Monday, June 15, 2020

Gravity is different, but not metaphysically different, from other forces

In the Quanta Magazine, Natalie Wolchover asked four physicists
Why Gravity Is Not Like the Other Forces.
Well, I think that in an impartial setup, the first question should be whether gravity is different, not why is different, but thankfully, the implicit statement is "largely true" in this case. (It's "largely true" but a complementary article "why gravity is basically the same as other forces" would still be totally desirable, too.)

Well, in our "theory of nearly everything" (TONE), there are four forces: gravity, electromagnetism, the weak nuclear force, the strong nuclear force. Gravity is largely described by Einstein's general theory of relativity – whose long-distance Newtonian limit is often enough – while the three other forces are described by the Standard Model i.e. by quantum field theory. In the Standard Model, electromagnetism and the weak force are actually mixed and rotated into two factors of a gauge group – and these two forces therefore cannot be considered quite separate even in the Standard Model: they are quasi-unified there.


That's one difference: we are normally OK with describing gravity classically while the non-gravitational forces are almost entirely quantum mechanical from the viewpoint of any modern enough description. However, this difference is largely sociological in character, not fundamental. We are often OK with Einstein's classical theory of gravity simply because gravity is so weak – by far the weakest force among the four, when considered at the level of elementary particles – that the combination of "quantum phenomena" and "gravity" is so hard to measure that we haven't really measured any of them.



Fundamentally speaking, however, gravity may be described within quantum field theory, too. Well, it must be an effective quantum field theory – because in \(D\geq 4\), gravity's quantum field theory is non-renormalizable. That may be claimed to be a big difference from the other three forces whose quantum field theory is normally renormalizable – like the Standard Model.

However, we may also adopt a more modern perspective and say that this "renormalizability" difference isn't so deep because even the non-gravitational forces should be described by an effective quantum field theory – an interpretation and refinement of the quantum field theory mathematics that automatically incorporates the assumption that the given quantum field theory breaks down at high enough energies. When we choose to describe our description of all forces as "effective field theories", renormalizability isn't really a well-defined or important condition (because renormalizability is only relevant when we want to extrapolate a quantum field theory to arbitrarily high energies, and that's exactly what we don't want to do when we "frame" a quantum field theory as an effective one).

Well, there is one "quantitative" difference between the gravitational and non-gravitational quantum field theories: the messenger particle, the graviton, has the spin equal to two (it arises from a field with two indices, \(g_{\mu\nu}\)) while the three forces have spin-one messenger particles (quanta of gauge fields \(A^a_\mu\)). This difference, \(1\neq 2\), is the root cause of many "qualitative" differences, too: the corresponding gauge symmetry is a diffeomorphism group in the gravitational case which is generated by the stress-energy tensor \(T_{\mu\nu}\) whose integral over the space is a vector (energy-momentum vector); while the Yang-Mills symmetry is the local (gauge) symmetry behind the three other forces and the integral over the current \(J_\mu\) is a scalar (spin-zero) charge.

Also, the higher spin of the graviton may be linked to the non-renormalizability of gravity. It's no coincidence that the higher-spin fields tend to be non-renormalizable or "less renormalizable". Why is it so? Because the products or "interaction terms" involving many high-spin fields have many indices (the number of indices and the spin are "almost" the same thing) and some of them want to be contracted with spacetime derivatives, thus increasing the power of the energy that makes the high-energy behavior "harder" and therefore "less renormalizable".

However, all these differences are rather technical and most laymen wildly overestimate the depth of this difference. In particular, they overlook the fact that e.g. the photon (or the W-boson, Z-boson, or gluon) and the graviton are "basically the same kinds of particles". They are quanta of the corresponding waves: electromagnetic waves are "composed of" photons and similarly the gravitational waves (e.g. those heard at LIGO) are "composed of" gravitons. The individual photons have been clearly seen experimentally (especially the high-frequency photons, e.g. gamma rays or X-rays etc.); the individual gravitons are much less strongly interacting so they haven't been seen experimentally (only their "big condensate" has been heard by LIGO/Virgo). But this difference is arguably just an artifact of our current state of experiments. Good theorists don't have any doubts that gravitons and photons work completely analogously. If you want to tell me that theorists don't have the right to feel certain in the absence of a direct experiment, I want to tell you "kiss my aß". Science would be utterly meaningless if we couldn't make a conclusion about things before we directly observe them. Yes, we can.

Another claim that is pretty much a fact by now – according to all good enough theorists – is that the non-gravitational forces cannot be added as a "last step minor decoration" to a "nearly completed" quantum theory of gravity. Gravity is actually the weakest force, as Wolchover argued in her previous nice article (according to the universal law we postulated) so the strength of gravity and other forces must be "linked to each other" in a similar way as in a unification theory that contains all these forces – even if we don't assume any unification.

Aside from the unification of the parameters describing the forces' strength, string theory indeed does like to predict a tighter unification of all the forces. The first simplest case of unification of gravity and Yang-Mills forces was already presented in 1919-1926, the Kaluza-Klein theory. In a five-dimensional spacetime, the electromagnetic gauge field \(A_\mu\) may be identified with components of the metric field \(g_{\mu 5}\). The unification of these forces in string theory may be said to be "either Kaluza-Klein theory or something whose overall effects are morally equivalent". In her new article, Wolchover tries to downplay string theory as follows:
They’ve found candidate ideas — notably string theory, which says gravity and all other phenomena arise from minuscule vibrating strings — but so far these possibilities remain conjectural and incompletely understood.
OK, a fine slogan whose seed resembles the truth in some way. However, every theory in science is "conjectural" in a similar sense. The Standard Model is also conjectural. Also, the effective field theory of quantum gravity is "incompletely understood", similarly to string theory itself. The incomplete understanding of both is manifested by the fact that sometimes, while talking about gravity in string theory, we switch to an effective quantum field theory – and vice versa! At any rate, string theory seems to imply that all these forces must be described simultaneously and they are fundamentally unified: every approach to quantum gravity that denies this "need to incorporate all forces from scratch" is wrong.

OK, so why is gravity really different? Here, we should first ask "what we actually mean by gravity in general?". Since the lessons that Albert Einstein has taught us, we mean the dynamics of the degrees of freedom that self-evidently describe "the geometry of a large enough, smooth enough, and sufficiently flat spacetime". When these adjectives are appropriate, the degrees of freedom describe "quantum gravity"; when some of the adjectives are inappropriate, the degrees of freedom describe "matter". That's how we divide the degrees of freedom to "gravity" and "matter" ("matter" is "non-gravity" here; we also have "matter" in the sense of leptons and quarks that are coupled to the "electromagnetic gauge field" that is considered non-matter in the context of gauge theories).

But again, I must emphasize that the very separation of the degrees of freedom to "gravity" and "matter" is partially a human if not social construct; or, alternatively, just a bureaucratic trick to divide our knowledge about Nature to several pedagogically natural university courses. Nature doesn't like to segregate them – and fundamentally correct theories of Nature such as string theory (or at least any hypothetical theory that respects the Weak Gravity Conjecture and similar insights) fundamentally outlaws this kind of apartheid. Matter and gravity must exist together and they can't be considered cherries on a pie of the other "race". SJWs should like this aspect of the unification of forces (and matter) and you should appreciate my moral strength because I love the unification in physics in spite of my negative emotional relationship towards the SJWs.

Let's look at the four answers. Claudia de Rham – whose surname sounds rather cohomological ;-) – said that
Gravity Breeds Singularities.
Well, right, Einstein's general theory of relativity leads to the birth of singularities (e.g. at the center of the black hole or in the past, at the beginning of the big bang expansion) while other forces don't. But this really follows from our previous definition – gravity is the dynamics of the spacetime geometry. If the spacetime geometry is dynamical, this dynamics may also lead to the singular evolution of the geometry (and the birth of singularities is unavoidable under certain conditions, as the Penrose-Hawking singularity theorems have proved). But other forces may hypothetically lead to other singularities within a fixed spacetime geometry (e.g. the Landau pole).

For this reason, I think that the statement "gravity breeds singularities" is not adding anything nontrivial to the defining statement that gravity is the dynamics of the spacetime geometry. I would also emphasize that the singularities are not the "defining trait" of black holes. Singularities inside black holes are just some "dead ends" in the observer's life; it doesn't really matter physically what "exactly happens over there" because no observer can really survive to precisely observe such outcomes and many questions are therefore operationally meaningless. Instead, the defining traits of black holes are the event horizons (they're where the black hole starts and most phenomena may still be observed very well in the vicinity of the event horizons). At any rate, we are getting to Daniel Harlow's answer which is
Gravity Leads to Black Holes.
Right. Black holes are the culmination of the shocking modern physics of gravity, examples of objects that are extreme and "qualitatively different" in many ways. At some level, it tautologically follows from "gravity is the dynamics of the spacetime geometry", too. A black hole is something that has event horizons (and the interior behind them, the region from which nothing can escape to infinity). If we consider non-gravitational forces, the spacetime geometry is kept fixed, often trivial, and so is its causal structure. Black holes have an altered causal structure of the spacetime so they clearly need gravity for them to exist. Harlow correctly mentions that some violations of locality (and holography is a "positive" way to present a major example of a violation of locality!) are necessary once black holes are agreed to exist.

Well, I would still add – and I find it important – that at the level of quantum gravity, there is really no qualitative difference between a black hole microstate and a generic heavy elementary particles. Both are just some unstable objects (towards a decay, described as the Hawking radiation in the black hole case). The shocking new picture of a black hole and its "interior where you may survive for a while" is just an emergent explanation of some new observations that are possible in the context of some very heavy elementary particles – but with the name "heavy elementary particle", you would implicitly assume that there is "nothing else to see there".

Juan Maldacena added that
Gravity Creates Something From Nothing.
Nice. This has many aspects. First, because the spacetime is dynamical in a theory of gravity, by definition, its "total volume" is dynamical, too. And it's possible to have the cosmological creation of a Universe out of nothing (Coleman-deLuccia instantons describe this birth in the approximation of quantum field theory). Again, it's obvious that the "dynamical spacetime geometry" is a necessary condition for this creation of "space itself out of nothing", too. He also mentions the new space in the wormholes that may be created "out of nothing" or out of entanglement of pre-existing, ordinary degrees of freedom inside matter, as his and Susskind's ER-EPR correspondence showed in detail.

We may also mention that a curved spacetime leads to "particle production", like the Hawking and Unruh radiation. The vacuum depends on the slicing of the spacetime and a curved spacetime makes many slicings "comparably natural". However, here I need to emphasize that the particle production may exist even when the curved background is fixed and classical and in this sense "non-dynamical" (the parameters describing the curvature are not quantum mechanical degrees of freedom) and in this sense, the very existence of the particle production (and Unruh or Hawking radiation) does not depend on quantum gravity.

Finally, Sera Cremonini responded by saying that
Gravity Can’t Be Calculated
because it's non-renormalizable. As I have mentioned, the quantum field theory calculations (diagrams) lead to infinitely many kinds of infinities. However, as she also says, these infinities are removed once the point-like gravitons are replaced with closed strings, as string theory dictates. Gravity becomes calculable in that, correct, framework. So the "incalculability" is only a "real vice" of gravity if we remain restricted to the (approximately valid) framework of quantum field theory.

To summarize: The difference between gravity and the non-gravitational forces may be both overstated and underplayed. Much of our learning of these two categories of forces is highly separate and different in character – up to some level, students need to learn two "almost different courses at their universities". But when they understand things deeply enough, all the accumulated wisdom starts to unify once again and at the end, according to the best theories and principles that we know in 2020, gravity and the other forces are fundamentally connected, inseparable, their strengths are tied together, and they probably require a totally unified theory to be explained – a theory which we incompletely (but increasingly completely) understand.

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