Quantum theory is strange and counterintuitive, but it’s very precise. Lots of analogies and broad concepts are presented in popular science trying to give an accurate description of quantum behavior, but if you really want to understand how quantum theory (or any other theory) works, you need to look at the mathematical details. It’s only the mathematics that shows us what’s truly going on. 

Mathematically, a quantum object is described by a function of complex numbers governed by the Schrödinger equation. This function is known as the wavefunction, and it allows you to determine quantum behavior. The wavefunction represents the state of the system, which tells you the probability of various outcomes to a particular experiment (observation). To find the probability, you simply multiply the wavefunction by its complex conjugate. This is how quantum objects can have wavelike properties (the wavefunction) and particle properties (the probable outcome).

No, wait. Actually a quantum object is described by a mathematical quantity known as a matrix. As Werner Heisenberg showed, each type of quantity you could observe (position, momentum, energy) is represented by a matrix as well (known as an operator). By multiplying the operator and the quantum state matrix in a particular way, you get the probability of a particular outcome. The wavelike behavior is a result of the multiple connections between states within the matrix.

Okay, maybe not. Actually, as Richard Feynman showed, the behavior quantum objects are governed by a summation of possible histories known as a path integral. In order for a quantum particle to get from point A to point B, it has to take a certain path. Since quantum objects could take lots of different paths (unlike classical particles), you have to calculate each possible path and sum them in a particular way to determine the probability that the quantum object will arrive at point B.

It turns out that there are multiple ways mathematically to describe quantum behavior. These three are just the most popular, but there are at least nine versions of non-relativistic quantum theory, each with a different mathematical formulation. But despite these mathematical differences, each of them will make the same physical predictions. When you apply these different mathematical models to a real situation, they give the same answer every time. So which one is right? Experimentally, all of them. So it really comes down to using the version you want for a particular situation. It seems the reality of quantum theory is as indefinite as quantum theory itself.

You might think this overlapping magisteria is simply due to quantum theory’s counterintuitive properties, but that’s not the case. Take, for example, the most intuitive and clear theory of Newtonian physics. In the Newtonian model, objects are solid and real, and their motion is governed by vector forces. Push on an object, and it accelerates in that particular direction. Easy peasy. Of course this view (what we might call vector mechanics) isn’t the only mathematical formulation we have.

Another approach is analytical mechanics, where instead of looking at forces we look at constraining quantities such as the object’s energy. If I toss a baseball it isn’t free to move just anywhere. Its motion is constrained by the kinetic energy of its motion and potential energy it could gain or lose from gravity. Those constraints determine the baseball’s equation of motion. Two versions using this approach are Lagrangian and Hamiltonian mechanics. There are other mathematical formulations of classical mechanics as well, such as Gauss’ principle of least constraint, and Appell’s formulation of generalized work. Each of these uses different mathematical methods, starts with different governing principles, and still gives the same physical prediction. So which one is right? Is motion all about forces, or limiting energy, or something else?

One of the fundamental ideas of science is that scientific models must be testable. There must be a way to distinguish an accurate model from an inaccurate one, and the ultimate arbiter is experiment and observation. But nothing in this idea requires that there must be one unique valid theory. Since multiple mathematical formulations each agree with experiment, they are all equally valid.

That’s perhaps the most wondrous thing about all of this. Mathematics does contain truths about our physical universe. It allows us to develop an understanding and precision we can’t get any other way. But this mathematical truth isn’t a single absolute view. Instead, mathematics is so powerful and so subtly interconnected that there are multiple roads to understanding. Two areas of mathematics that seem widely separate often reveal elegant connections, and those connections are often central to our understanding of physical reality. It’s a wondrous and amazing thing.

So to really understand the subtle beauty of physics, you do need to understand mathematics. It isn’t always easy to understand, but it’s well worth the effort.

Miss the beginning of the series? It all starts here.

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Suppose I were to pick two numbers. I won’t tell you what those numbers are, other than to call them A and B. If I asked you what there sum is, A + B, you couldn’t tell me the answer without knowing A and B, but you could tell me one important thing: the answer will be a number. That might seem trivially obvious, but it actually says something important about numbers. Any two numbers when added together will give you a number. That’s because numbers under addition form what is known as a group. 

Vectors can be added to make another vector.

In the physical sciences, lots of things are groups, and this has real physical implications. For example, vectors are groups under addition, so if you add up a number of vectors, the result will be a vector. Since forces are vectors, if you have a number of forces acting on an object you can add them all up to find the total or net force which determines how an object will move. Newton’s laws of motion rely upon the fact that vectors are a group.

Basically, a group is any collection of things (numbers, vectors, etc.) that can be connected by some mathematical operation (addition, multiplication, etc.). There are some specific rules that have to be obeyed, but the key is how members of the group are related under their operations. While it seems trivial to say that numbers or vectors are groups, the idea of groups is much more abstract. Take, for example, rotations.

A Rubik’s cube demonstrates how rotations are a group.

Rotations form a group, where a combination of rotations is equivalent to other combinations of rotations. A Rubik’s cube is a good example of this. If someone scrambles a Rubik’s cube by rotating different parts of it, you don’t need to know what specific rotations they used. Instead you can use a process where you use different methods to return the cube to its unscrambled position. The fact that a Rubik’s cube can be solved is due to the fact the the rotations of the cube form a group. The combinations of rotations you use to solve it are equivalent to the original rotations used to scramble it.

Where things get interesting is when you look at the types of things that are always the same within a group. For example, on Earth, locations are defined by their latitude and longitude. Latitude is defined by the angle north or south of the equator, while longitude is defined as the angle east or west of the prime meridian. But suppose we used a different reference frame. Instead of the equator, suppose we started with a different circle around the Earth, for example. The original latitude and longitude are used because they are convenient, but we could use all sorts of coordinate systems if we wanted. In fact, since we could always shift from coordinate system to another, the set of possible coordinate systems is a group. If we shifted from the original coordinate frame to some new one, then our “latitude” and “longitude” would change, but the distance between any two points on Earth wouldn’t change. Mathematically, we could say that distance is invariant under a coordinate transformation.

These invariant quantities are related to the symmetry of a group. In terms of the physical universe, they describe the symmetries that exist in the cosmos. For example, imagine a star in the middle of empty space. With no other stars in the area to determine the star’s position, there is a symmetry of motion. If you could take that star and shift it some distance in a particular direction, everything would look exactly the same. This is known as translational symmetry. Mathematically, this means the linear motion of an object is invariant under a change of position. In physics we call this conservation of momentum. It turns out that the mathematical symmetry of space is connected to the physical quantity of momentum. Which brings us to the most brilliant mathematical physicist of all time, Emmy Noether.

Emmy Noether used the mathematics of group theory to show that every symmetry within a group that describes a physical phenomenon is connected to a conserved physical quantity. So, translational symmetry means there is a conservation of linear momentum. Rotational symmetry means there is a conservation of angular momentum. Conservation of charge, energy, the connection between electric and magnetic fields, are all the result of group symmetry. This relation is now known as Noether’s theorem, and its power is hard to overstate. What Noether showed was that all the conserved quantities that exist in the Universe exist because of a symmetry that exists within the abstract mathematical concept of a group. Some of the most beautiful and powerful mathematics is physically connected to the way our Universe works. Mathematics doesn’t just describe the world around us. The most basic connections within mathematics describe the bedrock of physical processes.

Everything we study within physics depends upon Noether’s theorem, from dark energy to the Higgs boson. It has transformed the way we view the cosmos, and it demonstrates the real power of mathematics when it comes to understanding the Universe.

Next time: Given that mathematics is so powerfully central to our understanding of physics, does math tell us what is true about reality? The last part of the series starts tomorrow.

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In science, physical quantities are represented by numbers. A block has a mass of 42 kilograms, a car moves at 25 meters per second, or a star is 142 light years away. In mathematics these are known as real numbers. One of the basic properties of real numbers is that any real number multiplied by itself is a positive number. For example, 3 x 3 is 9, and (-4) x (-4) is 16. But imagine a number where its product is actually negative. For example, what if some number i works so that i x i is -1. Such a number is known as an imaginary number. The term imaginary number often gives the impression that these numbers are merely mathematical abstractions, but when it comes to the physical universe, imaginary numbers are quite real. 

Complex numbers on a plane.

While any two imaginary numbers will always yield a negative number, when you add imaginary numbers to real ones, things start to get interesting. For example, 2 + 4 = 6 and 3i + 2i = 5i, but 2 + 3i can’t be simplified to a real or imaginary number. Such combinations are known as complex numbers, and they have a real number part and an imaginary number part. Because of this, complex numbers share a similarity with the vectors we talked about last time. Just as real numbers can be imagined as points on a number line, complex numbers can be imagined as points on a number plane. Any point on this complex plane can be represented by a distance (magnitude) and direction from your origin.

Circularly polarized light.

This makes complex numbers extremely useful when describing certain physical phenomena. For example, light is an electromagnetic wave that has properties such as amplitude and wavelength, but it also has an orientation known as its polarization. Complex numbers can describe light wave as it travels through space can be represented as a complex number, where the real and imaginary parts tell us the orientation of the wave. While we could describe the polarization with real numbers, complex numbers can do it much more simply and elegantly. The same is true with many other physical processes, such as oscillating currents in a circuit, or the vibration of a bridge under stress.

Given the oddness of imaginary numbers, it’s still tempting to think of them as merely a useful mathematical tool. But it turns out there are some physical processes where imaginary numbers are not just useful, they are downright required. Take, for example, the strange world of quantum theory.

The real (blue) and imaginary (red) parts of a wavefunction tell us where an object is likely to be.

The physics of quantum objects is difficult to wrap our heads around, and there are lots of interpretations to help us understand. One of the most popular versions is known as the Copenhagen interpretation. In this view, a quantum object is described by a wavefunction. Just as the function of motion for a baseball can be determined by Newton’s laws of motion, the wavefunction of an object can be determined by the Schrödinger equation. But there’s a fundamental difference. The wavefunction by itself doesn’t tell you what a quantum object is doing. Instead, if you multiply the wavefunction by itself (technically by its complex conjugate) then you get a real function that tells you what the object is likely to do. Mathematically, a solution to Schrödinger’s equation contains imaginary numbers. It has to in order to accurately describe the way a quantum object behaves.

So rather than simply being abstract mathematics, or merely a useful tool, the very fabric of matter depends upon imaginary numbers. At its heart, our world is (mathematically) an imaginary world.

Next time: Complex numbers are useful because of the relationship between real and imaginary numbers. It turns out that in mathematics the relationships between numbers is what really matters. As we’ve come to understand this, we’ve also come to realized that mathematical relationships are also at the very heart of the cosmos.

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Newton’s laws are a story of arrows. Not physical arrows (although that’s sometimes the case) but a kind of mathematical arrow known as a vector. 

The simple concept of a vector is something that has both a quantity (magnitude) and a direction. For example, if you’re traveling in a car, the velocity of your car is a vector that determines both your speed and the direction in which you are moving. Newtonian physics is largely concerned with forces, which are vectors representing the push or pull on an object in a particular direction. One of the key aspects of Newton’s laws is that forces are exerted between objects. If you push on an object to get it to moving, you can feel the object pushing back against your hand. That makes sense if two objects are physically touching, but what about if they aren’t touching. What about, say, the Earth and the Moon?

The forces on an object represented as arrows.

The Moon doesn’t drift through space, but rather orbits the Earth. Newton reasoned the Earth must be pulling on the Moon to keep it in orbit. A force of gravity, if you will. Newton’s breakthrough was to recognize that the gravitational force that pulls apples to the Earth is the same gravitational force that holds the Moon in orbit around the Earth, or the planets in orbit around the Sun. Every body in the universe must pull upon every other body with a mutual attraction. This law of universal gravity worked so well Newton was able to derive Kepler’s laws of planetary motion as a consequence of gravity. It connected earthly motion with heavenly motion and laid the foundation for modern astrophysics.

A magnetic field lines revealed by iron filings near a magnet.

But there was still the issue of how distant objects can feel such a force. How does the Moon “know” the Earth is nearby and “know” to be pulled towards it? For a time you could brush aside the question and focus on solving Newtonian problems, but the question raised its head again in the study of electricity. By the 1700s we understood that electricity came in two forms of charge, positive and negative. Like gravity, charges exerted forces on other charges, even separated by some distance. Sure, the force got weaker with distance, but it was always there.

One proposed answer is that charges must reach out to each other with some kind of energy. That is, a charge is surrounded by a field of electricity, a field that other charges can detect. Charges possess electric fields, and charges interact with the electric fields of other charges. For this to work, the electric field must be a vector field. At each point in space the electric field must have a magnitude and a direction. When a charge detects an electric field, it therefore knows the magnitude and direction of the force acting on it.

For any vector field you can trace paths through the field known as field lines. If you start at a particular point in the vector field, you can take a small step in the direction the vector is pointing. At your new position take a step in the direction of the vector. Keep doing that, and you trace a line following the vectors of the field. If you trace field lines for an electric field, it turns out that they always draw a path from some positive charge and to some negative charge. This is because positive charges always push other positive charges away, while negative charges always pull positive charges towards them. As a result, electric field lines always spread out of positive charges and collapse into negative ones. In mathematics this property is known as divergence.

Divergence (left) and curl (right).

Field lines that have a beginning and an end will always be divergent. But what if you traced a path through space and eventually found yourself back where you started? Instead of a finite line, it would form a loop without end. In mathematics this is known as curl. It turns out that magnetic fields have exactly this property. While charges are always either negative or positive, magnets always come in pairs of poles (north and south). Magnetic field lines spread out from north poles, but they always loop back on themselves, coming together at a south pole. Because of magnetic curl, you can never have just a north pole or just a south pole, and in fact if you break a magnetic in half you will get two magnets.

So the electric field is a vector field with divergence, and the magnetic field is a vector field with curl. But mathematically a general vector field can have a combination of divergence and curl. The mathematics would therefore seem to imply that electric fields and magnetic fields are simply two parts of the same vector field. This is actually true. Electric and magnetic fields are each part of a more general electromagnetic field. Not only that, electric fields that change over time (say from moving charges) can induce magnetic fields, and changing magnetic field can induce electric fields. Together these changing fields can waves of electromagnetism we call light. Light isn’t a separate physical phenomenon, but is due to the fact that electricity and magnetism are connected both physically and mathematically.

Now you might be wondering about gravitational fields. Since the forces of gravity are similar to the forces between charges, doesn’t that imply that there should be a partner to gravity? Some sort of magnetogravity field? It would if gravity was a vector field, but it turns out gravity is a different kind of field altogether.

A metric tensor is often visualized as a rubber sheet, though it’s not a very accurate picture.

Although the simple representation of vectors is as an arrow, the mathematics is a bit more subtle. Not everything that has a magnitude and direction is a vector. For example, the Sun radiates light outward (direction) at a specific rate (magnitude), but that isn’t a vector. Mathematically, vectors are defined by things like how they add together, and how they transform in different frames of reference. While vectors can be represented by arrows in a physical space, that’s not the only thing they can be. The concept of a vector can also be generalized. You might remember from last time that gravity is actually a geometric property of bendable space and time. For this you need a mathematical quantity that can not only describe changes in quantity and orientation, but also volume and twisting and shear. This is done through a generalization of vectors known as tensors. Gravity isn’t a vector field, but the properties space and time that describe gravity are a tensor field known as the metric tensor. Since vector fields are a special case of tensor fields, gravity can be approximated as a vector field of gravitational force. Tensors are also useful in electromagnetism, since an electromagnetic field can also be described as a single tensor field.

So the arrows of simple forces pointed us to the general concepts of vectors and tensors, and it led us to a deeper understanding of both electromagnetism and gravity.

Next time: Just as geometry and vectors can be generalized into more abstract forms, so can numbers. By generalizing numbers beyond the familiar we’ve been able to understand the imaginary aspects of quantum theory.

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Stretch a string between two points, and you get a straight line. Walk one end of the string around the other and you get a circle. These two shapes, circles and lines, form some of the basic shapes of geometry. 

While early astronomy focused on the marking of time, such as when the rising Sun would be most northerly, or the number of days until the crescent Moon returned, the simple geometry of curves gave us a connection to the heavens. The Sun, Moon, and even the stars seemed to trace circles around the Earth. Likewise, a dropped ball traced a linear path to the ground, and fire seemed to rise linearly upward. The geometry of circles and lines was nothing less than the sacred geometry of Heaven and Earth.

Around 300 BC, Euclid of Alexandria formalized this geometry in a 13 book opus known as The Elements. He started with five basic assumptions (axioms) about lines and circles:

1. You can always draw a straight line segment between two points.
2. You can always extend a straight line segment indefinitely
3. For any line segment you and draw a circle where the line segment is the radius and one end is the center.
4. All right angles are the same.
5. If two lines aren’t parallel, they will eventually cross if you extend them far enough.

From these axioms he developed a formal method of proofs and theorems (elements), showing that if these initial axioms are true, then other aspects of geometry must also be true. Euclid gave us the language of geometry, and by extension the language of much of modern mathematics. This geometric language could then be used to describe the motion of heavenly bodies. If you wanted to know when Mars and Jupiter would appear close in the sky, or when Venus would appear as the morning star, you could calculate it with geometry.

That same geometry also gave us tools to measure heavenly motions more accurately. We could triangulate the positions of planets against the stars, and calculate their true paths about the Earth. It soon became clear that planets did not move in circles. Lines and circles were so incredibly useful that many proposed solutions still focused on them. Perhaps a planet’s motion about the Earth was circular, but not centered on Earth. Perhaps it was a combination of circles (epicycles) on top of other circles to trace out a planet’s path. Perhaps the planets move in circles about the Sun rather than the Earth. All of these improved upon the simple model of circular motion about the Earth, but they were always just a bit off from the true motion of the planets. The solution came from Johannes Kepler, who proposed not circular orbits about the Sun, but elliptical ones.

Conic sections as slices of a cone.

An ellipse is part of a family of curves known as conic sections. If you take a line segment and trace one end about a circle, the line traces out a cone. If you then slice the cone with a plane, you can form four different types of curves. Straight through, and you get a circle. At an angle, and you get an ellipse. Parallel to the edge of the cone and you get a parabola. Steeper than the edge of the cone and you get a hyperbola. In this way a circle is just a special case of a larger geometric family. By generalizing circles to ellipses, Kepler devised a set of three basic rules for planetary motion that came to be known as Kepler’s laws. They were extremely accurate, and were much simpler than offset circles and epicycles.

Each point in space has a unique coordinate.

Around the same time as Kepler, René Descartes was developing a new approach to geometry. Although our understanding of geometric forms had improved over the centuries, geometry still followed the methods of Euclid. Imagine a line bisecting a circle, envision a sphere enclosed by a cube. Geometry was about lines, curves and shapes in relation to each other, and this could be complicated. Kepler’s second law for example, imagined a line connecting a planet to the Sun to sweep out an equal amount of area for equal amounts of time. Descartes imagined that space could be filled with a grid as a reference frame. In this way, each point in space can be represented by a unique set of numbers (coordinates), and a curve can be represented as a function related one coordinate to others. With this analytic geometry, Descartes connected geometry to algebra, giving us even more tools to describe curves and forms.

Newton’s geometry of space and time is still used in introductory physics classes.

Analytic geometry also allowed us to look at motion not just as a path through space, but also as a path through time. Each position in space has three coordinate numbers marking its location, and by adding a fourth coordinate representing time we can create a geometry of where and when. When Isaac Newton developed his laws of motion, he described motion in terms of speed and acceleration. Using analytical geometry he could connect these functions of time to curves in space, tracing an object’s path through space and time. This same approach also allowed Newton to prove that Kepler’s laws of motion were the result of a universal force of gravitational attraction, ushering in the age of astrophysics.

The Euclidian geometry of space and time was so powerful that it’s validity seemed unquestionable. What else could the cosmos be if not an extent of space existing in time? Combined with the accuracy of Newtonian physics, it felt as if we’d reached the pinnacle of understanding. But in the 1800s Bernhard Riemann began to explore alternatives to Euclidian geometry. The coordinates of Descartes were a way of mapping out Euclid’s geometric space, but what if the relations between these coordinates could be distorted. We might imagine a Euclidian surface as a sheet of paper marked with a grid. If the sheet were made of rubber, stretching or bending the sheet would distort the shape of the grid. Some rules of geometry would still apply on the sheet, but not necessarily the five axioms of Euclid. Just as circles are just one example of a conic section, Euclid’s geometry is just one member of a much larger geometric family.

Riemannian manifolds come in a variety of shapes.

This gave rise to a more general form of geometry known as Riemannian geometry, where space could be a malleable manifold rather than a rigid background. The connections between points in space are determined by the structure of the manifold, and the old rules of Euclid can be stretched or even broken. Two circles of the same circumference might have different lengths radii. Parallel lines might eventually cross. Two right angles might not be the same when compared to each other. Just as Descartes connected geometry to algebra, Riemann connected geometry to topology. Geometry was no longer limited to a fixed background grid.

But surely none of this applied to the Universe at large. Sheets of paper and rubber balls can be distorted into different shapes, but space is not a physical material. Surely it must be rigid and absolute. Surely space and time must be Euclidian.

But it’s important to note that Euclid’s axioms were assumptions. They seem intuitively true for space and time, but assumptions can be wrong. One of the big assumptions about time in the Universe, was that it’s the same everywhere. If we sync two clocks, they should always read the same time even if they are speeding on a starship or light years away. But if space time were the absolute grid against which everything is measured, then the speed of an object must always be relative to that grid, even the speed of light. If you were speeding along relative to the cosmic spacetime grid, you would measure a different speed for light than if you were standing still. But it turns out space and time aren’t absolute frames, light is. Light forms a geometric connection between space and time, and the geometric rule that connects space and time is that its speed will always be a universal constant.

This is the insight Albert Einstein brought to physics. Riemann was right. The key to geometry is how a manifold is topologically connected. For our Universe, light is the connection, and space and time distort in whatever way necessary to preserve that connection. It is the general relativity of space and time.

The geometry of space is not Euclidian.

Perhaps the most amazing aspect of Einstein’s theory is that gravity — the force that causes the planets to trace their elliptical geometry around the Sun — is itself simply a consequence of geometry. The distortions of space and time mean that objects don’t always move in a straight line. Their path can be distorted, making it look like they are being pulled by a gravitational force. Newton’s law of gravity was an intellectual triumph, but it also represented an incomplete understanding of geometry. Our Universe does have a sacred geometry after all. It’s not the fixed geometry of a rigid and invisible grid, but the luminous geometry of light.

Next Time: Newton’s laws of motion brought vectors to physics. These mathematical arrows pointed us to a subtle property of nature leading to revolutionary new understanding of light.

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Mathematics is the language of science. From arithmetic to group theory, mathematics builds the very foundation of scientific models. We might be inspired by an idea or analogy, but the precision of science requires a mathematical structure. Perhaps the most fundamental thing we’ve learned about the cosmos is that it has a deep connection to mathematics.

This connection has raised the question about just why mathematics is so effective. Perhaps it’s simply due to the fact that we follow mathematical models where they are useful, and discard them where they are not, making applied mathematics self-selecting. Perhaps it’s because as evolved primates within this physical universe the mathematics we think is “pure” is simply a reflection of how our universe works. Regardless of the cause, mathematics seems unreasonably effective as physicist Eugene Wigner once argued. It’s so powerfully useful that some folks such as Max Tegmark have proposed that the structure of the Universe could simply be the structure of mathematics itself. Gallons of ink have been spilled on all sides.

But the very power of mathematical models within science also raises a wall separating those with the mathematical training to understand these models from those without. This is particularly seen in the popularization of science where (with some exceptions) equations never appear, and the focus is on broad analogies rather than the underlying maths. This reinforces the misconception that established science can be overturned simply be a new idea, and that the mathematics is merely a minor detail. The reality is that an idea can be a spark, but fire of knowledge is only captured with specific mathematics, and it’s in the mathematics where much of the beauty an elegance of a scientific model lies. The mathematics of science can be deep and subtle, and its nature is not often discussed.

So for the next several posts I’ll try to present some of the mathematical beauty behind several scientific theories:

1. Geometry – From Aristotle’s conception of earthly lines and heavenly circles, to Kepler’s elliptical forms, geometry has played a central role in astronomy. As we followed geometry into more abstract concepts, it opened the doors to the beginning of time.
2. Vectors and Fields – Forces have both a quantity (magnitude) and direction. In mathematics we call them vectors. From that simple mathematical concept arose the first unified field theory.
3. Complex Numbers – It was long thought that any number multiplied by itself is a positive number. But what if a number multiplied by itself was negative? This idea was so odd it came to be known as imaginary. It turns out that imaginary numbers open the door to very real physics.
4. Group Theory – We often think of mathematics as numbers, or at least equations. But it can also be about relationships and connections. Often how different parts of a model connect is the key to understanding the model on a deeper level.
5. Formalism – Is mathematics simply a set of connected rules, or is it something more? Does mathematics limit what we can know about the physical universe?

We’ll start with geometry, and how a simple curve tracing the motion of a planet led us to an exploration of space and time itself. It all begins tomorrow.

Paper: Wigner, E. P. The unreasonable effectiveness of mathematics in the natural sciences. Communications on Pure and Applied Mathematics. 13: 1–14 (1960). DOI:10.1002/cpa.3160130102

Paper: Max Tegmark. Is “the theory of everything” merely the ultimate ensemble theory? arXiv:gr-qc/9704009 (1998).

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Take a mass, any mass. Compress it into an ever smaller volume. As its density rises, the gravity near its surface with increase. Squeeze it into a small enough volume and the surface gravity will become so strong that nothing can escape, not even light. Squeeze anything into a small enough volume at it will become a black hole. The defining feature of a black hole is its event horizon, which defines the volume of no return. But the event horizon also marks a region where our basic understanding of physics breaks down. It is perhaps the greatest paradox of modern astrophysics.

The event horizon of a black hole is often defined as the point where the escape velocity becomes greater than the speed of light. It turns out the truth is a bit more subtle. Mass curves space around it, and for a black hole space is curved to the point where it basically folds into itself. The event horizon doesn’t mark an escape velocity, it marks a region that is isolated from the rest of the Universe until the end of time. The laws of physics conspire to keep you trapped, and you could no more escape a black hole than you could walk backwards in time.

However the existence of a one-way path to oblivion flies in the face of the most basic tenets of physics: phenomena should be predictable. If you throw a baseball in a particular direction at a particular speed, you can figure out where it’s going to land. Just determine the initial speed and direction of the ball, then use the laws of physics to predict what its motion will be. The ball doesn’t have any choice in the matter. Once it leaves your hand it will land in a particular spot. Its motion is determined by the physical laws of the universe. We can also work backwards. Knowing the speed and direction of the ball we can work out where it was in the past. If that’s true, then knowing something about the Universe now allows us to determine its past and future. But an event horizon breaks that rule. Once something crosses the event horizon, all you can possibly know about the object is its mass, charge and rotation. Was it a car or a spaceship? No idea. What path did it take to enter the black hole? No idea. All that information we’re supposed to know about the object, seems to simply disappear. This is known as the information paradox.

Now some of you might point out that quantum mechanics isn’t deterministic like a baseball, so perhaps information isn’t conserved after all. But it turns out that quantum theory does conserve information, it simply conserves the probabilities of certain outcomes. Knowing the state of an object we can still predict what it’s likely to do next, and what it likely did in the past. But it’s possible that quantum theory might provide a way out of the information paradox. After all, Stephen Hawking showed that quantum theory allows matter to escape a black hole through Hawking radiation. If matter radiates from a black hole, perhaps it also allows information to escape the black hole.

Unfortunately quantum theory isn’t an easy fix. Hawking radiation as it is typically defined is completely random, so while matter and energy can escape a black hole, information can’t. Theoretically you can make Hawking radiation non-random, but doing so turns it into an intense firewall near the event horizon. This flies in the face of the principle of equivalence, which says that a small region of space near an event horizon shouldn’t be any different than a small region of space anywhere else. Thus trying to solve the information paradox gives rise to another problem known as the firewall paradox.

So how do we solve this problem? The short answer is we don’t know. Lots of very smart people have tried to crack this problem, and while there are some interesting ideas there is no definitive solution. To really address this issue will require a quantum theory of gravity, which we don’t yet have. There have been some arguments that the way around the paradox is to simply declare that black holes can’t exist, but now that we’ve detected gravitational waves we know they absolutely do exist.

There’s no easy way around these paradoxes, and until there is, event horizons will remain a clear marker of the great unknown.

Miss the beginning of this series? It all starts here.

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Earth is showered with cosmic rays. They are protons, electrons and atomic nuclei traveling at nearly the speed of light, and strike our atmosphere to create the most power particle collisions ever observed. As a particle approaches the speed of light, it’s energy increases exponentially, so it might seem that there is no upper limit to just how much energy cosmic rays can have. But it turns out there is a limit, at least in theory. 

The limit is imposed by the cosmic microwave background (CMB). This thermal remnant of the big bang fills the Universe with a sea of microwave photons, which is why we observe the CMB from all directions in space. But because of relativity, a cosmic ray moving at nearly the speed of light will observe this radiation greatly blue shifted. Instead of a sea of faint microwaves, these cosmic rays observe CMB photons as high energy gamma rays. Occasionally the cosmic ray will collide with a photon, producing particles such as pions and taking some of the energy from the cosmic ray. This will continue until the cosmic ray isn’t powerful enough to produce pion collisions. As a result, over the vast expanse of intergalactic space any really high energy cosmic ray will be lowered to this cutoff energy.

High energy protons collide with CMB photons, producing pions while losing energy. Credit: Wolfgang Bietenholz

This cutoff is known as the GZK limit, after Kenneth Greisen,Vadim Kuzmin, and Georgiy Zatsepin, who calculated the limit to be about 8 joules of energy (a proton traveling at 99.999998% of light speed), and that any cosmic ray traveling at least 160 million light years will have dropped below this limit. While that’s a huge amount of energy, there have been observations of cosmic rays with even higher energy. The highest energy cosmic ray had an energy of about 50 joules. So how is this possible?

The short answer is that we aren’t sure. High energy cosmic rays are more powerful than any particle accelerator we have, so these kinds of particles can’t be recreated in the lab. One possibility is that our measurement of high energy cosmic rays is somehow wrong. We don’t observe cosmic rays directly, but instead observe the shower of particles they create when striking our atmosphere. From this we infer its energy and composition. While that’s certainly a possibility, the observations we have seem pretty robust.

Another solution is that these cosmic rays are produced locally (in a cosmic sense). Most cosmic rays have traveled billions of light years before reaching us, but if a cosmic ray was produced less than 160 million light years away it could have more energy than the GZK limit. The problem with this idea is that there is no known source of high energy cosmic rays within 160 million light years, so this answer simply replaces the GZK paradox with the mystery of their origin. Another possibility is that the highest energy cosmic rays might be heavier nuclei. About 90% of cosmic rays are protons, and another 9% are alpha particles (helium nuclei), with the rest mostly electrons. It’s possible that a few cosmic rays are nuclei of heavier elements such as carbon, nitrogen, or even iron. Such heavy nuclei might be able to sustain their energy over greater cosmic distances, thus overcoming the GZK limit.

But one other option is perhaps the most tantalizing. Since these cosmic rays have more energy than anything we can create in the lab, they are a test of really high energy particle physics. It’s possible that the GZK limit is simply invalid. It’s based upon our current understanding of the standard model, and if the standard model is wrong so could the GZK limit. The answer to the GZK paradox might be new physics we don’t yet understand.

The energy of the most powerful cosmic rays might just be too big to fail.

Next time: The event horizon of a black hole marks a one way trip to oblivion. It also seems to defy some of the most foundational ideas of physics. We look at the hottest paradox in physics tomorrow.

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Although the Sun seems ageless and never changing, it is a star like any other. It’s only a bit older than the Earth itself, and like every star it formed from the gas and dust of a stellar nursery. As we’ve come to understand stellar evolution, it has become clear that stars get warmer as they age. Billions of years ago, our Sun was about 70% as luminous as it is today. That means young Earth received less heat from the Sun than it does today. So much less heat that it wasn’t enough to sustain liquid water. But geologic evidence clearly shows that there were oceans of water in Earth’s youth. 

The luminosity of the Sun has changed over billions of years.

This is known as the faint young Sun paradox, and it remains a big challenge. Over the past few decades we’ve learned how atmospheric composition can drastically affect surface temperature on a planet. While Venus is warmer than Earth, it’s thick atmosphere makes it even hotter than Mercury. Mars, on the other hand once had liquid water on its surface due to a thicker atmosphere. But while Earth did have a thicker atmosphere in its past, that can’t fully account for young Earth’s oceans. It’s not just the amount of atmosphere, but its composition that plays a vital role in surface temperature. Greenhouse gases like methane and carbon dioxide are far more effective at trapping solar heat than other compounds. Measurements of Earth’s young atmosphere taken from air trapped in rocks show that methane and carbon dioxide levels weren’t high enough to maintain liquid water on Earth.

One possible solution to the problem is that Earth’s early atmosphere had high quantities of molecular hydrogen. Today our atmosphere has very little hydrogen. It’s so light that it can escape Earth’s atmosphere pretty easily. But it does so with the help of ultraviolet light. Since Earth’s young Sun was cooler it produced less ultraviolet light, making it more difficult for hydrogen to escape. Hydrogen is not a particularly strong greenhouse gas, but it can trap heat. As part of a thicker nitrogen atmosphere it might have been enough to maintain Earth’s early oceans. Other ideas propose that solar flares from our young Sun helped heat our atmosphere, or that tidal heating from a closer young Moon contributed to Earth’s warmth.

As it stands there is no definitive answer. So the faint Sun paradox remains a challenge, as it has since the dawn of time.

Next time: Cosmic rays are powerful. Too powerful, in fact. The discussion heats up tomorrow.

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The discovery of white dwarfs in the early 1900s was deeply perplexing for astronomers. From their temperature and brightness it was clear white dwarfs are roughly the size of Earth. Since some white dwarfs orbit other stars, we can also determine they are about as massive as the Sun. How is it possible for so much mass to be compressed within such a small volume without collapsing on itself? 

The most popular idea at the time supposed that under great pressure electrons would become free from atoms, producing a super dense plasma of free electrons and atomic nuclei. Since electrons are extraordinarily tiny, they would act like an ideal gas with the usual temperature and pressure relations. The “electron gas” of a white dwarf would therefore have enough pressure to keep the star from collapsing.

The Boomerang Nebula hovers just above absolute zero, with a temperature of just 1 K.

While that seems reasonable, Arthur Eddington noted it gave rise to a paradox involving thermodynamics. A fundamental law of thermodynamics states that nothing can be cooled below absolute zero. This applies to a gas of electrons as well. Since white dwarfs emit heat and light, over time they would cool. But Eddington noted that white dwarf matter only existed because it is under pressure. If you removed the pressure the material should expand back into regular atomic matter. So suppose you found a particularly cold white dwarf. The gas of electrons and nuclei would be above absolute zero, but it’s energy per mass would be less than that of regular matter at absolute zero. If you scooped up a bit of that white dwarf and remove the pressure, what would happen? Theoretically it should be colder than absolute zero, which isn’t possible.

The paradox was finally solved in 1926 by R. H. Fowler. The problem, he argued, stemmed from treating electrons as classical objects like atoms. Electrons follow the rules of quantum theory. Because of the Pauli exclusion principle there is a limit to how closely they can be pushed together. A gas of electrons in a white dwarf therefore can’t cool below absolute zero because the laws of quantum mechanics don’t allow it. Within a few years Subrahmanyan Chandrasekhar expanded upon this idea to show that white dwarfs can never have more mass than about 1.4 Suns. This upper limit on size became known as the Chadrasekhar limit.

What began as a paradox of thermodynamics became the first demonstration of the quantum connection between the very large and the very small. It pointed us toward the direction of modern astronomy.

Next time: Stars get warmer as they age, which means there was a time when our Sun was too cool to liquify water on Earth. But the evidence is clear water existed on Earth for much longer. What gives? The paradox of the faint Sun heats up tomorrow.

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No matter what direction you look in the night sky, it looks basically the same. In astronomy terms we say the Universe is homogeneous and isotropic. Sure there are areas where galaxies cluster together, and other areas where galaxies are rare, but on average the distribution of stars is pretty even. Because of this, an early idea for the cosmos is that it is the same everywhere forever. It seems both ageless and infinite in expanse. But if that’s the case it raises a few troubling paradoxes. 

Olber argued the sky should be bright as the Sun. Credit: Wikipedia user Htkym

The first paradox is perhaps the most famous. Known as Olber’s paradox, it questions how an infinite ageless universe could be mostly dark. At first glance it might seem obvious. The more distant a star, the dimmer it appears, so stars very far away are simply too dim to be seen. But the apparent brightness of a star follows a specific relationship known as the inverse square law. A single star some distance away is as bright as four similar stars twice as distant, or nine three times farther away. But if stars are distributed fairly evenly, then there are four times the number of stars twice as far away, and nine times more that are three times away. So while stars appear dimmer with distance, there are more stars at greater distances. So an infinite ageless universe should have a sky as bright as the Sun.

Thermodynamics requires that your coffee and the Universe are getting cold.

On the other hand, Clausius’ paradox argues that the sky should be completely dark, with no stars in the sky at all. First postulated by Rudolf Clausius, the paradox is based upon thermodynamics. One of the basic laws of thermodynamics is that heat flows from hotter regions to colder regions until they equalize in temperature. In other words, your morning coffee will always cool down until it reaches room temperature. It will never spontaneously heat up by cooling the surrounding room slightly. According to thermodynamics, even the stars will eventually cool. In an ageless universe the stars should have faded long ago, and the vast cosmos should be a sea of completely uniform temperature. So why is the universe not cold and dark?

Even Einstein thought the Universe was static.

Of course you might argue that stars still shine because gravity causes clouds of gas and dust to collapse in on themselves. New stars are being formed all the time, so naturally the Universe won’t be completely dark. But this raises another paradox: why does gravity work at all? As with light, gravity obeys the inverse square law. An object some distance away pulls upon you gravitationally with a force four times larger than an object of the same mass twice as far away. With distance a gravitational force gets ever weaker, but it never completely goes away. In an infinite universe the amount of mass at a particular distance also follows the square law. For every gravitational pull in one direction, there will always be enough mass in the other direction to balance it out. This is known as Seeliger’s paradox, and it means that gravity shouldn’t be able to act on anything. Gravitational forces should always balance out, so stars shouldn’t form and planets shouldn’t orbit stars. And yet they do.

The solution to these paradoxes is pretty clear. The Universe is not ageless, nor is it stationary. We now know it is only about 13.8 billion years old, and ever expanding. Because of expansion and a finite age, the observable universe doesn’t extend to infinity, so Olber’s and Seeliger’s arguments don’t apply. Since the Universe is finite in age, Clausius’ argument is also invalid. It seems an obvious solution to us, but it’s an excellent example of how incorrect assumptions are difficult to overcome. Before Hubble’s observation of cosmic expansion, it seemed obvious that the Universe must be ageless and stationary. The idea that it might begin with a primordial fireball seems downright creationist in comparison. But in the end, evidence for the big bang became overwhelming, and the paradoxes of an infinite cosmos were finally solved.

Next time: Nothing can be colder than absolute zero, or can it? Consider an ancient cold white dwarf. It’s temperature is near absolute zero, but it’s matter is tightly squeezed by gravity. If you took a chuck of the white dwarf away, would that chunk expand and cool even further? Arthur Eddington wrestles with stellar thermodynamics in tomorrow’s post.

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Anyone practicing science needs to get comfortable with uncertainty. Often the questions raised lead to an answer that is simply “we don’t know.” But there are times when we are instead faced with a contradiction. One set of evidence and theoretical reasoning leads to a conclusion in contradiction with another set of evidence. Usually these contradictions resolve themselves pretty quickly, but there are times when these contradictions grow into a paradox. While some of the most famous astronomical paradoxes are now used to demonstrate where our reasoning went wrong, others still challenge us with no clear resolution. 

What makes paradoxes so powerful is that they force us to reconsider both the evidence and our reasoning. If the Universe is self consistent (and we assume that it is) then there must be a solution to the paradox. So this week we’ll look at five major astronomical paradoxes. A couple have been solved, but most challenge even the most cutting edge research.

1. To Infinity And Beyond – Olber’s paradox is perhaps the most famous example, but there are similar paradoxes involving gravity and thermodynamics. They all raise the same question: How can our Universe possibly be infinite?
2. Cold Equations – When a white dwarf cools over time, can it actually get colder than absolute zero?
3. Icy Sunrise – Our Sun was much cooler in its youth. So how is it that liquid water existed on a young Earth?
4. Bigger Bang – There is an upper limit to the amount of energy a cosmic ray can have. So why do we observe cosmic rays that have even more energy than that limit?
5. Over The Edge – The event horizon of a black hole is a point of no return. But if nothing can escape a black hole, isn’t the fundamental nature of physics violated?

We’ll start by confronting the assumption that even Einstein failed to challenge. In an infinite and ageless cosmos, how is it possible that the Universe is cold, dark and dominated by gravity? The paradox series starts next time.

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