Galileo Almost Discovered General Relativity

Hipster Galileo
Figure 1. Galileo, he did it before it was cool. (Original image from wikimedia commons:

We all know the (probably apocryphal) story. Galileo Galilei, all around physics bad-ass, went up to the top of Leaning Tower of Pisa and dropped stuff off the top. He found that objects of vastly different weights, like bowling balls and feathers for example, would fall at exactly the same rate and hit the ground at exactly the same time. Air resistance gets in the way, of course. But if you perform the experiment in vacuum, as these guys did, then you do find the bowling ball and the feather land at exactly the same time:

This leads to a fundamental truth we’ve all memorized in school: The acceleration due to gravity is constant. But there’s a more fundamental truth underneath that one, a truth that sat unrecognised until the time of Einstein: Gravity is not a force. To get the full story, you’ll need to wait until next time, when I start to describe general relativity. But for now, let’s explore how Galileo’s experiment shows that gravity is incredibly special.

Electric Bowling Ball, Electric Feather

To understand why gravity is weird, we have to understand how the other forces work. So let’s set up an experiment analogous to Galileo’s, but with electricity, and see what happens. So here’s the experiment (shown in figure 2).

electric galileo
Figure 2. An electromagnetic analogue to Galileo’s experiment. In a constant electric field, all things being equal, a particle with bigger electric charge (right) will move faster than a particle with less electric charge (left).

We take two metal plates out into space, far enough away that there’s no gravity. Then we connect the plates to a battery so that one plate gets a positive charge (red) and one gets a negative charge (blue). This creates a constant electric field, much like the constant gravitational field near the Earth. Finally, we place two particles of equal mass at rest at the same position between the plates. We give one particle a very large positive charge (right), and one particle a smaller positive charge (left). Like charges attract and opposite charges repel, so both particles will move towards the blue plate.

The particle on the right will absolutely reach the plate before the particle on the left.

Okay, that’s strange. In this experiment, electric charge played the role of “mass” in the sense that it controls how strong the electric force that acts on the particles is. In electromagnetism, unlike in gravity, the stronger force resulted in a bigger acceleration. But is that the whole story? To find out, let’s try the same experiment, but with a twist.

 Electric Dumb-bell

Let’s take the same battery-powered metal plates into space. But this time, we put two particles of equal electric charge between the plates, as shown in figure 3. But we give one particle (on the left) much more mass by attaching some extra weights to it. Now the electric forces on the two particles are the same. Should they fall at the same rate?

analogue of galileo 2
Figure 3. An electric analogue of Galileo’s Pisa experiment. Now the particles have the same charge but different masses.

Not so fast. Now the less massive particle (on the right) falls faster.

There Are Two Types of Mass

What we’ve discovered with these two experiments is that there are two types of mass. There’s the gravitational mass, which controls how strong the force of gravity is for a particular object. This is directly analogous to electric charge. The bigger an electric charge, the bigger the electric force. The bigger a gravitational mass, the stronger gravity is. This is what we saw in our first experiment.

But there’s also the inertial mass. This mass controls how difficult it is to change an object’s motion. This is the mass in Newton’s second law of motion, F=ma. The bigger the mass is, the more force is required to move it. We all know this intuitively: the heavier an object, the harder it is to push. This is what we observed in our second experiment. More mass means less motion.

But when Galileo performed his experiment at Pisa, he discovered something incredible: for gravity and only gravity, these two types of mass are the same. This is unlike every other force. The electromagnetic force, the strong force, and the weak force, all have a charge, which controls how strong the force is, that is separate from their inertial mass. But the gravitational “charge” is the same as the inertial mass.

And if you allow inertial and gravitational mass to be the same, something changes. Instead of “the gravitational field due to the Earth is the same everywhere,” we get “the acceleration due to the gravity of the Earth is always the same everywhere.”

Gravity becomes the same as acceleration.

That’s very weird. And very special. And it is this fact that lead Einstein to develop general relativity. But through his experiment, Galileo almost found it first.

To learn how this leads to general relativity. Tune in next week.

Related reading

The thought experiments I described here are modifications of Einstein’s famous elevator thought experiments. You can find many descriptions of those thought experiments online. Here’s a few:


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Type 1a: The Other Type of Supernova

Supernova 1994D
Figure 1. For just a moment, a type 1a supernova (bright spot in the lower left) outshines an entire galaxy. (Image due to the Hubble Space Telescope. Copyright owned by NASA and the ESA.)

When people hear “supernova” they usually think of a star that runs out of fuel. Without the engine of nuclear fusion to heat it, the star collapses under its own weight, which triggers a huge explosion. This is a “core-collapse supernova,” one of the most energetic events in the universe. The result is usually a neutron star or a black hole.

However, there’s another type of supernova, one in which a star whose nuclear fires long ago petered out is reignited, causing a catastrophic explosion. This is the type Ia supernova. We start our story with the type of star that explodes: the white dwarf.

White Dwarfs

A star is a balancing act. On the one hand, these massive objects exert an enormous gravitational pull on themselves, driving all the gas to collapse towards the centre of the star. On the other hand, the nuclear fusion reaction at the core of the star heats it up, and hot gas likes to expand, holding the star apart. Paradoxically, the driver of this nuclear reaction is the gravitational pull of the star itself. The weight of the star pushes the stuff in the core together so much that the atoms fuse together, releasing huge amounts of energy.

(Surprisingly, stars need quantum mechanics to burn. When atoms fuse together in a star, the fusion only occurs because the atoms quantum tunnel together. Astrophysicist Brian Koberlein has  a nice article on this.)

The eventual fate of a main sequence star like our sun depends on its mass. If the star is more than about 1.4 times the mass of our sun (this is called the Chandrasekhar limit) then, once the nuclear reaction stops, the star collapses under its own weight, triggering a core-collapse supernova explosion. However, if the star is less massive, something amazing happens: the star collapses down to a tiny fraction of its original size–a white dwarf star might have a radius only 4 times or so larger than that of the Earth–but it doesn’t explode. Now the star isn’t held up by heat or nuclear fusion. It’s held up by a quantum-mechanical effect called Pauli exclusion principle.

Basically, a white dwarf is a hot, ultradense fluid made of electrons and atomic nuclei, packed together so tightly that the only thing holding them apart is their inability to occupy the same physical space. This means white dwarfs are incredibly dense. A tablespoon white dwarf starstuff would weigh about 100 tonnes. Figure 2 shows a white dwarf star next to a larger type A main sequence star on the left and our sun on the right. Keep in mind: that tiny little white dwarf star has the same amount of mass as our sun.

a white dwarf next to two other stars
Figure 2. The size of a white dwarf star (center) next to a type A main sequence star (left) and our sun (right). (Image due to RJHall on Wikimedia commons.)

(Neutron stars are very much like white dwarfs, and they are held apart by similar principles. However neutron stars are, unsurprisingly, made mostly of neutrons, and can be about ten times denser and smaller than white dwarfs.)

But sometimes, a white dwarf can reignite. And the results are explosive.


The nuclear fires of a white dwarf have died down. But these fires were first produced by intense pressure. So if the pressure in the core of the white dwarf is ever high enough, then the carbon atoms in the core of the star will start fusing and, temporarily, the nuclear furnace will reignite. Figure 3 shows a computer simulation of the beginning of this process. The core of the star becomes hot due to nuclear fusion and this spreads across the star.

the first part of a type 1a supernova explosion
Figure 3. Snapshots of a computer simulation of the first part of a white dwarf reigniting. Time flows sequentially from frame a to frame f. The color shows temperature ranging from one to five trillion degrees. Image due to G.C. Jordan et al.

The end results of stellar nuclear fusion are carbon and oxygen. So a white dwarf is made up of carbon and oxygen nuclei… and as we know, oxygen reactions are what make fire. So once the nuclear fires reignite, the star doesn’t just become hotter or expand. The entire star literally burns. That’s what figure 3 is showing. The bright orange stuff in the images is actually ash.

Rocket Star

Although the fusion reaction ignites the star, it doesn’t produce enough energy to make the star explode completely. Instead, all of the fire that spread across the star eventually concentrates on one side of the star in a concentrated burst, which can accelerate the star up to thousands of kilometres per second like a rocket. Stars moving this fast are, awesomely, called hypervelocity stars. Figure 4 shows the next part of the simulation in figure 3, where now one side of the star explodes in a pulse.

a supernova explosion continues
Figure 4. The rest of the simulations shown in figure 3. The flames that spread over the whole star converge and form an asymmetric jet on one side of the star. The temperature scale is the same as before. Image due to G.C. Jordan et al.

After the burning in figure 3 and the explosion in figure 4, things calm down. The nuclear fusion in the star stops, and it returns to normal… albeit with a very different velocity.

 Before the Explosion

So now I’ve described how the star explodes… but I still haven’t told you why it explodes. I said that if the pressure in the core of the star becomes high enough, it can re-ignite. But how does that happen? Quite simply, the star has to put on weight. Usually, this means that the white dwarf in question has a companion star–another star nearby such that the two stars orbit each other. And over time, the white dwarf steals material from the companion until it gains enough mass that the weight of the star on the core causes it to reignite.

It’s not known what type of star the companion must be. One possibility is that it must be a massive star near the end of its life. Figure 5 shows the stellar evolution process that might result in a white dwarf stealing from a massive companion. Another possibility is that two white dwarfs might collide. Distinguishing between these models, or perhaps some combination of the two, and identifying which stars will become supernovae is a long-standing problem in astrophysics.

type 1a supernova progenitors
The stellar evolution process that might result in a white dwarf stealing starstuff from a large companion. Image credit due to NASA, ESA, and A. Field.

Different Models and The Ignition Problem

It is worth noting that the precise mechanism by which the nuclear fusion restarts in the star is not completely known. There are also a number of models that describe the details of the supernova explosion. The simulation I showed is one such model, but there are others. However, all models are qualitatively the same and they all produce predictions that match the supernovae we observe in the sky.

Further Reading

As we’ve learned, core-collapse supernovae are not the only kind of supernovae. Indeed, the study of white dwarfs and type 1a supernovae is an active field of research with a rich history. Here I include some resources for your reading enjoyment.

Related Reading

If you enjoyed this post, you might enjoy some of my other posts on astrophysics.

  • In this post, I discuss how planets are formed.
  • In this post, I describe simulations of what happens when a black hole eats a neutron star.
  • In this post, I describe one speculative idea that says black holes can explode.
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Speculative Sunday: Can a Black Hole Explode?

Cassiopeia A Spitzer Image
Figure 1. The Supernova Remnant Cassiopeia A, all that remains of a star that ran out of fuel. It contains either a neutron star or a black hole… could that black hole someday explode? (Image created using data from the Hubble, Spitzer, and Chandra space telescopes.)

Nothing can escape the gravitational pull of a black hole, not even light. That’s why they’re, well, black. (Of course, as I’ve described before, black holes can glow very brightly, thanks to all the in-falling matter. Sometimes they even produce gamma rays. I’m also ignoring the negligible amount of Hawking radiation that black holes theoretically produce.) Once you pass the event horizon of a black hole, you cannot ever escape. Escape is simply forbidden by the laws of physics.

That is, of course…if there actually is an event horizon, not just something that looks like one. Carlo Rovelli , one of the founders of loop quantum gravity, recently proposed something crazy: Not only do black holes not really have event horizons, they eventually explode.

The conclusion is crazy, but the reasoning is surprisingly elegant. Let me walk you through it.

Carlo Rovelli
Figure 2. Carlo Rovelli, one of the founders of loop quantum gravity. (Source wikipedia.)

(DISCLAIMER: I want to emphasize that, although the science in this post is peer-reviewed, it’s extremely speculative. The quantum gravity predictions I describe in this post are not guaranteed or even likely to be true.)

Stellar Collapse

The typical story of black hole formation (at least for stellar-mass black holes) goes something like this: A massive star runs out of nuclear fuel, and the fusion reaction keeping the star alive peters out. Without the energy from the fusion, the star can no longer resist its own gravitational pull and collapses in on itself. The resulting compression of its gases triggers a catastrophic explosion, ejecting a fair amount of the gas to leave behind the stellar core, which becomes increasingly dense. If the star is massive enough, the collapsing core squeezes into such a dense ball that it forms an event horizon and becomes a black hole. (If the star isn’t quite massive enough, the core remnant is pushed outward by the Pauli exclusion principle and becomes a neutron star.) This is called a core-collapse supernova. Here’s a video of a simulation of a supernova that results in a neutron star:

(I am, of course, glossing over a huge number of details. Core-collapse supernovae are not fully understood and there is a rich body of work devoted to understanding them…which many of my friends and collaborators are contributing to. See the bottom of the article for a small, hopefully accessible sampling of current research in core-collapse supernovae.)

The Singularity

Once the event horizon forms around the collapsing matter, no information can emerge from the black hole, so we don’t know what’s going on inside. General relativity predicts that the matter will keep collapsing until it forms an infinitely dense singularity. But the modern view among physicists is that this isn’t what actually happens. Rather, the singularity is a sign that the theory of general relativity is incomplete. What happens inside the black hole can only be described by quantum gravity. We don’t have a theory of quantum gravity, but we are actively searching for one and making (slow) progress.

The Quantum Bounce

Rovelli and his collaborators speculate that these quantum gravity effects not only prevent the singularity from forming, but may in fact cause the black hole to explode.

One generic property of quantum mechanics is that it is probabilistic. Any self-respecting theory of quantum gravity will be probabilistic, too. Therefore, just as a proton has some probability of quantum tunnelling out of an atomic nucleus, a collapsing stellar core has some (admittedly tiny) probability of quantum transitioning into an explosion. But this will only happen when quantum gravity dominates–i.e., when the matter is so compact that it’s almost a singularity.

Loop quantum gravity makes an analogous prediction about the early universe. Instead of a Big Bang singularity at the beginning of time, we had a “Big Bounce,” where a collapsing universe transitions into an expanding one just in the regime where quantum gravity dominates. So why can’t a black hole experience a similar quantum bounce?

Indeed, Rovelli and collaborators performed an ad-hoc calculation in the context of quantum gravity to find the amount of time it should take for a collapsing star to quantum tunnel into an explosion. They found that it happens quickly enough to avoid forming a singularity.

What about the Event Horizon?

Some of you may be asking, “What about the event horizon?” If nothing can escape the event horizon, then doesn’t that mean the stellar material that makes up the black hole also can’t escape?

You’re absolutely right! If an event horizon forms, nothing can escape it. However, in Rovelli et al.’s proposal, an event horizon never forms. Instead, an apparent horizon forms. Like an event horizon, this is a region that light cannot escape from. But unlike an event horizon, it is only temporary. The details are technical, but Rovelli and collaborators have cooked up a model spacetime in which the horizon is not a true event horizon—only an apparent horizon.

 So Where Are All the Exploding Black Holes?

If black holes do experience a quantum bounce and form neither singularities nor event horizons, and if the bounce happens at the end of collapse, where are all the explosions? Surely we would have seen them!

Well, not so fast. Time warps in the presence of a strong gravitational field. In fact, near the collapsing star, time will distort so much that a tiny amount of time near the star will appear to be billions of years to a distant observer. Therefore, all the black holes we observe out in the universe—and we observe many—are in the process of collapsing and bouncing into an explosion behind their apparent horizons.

X-Ray imagery of the supermassive black hole at the centre of the Milky Way, Sagitarius A*, captured by NuSTAR.
Figure 3. X-Ray imagery of the supermassive black hole at the center of the Milky Way, Sagittarius A*, captured by NuSTAR.

Primordial Black Holes and Fast Radio Bursts

If Rovelli and collaborators are right, the first black holes that formed in the universe, which formed many billions of years ago, should be exploding about now. And when they explode, they should release a huge amount of energy. Some of this energy will be emitted as light, which we can detect.

The earlier the exploding black hole formed in the history of the universe, the less massive it will be. And this corresponds to a shorter wavelength of the emitted light. But, because the speed of light is constant, looking further away from Earth means looking back in time. So the wavelength of light emitted by exploding black holes should change depending on how far away the black hole is. After correcting for cosmological redshift, this results in a very peculiar and distinct wavelength of light as a function of distance, shown in figure 4.

The wavelength of light produced by exploding black holes (in arbitrary units) as a function of distance from us (in redshift).
Figure 4. The wavelength of light produced by exploding black holes (in arbitrary units) as a function of distance from us (in redshift). Source: Rovelli and Vidotto

So all we have to do is look for some light coming from outside the galaxy and see if we can compare the wavelength of the light to its distance from us. If it matches the curve in figure 4, then Rovelli and collaborators are right. Otherwise, they’re not.

Rovelli and collaborators suggest using fast radio bursts, which have approximately the right wavelength and may be of extragalactic origin, to test the model. So far, we don’t know very much about fast radio bursts. If they turn out to come from exploding black holes, this would be very exciting, because it would be a real probe of quantum gravity.

Black Hole Information Loss

Rovelli and collaborators aren’t the first to propose that black hole event horizons don’t exist. Previously, Rovelli proposed that black holes evaporate into so-called “Planck stars” that remain after a black hole disappears. Stephen Hawking recently argued that black holes only appear to have event horizons because the spacetime around them is turbulent. There is a rich history of such proposals.

These proposals are all motivated by the so-called black hole information paradox. Basically, we believe that information in the universe is conserved. It cannot be created or destroyed. When information falls into a black hole, it is irretrievable. This wouldn’t be so bad, except that the black hole eventually disappears because it gives up its energy to Hawking radiation, which doesn’t transmit all the information in the black hole. Therefore, once the black hole evaporates, all the information that fell into it is lost forever…simply gone from the universe. But that seems to break the law of conservation of information.

Rovelli’s proposal gets around the paradox by proposing that black holes explode and eject all information they contain. And this is certainly one motivation for him considering it.

Spherical Cows

I want to emphasize that Rovelli’s proposal is ridiculously speculative. He is relying on arguments from quantum gravity, which we don’t even remotely understand. And even the arguments that don’t use quantum gravity are rather contrived.

Rovelli writes down a quantitative model of the collapsing and bouncing star, but it’s very simplistic…in fact, I’d call it the general relativity version of a “spherical cow.” The spacetime has a region in which quantum gravity is non-negligible, which means a region in which physics we don’t understand take place. And the collapsing star is modelled as a thin spherical shell of matter, which is way too simple. (Furthermore, spherical shells of matter are known to have pathologies.) Worse yet, the expansion of the matter post-bounce is modelled as a white hole, which is known to be intrinsically unstable.

Yet, despite all that, Rovelli’s proposal is a cool idea. And I like it.

 Further Reading

  • You can find Rovelli and collaborators’ first paper on the bouncing black holes here. The paper where they predict that fast radio bursts come from exploding black holes is here.
  • For a review of the physics of core-collapse supernovae, first published in Nature, check out this article.
  • The physics of core-collapse supernovae are very complicated, and accurately modelling this phenomenon is an open problem in the numerical relativity community. Professor Christian Ott wrote an awesome article about some of the challenges the community faces (revealed by his and his collaborators’ research), which you can find here.
  • This is a nice article by PBS on Hawking’s recent claim that black holes don’t exist and how it relates to the black hole information paradox.
  • This is a great article by Sabine Hossenfelder about what we hope to gain from a theory of quantum gravity.
Posted in Astrophysics, Geometry, Mathematics, Physics, Quantum Mechanics, Relativity, Science And Math | Tagged , , , , , | 1 Comment

The Curvature of Spacetime

Abell 2218
An Image of the Abell 2218 galaxy cluster, taken by the Hubble Space Telescope

Spacetime is curved. We’ve all heard the line. But what does it mean? Well on the largest scales, the curvature of spacetime is abundantly clear as the warped fabric of the universe distorts images of distant objects.

The image below is of the Abell 2218 galaxy cluster, taken by the Hubble Space Telescope. The cluster is very massive so it warps the spacetime around it. This warped spacetime acts as a lens so that light light coming from galaxies behind Abell 2218 is spread out much more than it should be. The result is that images of galaxies behind Abell are very distorted. In the most extreme cases, the galaxy becomes the rings you see in the image, called Einstein rings.

Astronomers use gravitational lensing to view distant objects they wouldn’t otherwise be able to, and to search for objects that are invisible except by the lensing effect they produce on the starscape behind them, such as dark matter.

The gravitational lensing of light by our own sun was also the first major confirmation of the validity of general relativity.

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The Men Who Weighed Mountains

Title cover of discourse on the attraction of mountains
Figure 1. The original cover of the book “A Discourse on the Attraction of Mountains.” From Google Books.

 In 1687, Sir Isaac Newton published the Philosophiæ Naturalis Principia Mathematicahis magnum opus describing the laws of motion and the secrets of the universe. One such secret is Newton’s law of universal gravitation, which states that the same gravitational force that pulls us down to the Earth holds the planets in their orbits around the sun. Indeed, every mass attracts every other mass through gravity.

This means that not only are we pulled downwards towards the Earth, but we are pulled towards pieces of the Earth. We are all gravitationally attracted to mountains. In fact, this is an excellent test of Newton’s theory: if we could measure the gravitational attraction of a test mass to a mountain, we could confirm whether or not gravitation is indeed universal. And thus men began to weigh mountains.


Newton rejected the possibility of weighing mountains… he felt that the effect would be much too weak. However, others were convinced it should be possible. In 1738, French astronomers Pierre Bouguer and Charles Marie de La Condamine travelled to the mountain Chimborazo in Ecuador. They were there for other reasons, but they took the opportunity to test Newton’s theory. They brought a large, heavy pendulum with them, with a heavy mass suspended on a string. If the mass felt no force, other than the pull straight down due to the Earth, the string of the Pendulum should stand perpendicular to the ground. However, if the mountain attracted the mass, it should be deflected towards the mountain and the string wouldn’t quite be perpendicular to the ground, as shown in figure 2. Moreover, the amount the pendulum was deflected would be proportional to the ratio of the density of the mountain to the density of the Earth itself.

Schiehellion experiment
Figure 2. The forces on the mass at the end of a pendulum near a mountain. F is the attraction to the mountain, W is the pull of the Earth towards the ground, and T is the string holding the mass up. (Source: Wikimedia commons)

The conditions were difficult and the measurement wasn’t very precise. However, Bouguer and Condamine believed they had detected a deflection. They argued that this confirmed Newton’s theory. Moreover, they said, it showed that the Earth was not a hollow shell, disproving the belief held by several major thinkers of the day.


Given the tentative success of Bouguer and Condamine’s experiment, in 1772, Nevil Maskelyne, Astronomer Royal (that is a real title, I promise!) proposed a more careful repeat experiment. His proposal gathered quite a lot of enthusiasm and it became something of a political endeavor. The Royal Society of London formed a committee, the Committee of Attraction, whose members included Benjamin Franklin, to pick a mountain to use. And for political reasons, they wanted the mountain to be part of the United Kingdom.

Schiehallion viewed from Loch Rannoch
Figure 3. The mountain Schiehallion, where the second mountain attraction measurement was performed. Source.

With the aid of surveyor Charles Mason, the committee eventually settled on Schiehallion, shown in Figure 3 in Scotland. Then, with the aid of Charles Hutton and Reuben Burrow, Maskelyne performed the pendulum experiment. This time, thanks to the political enthusiasm, the experimenters had the time and money to do things right. They carefully surveyed the mountain and its terrain to measure any effects due to the curvature of the Earth and they purchased expensive surveying equipment.

The Maskelyne team’s endeavours were a complete success. After several years of work, they measured a deflection of the pendulum by 11.6 arc seconds. This told them that the density of the mountain was approximately half that of the Earth. By measuring carefully measuring the volume of the mountain and the density of the rocks composing it, they found that the density of the Earth was 4.5 times the density of water. This is different from the modern value by 20% or so! An amazing triumph.

Because of the Maskelyne team’s triumph, both Bouguer and Condamine’s and the Maskelyne team’s measurements are called the Schiehallion experiment.

One page of the Maskelyne team’s first report of their findings is shown in figure 4.

Figure 4. The first page of a report on the Schiehellion experiment. Source.

Related Reading

If you enjoyed reading this, you might enjoy reading about other historical experiments.

  • In this post, I describe the famous experiment that told us that the speed of light is constant.
  • In this post, I describe the photoelectric effect, which told us that particles are waves.
  • In this post, my good friend Michael Schmidt describes the Aharanov-Bohm effect.
  • In this post, I describe the experiment that discovered quantum spin.
Posted in History, Physics, Relativity | Tagged , , , | 4 Comments

Simulating Gamma Ray Bursts

A black hole eating a star.
Figure 1. Artist’s conception of a black hole eating a star. (Source, Wikipedia.)

It was the mid 1960s. The United States and the Soviet Union had recently signed the Partial Nuclear Test Ban Treaty, which forbid the detonation of nuclear weapons except underground. Since neither nation trusted the other, each was carefully monitoring the other for non-compliance. In particular, the United States feared that the soviets might be, I kid you not, testing bombs behind the moon.


The United States solved this problem with the Vela satellites. When a nuclear bomb goes off, it emits a short burst of gamma rays, which are rays of extremely high energy light. The Vela satellites were gamma ray detectors in space, orbiting the Earth 65,000 miles above the surface. Figure 2 shows one of these satellites in a clean room.

A Vela satellite
Figure 2. One of the Vela satellites in a clean room. (Source: NASA and Los Alamos)

The Vela satellites did detect gamma rays all right, but they didn’t come from nuclear weapons… they didn’t even come from the solar system. The satellites repeatedly detected short, very intense bursts of gamma radiation that nevertheless took too long to be from nuclear weapons blasts.

Gamma Ray Bursts

For a long time, we didn’t know anything about these events or what caused them. So we gave them the enigmatic name gamma ray bursts, and made up many models for what could cause them.

This changed in the late nineties, when we were able to measure X-rays and visible light emitted from the same source after the burst, which we call an afterglow. We now know that there are many causes for gamma ray bursts. Some bursts take a relatively long time and we’ve linked them to supernovae in distant galaxies.

The relatively shorter gamma ray bursts (creatively called short gamma ray bursts) are less common and less extensively studied. And we therefore know a lot less about them. One popular theory is that they’re caused by the merger of a black hole and a neutron star.

A Quick Aside on Neutron Stars

Neutron stars are the densest stars we know of. Figure 3 shows that a neutron star twice as massive as our sun might have a radius smaller than Manhattan. Indeed, the only thing preventing a neutron star from forming a black hole is the Pauli exclusion principle.

A neutron star superposed on Manhattan Island
Figure 3. A neutron star superposed on top of Manhattan island. In case you’re wondering, if this actually happened, the star would destroy the Earth. (Image source.)

Ordinary matter is made up of mostly empty space. The radius of an atomic nucleus is about a picometer, while the radius of an atom is about an angstrom. This means that, on average, 99.9999 % of matter is empty space. Not so with a neutron star. A neutron star is made up of neutrons packed as tightly as possible, like spheres. This means that in a neutron star, only about 25 % of a neutron star is empty space. (Obviously take this analogy with a grain of salt. The properties of a neutron star depend heavily on quantum mechanics and nuclear physics… so the neutrons aren’t actually packed like spheres. They’re waves.)

Anyway, neutron stars are incredible.

Black Hole-Neutron Star Mergers

When a neutron star gets too close to a black hole, the black hole can eat it. But as I’ve discussed before, black holes are messy eaters. The matter in the neutron star gets distorted and forms an accretion disk around the black hole, which glows incredibly brightly.

As the accreting matter falls the black hole, that matter can be accelerated to incredible velocities and launched out the poles, forming an ultrarelativistic “jet.” These jets are common in many circumstances, but we believe that the jet from a black hole-neutron star merger might be the source of short gamma ray bursts.

There’s a long and beautiful history of studies of accretion disks and the jets they drive. And we’ve known that black-hole neutron star mergers produce accretion disks of the right type. But there’s still a lot we don’t understand about jets, accretion physics, and neutron stars.

The Jet Emerges: A Piece of the Puzzle

Recently, Vasileios Paschalidis, Milton Ruiz, and Stuart L. Shapiro, of the University of Illinois at Urbana-Champaign numerical relativity group, helped add a bit to our understanding. For the first time, they simulated a black hole-neutron star merger, watched as the accretion disk formed, and the relativistic jet emerged. This provides additional evidence that black-hole neutron star mergers might be the progenitors of short gamma ray bursts. Figure 4 shows snapshots of the simulation as the black hole disrupts the star, accretes the matter, and finally drives the jet.

a black hole eating a neutron star
Figure 4. Snapshots of the simulation as the black hole disrupts the neutron star, accretes the matter, and finally drives the jet. The white lines are the magnetic field lines, which drive the jet of matter. Image source.

Now, studies like this have been attempted before. Researchers routinely run simulations of black hole-neutron star mergers to make predictions about gravitational waves. And many groups around the world have run simulations of the jets driven by black holes. However, no previous simulation has successfully observed a jet after merger. All the previous jet simulations started with an accretion disk already in place.

Paschalidis, Ruiz, and Shapiro got their jet to emerge by correctly configuring the magnetic field of the neutron star before merger. Previously, all magnetic fields were assumed to be confined only within the star, and not exist outside it. Paschalidis, Ruiz, and Shapiro argue that this isn’t particularly realistic and, by including the exterior magnetic field, the jet emerges naturally.

This is a pretty cool piece of science!

Related Reading

If you found this interesting, you might enjoy my other posts on astrophysics.

Posted in Astrophysics, Physics, Relativity, Science And Math | Tagged , , , , , , , , , , | 1 Comment

Lightning Detection

Since I’ve been very busy lately my good friend Michael Schmidt agreed to do another guest post! Mike has a masters degree in physics from the University of Colorado at Boulder. You can check out Mike’s own blog at or his personal website Mike’s Personal Website. Without further ado, here’s Mike:

Lightning Detection


Currently, in the mid-west of the United States the first thunderstorms of the year have begun. Because I am a giant geek, I love lightning and I think tracking lightning is quite interesting. My personal favorite site is LightningMaps. On LightningMaps website you’ll see Google Maps overlaid with dots representing lightning strikes and circles emanating from them. The circles represent the leading edge of the thunder as it propagates away from the strike. Seeing this I immediately started to wonder how they do this and, appropriately, started to investigate.


The first step in any scientific endeavor is collecting data (Yes, yes. I Know. The scientific method starts with the generation of a hypothesis but in this case we know what lightning is and we simply want to monitor it). Lightning is a large surge of current, that is electrons, flowing between the Earth and the clouds above or vice-versa. Any time electrons are accelerated they emit photons which we usually see as visible light. In addition to the visible light we see, the lightning emits all sorts of other frequencies which include infrared (aka heat), ultra-violet, and radio waves. You can hear these radio frequencies if you happen to be listening to the radio when lightning strikes. They will sound like static. We can detect these radio waves and keep track of the time we received each burst of radio waves.


If we have only one radio station detecting these bursts of radio waves, we wouldn’t be able to tell where it came from since we would only be able to tell when we received the signal. Now, if there are two stations we can keep track of the difference in arrival times. We know light travels at a constant speed, c \approx 2.99 \times 10^8 \frac{m}{s} . If the first receiver picks up the lightning strike at t_1 and the second at t_2 we know the distance between the strike and the receiver 1 and the strike and receiver 2 is |t_1 - t_2| . This is known as the time difference of arrival or TDOA. Using this information we can only restrict the possible location of the strike to a hyperbola. A hyperbola is a conic section–a slice of a cone–which is the locus of points for which all points are a constant difference in distance between two fixed points or foci. This is exactly the case we have.

The form of any hyperbola is the following:

    \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]

When plotted on a graph we see the following:

A Hyperbola
A hyperbola with two foci

We can fit this to the case we have where the two foci represent the locations of the stations and the time difference between arrival is fixed by setting a and b . This gives us both side of the hyperbola but only one side represents the possible locations. The otherside this the set of points where the TDOA is opposite that of what we observe. The plot looks like:

Possible strike locations if only two stations (red) are used. The yellow represents the actual lightning strike.
Possible strike locations if only two stations (red) are used. The yellow represents the actual lightning strike.

If we want to locate the position of the strike we will need a third station. With this third station we can create a second hyperbola which will intersect with the first exposing the location of the strike. In our example the second hyperbola has been drawn and it intersects at the location of the strike.

Two hyperbolas for two different pairs of stations showing the location of a strike.
Two hyperbolas for two different pairs of stations showing the location of a strike.

Solving the Equations

Unfortunately the equations used to describe these hyperbolas don’t lend themselves to a simple solution. There are solutions, for example Ryan Stansifer’s paper from Florida Institute of Technology, but they will fail if there is any error in your measurements. Instead, systems use a method called Gradient Descent. In gradient descent we create an error function which tells us how far off, in some way, from the solution we are and move in the direction which lessens that error. The the case of the lightning strike detector we know the time difference between two sites should be the measured TDOA so we can set the error for two stations to be

    \[err = [ (|\vec{x} - \vec{s}_1| - |\vec{x} - \vec{s}_2| ) - TDOA]^2.\]

Here (|\vec{x} - \vec{s}_1| means the distance between the position x and the first station s_1 . We square whole thing because we need positive and negative error to count against us. For our full error function we take every pair of stations and add up their error. What see what this looks like for our example above for the first pair of stations:

Error for two stations.
Error for two stations (red) for a lightning strike (Yellow)


We can see here there is a depression shaped like a hyperbola. This like the case with only two stations above only tells us the strike happened somewhere along that hyperbola.

If we add in the third point and account for its contribution to the error we get the following error plot:

With multiple stations the error function reveals the minimal error around the true location.
With multiple stations the error function reveals the minimal error around the true location.

Now we can find an approximate solution by letting a computer walk downhill and narrow in on the solution. To see this in action see this demo. At some point you tell the program to end when the error gets low enough. This is how sites like work where this process is repeated for every event they detect.

Hopefully this sheds light on this wonderful tool.

Further Reading

If you’re interested, here are some of Mike’s other guest posts:

  • Mike’s first two guest posts were on probability. In the first post, he describes how probability works. In the second, he covers some more advanced topics like probability distributions.
  • Mike also wrote a wonderful post on parallel computing.
  • Finally, Mike recently wrote a post on the Aharanov Bohm effect.
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The Long Arms of the Black Hole

The Galaxy Centaurus A and the jets from its supermassive black hole.

Black holes are incredibly messy eaters. As matter falls into a spinning black hole, that matter can be accelerated to incredible velocities and launched out the poles. In the case of the supermassive black holes at the centers of galaxies, these are the most energetic events in the universe since the Big Bang.

The exact mechanism for the creation of these jets is unknown. There are two competing theories, one called the Blandford-Payne mechanism, and one called the Blandford-Znajek mechanism. The details are too fiddly to get into here, but the former has more to do with the in-falling matter and the latter has to do with how magnetic fields interact with the spinning black hole.

The image above is of the galaxy Centaurus A and the jets produced by its super-massive black hole, which is fifty five million times the mass of our sun. The white glow and brown disk are the galaxy itself and associated dust cloud respectively. The blue line is the ultrarelativistic jet of material emitted by the black hole. (Actually, it’s the X-rays emitted by the fast-moving matter in the jet.)

You can’t see the black hole at all. Even on the scale of a galaxy, it’s just a dot, smaller than a pixel. But it has a wide wide reach, extending far beyond the galaxy and influencing the growth and evolution of the galaxy profoundly.

(The image is actually the composite of three images. From Wikipedia: This is a composite of images obtained with three instruments, operating at very different wavelengths. The 870-micron submillimetre data, from LABOCA on APEX, are shown in orange. X-ray data from the Chandra X-ray Observatory are shown in blue. Visible light data from the Wide Field Imager (WFI) on the MPG/ESO 2.2 m telescope located at La Silla, Chile, show the background stars and the galaxy’s characteristic dust lane in close to “true colour”.)

Image Source:

Related Reading

These objects are called active galactic nulcei. See:

Black holes glow for other reasons too. They have so-called accretion disks, which glow incredibly brightly. I wrote about this a while back:

Posted in Astrophysics, Physics, Science And Math | 1 Comment

Aharonov-Bohm Effect

Since I was busy last week and I’m feeling ill this week, my good friend Michael Schmidt has agreed to write a guest post for me this week. Mike has a masters degree in physics from the University of Colorado, an interest in teaching, and a passion for math and physics.  You can find out more about him on his personal website or read more on his blog,

So, without further ado, here’s Mike’s article.

Force Vs. Energy

When we teach physics, usually force is one of the first concepts. Force is easy to understand. I can have you imagine riding in a car riding around a curved road. As the car accelerates, the seat pushes you along. When the car turns you can feel the seat push you in the direction of the curve. In fact, force is such an understandable notion we often neglect to ask what force is or if there may be a better way to talk about the world.

What is force?

Newton’s notion of force is the method which physically exchanges momentum. If two objects interact, they change each other’s momentum. Think of a two billiard balls bouncing off each other. If you placed your finger between between the balls you could feel a considerable force (don’t really do this, it will hurt). The billiard balls feel force due to the other and bounce off each other.

Now, this is how we speak about interactions for the most part. We draw force diagrams and use them to create equations we can solve. This, however, is not always so simple.

Let’s consider two similar examples where a ball bearing rolls (frictionlessly) down a slide: one where the slide is a straight slope and the second the slide is curved. Now suppose you want to find out how fast the ball will be moving when it gets to the bottom of the slide assuming it was nearly stopped at the top. In the first example at every point on the slide the effective force on the ball is constant. This is due to the slope being the same, what is true for one part is true for any other. Since the force is constant we can use the constant force equations to solve this.

Two similar cases of a ball bearing sliding down a slide, one a simple straight slide, and a more complicated slide second.
Two similar cases of a ball bearing sliding down a slide, one a simple straight slide, and a more complicated slide second.

Now, the second case. This situations is substantially more difficult, we need to recompute the force for every point along the slide.
We can’t use any convent equations we have to derive them. This can certainly be laborious and is not preferred.

Wondrously, there is a better way: energy.


Energy is a strange notion; unlike force, you can’t feel energy.
The rules of Newtonian mechanics can be used to create two quantities: kinetic energy (or KE) and potential energy (or PE). Energy, unlike force which has a strength and direction, is just a number. Kinetic energy, roughly, is how much work it takes to accelerate an object up to some speed, whereas potential energy is the capacity for an object to acquire kinetic energy. In other words, potential energy is energy that can become kinetic energy in the future.

Energy may flow between each of these types of energy but their total must always remain the same. To illustrate this imagine a spring fixed to a table on one end and let there be a weight on the other.

An oscillating weight on a spring. (From Wikipedia En:User:Svjo)
An oscillating weight on a spring. (From Wikipedia En:User:Svjo)

If you pull the spring to one side, stretching the spring, and release the weight it will move back and forth. When the weight is at the resting position of the spring, the weight will be under no force and will be traveling as fast as it can go, since as it continues to move it will be slowed again by the spring. It’s at this point the weight has it’s maximum kinetic energy and it’s minimum potential energy since the weight will not be sped up anymore. In contrast to this point, at both ends of the oscillation, the weight will stop. Here, we say the kinetic energy is zero and the potential energy is maximum.

If we use the notion of energy, we can make any situation like the bearing on the ramp nearly trivial to solve. This works since energy is allowed to be either kinetic or potential and the total must always be same. For the ball bearing on the slide example, the ball has only potential energy at the top of the slide and only kinetic energy at the bottom. We can represent this in an equation by

    \[PE + KE = C\]

    \[mgh + \frac{1}{2} m v^2 = C\]

Since C is an constant, you can make both sides equal for the beginning and end:

    \[mgh = \frac{1}{2} m v^2\]

We can then solve for the v and we would know the final speed of the ball. This method is has some obvious advantages, but all it seems we have done is find a quantity which hides the force.

What is Energy?

Potential and kinetic energy seems just to be abstractions of force. In other words, energy isn’t real, the force is. We just made up energy to make the math easier.

This certainly seems like the right answer, especially in the light of how we can actually feel force and energy can only be referred to in equations. Of course, I would not have said that if it were so simple. Quantum mechanics seemed to turn the scientific world on it’s head but could the notion of force be false too?

The Aharonov-Bohm Thought Experiment

Double Slit Waves/Interference
From Wikipedia’s en:User:Lacatosias and en:User:Stannered

This experiment begins with the double slit experiment, which shows the wave-particle duality of electrons. The double slit experiment has three elements to it: an electron emitter, a solid panel with two parallel cuts or slits in it, and a phosphorescent screen all arranged in this order. The setup is shown in the following image:

Double Slit Experimental Setup

As the particles move away from the emitter they pass through the slits and interact to create the interference pattern show here:

From Wikipedia's  en:User:Lacatosias and en:User:Stannered
From Wikipedia’s  en:User:Lacatosias and en:User:Stannered

The additional element added by the Aharonov-Bohm experiment is a very long solenoid encased within an impenetrable shell. The solenoid is place between the screens and the slits. A diagram for this is here:
Double Slit Setup (solenoid)

This solenoid will create a magnetic field inside itself but not outside. This means under our view of things that there ought to be no changes to the setup outside the solenoid as the magnetic field cannot possible be exerting forces on the electrons. Interestingly, as you change the magnetic field strength the interference pattern on the screen will move. This effect is named the Aharonov-Bohm Effect after its discoverers. How could this be though, there is no force on the electrons! In fact there is no magnetic field anywhere the electrons are. The answer is there is another field present, the vector-potential. The vector-potential is a way to abstract the notion of a magnetic field and it is non-zero outside the solenoid. If it were just a mathematical trick, we would say it being non-zero outside is a side-effect of the math and is inconsequential. However, as we see the strength of this field has a direct impact on our observable world.

This questions our assumption that force is the most primitive or basic of interactions. Perhaps our mathematical trick is the real thing. There is much debate about this and there is likely no simple answer. The notion of force isn’t useless but it does have it’s limits. Maybe at some point a future experiment will help out understand more. For now, we’re stuck without an easy answer.

Further Reading

If you would like to learn more about quantum mechanics Jonah has written a number of articles you might find interesting.

Jonah has a three-part series on quantum mechanics:

  • In the first part, he introduces particle-wave duality.
  • In the second part, he describes matter waves using the Bohr model of the atom.
  • In the third part, he describes how one should interpret matter waves.

Jonah also wrote a post describing quantum tunneling.

More recently, Jonah wrote a two part series exploring the relationship between particles and waves.

  • In the first part, he shows how you can build a particle up from a bunch of waves.
  • In the second part, he shows that this isn’t always possible.
Posted in Physics, Quantum Mechanics, Science And Math | Tagged , , , | 1 Comment

Spin and the Stern-Gerlach Experiment

Figure 1. A plaque at the University of Frankfurt commemorating the Stern-Gerlach Experiment. (Source)

The word “quantum” means a single share or portion. In quantum mechanics, this means that energy comes in discrete chunks, or quanta, rather than a continuous flow. But it also means that particles have other properties that are discrete in a way that’s deeply counterintuitive. Today I want to tell you about one such property, called spin, and the experiment that discovered it: the Stern-Gerlach experiment.

(The goal of the original experiment was actually to test something else. But it was revealed later, after the discovery of spin by Wolfgang Pauli, that this is in fact what Stern and Gerlach were measuring.)


The Stern-Gerlach experiment involves magnetic fields. So before I tell you about the experiment itself, I need to quickly review some of the properties of magnets.

As you probably remember, the north pole of a magnet is attracted to the south pole of other magnets and repelled from their north pole, and vice versa—a south pole is attracted to north poles and repelled by other south poles. In other words, opposites attract.

Suppose we generate a very strong magnetic field (say, with a very big magnet or with a solenoid) and put a small magnet in the field, as shown in Figure 2. What happens to it? The north pole of the big magnet will attract the south pole of the small magnet, and the south pole of the big magnet will attract the north pole of the small magnet. Since the north and south pole of the big magnet are are equally strong, these attractions will be equal and opposite, and they’ll cancel each other out so that the little magnet feels no net force. As a result, it doesn’t move up or down—it just hovers in place.

Figure 2. We put a little magnet in a big magnetic field and see what happens.

Now suppose we create a big magnet whose north pole is more powerful than its south pole, as shown in Figure 3. (It’s not actually possible to make a magnet with a stronger north pole than south pole. However, we can create the same effect by using multiple smaller magnets.) What happens now?

Figure 3. If we create a magnetic field where the north pole is much stronger than the south pole, the behavior is different.

To answer this question, we must understand that the strength of a magnetic force depends on the distance between the interacting poles; the closer the poles, the stronger the force. This means that the net force the little magnet feels depends on its orientation, as shown in Figure 4. If the south pole of the little magnet is close to the north pole of the big magnet, the little magnet will be pulled upwards. If, on the other hand, the north pole of the little magnet is close to the north pole of the big magnet, the little magnet will be pushed downwards. If the poles of the little magnet are the same distance from the poles of the big magnet, the little magnet will feel no force. And of course, anything in between is possible. A little magnet whose south pole is just barely closer to the big north pole will feel a weaker pull than a little magnet whose south pole is very close to the big north pole.

Figure 4. If we put a little magnet in a magnetic field that’s stronger near the north pole and weaker near the south pole, the force on the little magnet depends on its orientation.

The Stern-Gerlach Experiment

The Stern-Gerlach experiment, performed by Otto Stern and Walther Gerlach, tested whether subatomic particles behaved like little magnets. To do this, Stern and Gerlach created a magnet with a bigger north pole than south, just like the one described above, and shot a beam of electrons with random orientations through the resulting magnetic field. If electrons behaved like little magnets, then the beam would be spread out by the magnetic field, as shown in Figure 5. Some electrons would be pulled upwards, some would be pushed downwards, and some wouldn’t change direction, depending on the orientations of the individual electrons. But if electrons didn’t behave like magnets, then none of them would be affected by the magnetic field, so they would all just fly straight through.

Figure 5. What we expect to happen in the Stern-Gerlach experiment  if electrons behave like little magnets.

Surprisingly, although the electrons were affected by the magnet, they didn’t spread out as in Figure 5. Instead, the electrons split cleanly into two beams, as shown in Figure 6.

Figure 6. What actually happens when the Stern-Gerlach experiment is performed on electrons. The electron beam doesn’t spread out like a fan, but splits neatly in two!

That’s very weird! It implies that electrons behave like little magnets, but only sort of. A magnet can be oriented any way it likes. But an electron can only have two orientations: either aligned with the big magnet or aligned against it. So the electron can travel up or down, but it can’t stay in between. This is a distinctly quantum phenomenon—the electrons behave like magnets fixed into a pair of discrete orientations, or states, as opposed to a continuum of possible orientations. An electron’s spin is what describes which of those two states it’s in.


A Cool Video

Here‘s a cool video I found on Wikipedia that shows what I just explained.

Where Does Spin Come From?

I won’t discuss it in detail here, but we can understand spin as emerging from the structure of the underlying quantum field theory that describes the behavior of a given particle. For those of you who know the lingo, it has to do with whether the underlying field is a vector or scalar field, and how large that vector is. (Among other sources, see Quantum Field Theory in a Nutshell by Anthony Zee.)


The Stern-Gerlach experiment reveals a dramatic difference between the quantum world and the world we’re used to. It’s not possible for a particle to have any old orientation; it must be oriented either with the external magnetic field or against it.

But what if there is no external magnetic field? How is the particle oriented? Somehow the act of measuring the system changed how it behaves, or at least how we perceive it. These are questions that physicists struggled with in the early twentieth century as quantum mechanics was being discovered. Indeed, to some extent, physicists are still struggling with them.

In the next few weeks, I’ll address some of these issues. Next time, I will talk about an extension of the Stern-Gerlach experiment that helps us explore, if not answer, some of these questions.

Related Reading

This is only the latest in a number of articles that I’ve written about quantum mechanics. For example, I wrote a three-part introduction to the field:

  • In the first part, I describe some of the experiments that first revealed particle-wave duality.
  • In the second part, I use the Bohr Model of the atom to explain how packets of energy emerge from the wave nature of matter.
  • In the third part, I describe how we can interpret matter waves as probability waves.

More recently, I wrote a pair of posts exploring particle-wave duality.

I’ve also written a number of stand-alone articles on quantum mechanics:

  • Quantum mechanics uses complex numbers, so I wrote a short explanation of imaginary and complex numbers here.
  • I explain the Feynman path integral, which is a way of understanding quantum mechanics, here.
  • I use particle-wave duality and matter waves to explain quantum tunneling here.
  • I use quantum mechanics to describe how atoms form covalent bonds here.

Further Reading

Here are some additional resources on the Stern-Gerlach experment:


Thanks as always to Alexandra Fresch for her line-editing.

Recently I’ve had a lot of discussions on Google+  about the interpretation of quantum mechanics. (In particular, I’ve spent a lot of time talking to +Charles Filipponi and +David R.) This article was partly inspired by those conversations. Thanks, guys!

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