The shape of things to come

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Physical scientists have a healthy attitude toward the history of their subject: by and large we ignore it. 

PJE Peebles (From an article on the history of cosmology, in The Big Bang and George Lemaitre, 1984.)

Most physicists learn a kind of folk history of their subject, told as anecdotes or asides during lectures, or passed on by word of mouth over lunch or beers, in a kind of narrative telephone chain. One piece of folk history I heard as a student relates how, as he was struggling to generalize his Special Theory of Relativity, Albert Einstein was completely unaware of Bernhard Riemann’s 1857 thesis and its theory of curved n-dimensional spaces, a difficult treatise that eventually provided Einstein with the mathematical language he needed to express his physical ideas. This much of the story is true, if a bit misleading.

By 1912 Einstein had already deduced from physical reasoning that the geometry of space-time in general settings had to be what is called ‘non-Euclidean’, and he understood the significance of that fact. But what does this mean?

First, let’s follow Einstein and consider a thought experiment, applying the Special Theory of Relativity to the case of a rotating disk. Special relativity predicts that moving measuring rods will contract along the direction of motion, but not in directions perpendicular to that motion. For example, if someone carries a frame of measuring rods moving in the x-direction, then the rods will contract along x but they will not contract in the y- or z-directions. Now consider the rotating disk: rods laid along a direction tangent to the disk would contract due their motion, but those laid in the radial direction do not, because the motion of those meter sticks are perpendicular to their length. Therefore, the circumference of the rotating disk, as measured by laying meter sticks end-to-end along its edge, will be greater than 2π times its radius. This surprising result is a clear indication that the geometry of any coordinate system attached to the moving disc has to be non-Euclidean, because Euclidean circles always satisfy the relation: circumference = 2π times the radius.

Meanwhile, on the frame attached to the disc, things appear at rest, and yet the relationship between the circumference and the radius must come out the same. An observer on the disc, or one standing to the side watching it rotate, will agree on the counting of measuring rods. But, the observer attached to the disc will interpret the result differently. Einstein argued that this observer will feel an effective gravitational field, familiar to anyone whose played on a merry-go-round, that pulls outward as you go round. He realized this was a key insight. Because of the Equivalence Principle, accelerated reference frames and frames with gravitational effects would all require non-Euclidean geometries. This short film by Derek Muller of Veritasium, does a good job of summarizing the basic idea:

Folsing relates [Albert Einstein, pps. 312-314] how Einstein believed that earlier work by Gauss on non-Euclidean spaces could be applied in these more general settings. Thus Einstein was already on the right track toward his generalized theory of relativity, and he was also aware of Gauss’ theory of non-Euclidean geometry. But, Gauss’ theory (described below) applied only to 2-dimensional spatial surfaces. Einstein hoped to generalize the theory to higher dimensions, and to relativistic space-times. This is the point where Einstein was alerted to Riemann’s work by his mathematician friend, Marcel Grossman. Therefore, the folk history I learned is only half-right: Einstein did not know about Riemann’s work, but he was well aware of Gauss’.

But the folk history went further, and claimed that Einstein was unaware of Riemann’s theory because it was an obscure piece of arcana, hiding unused on a dusty shelf in the library. Physics folk legends describe this piece of the history of general relativity as a rediscovery of something long forgotten, an almost miraculous thing, a piece of extreme serendipity. Einstein needed to know how to generalize the work of Gauss, he happened to mention this to Grossman, and Grossman just happened to know where to send him. Grossman and Einstein were old friends from their undergraduate days. So it’s as if you were an explorer, looking for a hidden city in the mountains, and by accident one day you found out that your best friend from childhood was in fact the only guide who knew the secret path. A novelist would reject this plot as an outlandish coincidence.

In fact, the folk history’s insistence on the “obscurity of Riemann’s work” is untrue. Riemann’s ideas were well known to pure mathematicians even if his work was considered difficult. Riemann believed he was doing physics, not math, and he therefore tended to “prove” mathematical results by appeals to physical intuition. This meant that for several decades some of his greatest work in complex function theory, algebraic geometry, and what we now call ‘Riemannian geometry’ were considered as lacking in rigor by pure mathematicians. The search for those rigorous proofs proved a useful goad to a lot of important work. (For those with a background in math and physics who want to learn more about Riemann and the enormous impact his ideas had upon mathematics, I highly recommend the wonderful mathematical biography of Riemann by Detlef Laugwitz.) Beyond the work of Riemann, it’s important to emphasize that the study of n-dimensional spaces and non-Euclidean geometries was a hot topic in the latter half of the 19th century, with thousands of papers in print by the early 20th century. This extensive literature is even more remarkable given that the number of active research mathematicians was far smaller than it is today.

The failing of the folk history I learned can be laid squarely at the door of us physicists, who perhaps prefer our folk history because a more accurate telling is unflattering. The fact is, by Einstein’s time physicists had been ignoring the mathematical research on n-dimensional geometries and curved spaces for several generations. Physicists are not the only one’s guilty of such lapses, of course. Most scientists tend to classify whole fields into the categories of ‘X’ and ‘Not X’, where ‘X = my field’ and ‘Not X = safe to ignore’. The title of Freeman Dyson’s 1972 article ‘Missed Opportunities‘ says it all.

So, what was all the buzz about around n-dimensional geometry, and what were those non-Euclidean ideas that Einstein eventually found so useful? Let’s begin with a (very brief) review of Euclidean Geometry. If you’re familiar with that topic, you can skip to the next section, where the fun begins.

Euclidean Geometry

Euclid’s Elements of Geometry (ca. 300 BCE) was considered the paragon of clear thinking for over two thousand years. (An online version of it can be found here.) The Elements was used as a textbook by students in the English school system well into the late 19th century. Bertrand Russell learned his geometry by reading Euclid and he notes in his History of Western Philosophy that when the US Declaration of Independence begins with the assertion that ‘We hold these truths to be self-evident,’ it models its rhetorical style on Euclid.

Like many books in pure mathematics today, Euclid begins with a short list of postulates or axioms that are viewed as intuitively obvious, and therefore not requiring proof. Given the axioms and definitions, the entire theory of planar geometry follows by logical deduction. Here are the first four axioms:

  1. A straight line segment can be drawn joining any two points.
  2. Any straight line segment can be extended indefinitely in a straight line.
  3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
  4. All right angles are congruent.

Clean and crisply worded, these four postulates seem quite clear and non-problematic. The fifth, and final, postulate is less obvious:

5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. (This postulate is equivalent to what is known as the parallel postulate.)

These five postulates lead by logical deduction to a number of very important results, and their deduction is delightful to those of us who love geometry. For our purposes, we highlight the following two results:

  1. The circumference of a circle (C) is 2π times its radius (R): C=2πR. This follows from the definition of 2π as the circumference of a circle with unit radius, R = 1, and then invoking scaling and congruence with all other circles. That is: if we double the radius, by scaling we also double the circumference. If we increase the radius by a factor of 1,000, the circumference scales by 1,000, etc. Rigidly moving the circle to any other point in space does not change this fundamental relationship between the radius and the circumference.
  2. The sum of the interior angles of a triangle is π. This follows by extending the line segments that form the triangle into lines, then erecting parallels at each vertex. We can then use congruence, and the fact that at each vertex the sum of the angles between the intersecting lines is 2π.

Any space that does not satisfy these two properties is non-Euclidean. For over two thousand years, mathematicians believed that it was not possible to construct a self-consistent geometry that was non-Euclidean. They were wrong.

Euclid’s fifth postulate is much less ‘self-evident’ than the previous four. It doesn’t have the same crispness to it, and feels contrived. Yet it cannot be proven as a theorem from the other four, even though many people tried for over two thousand years. Finally, in 1823, Bolyai and Lobachevsky independently realized that entirely self-consistent “non-Euclidean geometries” could be created in which the parallel postulate did not hold. Their geometries were special, isolated, examples but they proved that Euclid’s theory wasn’t the only self-consistent geometry.

Enter Gauss

Carl_Friedrich_Gauss_1840_by_Jensen
Carl Friedrich Gauss (1840), portrait by Jensen. Source: Wikimedia Commons.

Carl Friederich Gauss [1777-1855], a child prodigy born of poor parents, attracted the attention of the Duke of Brunswick who sponsored his education. This opportunity to have a good education made all the difference, and it explains Gauss’ deep and abiding loyalty to the Duke.

By his late teens, young Carl was making new discoveries in pure mathematics and he wrote his great book on number theory Disquisitiones Arithmeticae by the age of twenty-one. Number theory was an ancient subject, full of fascinating oddities and gems of patterns, but Gauss systematized number theory like never before, partly by introducing what are now called ‘complex numbers’. These ideas provided a powerful new set of tools for attacking old problems, and their adoption opened the door to a new world of mathematical delights. Much of 19th century mathematics can be understood as an effort by mathematicians to dig deeply into these astonishing new ideas, and to revisit the foundations of their subject in light of them. Something truly new was afoot.

Gauss also went on to derive what is now called the ‘Fundamental Theorem of Algebra’, a result he believed was so important that he proved it several different ways. This theorem says that a complex polynomial of order n will always have n roots. These are marvelous results, showing the power of complex numbers to organize results in algebra, number theory, the theory of functions, and geometry. The young Riemann later took to this set of interrelated ideas with enthusiasm.

In the same year that Gauss published the Disquistiones (1801), the asteroid Ceres was discovered (this is now classed as a dwarf planet). [N.B. the first unmanned probe to visit Ceres in early March 2015.] New celestial bodies were discovered fairly often at the time. Priority for new discoveries required an observer to characterize the orbit, so that other astronomers could find it in later observations. The astronomer who first observed Ceres, Giuseppe Piazzi, lost it when it went behind the Sun. These first few observations were taken over a short period of time (a month), hence the orbit was poorly characterized. Gauss, still a relatively unknown young scholar laboring in obscurity, reduced the problem of reconstructing the orbit from the known data to a problem in analytic geometry and conic sections. This required him to discover approximation methods for polynomials of eighth degree (which in general have no closed-form solution). The method of solution proved powerful, and he applied it to other orbit calculations.

Using his new methods, Gauss was able to solve for orbits in days when others took weeks. He became a celebrity among astronomers. His methods for pencil-and-paper numerical computation still remain an important foundation for modern computations and are summarized in his essay: “Theory of motion of the celestial bodies moving in conic sections around the Sun”. These methods form the basis of what are now called Fast Fourier Transforms (FFTs) and the method of least squares. Scientists use these methods all the time to analyze data, to perform complex numerical simulations, and to find patterns in large data sets. You use them everyday, too, when you look at a .jpeg or .gif image on your computer, stream videos via Netflix, watch cat videos, or do a Google search. The kernel of the necessary ideas flow from Gauss, though of course many additional improvements have been made in the intervening years, especially with the advent of digital computers and the access to data middens of such size that Gauss could only gaze in wonder.

Gauss was still supported by the Duke of Brunswick during this period. But, he worried about his financial security long term. Since he was now famous among astronomers, he decided to seek a position. Gauss was appointed Professor of Astronomy and then director of the astronomical observatory at Göttingen in 1807, at age thirty. From that time until his death in 1855, about half of the entries in the observatory log books are in Gauss’ hand. So, he was up there most nights assisting in observations. Though brilliant at math, he was a diligent worker and took his formal duties seriously.

Notice that Gauss is moving from pure mathematics into more applied areas, and dealing with observations. Pure mathematicians lament this, but scientists are grateful.

Gauss was one of those rare scientists who used very practical problems as a goad to think deeply about the — far more abstract — foundations of their subject. Around the same time that Lobachevsky and Bolyai had proved the existence of certain special non-Euclidean geometries, Gauss had proved there were an infinite number of them. We don’t know the exact date of his discovery because he didn’t publish the result. The likely prompt for Gauss’ theory of non-Euclidean geometry were the long days of field work, and nights spent in a camping tent, during his leadership of a multi-year campaign to survey the kingdom of Hanover. The Duke was deeply worried about the boundaries of his kingdom, located as it was on a continent that, one generation later, still reverberated with the echoes of the Napoleonic Wars.

The land of Hanover was returned to King George III after the Napoleonic Wars. (Yes, that King George III, villain of the American Revolution. It’s complicated.) This region remained part of the House of Hanover until 1837, when Queen Victoria rose to the English throne. The line of succession in the Kingdom of Hanover did not allow a woman to take the throne if a surviving male heir was in waiting, so the region passed from English influence, and was eventually conquered by Prussia in 1866. Here is a map from around the time of Gauss’ surveying project, which took from 1818-1832.

A map of the Kingdom of Hannover around the time that Gauss undertook his surveying mission. Source: Wikimedia Commons. See http://en.wikipedia.org/wiki/File:KrkHannover.png for full attribution and information.
A map of the Kingdom of Hannover around the time that Gauss undertook his surveying mission. By kgberger [GFDL (http://www.gnu.org/copyleft/fdl.html), CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/) or CC-BY-2.5 (http://creativecommons.org/licenses/by/2.5)%5D, via Wikimedia Commons

Hanover is a pretty complicated gerrymandered type of province, and establishing clear boundaries was clearly of political importance. Gauss was asked to oversee the survey. He was not someone to do a slipshod piece of work, so even though this work took him away from his home during many long summers of effort, he did a competent job. He invented new instruments to improve the results, and along the way he pondered the nature of measurement in geometry. Ultimately, this led him to rethink the ‘extrinsic’ geometry in standard use, and to develop what is now called ‘intrinsic geometry’. This is far more radical an idea than simply introducing ‘curvilinear coordinates’, as we’ll see. One can introduce curvilinear coordinates on flat surfaces, after all. The relationship between coordinates and surface shape is a subtle one, as we’ll now explore.

Extrinsic vs intrinsic geometry

How does ‘intrinsic’ geometry differ from ‘extrinsic’? And, why do surveyors use the extrinsic form? In surveying, you are measuring angles and distances in three spatial dimensions in order to understand the shape of a two-dimensional object, the surface of the Earth. Surveying is based upon Euclidean geometry, extended to three-dimensional space. This is the way surveying has been done since antiquity. The advance of technology and improvements in precision doesn’t change the fact that the underlying mathematical principles are unchanged in over two thousand years.

Gauss asked some radically new questions: Why do I need three-dimensional measurements to uncover the shape of a two-dimensional object? Why can’t I carry out measurements restricted to that two-dimensional surface and still uncover its shape? He eventually found that he could describe the shape of any smooth two-dimensional surface using only measurements taken within the surface itself. Where Bolyai and Lobachevsky had shown it was possible to create special non-Euclidean geometries by example, Gauss provided a general means to study and characterize all smooth surfaces.

Whether or not Gauss would have tackled non-Euclidean geometry if he had not been involved in the survey of Hanover is unknown. But, he is found pondering the idea in letters, and already by 1828 his research had led to what he called his ‘remarkable theorem’, which states that you can determine the shape of a smooth surface by measuring distances and angles entirely within the surface. In order to prove this result, Gauss introduced the notion of a metric, which allowed him to generalize the Pythagorean Theorem to non-Euclidean spaces. The metric allows one to convert from coordinate differences to physical lengths, and it provides a kind of rule-book for doing so at each point on the surface. Gauss understood that coordinates were a kind of intermediary that allow you to smoothly assign numerical labels to points in space, but that physical things — like distances — could not depend upon the choice of coordinates. Coordinates are like the words of a language, in that we can use different languages to describe the same thing in the world. I can talk about elephants in English, French, or Urdu, but changing the word doesn’t change the underlying elephant. In the same way, I can lay down many different kinds of coordinates on any given surface, but that doesn’t change the underlying shape. Some more mathematical details are given below.

This set of ideas was a bombshell. Gauss realized it meant that Euclidean geometry was not the only possible one, and that the actual geometry of our world might not be Euclidean. Therefore, the geometry of our world was something that had to be determined empirically. This was a huge change in outlook, likely considered radical by some of Gauss’ colleagues at Göttingen, who were strongly influenced by Kant’s philosophy, with its assumption that the Euclidean nature of space was an example of a priori knowledge, i.e. something we know prior to experience.

The basic ideas, how to go from coordinates to shapes, can be understood by first considering the example of determining the shape of a smooth one-dimensional curve. Let’s restrict the one-dimensional curve to lie in a two-dimensional (Euclidean) plane for simplicity, and begin by considering the case extrinsically. That means, we draw the curve on the [x,y]-plane and start by using the pair of coordinates x and y. We can also label each point on the curve by a single smoothly-varying parameter, t, so the one-dimensional curve is completely described by the pair [x(t),y(t)]. So far, so good. But, given that the curve is one-dimensional, why do we need to use two numbers to describe its shape? Why can’t we describe the shape of the curve using only quantities we measure on the curve itself?

In 1847 Jean Frederic Frenet [1816-1900] and in 1850 Joseph Albert Serret [1819-1885] independently discovered how to do this. Frenet-Serret theory is for curves in three dimensions, but the essential idea is already present in the two-dimensional case. The work of Frenet and Serret came after that of Gauss’ 1828 ‘remarkable theorem’, which concerns two-dimensional surfaces, not one-dimensional curves. But, when studying differential geometry, you start with Frenet-Serret because it is mathematically and conceptually simpler. Gauss leapt ahead and solved the harder problem first.

At each point on a smooth curve, we can introduce best-fit circles. These allow us to measure the local 'curvature' of the curve. We adopt a convention where if the best-fit circle lies on the top of the curve, it has 'positive' curvature at the point, and negative if the best-fit circle lies below. Figure by Kiatdd (Own work) [CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons
At each point on a smooth curve, we can introduce best-fit circles. These allow us to measure the local ‘curvature’. Suppose we traverse the curve from left to right, with the curve parameter t increasing in that direction. According to the sign convention outlined in the text, if the best-fit circle lies on the top of the curve, it appears to curve ‘to the left’ as we traverse it, so it has ‘negative’ curvature at the point, and the curvature is positive if the best-fit circle lies below. Figure by Kiatdd (Own work) [CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0)%5D, via Wikimedia Commons

The key idea is that to describe the shape of the curve at a point on the curve, p(t), you fit the curve to a circle at that point. This is still extrinsic, however, since the circle lies off the curve, but let’s go with it and see where it leads. Orient yourself so that increasing t defines a direction along the curve. For example, imagine you are are one-dimensional being, walking along the curve carrying a calibrated rope with you, with distances marked off at regular intervals. Suppose, also, that you carry a clock, and that you walk along so that your speed, the change in distance with respect to the clock time, is constant. At the position where the rope is marked s, you are at the point p(s). At that point you measure the curvature at p. This is the signed radius of the best-fit circle at that point, R(p). The curvature at p is the inverse of the radius, i.e. 1/R(p), of the best-fit circle (this ensures that the curvature is zero for a straight line, for which the best-fit circle has an infinite radius). Define the curvature to be positive if you are curving to the left as you move along the curve (increasing t), and negative if you curve to the right.

But why can’t I avoid referring to the ‘circles’ that lie outside the curve? You can, if you now grant me the possibility that I can measure my acceleration as I move along the curve at fixed speed. Suppose we are one-dimensional beings, restricted to have knowledge only of the world of the curve, and we move along that curve as if in a kind of tunnel, where we can only see what is in our immediate surroundings. The velocity is always tangent to the curve, so even though my speed along the curve is constant, the velocity has to change in order for me to stay on the curve. Just like rockets and submarines that carry onboard inertial guidance systems, let’s allow our one-dimensional actor to have the one-dimensional equivalent of an accelerometer. At each point on the curve, if the extrinsic description sees a smaller local circle, the one-dimensional being will measure a larger acceleration required to move at fixed speed yet stay on the curve. Larger radii of curvature imply more gentle accelerations. A straight line implies zero acceleration.

In summary, by carrying along a rope with physical distances measured out, and a clock I have the means to move at fixed speed along the curve. By measuring my local acceleration I am able to infer the local shape of the curve. By calling this a ‘curvature’ it seems to imply that we must refer to those circles outside the curve, but in fact that’s merely a smiling Cheshire Cat remnant of our original extrinsic description. By moving along the curve with our inertial guidance system onboard, we can imagine we move along a long tunnel feeling ourselves buffeted, pulled by unseen forces, to which we infer that the one-dimensional space we live in is curved.

Frenet and Serret showed that if you invert the situation, and instead of starting with a curve of known shape, you start with the radius of curvature at each point on the curve, R(p), this information completely determines the shape of the curve up to a rigid translation or rotation. We can now ignore the two-dimensional space we started with, and in fact we can in principle ignore the notion of ‘best-fit circle’. The two-dimensional [x,y] space can be used to do what mathematicians call embed the curve, but we don’t have to do that. Essentially all the embedding does is provide us with an easy means to view the curve from ‘outside’. But the curve has a separate existence from the embedding space. We could embed the same curve in lots of different spaces: three-dimensional spaces, four-dimensional spaces etc. This emphasizes that the curve is the primary object, not the embedding space, and that the ‘radius of curvature’, R(p), which is inverse to the ‘acceleration,’ is fundamental to understanding the shape of the curve.

In three-dimensional spaces, one-dimensional curves can get a lot more twisty-turney than in two-dimensions, and the Frenet-Serret Theory deals with this by introducing a concept called the torsion. This is related to how much the acceleration is changing at each point along the curve. However, this takes us beyond the scope of the current discussion, and we want to get back to two-dimensional surfaces and Gauss’ theory.

How can we generalize Frenet-Serret theory to define the shape of two-dimensional surfaces intrinsically? At each point p on the surface, we now fit two circles. Gauss showed that there will be two principle directions, which are orthogonal to one another, where the fit to a circle is good.

Gauss showed that locally, a two-dimensional surface had two 'principle directions'. In each of these directions, a best-fit circle can be found. In this figure, the two circle lie on opposite sides of the surface, hence the 'Gaussian curvature' is negative (by convention). Illustration by Eric Gaba (Sting) (Based upon a drawing in a book) [GFDL (http://www.gnu.org/copyleft/fdl.html), CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/) or CC-BY-SA-2.5-2.0-1.0 (http://creativecommons.org/licenses/by-sa/2.5-2.0-1.0)], via Wikimedia Commons.
Gauss showed that — locally — a two-dimensional surface had two ‘principle directions’. In each of these two directions, a best-fit circle can be found. In the above figure, the surface locally has a saddle shape, so the two circles lie on opposite sides of the surface. Hence, the ‘Gaussian curvature’ is negative (by convention). Illustration by Eric Gaba (Sting) (Based upon a drawing in a book) [GFDL (http://www.gnu.org/copyleft/fdl.html), CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/) or CC-BY-SA-2.5-2.0-1.0 (http://creativecommons.org/licenses/by-sa/2.5-2.0-1.0)%5D, via Wikimedia Commons.

Choose a sign convention, so the curvature is positive if the circle is on one side of the surface and negative on the other. The radii of these two best-fit circles are inversely related to the two principle curvatures at the point p. The Gaussian curvature is the product of the two principal curvatures. The Gaussian curvature will be positive if both circles are on the same side of the surface, meaning it locally looks like part of an ellipsoid. The Gaussian curvature is negative if the two best-fit circles are on opposite sides of the surface, meaning it locally looks like a saddle.

Three surfaces with different Gaussian curvatures. The one on the left has negative Gaussian curvature, while the sphere on the right is positive. The cylinder in the middle has zero Gaussian curvature. Figure by Jhausauer (English wikipedia,) [Public domain], via Wikimedia Commons
Three surfaces with different Gaussian curvatures. The one on the left has negative Gaussian curvature, while the one on the right is positive. The cylinder in the middle has zero Gaussian curvature. Figure by Jhausauer (English wikipedia,) [Public domain], via Wikimedia Commons

A remarkable theorem

Recall that with Frenet-Serret Theory, we moved from an extrinsic to an intrinsic description by introducing the ability to measure distances using rope, and a clock so we could define what it meant to move at constant speed along the curve, and then the ‘acceleration’, which was inverse to the ‘curvature’. How can we generalize this to two-dimensions?

Following Gauss, we try to stay as concrete as possible. We start by defining the analog of ‘straight lines’, ‘circles’, and ‘triangles’ on a curved surface. These are the building blocks of any geometry, and without an understanding of what these concepts might mean on a curved surface we can’t really begin.

  • First, choose a point p on the surface. Next, fix one end of a string to p, and leave the other end free. Walk away from p a short distance to the point p‘. Pull the string taut. (The string must remain within the surface everywhere along its length!) The string now forms a curve called the geodesic curve connecting p and p’. The  length of this string is defined to be the distance between p and p’.  This is the equivalent of a line segment from Euclidean geometry, and is the shortest curve between the two points.
  • By walking away from p’ following the local tangent to the geodesic at that point, we can extend the geodesic further, just like extending the straight line in Euclid’s theory. We carry a clock, so we can define a constant speed motion along this curve if needed. If our curved surface is bounded we will not be able to extend our geodesic to infinity.
  • Now keep the length of our string fixed, and keep one end of the string pinned at point p. We can now walk around p while keeping the string taut. The set of points formed by the free end of the string is a non-Euclidean circle. We can measure the length of its circumference of this circle, and in general it will not equal 2π.
  • A non-Euclidean triangle is any region on the surface formed by the intersection of three geodesics. The sum of the interior angles of this shape, formed at the three interior intersections, will not equal 2π in general.
Geodesics (great circles) on a sphere, forming a triangle. By derivative work: Pbroks13 (talk) RechtwKugeldreieck.svg: Traced by User:Stannered from a PNG by en:User:Rt66lt (RechtwKugeldreieck.svg) [Public domain], via Wikimedia Commons
Geodesics (great circles) on a sphere, forming a triangle. The point O is the center of the sphere. The vertices of the triangle are the points A, B, and C. The geodesic line segments forming the edges of the triangle are the curves a, b, and c.
By derivative work: Pbroks13 (talk) RechtwKugeldreieck.svg: Traced by User:Stannered from a PNG by en:User:Rt66lt (RechtwKugeldreieck.svg) [Public domain], via Wikimedia Commons

Gauss proved that at any given point p on the surface, by measuring the local ratio of the circumference to the diameter of ‘circles’, and the included angles of ‘triangles’, properly defined, we can uncover the values of the principal curvatures at that point. In other words, we can deduce the shape of the surface without having to refer to anything off the surface. He was so taken by this result that he called it the Theorema Egregium, or the “Remarkable Theorem”.

This is a coordinate-free approach. How do we bring coordinates into this? The great advantage of coordinates is that they make it easy to specify the position of any point on the 2D surface, just by giving a pair of numbers. The system of coordinates on a curved surface cannot be Cartesian, of course, so this motivates the study of non-Cartesian coordinate systems. The problem here becomes: how do we relate fundamental physical quantities, like the distance between two points, in terms of their coordinate labels? If we know how to do that, then doing physics involves simply reading off coordinates, rather than having to carry out a surveying mission every time we want to know “How far apart are the points A and B?”.

What we need to know is how to convert coordinate difference to distances. The basic idea, due to Gauss, is to generalize the Pythagorean formula

c^2 = a^2 + b^2,

which relates the lengths of the sides of a right triangle (note that the carat ^ notation means a power here, since I can’t do superscripts in WordPress). This can also be viewed as a rule concerning coordinate differences between the two points on the hypotenuse of the triangle, provided we have introduced Cartesian coordinates [x,y] and now wish to use coordinate differences to infer physical distances. That is, suppose a is the difference in x-coordinates and b the difference in y-coordinates:

a = Δx, and b = Δy,

then by Pythagorean theorem the distance Δs between the two points on the plane satisfies

Δs^2 = Δx^2 + Δy^2.

On a curved surface, we cannot use Cartesian coordinates, and we should only expect to relate the distances of nearby points to small coordinate differences. But this is okay, because we can then measure the distance between widely separated points by summing up the small distances along the way. That is, suppose Δx and Δy are the coordinate changes between two neighboring points p and p‘ on the curved surface, and on the surface they are separated by a small physical distance Δs. The Gaussian generalization of Pythagorean theorem is:

Δs^2 = g_xx Δx^2 + 2 g_xy ΔxΔy + g_yy Δy^2.

On the right are coordinate objects (specifically, the changes in the coordinates x and y between the two points), and the coefficients of a new object called the metric tensorg(p). This set of coefficients tell you how to convert coordinate differences to a distance. It’s a kind of translator from a particular coordinate language into a physical quantity that does not depend on coordinates. The metric tensor depends upon the point p, and using it, we can deduce what the shape of the surface is without ever referring to anything off the surface. The mathematics is difficult, and beautiful, and would take us well beyond the level of this simple exposition. But, I hope you’ve now got some idea of what’s involved.

Non-Euclidean geometry and theoretical physics prior to Einstein

Gauss presented these results in a seminar at Göttingen. Luckily for us, a young mathematician named Riemann was in attendance. Riemann was seduced by these ideas. He chose the topic for his 1854 “Habilitationsvortrag”. This was to be his ‘qualifying lecture’, and success would provide an endorsement by the university that would allow the impoverished Riemann to take on paying students. He chose the topic at Gauss’ suggestion, and he produced a generalization of Gauss’ ideas, suitable for any number of spatial dimensions. He wrote the thesis in honor of Gauss, and even though Gauss was reportedly excited by the result, it remained a curiosity for a few years, known only to mathematicians working in the area. In Helga Kragh’s review “Geometry and astronomy: Pre-Einstein speculations of non-Euclidean space” [available at the physics ArXiv here], she notes that even among mathematicians there was little notice of Riemann’s work for the first few years. This may have been partly due to Riemann being somewhat indirect in his lecture and the related paper. This, in turn, might have been due to the presence of the philosopher Lötze on his defense committee, a confirmed Kantian who believed that Euclidean geometry was self-evident.

The figure below shows the number of references (in five-year intervals) to non-Euclidean geometry from about the time of the first published work by Lobachevsky and Bolyai (recall that Gauss did his work slightly earlier, but did not publish). Riemann’s thesis was defended in 1857, which means it falls into five-year bin Number 6.  Note the decided lack of impact it had at first. [The figure is taken from Kragh’s article.]

Page 8 of Kragh’s paper shows a time series of the number of publications on the subject of non-Euclidean geometry, taken from Sommerville’s 1911 bibliography. The takeoff point is in the late 1860’s, around bin Number 10 or so, which Kragh argues corresponds with the publication of papers by Beltrami on the subject. After that, the number of articles grows dramatically, rising to almost 900 articles in the five-year period around the turn of the century. So, there was ferment in mathematics on this topic, but Riemann’s work still had not penetrated into physics. As already mention, Einstein was aware of Gauss’ work, but not Riemann’s generalization [Folsing, Albert Einstein: A biography, (Viking, 1997)].

Riemann believed he was doing physics by asking the question: What is the most general shape of a space, starting only from the notion of smoothness? What we now call Riemannian geometry provides the answer. He starts by noting that in a space of N dimensions, we can always slice through it with a smooth surface of two dimensions, constructed by launching geodesics in a fan about any given base point. On that two-dimensional surface, things reduce to Gauss’ theory, and the shape of the two-dimensional space could be deduced by the same means that Gauss used: by using those geodesics to defines circles and triangles, then locally measuring the deviations from Euclidean behavior. There are, of course, an infinite number of two-dimensional surfaces that pass through any given point in an N-dimensional space. What Riemann managed to do was show that you can introduce a kind of master bookkeeping scheme using an NXN version of the metric tensor, g_mn(p), and from this object the ‘curvature tensor’ can be derived. The metric tensor is the bookkeeping storehouse of information. From it, the projection to any two-dimensional subspace can be carried out, and the Gaussian curvatures derived.

All of this was beautiful mathematics, it but had no impact on the development of late 19th century science, only mathematics. Aside from the mathematicians laboring in their separate vineyards, Riemann’s work made no impression on physics for many years, until around 1912 when Einstein and Grossman had their conversation, almost sixty years after the publication of Riemann’s thesis. It is also important to emphasize that Riemann’s paper concerned space, not spacetime. The generalization to non-Euclidean spacetime was due to Einstein.

It’s important to note that Riemann’s work was translated into English in 1870 by the mathematician Clifford, who speculated that the ideas might have application in physics, and that the motion of massive bodies under gravity might instead be understood as bodies moving in a non-Euclidean space. These ideas were presented in a talk to the Cambridge Philosophical Society in 1870, and only a two-page abstract survives. It reads in part [quoted in “The Fourth Dimension in Nineteenth-Century Physics“, Alfred M. Bork, Isis Vol. 55, No. 3 (Sep., 1964), pp. 326-338]

“I hold in fact

(1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.

(2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the matter of a wave.

(3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial.

(4) That in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity.”

These ideas were prescient. But, Clifford apparently either did not succeed in this program, or he did not pursue it beyond the original speculation.

The mathematics of Riemannian geometry is difficult, the algebra is daunting, and there seemed to be no application (aside from Clifford’s hint). This might explain the lack of interest in the formal physics journals during this period. Although, Fayter notes there are opinion pieces in Nature magazine, one of the leading scientific journals of the day, regarding speculations on a ‘fourth dimension’ by various scientists. [Fayter, “Late Victorian science and science fiction”, in Lightman’s Victorian Science in Context (U Chicago Press, 1997).]

On the subject of whether we can decide what the geometry of our world is empirically, in 1902 the mathematical physicist Henri Poincare wrote:

“If Lobchatschewsky’s geometry is true, the parallax of a very distant star will be finite. If Riemann’s spherical geometry is true, it will be negative. These are the results which seem within reach of experiment, and it is hoped that astronomical observations may enable us to decide between the two geometries. But what we call a straight line in astronomy is simply the path of a ray of light. If, therefore, we were to discover negative parallaxes, or to prove that all parallaxes are above a certain limit, we should have a choice between two conclusions: we could give up Euclidean geometry, of modify the law of optics, and suppose that light is not rigorously propagated in a straight line. It is needless to add that every one would look upon this solution as the more advantageous. Euclidean geometry, therefore, has nothing to fear from fresh experiments.” [Poincare, Science and Hypothesis, Dover Books, p. 72-73.]

This is the state of the field when Einstein began to cast around for the appropriate mathematical language to use for his new Generalized Theory of Relativity. He knew he needed the geometry to be non-Euclidean, but he was unfamiliar with the prior literature.

Einstein’s General Theory of Relativity

[The following draws upon pages 312- 315 of Folsing’s 1997 biography of Einstein.]

By about 1912, Einstein had realized that he would need a non-Euclidean geometry to allow for accelerating frames. This led him back to Gauss’ original work. In an address in Tokyo in 1922 he said that “Describing the physical laws without reference to geometry is similar to describing our thought without words. We need words in order to express ourselves. What should we look to to solve our problem [of describing physical laws]?” He generalized Gauss’ work to four dimensions of spacetime, and found that it allowed for the inclusion of the equivalence principle, the bending of starlight, gravitational redshifts, all of which he had convinced himself needed to be part of theory. He had at that point been pondering the nature of generalized relativity since 1907. But, by 1912 he had not pushed the theory all the way through yet. He did not have a dynamical theory, which included the effects of matter on the metric. The key problem that Einstein was trying to solve, and for which he needed Riemann’s posthumous assistance, was that of finding a way to relate the geometry of space-time to the distribution of matter and energy in that space-time. He had already arrived at a good idea of what the theory should look like on physical grounds.

  • The equivalence principle led him to conclude that the effects of acceleration and gravity are (locally) equivalent, and that this would cause space to be non-Euclidean. This result follows from the thought experiment on the merry-go-round and special relativity, as we described earlier in this post.
  • In a freely-falling reference frame, the special theory of relativity applies locally.
  • The mathematical form of the laws of physics must be the same in all reference frames, including accelerating frames and frames with gravitational fields.
  • In weak gravitational fields, and for speeds small compared to the speed of light in vacuum, c, the theory should reduce to Newton’s theory of gravity.

Although Einstein knew these were the properties the theory must possess, he was having trouble pushing the mathematics through. He was aware of Gauss’ work on non-Euclidean geometry, and so had some of the tools he needed, but he was unable to solve the problem of covariance, i.e. of making sure that the mathematical form of his equations were the same in all coordinate systems. This is when he sought help from his friend Marcel Grossman, who pointed him to Riemann, and Einstein found the ideas he needed there. Combining Riemann’s work, which was purely for curved spaces, with a more general metric, which was needed for spacetime, he was finally able to work his way toward the solution.

He had worked on this all-out, and had almost succeeded when he went to Göttingen to give a talk at the home of Gauss and Riemann. This is where a famous exchange with the mathematician David Hilbert happened, an interaction which led to the now-famous ‘Einstein-Hilbert action’.

Einstein presented his summary of the desired properties of the theory, and admitted his struggles. This is something Einstein tended to do. He was famous later for publishing his mistakes, because he felt that others might learn from them, and to help them avoid wasting time. In any event, his openness gave Hilbert ideas to work with, and the claim is that shortly after Einstein’s seminar Hilbert resolved the problem Einstein was struggling with. Meanwhile, Einstein carried by himself for a few more weeks, and arrived at the same solution in his own way. This is why Einstein is given priority for the theory overall, but Hilbert’s name goes on the ‘Einstein-Hilbert action’, too. Here are the equations that result, which I include only so we have something to point to:

R_mn – 1/2 g_mn R + g_mn L = (8 π G/c^4) T_mn

On the left of the equal sign is geometry, i.e. quantities that capture geometric information, on the right is the distribution of mass-energy. In conceptual terms, therefore, the Einstein equations look like:

Geometry <—> Mass-energy

If we know the right-hand side of this relation, the left-hand side consists of a system of ten nonlinear partial differential equations which must be solved to find the metric tensor g_mn at each point of space-time. The object R_mn is the ‘curvature tensor’, which is a complicated function of g_mn and its derivatives. Given g_mn, the ‘shape’ or ‘geometry’ of the space is known because the non-Euclidean deviations of circles and triangles can be deduced. What this means in practice is that the paths of freely-falling particles or light rays can be computed.

On the right side of the equation is mass-energy. This can be computed if we know the distribution of matter and light, specifically how that matter is moving, how the light is distributed, and how the potential energy of interactions is distributed. Given the metric and the forces acting between particles, mass-energy knows how to evolve. Given the mass-energy, the shape of the space is (largely) determined. Details are left for the gentle reader.

[Sources: Most of the biographical information about Gauss is drawn from the 1955 biography Carl Friedrich Gauss, Titan of science: a study of his life and work by Dunnington, and the Britannica and Wikipedia entries for Gauss. The sources for Riemann are primarily the 2008 book Bernhard Riemann 1826-1866: Turning points in the conception of mathematics, by Detlef Laugwitz (english translation by Abe Shenitzer; for Einstein I refer the reader to Albrecht Folsing’s Albert Einstein; and for the review of the publication history prior to Einstein, see Helge Kragh’s review “Geometry and astronomy: Pre-Einstein speculations of non-Euclidean space“.]

Copyright © Eugene R. Tracy

All rights reserved.

 No part of this essay may be reproduced in any form or by any electronic or mechanical means including information storage and retrieval systems, without permission in writing from the author.

2 comments

  1. Wow, this article is packed full of interesting facts and even more interesting commentary/perspective! I really like how we are transported from the view from outside to the view from inside. Also, I will definitely start using the term “data middens” 🙂

    Liked by 1 person

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