield the same result.  In fact, NASA engineers will use Newton’s version of gravity when they can as it is easier to work with than relativity.  The Apollo program, for example, sent humans to the Moon using Newton as a guide.  Replicating Newton’s results where they are accurate was a key stepping stone for Einstein when devising relativity theory.

Another key stepping stone for relativity was making successful predictions where Newton could not.  One such example is the orbit of Mercury.  The perihelion (spot closest the Sun) of Mercury’s orbit advances 43 seconds of arc per century (43/3600th of a degree) more than predicted by Newton.  This advance is visualized in exaggerated form below.

Credit: Cornell University

When Einstein found out that his theory’s solutions predicted Mercury’s orbit perfectly, he was so excited he experienced heart palpitations.  As opposed to being a force, general relativity views gravity as a bending of space-time.

Earth bends space-time. Credit: NASA

As an object bends the space around it, another object will travel along the path of that curvature.  Also, electromagnetic radiation such as light will follow the curvature as well.  If an object accelerates, as when happens when black holes are colliding, it will generate ripples in space time.  And it is these ripples that LIGO detected.

A 3-D visualization of gravitational waves generated by colliding black holes. Credit: Henze, NASA

The universe is not very pliable and it took a tremendous amount of energy to create these waves which are very small, only 1/1000th the size of a atomic nucleus.  How much energy?  Matter in the amount of 3 solar masses was converted into energy in the collision.  Using Einstein’s famous equation:

E = mc2

E = 3(1.99 x 1030 kg)(3.0 x 108 m/s)2

E = 1.79 x 1039 J  where J = Joules

The Hiroshima atomic bomb released about 1014 J of energy.  This means the black hole collision detected by LIGO released 1.79 x 1025 times the amount of energy as the 1945 atomic bomb.  When you see the amount of energy involved, and how small the gravitational waves detected were, its easy to understand how difficult it is to observe these waves.  In fact, Einstein was doubtful gravitational waves could ever be detected as they are so faint.  The announcement today is a result of an effort started in the 1980’s to build the LIGO facility.

LIGO’s two gravitational wave detectors. Credit : LIGO

In 1992, the NSF granted funding for the LIGO project to commence.  It consists of two facilities, one in Livingston, LA, and the other in Hanford, WA.  As a sidenote, Hanford was the site of a key plutonium production plant during the Manhattan project.  Each facility has two 4 km tubes where a laser is sent through.  The mirrors in the interferometer are calibrated so when the two light beams reach their final destination, they cancel each other out so no light is recorded at the photodetector.  This is known as destructive interference and is pictured below.

Credit: NASA

If a gravity wave passes through LIGO, the ripple in space-time moves the mirrors just enough to cause the laser to captured by the photodetector.  This movement is much too slight to be felt by humans and thus the need for sophisticated equipment to catch it.

Credit: LIGO

LIGO has been operational since 2002.  During its first run, no gravitational waves were detected.  LIGO underwent a recent $220 million overhaul to increase its sensitivity.  As mentioned in the press conference today, LIGO is only at a third of its final expected resolution capability.  This bodes very well for more discoveries at LIGO over the next decade.  In all, LIGO has cost $650 million since its inception in 1992.  That is 1/10th the cost to rebuild the San Francisco-Oakland Bay Bridge.  This discovery has the potential to open a new window of observation for astronomers.

To the general public, astronomy for the most part means the classic image of an astronomer peering through an optical telescope or the famous imagery from the Hubble Space Telescope.  What is not as well known are telescopes that observe other forms of radiation.  This includes Earth-bound radio telescopes and space telescopes such as the infrared Spitzer Space Telescope and the Chandra X-ray Observatory.  Why bother with these other forms of radiation?  Think of it this way, imagine a tower located a mile away on a foggy day.  The tower has both a light beacon and radio transmitter.  The fog blocks out the light, making it invisible.  However, if you have a radio receiver, you’ll be able to pick up the radio transmission as fog is transparent to radio waves.  In this manner, astronomers use different types of radiation to detect objects not visible in the optical range.

Besides the continuing upgrade at LIGO, there are future gravitational wave observatories anticipated in India, Japan, and it is hoped, in space.  Today’s result overcomes the most important hurdle.  When LIGO was funded, many scientists were skeptical it could actually detect gravitational waves.  Now that we know it can be done, that clears a major obstacle for funding.  The opening of the radio window allowed the discoveries of pulsars and the cosmic microwave background radiation.  The x-ray window allowed us to view accretion disks around black holes.  The next decade should provide us with additional surprises about the universe as the gravitational wave window opens up.

Credit: LIGO

Above is the LIGO gravitational wave detection result announced today.  The strain is the distance space-time was stretched during the event.  At 10-21 m this is, as mentioned before, about 1/1000th the size of an atomic nucleus.  What gives the LIGO team confidence this is not a false detection as the one produced by the BICEP team two years ago is the gravitational wave was detected by both the Livingston and Hanford observatories.  You’ll also note how closely the observed wave matches with the predicted wave.  The hallmark of progress in science is when theoretical prediction matches observation.  If Einstein were around to see this, I suspect he may have had heart palpitations just as when he found a match between relativity and the orbit of Mercury 100 years ago.

*Image on top of post displays how the colliding black holes produced the gravitational waves discovered by LIGO.  Credit:  Credit: LIGO, NSF, Aurore Simonnet (Sonoma State U.)

Elementary Einstein

While I was in grade school, a teacher wrote the equation E = mc2 on the board and flatly stated, “less than ten people in the world understand this equation.”  In retrospect, that really seems an odd statement to make about a rather simple algebraic equation.  However, it did speak to mystique relativity has among even the educated public.  Nonetheless, this classic equation, which demonstrates the equivalency between matter and energy, is perhaps the easiest aspect of relativity theory to understand.

Relativity typically deals with phenomena that we do not experience in our day to day lives.  In the case of special relativity, most of its esoteric quality deals with objects as they approach the speed of light that represents the highest velocity possible.  As an object approaches this upper bound, it’s clock runs slower compared to stationary observers and its mass approaches infinity.  The fastest speed we approach for most of us is when we fly a jet airliner at about 700 mph.  While that seems fast, it is only 0.000001 the speed of light, much too slow for relativistic effects to be noticed.  Thus, relativity has a strong counter-intuitive sense for us.

That alone does not explain relativity’s fearsome reputation as expressed by my teacher some forty years ago.  Some of that reputation can be attributed to how the media reported the experimental confirmation of general relativity during after the eclipse of 1919.  General relativity provides a more comprehensive theory of gravity than Newton’s Laws.  During the eclipse, astronomers were able to measure the Sun’s gravity bend light, something not predicted by Newton but is by general relativity.  The New York Times reported that:

“When he (Einstein) offered his last important work to the publishers he warned them that there were not more than twelve persons in the whole world who would understand it.”

That was referring to general relativity, which is very complex mathematically and was only four years old in 1919.  It is understandable for those not trained in modern physics to conflate special and general relativity.  Add to that the equation E = mc2  was most famously associated with Einstein and you got the perception it could not be understood unless you were a physicist.  As we will see below, that perception is most assuredly false.

To begin with, lets start with a hypothetical situation where mass can be completely converted to energy.  A science fiction example of this is the transporter in Star Trek that converts a person to energy, transmits that energy at another location, then reconverts the energy back into matter in the form of that person.  How much energy is present during the transmission stage?  Einstein’s famous equation gives us the answer.

Lets say Mr. Spock is about 200 pounds.  Converted to kilograms that comes out to 90 kg.  The speed of light is 3.0 x 108 m/s.  The mass-energy equation gives us:

E = (90 kg)(3.0 x 108 m/s)2

E = 8.1 x 1018 kg*m2/s2

The term kg*m2/s2 is a unit of energy called a Joule (J).  So as Mr. Spock is beaming down to the planet surface, his body is converted to 8.1 x 1018 J of energy.  Exactly how much energy is that?  Well, the average amount of energy consumed in the United States each month is 8.33 x 1018 J.  That’s right, if you converted your body to energy, it would almost provide enough to power the United States for an entire month.  As you can see, a small amount of matter has a whole lot of energy contained with it.

However, most nuclear fission and fusion processes convert a small fraction of matter to energy.  For example, lets take a look at the fusion process that powers the Sun.  It’s a three step process where four hydrogen atoms are fused to form a single helium atom.  The four hydrogen atoms have four protons in their nuclei whereas the final helium atom has two neutrons and two protons in its nucleus.  A proton becomes a neutron by releasing a positron and a neutrino, thus a neutron has slightly less mass than a proton.  In the solar fusion cycle, this mass is converted to energy.

The mass of four hydrogen atoms is 6.693 x 10-27 kg and the mass of the final helium atom is 6.645 x 10-27 kg with a difference between the two being 0.048 x 10-27  kg.  How much energy is that?  Using the famous Einstein equation:

E = (0.048 x 10-27 kg)(3 x 108 m/s)

E = 4.3 x 10-12 J

By itself, that might seem like a small amount of energy.  However, the Sun converts some four million tons of mass into energy each second for a total of 4 × 1026 watts (one watt = one J/s).  Worry not, although average sized for a star, the Sun is still pretty big.  In fact, it constitutes over 99% of the mass of the Solar System.  The Sun will burn up less than 1% of its mass during its lifetime before becoming a planetary nebula some five billion years from now.

Albert Einstein, 1904.

Einstein published this equation in 1905, what would later be called his Annus Mirabilis (Miracle Year).  During this year, Einstein would publish four groundbreaking papers along with his doctoral dissertation.  These papers would describe the photoelectric effect (how light acts as a particle as well as a wave-a key foundation of quantum mechanics), Brownian motion (heat in a fluid is caused by atomic vibrations-helped establish atoms as building blocks of matter), special relativity, and finally, the mass-energy equivalence.  Ironically, it was the photoelectric effect and not relativity that was cited when Einstein was awarded the Noble Prize in 1921.

Information traveled a lot slower back then, and the fame that awaited Einstein was more than ten years away.  The major news story that year would be the conclusion of the war between Russia and Japan as well as the election of Theodore Roosevelt to another term as president.  The New York Times would not mention Einstein at all in 1905.  Even in 1919, when Einstein became a famous public figures, some were mystified at the attention.  The astronomer W.J.S. Lockyer stated that Einstein’s ideas “do not personally concern ordinary human beings; only astronomers are affected.”  As we now know, the public was ahead of the curve in discerning the importance of Einstein’s work.

And that interest remains today.  Yet, there is very little opportunity for students to take a formal course in relativity (or quantum mechanics) unless they are college science majors.  Does the mathematics of relativity make it prohibitive for non-science majors to study relativity?  It shouldn’t.  A graduate level course in electromagnetism contains higher order mathematics that is very complex.  Yet, that does not stop us from presenting the concepts of magnetic fields and electrical circuits in grade school.  As educators, we should strive to do the same for relativity.  And I can’t think of a better place to start than that famous equation E = mc2.

*Photo on top of post is sunset at Sturgeon Point 20 mile south of Buffalo.  The light photons recorded in this image were produced via a nuclear fusion reaction in the Sun’s core that occurred 1 million years ago when only 18,500 humans lived on Earth.  Once the photons were released at the Sun’s surface, it took only an additional eight minutes to end their journey on Earth in my camera.  Photo:  Gregory Pijanowski