Albert Einstein is easily one of the most brilliant physicists who ever lived. His theories of general relativity changed our understanding of the cosmos, as did his work on quantum theory. But his genius has also led many to hold him up as a poor stereotype of science. The lone genius who ignores the science of his day to overturn everything with a simple brilliant theory. He’s become the icon of every crackpot who feels compelled to send emails to scientists about their idea that will revolutionize science if we only take the time to listen (and work out all the math for them). But as revolutionary as Einstein’s ideas were, they weren’t entirely unexpected. Other scientists had similar ideas, and developed similar equations. Take, for example, Einstein’s most famous equation, E = mc². 

The equation appears in Einstein’s 1905 paper “Does the Inertia of a Body Depend Upon Its Energy Content?“, and it expresses a fundamental connection between matter and energy. Energy was long known to be a property of matter in terms of its kinetic motion, heat and interactions, but Einstein’s equation proposed that matter, simply by having mass, has an inherent amount of energy. It allowed us to understand how radioactive particles decay and how stars create energy through nuclear fusion. But the idea had been proposed by others before.

Like Einstein, J. J. Thompson wondered about the connection between light and matter. He thought that electromagnetism was more fundamental than Newton’s laws of motion, and tried to figure out how mass could be created by electric charge. In 1881 he showed that a moving sphere of charge would create a magnetic field, and this caused a kind of drag on its motion. This acts as an effective mass of the charge. Thompson found that the electromagnetic mass of the electron is given by m = (4/3) E/c², which is surprisingly close to Einstein’s equation. Thompson’s derivation was rather cumbersome, but other researchers found the same result with more elegant derivations.

Thompson’s model was not without it’s problems. For one, it only applied to objects that have charge, and only when they are moving. Another problem came from Thompson’s assumption of a uniform sphere of charge. If an electron were an extended sphere of charge, some kind of force or pressure must keep the electron from flying apart. This pressure would obviously have some energy. This led Henri Poincaré to propose non-electromagnetic stresses to hold the electron together. When he calculated the energy of these stresses, he found it amounted to a fourth of an electron’s total mass. Thus, the “actual” mass of the electron due to its electric charge alone must be  m = E/c². Poincaré’s paper deriving this result was published in June of 1905, just a few months before Einstein’s paper.

Although the equation is often attributed to Einstein’s 1905 paper, Einstein didn’t actually derive the equation from his theory of relativity. The paper is only two pages long, and only shows how the equation can arise from approximations to relativity. It’s more of a proof of concept than a formal derivation. It took other scholars to definitively prove that the equivalence between mass and energy is a consequence of special relativity.

None of this detracts from Einstein’s brilliance, but it does demonstrate that even radical ideas in science rarely come from a single individual. The ideas of Thompson,Poincaré, and others were on the right track, as were the ideas of Einstein. Over the decades the scientific evidence we’ve gathered has further confirmed Einstein’s theory as the best representation of reality. And in the end it’s the best models that win, regardless of who first thought of them.

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Albert Einstein is perhaps the most famous scientist in history. He was a true “rock star” scientist, known around the world for his theory of general relativity, which revolutionized our understanding of gravity. Not surprisingly, he was awarded the Nobel prize in 1921, but it wasn’t for general relativity. It was for a completely different work he published in 1905, the year known as Einstein’s annus mirabilis, or wonderous year.

Publishing research is a challenge for any scientist. Most of us might publish a few to several papers a year, collaboratively with other scientists. While our work is interesting and innovative, it isn’t typically revolutionary. Publishing a truly revolutionary, groundbreaking paper is rare, and something most scientists won’t achieve in their lifetime. But in 1905 Einstein published four groundbreaking papers. Each one was a revolutionary work that changed our understanding of the universe. None of them were about gravity. Einstein’s most famous work wasn’t published until 1915, and one could argue that it wasn’t nearly as revolutionary as his 1905 papers.

So this week we’ll look at Einstein’s annus mirabilis papers:

Brownian Motion, which settled the debate over the existence of atoms, and laid the foundation for a new field of work known as statistical mechanics.

The Photoelectric Effect, which demonstrated the particle aspects of light, and led to the quantum theory of matter.

Special Relativity, which overturned a model of space and time that had stood for millennia.

Mass-Energy Equivalence, which connected matter and energy, and led us to a true understanding of the stars.

Although the photoelectric effect is specifically noted in his Nobel prize award, one of these papers would have been worthy of note. We’ll find out why starting tomorrow.

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I’ve been seeing a lot of Albert Einstein quotes recently. You know, the ones like “Everyone is a genius. But if you judge a fish by its ability to climb a tree, it will live its whole life believing that it is stupid.” Or the one where he says “Everything is energy and that’s all there is to it. Match the frequency of the reality you want and you cannot help but get that reality. It can be no other way. This is not philosophy. This is physics.” As Abraham Lincoln said, “95% of all quotes on the internet are false,” so you can probably guess that neither of these were ever said by dear Albert.  But it struck me how the quotes are attributed to Einstein as if it gives them more power.  Einstein was such a genius that his views on education or new-age philosophy must be genius as well.  Of course that’s not how it works. Being very talented or knowledgable in one area doesn’t make one an authority in others. And as history shows, Einstein even got things wrong in his own field.

Take, for example, Einstein’s very first publication. It was on capillary action. which is the effect that allows water to wick into materials among other things. Einstein proposed a connection between the amount of capillary action and the atomic weight of different liquids. If you’ve never heard of Einstein’s work on capillary action, that’s because it isn’t talked about much these days. The idea was completely wrong.

Perhaps his most famous errors involved general relativity. In his first version of gravitational theory, Einstein equated energy and spacetime in a way that violated conservation of energy. It was latter corrected to the version we use now. One of the things general relativity predicts is an expanding universe. Einstein didn’t like the idea, and so rejected it as a conclusion. When it was made clear that a relativistic universe cannot be stationary, Einstein added a term to his equations (now known as the cosmological constant) in order to make his model stationary. He was wrong, and what could have been a prediction turned out to be his greatest blunder.

Einstein also wasn’t keen on black holes. When it was found that black holes were a possibility within general relativity, he rejected the conclusion, calling the results a “disaster.” The same is true for the big bang. When Georges Lemaître demonstrated that cosmic expansion and GR led to the conclusion that the universe began as a dense “primeval atom,” Einstein called the result “abominable.” Einstein was wrong on both counts.

Then there was his opposition to quantum theory. Despite contributing foundational work to quantum theory (and being awarded a Nobel prize for the work), Einstein never fully accepted the ideas of quantum mechanics. A universe of particle-wave duality and probabilistic physics seemed nonsensical to Einstein. His opposition to quantum theory was made popular through the famous Bohr-Einstein debates, where Einstein tried to pick apart quantum theory, and Bohr clearly showed where Einstein went wrong.

Of course there were also a great many things Einstein got right. Revolutionary things that changed our view of the universe. Einstein’s genius is indisputable. But he wasn’t superhuman either. He made mistakes just like the rest of us. In fairness, when the evidence clearly revealed his errors, he did tend to accept them. Einstein’s goal was to have an accurate understanding of the universe.

But Einstein’s mistakes also demonstrate a power of science. Even the great Albert was wrong sometimes. Provably so by experimental and observational evidence. Instead of denying the evidence and declaring Einstein’s work to be unassailable, we celebrate the work Einstein got right, and discard the ideas he got wrong. In the end it is the evidence that leads us to better theories.

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Newton’s laws of motion and gravity predicted the motions of the planets almost perfectly. Newton’s laws are so accurate that we use them to accurately send robotic probes to Mars and other planets, but Newton’s laws aren’t perfect. The motion of some planets differ very slightly from Newton’s predictions. In the case of Uranus, its small deviation led to the discovery of Neptune. In the case of Mercury, however, its small deviation led to a completely new understanding of gravity.

Newton’s law of gravity states that objects are attracted to each other by gravity, and that the amount of attraction decreases the farther apart they are (what is known as an inverse square relationship). As a result of this force, the planets should move around the Sun in elliptical paths. Because of the gravitational pull of the other planets, the motion of each planet isn’t quite a perfect ellipse, but these small deviations can be calculated from Newton’s laws. With most of the planets, the gravitational pull of the other planets accounts for these small deviations.

But this was not the case for Mercury. Mercury is the closest planet to the Sun. It moves so quickly that it makes a complete orbit in about 88 days. As astronomers observed the motion of Mercury, they noticed that it was slowly changing over time. While Mercury essentially followed an elliptical path, just as Newton predicted, its orbit was also slowly changing its orientation. The orbit of Mercury was precessing over time. The rate of precession is very small, only about 43 arc-seconds every hundred years. In other words, the point at which Mercury is closest to the Sun (called its perihelion) was shifting by about 20 miles with each orbit. That might seem like a lot, but it means that Mercury would arrive at perihelion about half a second later than Newton predicted.

Newton’s prediction was only off by about a millionth of a percent, so it is tempting to simply say that is close enough. But in science, when you see a consistent deviation—even a tiny one—it’s worth looking for the cause. Since Newton’s laws were so incredibly accurate, the most likely possibility would be the existence of an undiscovered planet. After all, deviations in the orbit of Uranus had resulted in the discovery of Neptune in 1846.

In 1859, the French mathematician Le Verrier made a detailed analysis of Mercury’s orbit, and hypothesized the existence of a planet even closer to the Sun, which he called Vulcan. Several astronomers made a search for Vulcan, but no such planet was ever discovered. Then in 1915, Albert Einstein proposed a new model of gravity.

A decade earlier, Einstein had developed his theory of special relativity. In special relativity, nothing can travel faster than the speed of light. Time and space are not absolutes, but rather depend upon your motion. Newton assumed that space and time were absolute, and that gravity acted instantly. This contradiction meant that either Einstein or Newton had to be wrong about the way the universe worked.

Einstein started with the assumption that Newton’s gravity must be incorrect. It was a bold assumption given that Newton had worked almost perfectly for nearly 300 years. But Einstein was intrigued by the fact that objects fall at the same rate, regardless of their mass. This idea, known as the principle of equivalence, is why astronauts appear weightless in space. Gravity is still pulling on them, but because everything (including their spaceship and everything in it) falls at the same rate, the astronauts have the illusion of being weightless. Because of the principle of equivalence, an astronaut in a windowless spaceship wouldn’t be able to tell if she was in deep space where there is no gravity, or if she was simply falling in orbit around the Earth. Both cases would give the same feeling of weightlessness.

Announcement of Eddington’s discovery. Credit: Illustrated London News (1919)

Einstein saw a parallel between the relative nature of motion in his theory of special relativity and the relative nature of gravity, and so he worked to generalize relativity to include both gravity and motion. This theory of general relativity is what he published in 1915. It was a radical proposal. In his theory Einstein argued that gravity was not a force in the way Newton had thought. Instead, gravity was an effect of a curvature of space and time. Newton had assumed space and time never changed, but Einstein proposed that space and time could be warped by large masses such as the Sun. The Sun, he argued, curved the space around it. As a result, the path of the planets curved, just as if there was a force acting on it.

General relativity explained why everything falls at the same rate, and it agreed with Newton in most cases. The downside of Einstein’s new theory was that it was very difficult to understand. Einstein had used an obscure branch of mathematics to create his theory, and few scientists were able to follow it. One scientist who was able to understand Einstein’s work was an English astronomer named Arthur Eddington.

Eddington encouraged Einstein to use general relativity to calculate the precession of Mercury’s orbit. By 1919, Einstein finished a calculation which predicted a precession of 43 arc-seconds, exactly the amount observed. While this demonstrated Einstein’s theory could be correct, it wasn’t enough. As Carl Sagan once said, “Extraordinary claims require extraordinary evidence.” and Einstein’s theory was a very extraordinary claim. Newton had worked perfectly for nearly three centuries, and unlike Einstein’s model it was beautifully simple. Newton would not be toppled by confirming an observation that was already known.

To topple Newton, Einstein would need a prediction that hadn’t been observed. One that clearly showed his theory was correct, and Newton’s was wrong. It was Eddington who devised just such an experiment. All he needed was a total solar eclipse. Newton’s and Einstein’s theories predicted almost the same results for the planets. Einstein’s theory did account for Mercury’s precession, but this was not a huge effect. Where Einstein’s theory differed significantly was in the way light behaved. If space is truly curved, then that curvature would effect light as well as planets. If a beam of light passed near the Sun, the curvature of space would bend the beam, an effect now known as gravitational lensing. If Newton was right, and space wasn’t curved, there wouldn’t be such an effect. Eddington realized that one could observe this effect during a total solar eclipse.

In 1919, Eddington traveled to the island of Principe off the coast of West Africa to photograph a total eclipse. He had taken photos of the same region of the sky sometime earlier. By comparing the eclipse photos and the earlier photos of the same sky, Eddington was able to show the apparent position of stars shifted when the Sun was near, just as Einstein had predicted.

Einstein and Eddington had proven that space and time were malleable, that they could be bent and twisted by the presence of matter, and our view of the universe was changed forever.

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When someone mentions time machines, you might think of fantastical machines such as Dr. Who’s TARDIS or the DeLorean in Back to the Future, but several physicists have made a serious study of time machines. Most of this work focuses on “what if” scenarios, which are really about testing the limits of a particular theoretical model, rather than actually engineering a device that can travel to the past.

The physics of time travel is based upon general relativity. If you’ve ever taken a physics course you might remember that the motion of objects is due to forces acting on them. That is, by pushing or pulling on them—either directly or by gravitational or electric fields—you can cause them to move. This is Newton’s physics, where objects fall because a gravitational force acts upon them.

But Einstein had a different way of looking at things. Through his theory of general relativity, Einstein demonstrated that gravity occurs because matter and energy distort space and time. For example, the mass of the Earth curves space around it. The motion of anything near the Earth, such as a satellite, is changed because of this spatial curvature, as if there were a force of gravity acting on it. Since space and time are connected, the mass of the earth also distorts time, which means a clock on the satellite ticks at a slightly different rate than a clock on the Earth. This effect on a satellite’s time is small (on the order of microseconds) but it is a measurable effect. In fact the satellites of the global positioning system have to take this time distortion into effect in order to work properly. If you’ve ever used a GPS receiver to find your way, you’ve counted on Einstein being right.

Although the mass of the Earth really does distort time, it doesn’t allow you to create a time machine. The clocks in satellites tick at different rates because of their motion around the Earth, but they always still tick forward. It is only the rate of their ticking that changes relative to other clocks on Earth. According to general relativity you can change the rate at which time flows but you can never quite stop time completely, and you can never cause your clocks to tick backwards. If that’s the case, it would seem that a true time machine—one that would let you travel into the past—is impossible.

But general relativity leaves the time-travel door open just a little. In Einstein’s theory time is connected to space, which means time can be bent in ways similar to the way space is bent by the Earth’s mass. So in principle time can be bent into a loop in such a way that it connects with its own past. If you found yourself in such a wibbly-wobbly space-time it would be possible to meet your younger self. Such a loop of time would be an actual time machine. As strange as this seems, there are examples of these time loops—what physicists call closed timelike curves (CTCs)—in general relativity.

One place where CTCs appear is in a solution to Einstein’s gravity equations known as the Godel Universe. This is a general relativistic description of a universe with an inherent rotation to it. If this were an accurate description of our universe then we would observe a rotational effect where distant galaxies are not only moving away from us, but also appearing to rotate about us. We don’t see any cosmic rotation among distant galaxies, so the Godel model doesn’t apply to our universe. While it is an interesting model, it is non-physical.

However CTCs also appear inside a rotating black hole. In general relativity, a rotating mass causes space and time to swirl around it a bit. This effect is known as frame dragging, and it has been observed experimentally by a satellite known as gravity probe b. Near a rotating black hole this effect is larger, but still not large enough to make a time machine. However, once you are within the event horizon of the black hole there are CTCs. This would imply that a time machine might be possible inside a black hole. The problem is that though they might exist inside a black hole, you would have to go into a black hole to travel in time, and once inside the black hole you would be trapped there forever. You couldn’t travel to this cosmic time machine, go into the past, and arrive back on Earth in the 1700s. The other problem is that just because general relativity works outside a black hole doesn’t mean it applies inside a black hole. The matter inside a black hole is so small and dense that quantum mechanics and particle physics comes into play, and we don’t have a solid understanding of quantum gravity. There might be something that prevents CTCs from forming inside a black hole.

Most physicists figure this must be the case, because CTCs create all sorts of problems with traditional physics. For example, CTCs can violate the principle of causality (basically cause and effect). This is popularized by the so-called grandfather paradox. Suppose you have a time machine, travel to the past, and accidentally kill your grandfather before he has a chance to woo your grandmother. By preventing their offspring you have prevented your own existence. But that means you couldn’t have travelled back in time, so you couldn’t have killed your grandfather. But that means you didn’t kill your grandfather, which means you were born, which means you did kill your grandfather, which means…

So what would really happen in this case? The answer is unclear, because such a time loop violates causality. The cause and effect contradict each other. In many science fiction stories this is solved by simply declaring that history rewrites itself, or that there are parallel timelines and such. We’ll look at parallel universes later, but this doesn’t solve the problem. The CTCs that general relativity allows exist in a single universe. Following the physics, we can’t simply invoke parallel universes to solve a tricky problem.

One possible solution is to impose what is called the “self-consistency” principle. This requires that any “time machine” example must be self consistent. So the grandfather paradox mentioned above is forbidden because it is not self-consistent. What would be allowed is for you to go back in time and wound your grandfather. While in the hospital he meets a kindly doctor who turns out to be your future grandmother. So your trip back in time caused your grandparents to meet, which allowed you to be born. Perfectly self consistent.

But this solution doesn’t prevent every problem. Suppose when you were 16 a stranger gives you a book. As you read through the book you find it is a set of instructions for building a time machine. It even includes all the background physics necessary to make it work. You go to college, study physics, and your doctoral dissertation is on the physics of time travel (which you got from the book). This groundbreaking work wins you the Nobel prize, and with the prize money you build a time machine, travel back in time and present your younger self with the book on time travel you once received… from yourself.

This is self consistent, but we seem led to ask where the book came from. Yes, you got it from yourself, but that doesn’t seem to be a satisfying answer. Where did the knowledge of time travel originate? The only answer is that the book is itself a closed timelike curve. It doesn’t have an origin. It just is.

Various theories have been proposed to provide a more satisfying answer to examples such as this. They invoke aspects of quantum mechanics, thermodynamics, entropy, information theory, and on and on down the rabbit hole. None of these models provide a completely satisfying description of time travel that makes sense. This is why most physicists figure time travel is impossible. There isn’t a clear way for it to make physical sense. Stephen Hawking went so far as to propose a chronology protection conjecture, which proposes that all macroscopic CTCs are physically impossible.

Still, there are a few physicists who think time machines are possible. For example, Ron Mallett at the University of Connecticut has found a solution to general relativity that allows for CTCs without an event horizon. What Mallett has shown is that light can curve space and time in the same way as mass. By creating a rotating ring of laser light it is possible to distort space and time in a way similar to the way it is distorted by the rotating mass of a black hole, but without the black hole. This, he argues, opens the door to the possibility of creating a time loop you could step into. Critics have pointed out that Mallet’s solution still contains a singularity, so it isn’t a valid physical solution, but Mallett argues the singularity in his solution is an artifact that doesn’t affect the physics.

Even if Mallett is right, his time machine would not allow you to travel anywhere in time. The CTCs could only form in the span of time in which the time machine existed. So you can only go back as far as the moment the time machine was turned on, and you could only travel from the future in which it was still on. In other words, if you wanted to travel 10 years back in time, you’d have to turn on your time machine, keep it running continuously for 10 years, so that you could climb into the machine and arrive when you started. To go back to the future you’d simply have to hang around for another 10 years.

Of course the real question is whether it is possible to distort space and time strongly using only laser light, and whether that distortion could be made into a time machine. Mallet has proposed an experiment to test his model, but so far it hasn’t been performed. Until that happens (and succeeds) time travel is still very hypothetical.

In the end, time will tell.

Tomorrow: Warp Drive. NASA is rumored to be working on it, does that make it so? Find out tomorrow.

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In the mid-1800s, astronomy faced a serious problem.  No one knew how the stars shone.  Now a little mystery never hurt anyone, but in this case it was deeply perplexing.  By this time we had a solid understanding of energy, and that energy is conserved.  That is, energy must come from somewhere, and it must go to somewhere.  We also had a general understanding of thermodynamics, specifically that you could squeeze something to heat it up.  This gave us a mechanism which could cause the stars to shine, known as the Kelvin-Helmholtz mechanism.

The basic idea of this mechanism is that for a large body like a planet or star, gravity tries to compress the star or planet smaller and smaller.  This compression heats the core of the body, which is radiated as light.  In this way a star could be heated by its own weight.  This is a very real effect for gas planets and brown dwarfs.  Jupiter, for example, radiates more heat than it receives from the Sun because of this mechanism.

But for a star such as the Sun there is a problem.  Gravity can only squeeze a body so far, so there’s a limit to how much heat can be generated by the Kelvin-Helmholtz mechanism.  And by conservation of energy, once the Sun has radiated all its heat energy as light, it is done shining.  It’s fairly easy to estimate the rate at which the Sun loses energy given its brightness and size.  Its also fairly straightforward to calculate just how much energy the Sun could gain by gravitational compression.  If you apply conservation of energy, then you can determine how long the sun could shine before it runs out of energy, and you get a clear answer:  about twenty million years.

That’s quite a long time, but it disagreed horribly with geology, where fossil evidence demonstrated that life existed on Earth for several hundred million years, likely much longer.  How is that possible if the Sun could only be tens of millions of years old?  Kelvin actually argued that the geologists must be wrong, and the Earth was only millions of years old.  But by the early 1900s, we understood that life on Earth was around not only millions, but billions of years.  So the Sun must have shone for billions of years, but astronomers and physicists had no explanation for how that was possible.  Gravitational compression couldn’t provide enough energy, nor could chemical reactions, and what else was there?

Then in 1905 Albert Einstein published his paper on special relativity. Usually when people talk about relativity they mention that time slows down at high speeds, or as the figure above claims the faster you move the more mass you have (which isn’t technically true, but I’ll explain that in a couple days).  But central to all this wibbly-wobbly timey-wimey stuff is that energy and mass are two sides of the same coin.  They are connected.  Not only that, mass can become energy and energy can become mass.  This connection is summarized in equation  above.  In the equation E stands for energy, m for mass, and c for the speed of light.  What the equation says is that given a certain amount of mass, it is possible in principle to convert it into a certain amount of energy.  Just how much energy you can get is surprisingly huge.  The speed of light is about 300 million meters per second, and it is squared in the equation.  To give you an idea of just how big this is, if we could convert one paper clip entirely to energy, it would produce enough electrical energy to power the entire world for about 20 years.

Einstein gave us the key to understanding the stars.  Somehow the Sun was converting a bit of its mass into energy, and with it’s great mass the Sun could shine for billions upon billions of years.  But while special relativity showed it was possible, it gave no clue as to how it actually occurred.  There were, however, tantalizing clues.  By the late 1800s Marie Curie and others had begun to study radioactive decay.  By the early 1900s it was demonstrated that radioactive elements could transmute into other elements, which gave us another key to the puzzle.  For example, through radioactive decay, a thorium atom can break apart into a radium atom and a helium atom (also known as an alpha particle).  The mass of the radium and helium atoms is less than the mass of the original thorium, and the “missing mass” is converted to energy, as per Einstein’s equation above.

But this fission process only works for heavy elements.  The sun and stars are mostly made of the light elements hydrogen and helium, and those can’t be split into lighter elements.  But perhaps the tremendous heat and pressure of a sun’s core could fuse lighter elements into heavier ones.  In 1939, Hans Bethe demonstrated how four hydrogen atoms could be fused into a single helium atom.  A helium atom has less mass than four hydrogen, so the result was helium plus energy.  We then learned how helium could become carbon, nitrogen and oxygen, and on to heavier and heavier elements.

This gave us far more answers than we expected.  It not only gave us fusion as the source of a star’s power, but explained from whence the diversity of elements came.  Hydrogen and helium fusing into heavier elements.  Exploding stars scattering those elements across the cosmos.  New stars forming, with planets, one of which is home to us.  From a single, simple equation we came to know that we are the dust of stars.

Of course this equation is also a stark reminder that knowledge can be double-edged sword.  In understanding the source of the Sun’s power, we longed to wield that power ourselves.  By the mid-twentieth century we succeeded.  We can now drop the heart of a star over the cities of our enemies, or power the probes that travel to the furthest reaches of our solar system and beyond.

Which muse we follow is up to us.

Coming tomorrow:  how Newton united physics and astronomy, and brought us face to face with one of the most mysterious and terrifying objects in the universe.  All in part two of this six part series.

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Today is March 14, which many celebrate as Pi Day since the month and day mark 3.14, which is approximately pi.It is also Albert Einstein’s birthday, so it seems fitting to ask whether pi can exist in a universe as Einstein described it.  Just for fun, I’m going to outline why the answer is no, and then explain why that answer is wrong.

The value of pi (3.14159…) is defined as the ratio of the circumference of a circle to its diameter.  But in a physical universe where (as Einstein demonstrated) space is curved, the ratio of circumference to diameter isn’t pi.  For example, if you drew a circle around the Earth, the ratio of circumference to diameter would actually be a little less than pi.  This is because the mass of the Earth curves space around it, making the diameter of your circle a bit longer than it should be.

Geometry of curved space. Credit: The Airspace

This is actually a way you could define a region of space as being curved.  Draw a circle around a region of space, find the ratio of circumference to diameter, and if the value is less than pi then that region of space is curved.  The smaller the ratio, the more strongly that region of space is curved.  If you drew a circle around a black hole, its diameter would be infinite, so the ratio would be zero.

So if we define pi as the ratio of circle’s circumference to diameter in physical space, then pi would actually have lots of values depending the curvature of space around the circle, and none of them would be 3.14159…

Of course that isn’t how pi is defined.  The circumference/diameter definition applies for a mathematically ideal circle, where space isn’t curved.  It can also be defined in other ways, such as an infinite series pi = 4 – 4/3 + 4/5 – 4/7 + …  The geometry definition is just a simple (and perhaps the oldest) one. The key point is that pi is a mathematical concept, not a physical one. 

So even though physical space is actually curved, pi still exists and has the value we all know and love.  So celebrate the day, because it is a perfect excuse to have a slice of pie. 

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One of the more interesting astrophysical discoveries of the 20th century is the fact that the universe is expanding. The result was so unexpected that even Einstein discarded its prediction within general relativity. Einstein went so far as to introduce an extra constant in his equations specifically to prevent an expanding universe model. He would later call it his greatest blunder.

But how do we know the universe is actually expanding? For this we need to use the handy-dandy Doppler effect. You might remember that the observed color of light can be effected by the relative motion of its source. If a light source is moving toward us, the light we see is more bluish than we would expect (blue shifted). If a light source is moving away from us, the light is more reddish (red shifted). The faster the source is moving, the greater the shift.

We have measured this color shift for lots of stars, galaxies and clusters. We’ve also determined their distances (exactly how will be a post for another day). If we plot a graph of the distance of galaxies and clusters versus their redshift we find something very interesting. I’ve plotted such a graph below, and you can see there is almost a linear relationship between distance and redshift.

Distance vs speed for galaxies.

This means galaxies are not simply moving at random, as you would expect in a stable, uniform universe. Instead, the more distant the galaxy the faster it is moving away from us. This relation between distance and speed is the same in all directions, which means the universe seems to be expanding in all directions.

Since this relationship is linear, you can fit this data to a line. The slope of the line is known as the Hubble constant, named after Edwin Hubble, who was one of the first to observe this relationship. When I did a simple linear fit to the data (the dashed line), I got a Hubble constant of 68.79 km/s per megaparsec. This is in the range of the accepted value.

Of course if the universe is expanding, then it must have been smaller in the past. If we assume the universe expands at a constant rate, then we can trace its size back in time to a point where the universe would have zero volume. In other words, the universe has a finite age, and it began very small, very dense (and therefore very hot). We call that starting point the big bang. If you do the math, the age of the universe is simply the inverse of the Hubble constant. Given our value, this puts the age of the universe at about 14.5 billion years. More accurate calculations put the age at 13.75 billion years.

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An earlier post about black holes stimulated several questions about their nature. Of course to really understand black holes you need to understand Einstein’s theory of general relativity and the curvature of space and time. So today I’ll present Einstein simplified. Ignore the math, fudge a bit on the details, and you’re left with the central idea: no observer is special. The consequence of this is that gravity is not a force but the curvature of space.

General relativity is the relativity of gravity. Before we can talk about that, we must first review the idea of relativity without gravity, known as special relativity. For this, let’s imagine two astronauts (Sue and Joe) floating freely in space. There are no forces acting on either of them, so Sue and Joe both feel as if they are at rest. As they float past each other, Joe and Sue see things differently.

Sue says “I am at rest, and Joe is moving at a constant speed!”, while Joe says “Not so! I am at rest, and Sue is moving at a constant speed!” The reason they see things differently is that they assume their reference frame to be the absolute and correct one. The principle of relativity says that there is no absolute reference frame. Thus, we can only say that Joe and Sue are moving relative to each other. Although Joe and Sue have different perspectives they can agree that each is moving relative to the other.

So how does this idea extend to gravity?

To answer this question, we put Sue in a rocket ship. There are no windows in this ship, so Sue can only make observations within the ship. Joe is floating some distance away, and can watch Sue’s experiments. We are going to put Sue in four situations, and see what she observes. For the first situation, we put her rocket in the middle of deep space and accelerate it with the same rate as gravity on Earth. Since the rocket is accelerating, Sue will feel the force of of the floor upon her feet. If she drops a ball, it will appear to fall just as it would on Earth. Since Sue can’t see outside, she assumes she must be on Earth, and therefore declares “I feel gravity!”

For the second situation we place Sue’s rocket on Earth. She will observe the same behavior, and again declare that she feels gravity. In other words, Sue has no way to distinguish between sitting in a gravitational field on Earth and uniformly accelerating in empty space. For her they are exactly the same. Joe disagrees. He declares that Sue is being tricked, and only in the second case does she really feel the force of gravity.

In the third case Sue floats freely in her rocket in the middle of deep space. She doesn’t feel any forces on her, and if she throws a ball it appears to travel in a straight line at a constant speed. Thus, Sue declares “I feel no gravity!”

Finally in the fourth case we place Sue in orbit around the Earth. As she falls around the Earth, everything in her rocket falls at the same rate. Again Sue seems to float freely. She doesn’t feel any forces on her, and if she throws a ball it appears to move in a straight line. Again, Sue declares “I feel no gravity!”

Meanwhile, Joe begins to get frustrated. “She has it all wrong!” he says. “She is being tricked since she can’t see outside her rocket. I can see the whole thing, and I know gravity is acting on her when she is in orbit, since she is not moving in a straight line. Newton said that if the rocket doesn’t move in a straight line at a constant speed then there is a force acting on it. That force is gravity!” And since Joe can see what is really going on, surely his observations are correct.

This is where Einstein steps in. Einstein proposed that relativity should work in general. This means that if Sue and Joe disagree there must be some assumption they are making which is incorrect. For special relativity, the incorrect notion was that of absolute speed. What could it possibly be in this case? It turns out that the incorrect assumption is that objects which change directions must have forces acting on them. That is true in most cases, but not in the case of gravity.

To see how this could be, consider the path of an airline flight from New York to Tokyo. If you’ve ever taken such a flight you know the plane doesn’t fly due west, instead it flies over the north polar region. The flight takes the shortest distance between two points, but on a regular map the path looks like a longer curved line. The reason it looks curved on the map is due to the fact that the Earth is not flat. The map is flat, and assumes a flat Earth. There is no force causing the flight path to bend, but rather the curvature of the Earth that results in a curved path.

The same is true of Sue’s rocket. It follows the shortest path in space and time. The reason it looks curved to Joe is that he assumes space and time are flat, when in fact they are curved. The Earth bends space around it, thus near the Earth objects move in curved lines.

At last Sue and Joe can agree. In the first two cases Sue is in a non-inertial frame of reference, which is why Sue detects the appearance of a gravitational force, and in the last two cases Sue is in an inertial frame of reference, thus she feels no forces around her. Gravity is nothing more than an illusion. A trick of curved space.

Strange? Yes, but powerful as well.

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I’ve been preparing for an intro physics class Monday, and that means covering Newton’s laws of motion. Since it is an introductory class I don’t discuss the nature of mass too deeply. Essentially I tell my students that there is inertial mass, given by the second law of motion, and a gravitational mass given by Newton’s law of gravity. I then go on to say that since everything near the Earth falls at the same rate, the gravitational mass in the law of gravity must be proportional to the inertial mass in Newton’s second law. That is, the two masses are equivalent, which is the heart of the equivalence principle.

But things are never quite as simple as they seem, and the concept of mass is no exception. In Newtonian physics there are not two types of mass, but three. There is the inertial mass, which determines the acceleration due to an applied force; there is the passive gravitational mass, which interacts with the local external gravitational field; then there is the active gravitational mass, which creates the external gravitational field in which other particles interact. Newton assumed that all three types of mass were one and the same, and it is generally assumed that Newton’s was correct, but nothing in general relativity requires it, and there is (as yet) no experimental evidence to validate it.

Acceleration “looks” like gravity.

When Einstein first proposed the principle of equivalence as a foundation to general relativity, his basic argument was that, without some external point of reference, a free-floating observer far from gravitational sources and a free-falling observer in the gravitational field of a massive body each have the same experience. Likewise an observer standing on the surface of a massive body and an observer which uniformly accelerates at a rate equal to the body’s surface gravity have identical experiences. Thus, the free-float and free-fall frames can be considered equivalent. In the same manner, the uniform acceleration frame and the surface frame are equivalent. This is known as the weak equivalence principle:

All effects of a uniform gravitational field are identical to the effects of a uniform acceleration of the coordinate system.

In order to formulate general relativity in terms of general covariance, Einstein later strengthened this argument to yield what is known as the strong equivalence principle:

The ratio between the inertial mass of a particle and its gravitational mass is a universal constant.

It is this latter principle which was experimentally validated by the classic Eötvös experiment, which determined that objects fall at the same rate regardless of their material consistency.

The strong equivalence principle does not require that all masses are equal. It only requires that an object’s inertial and passive masses are proportional. Although the equivalence principle says nothing about active mass, conservation of momentum does. If you apply conservation of momentum to two gravitationally interacting objects, you find that momentum is only conserved is if the active mass of an object is proportional to its inertial and passive masses. Thus in order to relate all three masses, we need not only the equivalence principle, but also the conservation of energy-momentum.

Matter vs. antimatter. Source LBNL.

The constants of proportionality can be wrapped into the gravitational constant, so it would seem we can simply follow Newton, set all three types of mass equal to each other and be done with it. There is, however, a catch. Although we can arbitrarily set the magnitudes of active and passive mass equal to each other, it is possible for them to be opposite in sign. In other words, if there was some weird type of matter that gravitationally repelled other masses, the equivalence principle and conservation of momentum would still hold true. The equivalence principle has been tested between regular matter, which requires all three masses to be the same. Since ordinary matter is mutually attractive we can say that Newton’s assertion is correct for matter.

But what about anti-matter? No one has been able to test this assumption, so we can’t say for certain. It is possible that active mass is negative for antimatter, which would mean it falls upward in a gravitational field. If that is the case, then although general relativity would still apply to regular matter, it wouldn’t apply to matter + antimatter. Since general relativity is a powerful and experimentally validated theory, it is generally assumed that Newton’s assertion would hold for anti-matter as well. But the only way to know for sure is to test it.

Recently we’ve been able to create usable quantities of anti-hydrogen, which will finally give us the chance to put antimatter to the test. It’s generally thought that antimatter will fall downward just like regular matter, but if it doesn’t, it will be time for some new ideas for gravity.

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