- Quote: . . . the Higgs field exhibits the most inelegant of the known laws governing fields and particles. There’s an amusing tendency for those who tout beauty to ignore this, as though it were an inconvenient family member, and to focus instead on Einstein’s elegant theory of gravity. Yet even that theory has its issues.
- Endnote: Einstein’s theory of gravity is amazingly elegant as long as one ignores the puzzle of “dark energy,” which would have been easier to do had it been exactly zero, and as long as gravity is a very weak force, as its weakness leads to extremely simple equations. In string theory, Einstein’s equations become much more complex, and the elegant simplicity of the math shifts to the level of the strings themselves . . . perhaps.
I’ll expound below upon the second bullet point, hoping to draw attention to general questions concerning aesthetics in theoretical physics.
The Beauty of General Relativity
General Relativity, Einstein’s masterpiece, revolutionized human understanding of gravity. It’s widely seen by its practitioners as the most beautiful theory in all of physics (where “theory” means “a combination of math equations and concepts used to make predictions about nature”).
In Newton’s notion of gravity, isolated objects travel in straight lines across the flat space that makes up the universe, while gravity attracts objects to each other, causing their paths to curve. But in Einstein’s view, objects travel in straight lines (or the next-best thing, in lines called “geodesics”) even in the presence of gravity! Geodesics are quasi-straight lines across a curved four-dimensional space-time. When those paths are projected onto our ordinary nearly-flat three-dimensional space, they appear curved to us.
Space-time, meanwhile, develops its curves in response to the presence of physical objects. As John Wheeler is said to have famously put it, “spacetime tells matter how to move; matter tells spacetime how to curve.”
Einstein’s equation for how physical objects determine spacetime’s shape was the last piece of the puzzle in his years-long quest for a modern theory of gravity. He found the equation just in time, winning his race to beat the famous mathematician David Hilbert, who was just a half-step behind him. In its original form from 1915, it reads
This equation connects R, a quantity constructed from the shape of space and time, to another quantity T, constructed from the energy and momentum carried around by material objects. The other things that appear in the equation are
- g (another measure of the shape of space, called the “metric”)
- c (the cosmic speed limit, a.k.a. the speed of light)
- GN (Newton’s constant of gravity, the same one that appears in Newton’s law of gravity)
This is remarkable on multiple grounds. First, only the energy and momentum carried by the material objects matters — that’s the only thing that appears in T. There is no need to know the details of the types of material objects involved — whether they be atoms, photons, or neutrinos, stars or planets, gas clouds or black holes. Gravity doesn’t care; it is universal, just as Newton originally suggested.
(For Newton, gravity is created by mass, as we all learned in school. But rest mass, Einstein showed, is related to energy; and for slow objects gravitational mass and rest mass are the same. So even though, in Einstein’s theory, gravity arises from energy rather than directly from mass, it still gives the same answers as Newton’s gravity whenever the gravitating objects move slowly and aren’t too compact.)
Additionally, the only parameters appearing in the equation are the cosmic speed limit and Newton’s constant. These parameters were already known and well-measured by Einstein’s day. It is rare that a great step forward in scientific understanding requires no new parameters at all. This meant that Einstein’s theory could immediately make predictions without the need to measure any new quantities.
Finally, gravity is represented, on the left-hand side of the equation, as purely due to the geometry of space and time. For Newton, space and time were simple and static, and he stuck gravity in on top of them, without explanation for where it comes from. Einstein, at a stroke, explained gravity not by adding something but by taking something away; there’s just space and time, only they’re not simple and static, with gravity emerging from their collective properties and behavior.
From this perspective, you can see why general relativity is seen as an elegant conceptual idea, captured in an elegant equation — a truly elegant theory of nature.
Facing Reality: The Cosmological Constant
Unfortunately, the universe does not satisfy this equation. It satisfies this one:
where Λ is a new parameter, known as the cosmological constant and usually referred to as dark energy. No longer is Einstein’s theory so directly predictive; this parameter first has to be measured. This was done in 1998 (and awarded the Nobel prize in 2011), and the value of Λ is now nailed down to within a few percent.
(The above statement is glib. Strictly speaking, dark energy might vary slowly over time, in which case it would not be quite the same as a cosmological constant, and the equation potentially becomes yet more complicated. But let’s skip this distracting detail.)
Einstein himself introduced this modified equation in 1917. He did this to accommodate a universe that was static and eternal, as the universe was widely assumed to be. But later he called this alteration of his equation his “greatest blunder.” That’s because it was soon discovered, by 1929, that the universe is not static and eternal; it is expanding and, in its current form, has a finite age. Once this was known, there no longer seemed to be any need for Λ. If only Einstein had insisted that his original elegant equation must be right and that Λ must be zero, based on aesthetics, he might indeed have predicted that the universe cannot be static!
Today, however, we know that if Einstein had done so, he would still have been wrong, for the universe is expanding and has non-zero Λ. The resulting equation may be less elegant than Einstein’s original, but it’s the one nature actually obeys.
Quantum Effects on Gravity
The above equation isn’t the full story either. Gravity interacts with all other fields and their particles, including those we have observed in experiments, and also perhaps others we may not yet know of. These particles are definitely governed by quantum physics, and their quantum effects feed back on gravity, inevitably modifying Einstein’s equation further.
Similar modifications are well-known from electromagnetism. For instance, Maxwell’s elegant equations for electromagnetic waves, taught in first-year university physics classes, predict that light waves do not interact with each other. But the quantum effects from electrons [more precisely, from the electron field] change this: they cause light waves to have a small probability to scatter off each other (though the effect is small and hard to measure.)
For the same reasons, quantum effects from electrons change Einstein’s equation. In fact, corrections to the equation come from all particles [more precisely, from all fields]. The Higgs field, too, may modify the equation when it is switched on (i.e. has a non-zero average value.) So we already know that Einstein’s equation, even accounting for the cosmological constant, is not the full story.
Still, perhaps these quantum effects should not be viewed as fundamental. Perhaps, in a complete theory of quantum gravity, Einstein’s equation might again appear simple?
Through the Lens of String Theory
String theory is a candidate for a complete theory of quantum gravity. But Einstein’s equation in string theory isn’t so simple. [Here and throughout, by “string theory” I mean the famous string theory that’s been widely popularized; only the strings in that special theory behave as described below.]
The strings themselves, as they move through space and time, are described by a simple-looking set of equations. These can be written in various ways, including (as in this paper):
Here X represents the location of the string, while G (the metric, called g in Einstein’s equation above) and Γ represent aspects of spacetime. [τ is proper time (time as measured by an observer traveling with the string), σ is a coordinate along the string, and + and – subscipts represent σ+τ and σ-τ. The curly-d symbols measure how things are changing (i.e. they are derivatives in calculus.)]
In fact these equations generalize those of particles moving in curved space, which Einstein wrote as
Here m is the particle’s rest mass.
The stringy version of Einstein’s gravity equation, meanwhile, is obtained by requiring that the string equations written above, when treated using quantum physics, are self-consistent. The result is that strings tell spacetime how to curve… and even what equation to satisfy while curving. That’s something mere particles can’t do. (The reason: strings intrinsically include a graviton — a particle that is to gravity as photons are to electromagnetism — and so having a correct equation for gravity is a self-consistency condition, without which quantum strings would not make sense.)
The resulting equation for gravity is much more complex than Einstein’s original one. Instead of one or two terms on the left hand side of the equation, with only two parameters that have to be measured, it has an infinite number of terms on the left-hand side, each with its own parameter. In principle, string theory should tell you how to calculate all but one of those parameters. Unfortunately, that’s only true if you know which universe you are in… and string theory offers a gigantic variety of possibilities. Even if we knew that string theory is a correct description of the laws of nature, we would not immediately know all of the details of Einstein’s equation in our universe.
Where is Elegance Now?
None of this complexity matters if gravitational forces are weak, as they are in daily life and in most contexts across the universe. Then all of the extra terms can be dropped, and the string theory version of Einstein’s equation is the same as the modified one above, including the cosmological constant.
But this is relevant to the question of elegance. The simplicity of Einstein’s equation, from this perspective, isn’t intrinsic to it. It is instead merely a consequence of gravity around us being weak. Near a black hole’s core, with its apparent singularity, this wouldn’t be the case; all those terms in the string theory version of Einstein’s equation would potentially matter.
So Einstein’s original form of general relativity may look clean and simple, but in this example of a potentially realistic theory of quantum gravity, the original elegance of Einstein’s gravity equations is long gone. At best, the string theory as a whole may be viewed as elegant.
Is a theory elegant, though, if it predicts an immense number of possible universes, and yet doesn’t tell us which one we’re in? Actually, you could have asked Einstein this question; his theory of gravity predicts a tremendous number of possible stars with possible planetary systems, and is completely unable to tell us which one we live in. It has no way to know that we live around the Sun, and that we evolved on its third planet. Does this lack of specificity make a theory less elegant? I’ll leave that question for you to consider.
But the issues in this section suggest that elegance is somewhat a matter of taste and perspective, inherently subjective. That in turn raises questions as to whether elegance might be a risky guide for theoretical physicists. (Perfectly circular planetary orbits are far more elegant than approximate ellipses, but the latter are the reality.) There are many examples from scientific history that would argue against reliance on mathematical aesthetics.