- 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

**, constructed from the energy and momentum carried around by material objects. The other things that appear in the equation are**

*T*(another measure of the shape of space, called the “metric”)*g*(the cosmic speed limit, a.k.a. the speed of light)*c*(Newton’s constant of gravity, the same one that appears in Newton’s law of gravity)*G*_{N}

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

**is now nailed down to within a few percent.**

**Λ***(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

**must be zero, based on aesthetics, he might indeed have predicted that the universe cannot be static!**

**Λ**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.

Approximately.

### 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

**(the metric, called**

*G***in Einstein’s equation above) and**

*g***represent aspects of spacetime. [**

*Γ***is proper time (time as measured by an observer traveling with the string),**

**τ****is a coordinate along the string, and**

*σ***+**and

**–**subscipts represent

*σ*+*and*

**τ***σ*

*. The curly-*

**-τ***symbols measure how things are changing*

**d***(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.

## 37 Responses

Can dark energy vary across space as well as time? Wouldn’t that confuse astronomical measurements in interesting ways?

It certainly could! Dark energy may be a cosmological constant — literally constant — or it may reflect the vacuum energy of a field that might not be constant across the universe. However, variations of a significant size would leave many imprints on astronomical measurements. Observations show no evidence for this variation up to now… though I am having trouble finding a paper that says this explicitly.

There are a bunch of theories with cosmologically varying dark energy – one I like, and one that fits the existing data, is early dark energy. EDE remains viable because it “refers to a new form of dark energy active at early times (typically a scalar-field), that quickly dilutes away at a redshift close to matter-radiation equality.” (i.e. by the time of the CMB.) But, even with that early disappearance, it can be tested in relatively near-term experiments.

EDE variations could be in space as well as time – see, for example, the Acoustic Early Dark Energy in this paper, where you would expect that sort of spatial variation.

https://arxiv.org/abs/2302.09032

By taking GR as a curved space-time structure completely seriously, we conclude that the structure of spacetime has been formed by the accumulation of matter from gas to densifications. From the point of view of matter, such an outside-to-centers definition gives freedom and allows us to consider only realized material quanta as a contribution to the signals of gravitational changes without worrying about superpositions.

By the way, in my opinion, it doesn’t really matter where we put the Λ term ( left or right).

I don’t think that the elegance of the EFE is “affected”…

Actually, in GR the Λ defines which is the “default” spacetime, so to speak…

With Λ = 0 and total mass/ energy etc zero also, the default spacetime is Minkowski.

With Λ > 0 is DeSitter and with Λ < 0 AdS.

The point, from this perspective, is that Einstein’s original equation had no new parameters, but the modern equation has another parameter… even before quantum physics is accounted for. You can view that parameter as a parameter of the universe instead of a parameter of its equation, but that’s just math gymnastics. String theory is declared to be beautiful by some because it has only one parameter (the Planck scale, in the M theory version) but the universe that emerges from it has a huge number of parameters, rendering the string theory equations almost useless for making detailed predictions — except for experiments that currently are wildly impractical.

Some notes and comments about this interesting post:

The Stress Energy tensor Tμν, besides energy density Tοο, and the non diagonal terms ( momentum etc) also has the other three diagonal terms Tαα for pressure.

Interpreting the Cosmological Constant term as energy / pressure of the vacuum, we can move the Λgμν term on the right side of the Einstein equation ( as another commenter pointed out) with energy density equals minus the pressure density: ρ = -p ( in geometrized units).

So the term ρ + 3p = -2p, thus we have a simplified intuitive explanation ( there are three spatial dimensions vs one temporal, so the pressure terms dominate!) for the accelerated expansion of the Universe.

One crucial characteristic about the Einstein equation is its non linearity.

So , not only the matter fields ( Tμν) or the Λ term, but also the gravitational field itself is a source of gravitation! This property is very important in the case of strong gravitational fields, as in Black Holes, e.g.

We know from cosmology that our universe with its flat space and low energy slow roll inflation is a lot more like Newton’s than Einstein’s.

As long as we stay out of black holes* there doesn’t seem to be any apparent singularities and Einstein’s theory is an elegant classic theory approximation encapsulating tidal effects with non-linear first order derivatives. The corresponding effective quantum gravity field theory has derivatives of all degrees (if I understand loop diagrams correctly) and needs observations to estimate parameters at various scales.

And its quantum field has a physical reality since they have now measured its Aharanov-Bohm phase shifts from its potential – simplest part of dark energy if performed in vacuum [“Observation of a gravitational Aharonov-Bohm effect”, CHRIS OVERSTREET et al., Science. 13 Jan 2022 DOI: 10.1126/science.abl7152].

So I wouldn’t worry about that dark energy is predicted by Weinberg’s anthropic theory. Why else would the sum of potential energies of the vacuum be low enough but as high as possible in a livable universe!? That too seems to lend some elegance to the quantum theory of gravity, along Wyrd Smythe’s elegant definition of elegance, to use what is necessary but no more.

* Who may have a physics far from Planck scales as far as we know.

Maybe I should note “was predicted” since people seem confused over the publication record (Weinberg in the 80s, dark energy observations starting in the 90s).

One should be cautious about Weinberg’s prediction. He made assumptions, and one can change the details of the prediction by changing the details of the assumptions.

Our Universe seems to be spatially flat in large scales ( similarly, the surface of a table may look flat, but if you look it through a microscope, it is full of “rifts” and ” mountains”).

Moreover, our Universe being spatially flat , does not mean that its *spacetime* geometry is flat!

Actually it isn’t!

Have you written on the positronium atom? Is there any possibility the positronium is a piece of the puzzle of quantum gravity right up to GR as well?

OK! I agree the original GR eqn. is more elegant than the ST eqn. My question is : now that data are coming in from blackholes by gravitational waves, can we say that original GR eqn. is also correct when the gravity is strong. Or it is too difficult to solve for strong gravity, so you cannot say anything.

The original Einstein equation does a great job on strong gravity (see, as one of many examples, https://arxiv.org/abs/1010.5260) as long as (a) distance scales and time scales are small enough that the cosmological constant is relatively unimportant, and (b) the questions being asked are such that quantum physics doesn’t matter.

Elegance is indeed subjective, but I’ve long thought there are some objective qualities. I tend to equate elegance, in science, technology, or art, with “doing a lot with very little”. Something along the lines of the action principle, elegance involves taking the least-expensive path.

I like that definition. It’s elegant.

The hard part is that you don’t always know, in science, how much you can actually do with the very little you put in. Often experiment shows that it’s a lot less than you hoped.

Yeah, that happens in other fields, too. Sometimes it takes a while to find that sweet spot.

Whether it is elegant or not, whether it is the final “truth “ or not, Einstein’s theory works at the level of energies we are likely to function in. String theory is an unprovable pipe dream.

I’m not arguing otherwise, at least for the present day… though “unprovable pipe dream” is a bit strong — there were plenty of people who thought that general relativity was in that very same category, and dismissed the Mercury result as an accident and the measurement of the bending of light’s paths by the sun as unconvincing. Even among those who favored the theory, black holes and gravitational waves were thought to be completely unobservable. Things can change over decades, and perhaps this will even happen for string theory, hard as it is to imagine now.

But in any case, the question of elegance is an aesthetic one. It is quite separate from the question of practicality, which I was not addressing here.

If we are truly “in the dark”, is elegant relevant?

You notice that I remained completely agnostic about this question in my writing of this article.

Part of the agenda here is that what seems elegant to most people in 2023 may not seem elegant to those alive in 2123, who will be blessed with much greater experimental knowledge and probably broader mathematical knowledge as well. The fact that perspectives change over time is one of the great pitfalls of using elegance, or any other aesthetic viewpoint, as a criterion for choosing a theory. If you’d shown Newton the Standard Model, with all of its unexplained parameters, he would probably have expressed dismay.

In a similar way, what seems “theoretically motivated” in 2023 may not seem so in 2123 (or even in 2025). Novel ideas in particle physics are sometimes roundly criticized for being “unmotivated”; but if the work carries enough insight, the future often brings theoretical motivations, even if the idea itself turns out to be one that nature doesn’t exploit.

Thank you, but I was only referring to the overwhelming theoretical abundance of dark matter and dark energy in relation to The General Theory of Relativity within the scope of the observability of elegance. Sorry.

Okay then 🙂 but then I would say that elegance remains relevant to those who think that it is a guide to the right equations. For them, what we don’t know experimentally about dark matter and dark energy may just be a distraction. For instance, the logical contradictions between quantum physics and general relativity that emerge in the context of black holes and information may be much more important to the progress of theoretical physics than the details of what dark matter and dark energy might be. [I’m not endorsing this view, just explaining it.]

There has been a great effort for several decades for finding a consistent answer to this infamous Black Hole information problem.

Almost all these attempts take for granted that Unitarity is preserved, so they are somehow biased and there’s not any realistic way- observationally or experimentally ( analogue gravity experiments are not considered conclusive or even relevant anyway)- to find out if this the right direction .

The guidance is only from theoretical “anchors”.

And almost all these attempts for a solution in this direction ( e.g. that there are subtle corrections in the Hawking radiation) come with a cost : “Strong” non locality, as we need transformation of information in ( vastly) spacelike separated regions of spacetime, at the macroscopic level…

So, the resolution of one big problem opens Pandora’s box for another ( “strong” non-locality is essentially loss of Causality ).

I’m not sure which of the two is more problematic…

Agreed. (I wrote this post about the black hole paradox after Hawking made an argument, which went all across the media, that black holes don’t really exist.)

“These particles are definitely governed by quantum physics, and their quantum effects feed back on gravity, inevitably modifying Einstein’s equation further.”

By this, do you mean that the form of Einstein’s equations remain as they are, but the input information changes such as energy/momentum, space-time curvature etc?

No; the equations of gravity, measured at low energies E_0, are changed by the quantum physics of particles with mass m whose energy mc^2 is greater than E_0. In electromagnetism, this is also how scattering of light comes about: Maxwell’s equations for light at low frequencies are changed by the quantum physics of the electron and other such particles. The form of the equation actually changes… (for example, see equations 4.1 – 4.11 in https://repositorio.uniandes.edu.co/server/api/core/bitstreams/f705c951-262d-4a53-9441-2a06612a80db/content , which show (in 4.10 and 4.11) the first correction from the electron field on Maxwell’s equations.)

I think stringy equations are more elegant because they don’t have an explicit mass term…

The equations are elegant; but is the theory elegant? There are no parameters in the equation, as there would be in the equations for the Standard Model; the Standard Model makes no predictions until the 20 parameters in its equations are measured. But there are enormous numbers of parameters in the *solutions* to the string theory equations, and so, again, the equations make no predictions until those parameters are measured.

Prof. Strassler, you sound like you have dove deep into both the Standaerd Model and String Theory, so you would be one of the best physicist to ask this highly speculative but, I believe valid question.

Is there any evidence and/or do you think that as we get closer and closer to the truth, i.e. Grand Unified Theory (GUT), we should expect the number of parameters must reduce. Much like the fundamental particles, from composite down to the bosons and fermions.

I don’t know if this line of thinking, the reduction of parameters, will lead to a holographic and/or fractal universe but it seems to me, solution, the truth should be the simplest mechanism possible to satisfy the conversation laws.

This is why I keep asking the question, is “space” (“the vacuum”), all that is required. Of course this implies, space, fields, and energy, are all synonymous.

PS: When I referred to “space”, I don’t mean a coordination system like the cartesian and/or polar, but a vacuum filled with stuff, fields/energy. [ ΔEΔt ≥ ћ ]

I guess the intuition one might have had that the true theories were typically the elegant ones might be kind of like the early impression that the typical exoplanets were hot Jupiters. Elegant theories the fit the data pretty well are easier to find than the ugly ones.

Indeed, it’s very common for equations to look a lot simpler when your knowledge is still fragmentary. Circular orbits aren’t bad when your measurements are inaccurate… but better measurements make them look bad. Same even with the harmonic oscillator; real springs aren’t exactly harmonic.

I’m sure a lot of equations which look complicated using a Euclidean space could be made to look simpler by redefining part of the space to be non-Euclidean. In isolation.

Why don’t you just include the cosmological constant on the rhs of Einstein’s equation and include it as part of the energy tensor, especially as it seems to be widely called ‘dark energy’? Thus reverting to the ‘esthetic’ form.

Indeed, this is a fair objection… and in a sense, the question of whether the cosmological constant (or dark energy more generally) is an aspect of spacetime, an aspect of the energy-momentum tensor of some kind of matter, both, or neither, remains open. However, even in a pure gravity universe (where the only field in the cosmos is the gravitational field, and so T_{mu nu} = 0), Lambda must still be included and measured. This is the reason why Lambda should be on the left-hand side, even if there is also something Lambda-like on the right-hand side.

Either way, this would obviously not resolve the later issues in the post — those raised by the quantum physics of other fields and by the quantum physics of gravity itself.

Einstein’s field equations are a geometric interpretation of gravity which is a quantum phenomenon. The left side of the equations is a form of a laplacian which averages the paths a test particle would take in the presence of a localized mass-energy density.

A quantum theory of gravity should describe this phenomenon with relatively fewer terms than string theory

We don’t actually know that there are any quantum theories of gravity that are both simpler than string theory and mathematically consistent. String theory may be rather typical.

We also must remember that a quantum theory always makes its equations complicated through the renormalization group; the equations always have an infinite number of terms at any finite energy scale. It’s far from obvious there will be any meaning to “fewer terms” in quantum gravity.