[HN Gopher] Gravity is a double copy of other forces
       ___________________________________________________________________
        
       Gravity is a double copy of other forces
        
       Author : cjg
       Score  : 288 points
       Date   : 2021-05-09 08:06 UTC (14 hours ago)
        
 (HTM) web link (www.quantamagazine.org)
 (TXT) w3m dump (www.quantamagazine.org)
        
       | wyager wrote:
       | > Most theorists assume that gravity actually pushes us around
       | through particles
       | 
       | Is this true, and if so, why? The interpretation of gravity as
       | something that warps spacetime very elegantly yields its
       | "gravitational force" via its effect on the action integral, and
       | is easily understood using a path integral style framework. It
       | seems like a particle-based framework would necessarily be a lot
       | more complicated, although maybe it's necessary for some reason I
       | don't know.
        
         | zachf wrote:
         | So you're right that (low-energy) gravitational physics is
         | understood because you know how to write down the action. The
         | question is whether the sum over paths in the path integral
         | should also include a sum over metrics. That sum over metrics
         | is equivalent to saying that gravitons exist (in the same way
         | that the sum over electromagnetic potentials is equivalent to
         | saying that photons exist).
         | 
         | Now if you want to include gravitational effects and do it in a
         | consistent way, you have to sum over metrics, meaning you have
         | to have gravitons. That's because trying to treat gravity like
         | it's classical but treating everything else like it's quantum
         | mechanical is inconsistent. For example, classical gravity
         | could tell you which slit an individual electron passed through
         | in a double slit experiment, if you measured the gravitational
         | field accurately enough--you could say definitively that it
         | came through one slit or the other by measuring which way the
         | gravitational field that it generated is pointing. This would
         | destroy the interference pattern and you wouldn't be able to
         | conduct the double-slit experiment at all.
        
         | wizzwizz4 wrote:
         | Everything else (that we know about) in the universe is
         | quantised. (I don't know whether experiments have shown gravity
         | to be quantised yet.) If something's quantised, that means
         | there's a smallest possible unit of it (a "particle").
        
         | ssivark wrote:
         | Classically that's true, but if we want consistency with
         | quantum mechanics, the only starting point we know is to
         | quantize the force carrying fields (gravitons, just like photos
         | and gluons), but then it turns out we can't get far from the
         | starting point.
         | 
         | Why do we want to quantize gravity? Imagine you are doing a
         | scattering experiment. To predict the full results consistently
         | (in principle), you would need to include the gravitational
         | forces between the scattering particles. This would be
         | particularly important in early universe cosmology where you
         | might have lots of heavy stuff zipping around at high speed!
         | The classical GR picture is not too helpful in these
         | situations.
        
         | evanb wrote:
         | >It seems like a particle-based framework would necessarily be
         | a lot more complicated.
         | 
         | That's true for the other forces as well! And yet after very
         | careful study we know the electromagnetic field IS quantized,
         | particles of light are photons, and to get them to do the right
         | thing (exhibit interference, for example) you need a quantum
         | mechanical framework.
         | 
         | So why not for gravity too?
        
       | Google234 wrote:
       | If this is related to string theory then the claims of any
       | usefulness should be taken with a grain of salt. I remember all
       | the ADS/CTF people claiming that they could provide huge insights
       | into condense matter which later turned out to be very
       | underwhelming.
        
         | Cancan82 wrote:
         | Sorry - I laughed at your first sentence and spit coffee on my
         | keyboard; just thought you should know!
        
       | 123pie123 wrote:
       | sorry for this daft question
       | 
       | I imagine a very very slow moving rock in space - going at 1 m/s
       | (relative to earth) in a straight line, the earths mass causes
       | spacetime to bend making the rock head towards earth, but then
       | the rock starts to accelerate
       | 
       | the bit I don't understand is why does the rock accelerate
       | towards mass? Why does the bending of spacetime make it not carry
       | on at 1 m/s towards earth
       | 
       | and since this rock has increased its speed due to accelation,
       | where has this extra energy come from?
        
         | ItsMonkk wrote:
         | I really have no business answering this, but in the spirit of
         | Cunningham's law I'm going to try to give an answer how I think
         | of it. Would like some feedback and some guidance. This is
         | purely a intuitive answer and not at all an academic one.
         | 
         | As we know from relativity and time dilation, as our GPS
         | satellites need to be resynchronized as they move fast and at a
         | lesser mass level. The center of the Earth is very heavy and is
         | therefore going through time slower than the rock or surface
         | objects. Gravity doesn't really exist. Gravity is merely the
         | side-effect of heavy-mass objects going through time slower
         | than lighter objects. This is what is meant when it is said
         | that spacetime is curved.
         | 
         | So back to your question, where does the extra energy come from
         | to accelerate the rock towards the Earth? An easy way to think
         | of it, expanding on E=mc^2, Et=mc^2, where t is time. As the
         | ball is now falling further and further into the Earth's
         | gravity well, it is now going through time slower and slower.
         | With t going slower, E must increase to balance the equation,
         | which speeds it up.
        
           | 123pie123 wrote:
           | I've always thought that at slow speeds - ie less than 1/10
           | speed of light, then time can be ignored
        
         | cercatrova wrote:
         | > sorry for this daft question
         | 
         | No need to feel dumb about asking questions, that's how we all
         | learn :)
         | 
         | With regards to your question, looks like someone answered this
         | in another comment chain:
         | https://news.ycombinator.com/item?id=27096279
        
           | 123pie123 wrote:
           | thanks, so the rock that was heading towards earth at 1 m/s
           | will technically not accelerate - with in its own reference
           | point, but the rock will accelerate to the earth (from the
           | earths reference point)
           | 
           | slight change of question, if the rock was heading directly
           | in a straight line towards earth at 1 m/s, as the rock nears
           | the earth it will accelerate (from the earths perpective)
           | where has the extra energy come from? before it was heading
           | at 1 m/s, but as it hits earth it will be travelling much
           | faster
        
       | lukeplato wrote:
       | Anybody here know if this double copy technique is related to the
       | cobordism property found in Donaldson's Theory [1]. From wiki:
       | 
       | > Donaldson was able to show that in specific circumstances (when
       | the intersection form is definite) the moduli space of ASD
       | instantons on a smooth, compact, oriented, simply-connected four-
       | manifold X gives a cobordism between a copy of the manifold
       | itself, and a disjoint union of copies of the complex projective
       | plane CP^2.
       | 
       | [1]
       | https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_equations#D...
        
       | BlueTemplar wrote:
       | Very exciting ! Reminds me of the time when I learned how
       | imaginary numbers were related to rotations...
        
         | andi999 wrote:
         | Be prepared to get excited again when reading about
         | quarternions.
        
           | nimish wrote:
           | Clifford or geometric algebra generalizes this to spinors in
           | any dimension and signature, so you also get the dirac
           | equation for free in a sense
        
             | BlueTemplar wrote:
             | Yeah, quaternions didn't seem _that_ exceptional,  "just"
             | the generalization to rotations in 3D, but Geometric
             | Algebra _did_ get me excited because I finally managed to
             | understand what was the whole deal with those weird
             | "pseudo-vectors"...
             | 
             | https://www.av8n.com/physics/clifford-intro.htm
        
         | DigiDigiorno wrote:
         | "If +1, -1, and [?]-1 had been called direct, inverse and
         | lateral units, instead of positive, negative, and imaginary (or
         | impossible) units, such an obscurity would have been out of the
         | question." - Gauss (translated), a couple hundred years ago.
         | 
         | Of all the things to rename, this may be at the top of my list.
         | I think some kids get lost in the abstraction specifically
         | because the name "imaginary" isnt helpful in understanding the
         | underlying concept.
        
           | gumby wrote:
           | > Of all the things to rename, this may be at the top of my
           | list.
           | 
           | I think more confounding is that the electron carries a
           | negative charge.
           | 
           | This mistake goes back to Franklin but we can't blame him;
           | from the data he had he had a 50/50 chance of getting it
           | right.
        
       | k__ wrote:
       | I don't know much about physics, but I always wondered if gravity
       | is like a shadow.
       | 
       | Appearently shadows can move faster than light, breaking with the
       | rest of physic knowledge, but when you look into the details,
       | they adhere the laws of physics no problem.
       | 
       | Maybe, gravity is like that, not really a physical force, but the
       | shadow of physical forces.
        
         | formerly_proven wrote:
         | "Projections" can move faster than light in general (e.g. the
         | position of an electron beam can change faster than light).
        
           | detritus wrote:
           | I just happened to watch this, which says pretty much the
           | same
           | 
           | https://youtu.be/Bq9xR5PUs6s?t=388
           | 
           | "Is Infinity Real?" Sabine Hossenfelder, about 6:25 in.
           | 
           | Perhaps this is what OP was referring to?
        
         | oefnak wrote:
         | A shadow is not a thing that moves. What you see as a shadow is
         | just the lack of light.
        
           | alexnewman wrote:
           | That is his point.
        
           | k__ wrote:
           | Yes, that's what I meant.
           | 
           | Maybe, gravity is like that in some more complex (or
           | abstract?) way.
        
             | hellbannedguy wrote:
             | I get what you are saying completely. I have thought the
             | same, but never mentioned it.
        
             | lupire wrote:
             | Static Gravity (instantaneously, when nothing is moving) is
             | the curvy shape of the Universe, created by massive
             | objects. In that sense gravity not really there, it's just
             | a description of the shape. But when the universe changes
             | shape (due to particles moving or being created or
             | destroyed), those changes happen in gravitational waves
             | that travel no faster than the speed of light.
        
         | tux3 wrote:
         | Note that gravity is very much bound by the speed of light! If
         | you made the Sun vanish, the Earth would only stop orbitting
         | around it a few minutes later.
         | 
         | https://en.m.wikipedia.org/wiki/Speed_of_gravity
        
           | Y_Y wrote:
           | I'd like to add that you can't make the Sun vanish. Aside
           | from the moral and legal objections, physics won't let you do
           | anything that would just "turn off the gravity". You'll have
           | to be content with moving it away bit by bit in some
           | continuous flow of mass-energy, again limited by the speed of
           | light.
        
             | blueprint wrote:
             | Look... we filed notice. It was on display in the bottom of
             | a locked filing cabinet stuck in a disused lavatory with a
             | sign on the door saying "Beware of the Leopard".
        
           | k__ wrote:
           | Yes, I didn't mean to imply that the link between a shadow
           | and gravity is the speed of light.
           | 
           | I wanted to say they are both emergent phenomenons... or
           | something in that direction.
        
           | amelius wrote:
           | Yes, this breaks Newton's third law that every force should
           | have an equal and opposite reaction force, and thus the sum
           | of all forces are zero at all times.
        
             | WJW wrote:
             | Newtons laws are not accurate at the quantum and
             | relativistic scales anymore though.
        
               | PeterWhittaker wrote:
               | Yes, this. Of all the things I learned in SR, the example
               | of two particles of the same charge approaching one
               | another near c and the force vectors not lining up with
               | the line between them was the most mind blowing.
               | Instantaneously, conservation laws were violated, they
               | held only over a "long enough" consideration of the
               | interaction.
               | 
               | Newtonian mechanics is a useful approximation at certain
               | scales. At other scales, they are problematic.
        
             | codesnik wrote:
             | although nothing really vanishes anywhere, no? I wonder, if
             | sun or anything else would fully annihilate, would
             | resulting gamma radiation still produce a gravitational
             | pull just because of the energy contained in it?
        
               | ben_w wrote:
               | Correct, but this is a different hypothetical.
        
             | Iolaum wrote:
             | It doesn't. Conservation of momentum, which is an
             | equivalent of Newton's third law, exists in General
             | Relativity. The results of how a system would evolve under
             | GR and the special case like the one above just happen to
             | look strange on a superficial look. If one makes the effort
             | to calculate all the constituents of momentum in such a
             | system, from GR equations, they d verify their sum is
             | conserved.
        
           | blueprint wrote:
           | What if that's just the speed of the ability of the change in
           | gravity to have an effect, rather than the gravity change's
           | actual speed of propagation
        
         | withjive wrote:
         | Sorry, "apparently" shadows do not move faster than light.
         | Since they are... well... defined by light...
         | 
         | I am curious though, as to where you got such a strong notion
         | that shadows can move faster than light...
        
           | wyager wrote:
           | He's referring to the fact that if you sweep a laser pointer
           | across the moon, the dot can easily move faster than light.
           | This is not proscribed by physics, nor are similar phenomena
           | like the phase velocity of a wave exceeding the speed of
           | light. None of these things allow you to transmit information
           | faster than light.
        
             | cb321 wrote:
             | Another fun example besides projections like laser pointers
             | or the spot of a searchlight on the clouds is the junction
             | between blades of scissors. Almost all the examples one
             | sees of this relate to synchronization defined entities not
             | being physical objects. In my experience explaining this,
             | the larger scale of the searchlight on the clouds seems to
             | help people "get it".
        
               | lupire wrote:
               | To really kill the illusion, imagine a long line of
               | people, doing "The Wave". The wave can move arbitrarily
               | fast as the delay between people decreases, and even
               | infine speed! (everyone does their part at the same time,
               | assuming they have synced clocks and aren't dependent on
               | their neighbor for the trigger, which is the key feature
               | of the faster-than-light illusion.)
        
           | lenkite wrote:
           | https://en.wikipedia.org/wiki/Faster-than-
           | light#Light_spots_...
           | 
           | http://thescienceexplorer.com/universe/4-ways-travel-
           | faster-...
           | 
           | https://www.discovery.com/science/darkness-is-faster-than-
           | th...
        
           | stared wrote:
           | "Light thinks it travels faster than anything but it is
           | wrong. No matter how fast light travels, it finds the
           | darkness has always got there first, and is waiting for it."
           | - Terry Pratchett
        
             | deadite wrote:
             | >>>/r/iam14andthisisdeep
        
           | k__ wrote:
           | https://physics.stackexchange.com/questions/259732/speed-
           | of-...
        
             | withjive wrote:
             | Yes that is a great of example of reading something on the
             | internet and believing it completely?
             | 
             | You just linked me to a Stackoverflow post about a theory,
             | which turns out to just be completely wrong, while managing
             | to include a few valid facts.
             | 
             | Was a fun exercise, but please don't get your science this
             | way.
        
               | drooby wrote:
               | Hey dude, sorry our tiny brains are struggling with the
               | science. Do you mind sharing a link to the actual
               | science?
        
               | formerly_proven wrote:
               | Of course a shadow can move faster than light. Because a
               | shadow is not actually an object moving at all.
        
               | knome wrote:
               | The moon is a little more than 0.1 light seconds in
               | diameter. Any wag that took more than a tenth of a second
               | to traverse the moon's surface would be moving slower
               | than the speed of light across it. There's no way the
               | ponderous wag displayed in the video was moving so quick.
               | At best, it was moving half the speed of light across it.
               | 
               | What would really be interesting of the things mentioned
               | in the video linked from the stackoverflow question,
               | would be how a perceiver would see the closing scissors
               | occur.
               | 
               | If you were 12 light years from the handle end of the
               | scissors, and only 2 light years from the tips, if the
               | closing motion took a year to occur, you would see the
               | point move backwards from the tip to the handles, since
               | the light would take longer to reach you to see the
               | handles close than the tips.
        
               | d0mine wrote:
               | Claiming that something is wrong without an alternative
               | theory is not how science is done too
        
               | fastball wrote:
               | Not really true. You can definitely falsify someone
               | else's hypothesis without positing one of your own.
               | 
               | This is more of how it should be done in
               | business/politics - not very helpful to say "this is the
               | wrong way to do it" but not propose a better solution. In
               | science, however, falsify things is always useful.
        
               | d0mine wrote:
               | It is a common misconception. You can't falsify anything
               | without your own theory. Some people just can't (at the
               | conscious level) recognize their assumptions (if you
               | don't see air, it doesn't mean there is none).
        
               | fastball wrote:
               | I don't think this is correct.
               | 
               | Hypotheses make predictions. Experimentation can test
               | those predictions without providing an alternative
               | explanation for the phenomena you are testing.
               | 
               | For example, if I came up with a theory of gravity that
               | implied everything should fall towards the earth at the
               | same speed, all you need to do to falsify my theory is
               | show that this is not the case. You do not need to know
               | that wind resistance is the confounding variable to know
               | that the theory which suggests everything should fall at
               | the same speed is wrong.
        
               | d0mine wrote:
               | Again, you can't make an experiment without an underlying
               | theory.
               | 
               | I understand some concepts are so ingrained, that they
               | are perceived as theories but as reality itself
               | (Newtonian physics before the end of XIX century) but
               | physics is not constrained by our intuition (what we are
               | familiar with).
        
               | fastball wrote:
               | Could you provide an example? Because I did so, and you
               | repeating exactly what you said earlier doesn't really
               | invalidate mine. As far as I can tell, my example is
               | valid and invalidates your theory that you need a theory
               | to falsify another theory.
        
           | mortehu wrote:
           | I think they're talking about the edge of a shadow moving
           | perpendicular to the light, as opposed to a shadow moving
           | away from the light.
        
         | alfiedotwtf wrote:
         | I've always wondered if gravity could be seen as like a flux,
         | and measured by bounds vs time.
         | 
         | So if any physicist here could shoot this down, I'd be most
         | happy:
         | 
         | Let's say I ran between opposite ends of a basketball court.
         | Back and forth, back and forth... what would happen if I
         | started increasing my speed, slowly all the way to the speed of
         | light?
         | 
         | So my thinking is as you become significantly faster, you
         | become heavier, but to yourself, you don't see any change - so
         | it's not me who's getting heavier as I start to run faster and
         | faster to the speed of light, but the enclosed box of me
         | between opposite ends of the court - i.e gravity is the flux of
         | a moving object bound between a distance measured over some
         | time period? i.e some sort of integral of moving matters
         | between a distance, with respect to time.
         | 
         | Sorry for my incorrect science... I was just curious how
         | relevant my thinking is
        
           | behindsight wrote:
           | > but to yourself, you don't see any change
           | 
           | This hints to me that you might be thinking of the concept of
           | invariant vs relativistic mass[0]
           | 
           | > i.e some sort of integral of moving matters between a
           | distance, with respect to time.
           | 
           | It might be easier if you intuit it as total energy
           | (encapsulates the relativisic aspects) - but that description
           | you wrote seems to be another way to rephrase the curvature
           | of space-time analogy; you could say that what you are
           | describing is retreading how "steep" the bend is, yes?
           | 
           | 0: https://en.wikipedia.org/wiki/Mass_in_special_relativity
        
             | alfiedotwtf wrote:
             | Yes? To be honest, I don't know... despite 2 years of
             | university physics, we sadly never got to the fun physics -
             | relativity and particle physics
        
           | IX-103 wrote:
           | I don't think you can think of it as becoming heavier in the
           | traditional sense, as the higher speed does not increase the
           | force of gravity.
           | 
           | There are several (roughly equivalent) ways to describe what
           | happens as you approach the speed of light.
           | 
           | One is that instead of gaining mass, you gain extra inertia,
           | making changing the velocity require more force. This makes
           | it appears to outside observers like your acceleration slows
           | down. I don't particularly like this explanation as it
           | doesn't explain why your rate of energy usage appears to go
           | down (as that is more a function of your clock). I prefer the
           | other explanations:
           | 
           | Another is that your clock slows down relative to the outside
           | world so you have less time to apply your force so you need
           | more force to apply the same externally visible acceleration.
           | 
           | Another is that the outside world looks shrunk in the
           | direction of travel, so again you have less distance to a
           | apply your force to drive the externally visible
           | acceleration. This is effectively the same as previous since
           | distance=time*speed.
        
             | alfiedotwtf wrote:
             | Thanks for your explanation! Damn physics is fun since it
             | seems intuitive and yet bizarre at the same time
        
       | tomrod wrote:
       | So, this is a really cool concept and I look forward to the
       | determination if it is true or testable.
       | 
       | I find the "double copy of other forces" verbiage to be difficult
       | to follow. Walking through a formula example would be helpful,
       | and formaluae exist to communicate exactly this kind of clumpy-
       | wumpy-lumpy-timey-wimey awkwardness language clods through in its
       | quest to communicate mathematical structures.
        
         | alanbernstein wrote:
         | This is my complaint about most of what I read about advanced
         | physics. I have enough understanding of math and basic physics
         | that it seems within reach for me to follow an equation-driven
         | explanation, but the papers are a bit too dense for me. And I
         | suppose I'm not interested enough to invest the time to truly
         | understand all of the notation.
        
           | pishpash wrote:
           | Theoretical physicists are handwavy a.f. to an extent that
           | you can't tell a crank from a real one.
        
         | dmix wrote:
         | There's a ton of articles about this on every science news site
         | and youtube channel.
         | 
         | Here's an old one for example that explains what they are doing
         | at Fermi wit the G-2 machine:
         | 
         | https://www.youtube.com/watch?v=O4Ko7NW2yQo
         | 
         | And just check the science sites.
        
           | wrycoder wrote:
           | Your link is unrelated to the topic. Do you have something
           | pertinent to gravity "double cover" work?
        
         | wrycoder wrote:
         | Charlie Wood is no Natalie Wolchover. The article is
         | depressingly simplistic - it's something I'd expect from
         | Science Alert, not Quanta.
         | 
         | I looked for a better entry into the subject and found this:
         | 
         | https://arxiv.org/abs/1810.08183
         | 
         | And a longer review article:
         | 
         | https://arxiv.org/abs/2003.12528
        
         | canadianfella wrote:
         | >clumpy-wumpy-lumpy-timey-wimey
         | 
         | What the heck is this?
        
         | ssivark wrote:
         | (Passing comment from a phone, so forgive the brevity.)
         | 
         | The key predictions one typically calculates using quantum
         | field theory are scattering amplitudes. Let's consider both
         | gravity and one of the other forces in the "weak"/perturbative
         | limit -- flat space time and force carriers as (quasi)particles
         | Eg: photons, gluons, etc. Now let's compare the algebraic
         | expressions for the scattering amplitude in both cases (final
         | answer after pages of calculations).
         | 
         | For gluons it turns out there is one piece due to how the
         | gluons are moving when they collide (kinematic piece),
         | multiplied by another piece for the charge they carry (aka
         | color); surprisingly the two pieces have a somewhat similar
         | form if you squint (called "color kinematics duality"). Also
         | gluons are spin-1.
         | 
         | Now, gravitons don't carry charge, but they're spin-2. If you
         | look at the algebraic expression for the amplitude of
         | scattering gravitons (after a much more tedious calculation) --
         | lo and behold -- it looks like it just has kinematic-like piece
         | multiplied twice! (and no color piece.)
         | 
         | This is what is commonly referred to as "double copy" (of
         | kinematic term) or "gravity = gauge (force) squared".
         | 
         | This is one of the seminal papers; it's very short, and has
         | very few equations (but they're written in a very abstract
         | form) -- feel free to stare at them if you like:
         | https://arxiv.org/abs/1004.0476
        
           | naikrovek wrote:
           | so "double copy" means "squared".
           | 
           | no wonder no one can follow any of this. they intentionally
           | obfuscate everything.
        
       | geomark wrote:
       | This probably isn't the right place for this question. But the
       | right kind of people will be reading this thread so I will ask
       | it.
       | 
       | In college physics my teacher insisted that, despite gravity
       | being popularly referred to as a force, it is _not_ a force.
       | Weight is indeed a force, but gravity is more like a field. I
       | understood it like this: Say you have a point mass in an isolated
       | system. There is certainly gravity all around that mass, but
       | there are no forces anywhere in the system. Not until another
       | mass is introduced into the system do you have any forces. It is
       | apparent from the formula for force since then you have that new
       | mass times the acceleration of gravity.
       | 
       | Or is this just being pedantic and does it even matter?
        
         | cyphar wrote:
         | Seems needlessly pedantic to me, since by that definition
         | electromagnetism isn't a force either -- it's not possible to
         | have a force on a charged object without there being another
         | object that creates the field the first object is interacting
         | with (and vice versa). In fact, Newton's third law kind of
         | implies that it's impossible to have _any_ force in isolation.
         | The same formula argument would apply to Coulomb 's law.
         | 
         | Now, is it fair to point out that you can model these forces as
         | fields, and that such a model is very useful? Yeah. But it's
         | silly to argue that something is or is not a "force" but is
         | instead a "field" -- these are mathematical constructs used in
         | the models we use for physics. I mean, it's not really accurate
         | to say that "forces" and "fields" exist _in the first place_
         | (in the sense that the mathematical models themselves are a
         | real thing that exist in the world -- they are simply models).
         | And in the end it definitely doesn 't matter, neither the maths
         | nor the real world cares whether you personally consider
         | gravity a force or not.
         | 
         | (I expect this is also the case for the strong and weak forces,
         | but I haven't done post-grad physics.)
        
           | derefr wrote:
           | EM isn't a force. EM _flux_ is a force. The pedantry is
           | important because it tells you which formulae /units are
           | compatible with one-another.
           | 
           | Weight (~gravitational force), or EM flux (~electromotive
           | force), being forces, can be directly translated into impulse
           | in a dynamic system--reduced to an instantaneous acceleration
           | upon a mass, and thus into the fundamental units of meters,
           | seconds, and grams/mols.
        
         | monster_group wrote:
         | IANAP but wanted to say this - Same argument could be made for
         | EM force. There is electric field around a charge but only when
         | you place another charge the two charges attract or repel as
         | per electromagnetic force. That way gravity and EM seem to be
         | similar. But gravity does seem to be different as it can be
         | thought of as not a force (in agreement with your physics
         | teacher). It curves space and time and objects just move along
         | the geodesics (straight lines on curved surfaces). So gravity
         | gives rise to space time geometry. Once that happens, you can
         | stop thinking about gravity as a force. Worth emphasizing again
         | - IANAP.
        
           | geomark wrote:
           | Right. Just like the force between opposite charges. That's
           | how I've always thought of it.
           | 
           | Small nit - I think it is actually called electrostatic
           | force, at least when we are talking about charges at rest.
        
             | hexane360 wrote:
             | Is there a way to describe electrostatics as objects
             | following inertial trajectories through curved space?
        
           | DangitBobby wrote:
           | See, that makes sense. I've heard it said before that gravity
           | is not a force, and my thoughts were just that it obviously
           | is a force. And of course here we are, cutting through the
           | pedantry to learn that they were in fact not communicating
           | anything actually meaningful with the phrase. You could
           | simply say that potential energy and kinetic energy are
           | different and it would be just as meaningful as what people
           | mean when they say gravity is not a force. It's misleading. A
           | single particle creates a gravitational potential that exerts
           | nothing on its own. Okay. I say this as someone with a BS in
           | Physics.
           | 
           | Thank you for your explanation.
        
         | [deleted]
        
         | _Microft wrote:
         | Gravity is what we call it when objects _interact via their
         | masses_. Other _fundamental interactions_ are electromagnetism,
         | the weak or the strong interaction (often also called  "weak
         | (nuclear) force", "strong (nuclear) force").
         | 
         | In the static case (i.e. time-independent), there is a (let's
         | e.g. focus on electric / gravitational) _potential_. It 's
         | spatial change ("gradient" / "derivative in space") is its
         | electric / gravitational _field_ which is also the ratio of the
         | _force_ on an infinitesimally small charge /mass at the given
         | distance to the charge/mass (it has to be negligibly small so
         | that it does not influence the field).
         | 
         | So there are few different concepts that usually all get mixed
         | up and lead to some confusion. It hope the overview helps a
         | bit.
         | 
         | If you want to read more, look up the italic terms in
         | Wikipedia.
        
         | agumonkey wrote:
         | I thought it was an abstraction trick. Just like they derive
         | the electric field of a charge from abstracting away the second
         | charge from the classic electric force equation.
        
         | jessermeyer wrote:
         | In a Newtonian context, gravity is a force.
         | 
         | In an Einsteinian context, forces are not even discussed.
        
           | Enginerrrd wrote:
           | They are, in the context of F=ma where m is the inertial mass
           | and an object is deviating from a spacetime geodesic, which
           | explains why we feel a force from gravity as we sit here.
        
             | lupire wrote:
             | Do you feel a force _from gravity_? I thought you feel the
             | electromagnetic force (normal force) of your chair pushing
             | you away from the Earth and stopping your free fall.
        
               | DangitBobby wrote:
               | If not for the gravitational force, the normal force
               | would have nothing to exert. So I guess you feel the
               | magnitude of the force from gravity via the normal force.
        
             | tsimionescu wrote:
             | Well, if you are currently at rest in relation to the
             | surface of the earth, that means you are moving with speed
             | c towards the future (the ct direction in Minkowsky space
             | time). The shortest path from the past to the future, near
             | a large mass like the Earth, is curved towards the center
             | of the Earth. So, because of inertia, your trajectory in
             | spacetime will be curved towards the center of the Earth.
             | However, the surface you are standing on is preventing this
             | motion via an electromagnetic interaction.
        
         | im3w1l wrote:
         | You can have a gravitational field with waves in it without any
         | charges at all. Place a single test particle in this field and
         | it be affected by the field
        
         | climech wrote:
         | Veritasium made a cool video about this some months ago:
         | https://www.youtube.com/watch?v=XRr1kaXKBsU
        
         | alberth wrote:
         | This video really helped me understand why it's not a force.
         | Also awesome physics channel as well.
         | 
         | https://youtu.be/XRr1kaXKBsU
        
         | prof-dr-ir wrote:
         | Others have answered your question rather well. Let me offer a
         | perspective.
         | 
         | I believe it was Feynman who said that theoretical physicists
         | have a very simple goal: they just wants to predict the future.
         | Now this might be impossible in full generality, but in
         | controlled circumstances (which we usually call "experiments")
         | they can often do a pretty good job of predicting the outcome.
         | 
         | As far as we know, predicting the future requires mathematics
         | which is therefore the physicists' main tool. In doing so it is
         | practical to give a name, like 'force' or 'geodesic motion' or
         | 'particle' or 'wave', to certain mathematical concepts. But in
         | my view one should not get too hung up on the precise meaning
         | of any of those words, simply because it is not productive if
         | the mathematics is already clear enough.
         | 
         | In fact, I think this is the most common misconception for
         | people with a passing interest in physics. (See also
         | discussions elsewhere on this page...) So allow me to stress
         | this: these words really mean very little without the
         | equations.
         | 
         | Of course, mine is just a physicist's perspective. I imagine
         | your question would be the bread and butter for a philosopher.
        
         | forgotpwd16 wrote:
         | Mathematically all interactions can be described as fields.
         | Where gravity differs from other interactions/forces is that it
         | lacks, according to standard model but hypothesized to exist, a
         | carrier (boson). Instead gravity is described by general
         | relativity as the result of curved spacetime.
         | 
         | Edit: This is what article is about and how _double copy_ helps
         | with this. The previous is described in the article 's second
         | section.
        
           | DangitBobby wrote:
           | How does it being a consequence of curved spacetime make it
           | not a force? When we learn the more fundamental origins
           | behind the other forces, will they also cease to be forces?
           | It's not helpful, IMO.
        
             | nimish wrote:
             | It's a pseudoforce in the same way centrifugal acceleration
             | is "fictitious"
             | 
             | It's a consequence of massive objects following geodesic
             | motion on a curved space; e.g. To keep an ant on a sphere
             | you would need "something" to ensure the ant's velocity
             | doesn't lead the ant's position "off" the sphere. That
             | something is interpreted as gravity.
             | 
             | Here's a more technical explanation:
             | https://physics.stackexchange.com/questions/212167/what-
             | is-a...
        
               | lanstin wrote:
               | And massless. Light follows a geodesic in GR. Also i
               | don't think intrinsic curvature is well explained by the
               | ant analogy. The apparent force isn't to stay in the
               | manifold but to stay on the geodesic thru space time
               | (which is free fall).
        
               | nimish wrote:
               | Yep, though massless particles must follow null geodesics
               | specifically.
               | 
               | Yeah, the ant analogy isn't the best, though it does show
               | off that movement on a curved surface needs "corrections"
               | from the movement on a flat surface (and the degree to
               | which that occurs happens to entirely characterize
               | curvature itself).
               | 
               | If you force the ant to always travel on a great circle,
               | then it might be a little more precise :)
        
               | ars wrote:
               | It's neither massive nor massless object that follow
               | geodesics, it's objects with energy (including the mc^2
               | kind).
        
               | nimish wrote:
               | Yes, and "massive" and "massless" exhaustively covers
               | both, with the special highlighting of massless particles
               | which are further constrained to follow null geodesics.
               | 
               | Massive particles don't have that limitation but they
               | only travel time-like geodesics (as far as we know).
        
               | ars wrote:
               | > with the special highlighting of massless particles
               | which are further constrained to follow null geodesics.
               | 
               | Only to an approximation. Photons generate gravitational
               | fields, and the planets they travel near are attracted to
               | the photons (not much obviously). (See: Kugelblitz)
               | 
               | Gravity does not treat photons specially, the rules are
               | the same as for other particles.
               | 
               | For extra credit imagine a Kugelblitz traveling by at
               | light speed - it's moving so fast, it doesn't have time
               | to add any mass. Yet, mass transited inside the event
               | horizon of this object.
        
               | DangitBobby wrote:
               | Interesting, thank you.
        
             | forgotpwd16 wrote:
             | A force is considered an interaction mediated by a carrier.
             | 
             | When we learn, a _force_ will be described in a way that
             | adheres to the new /modified framework.
        
               | DangitBobby wrote:
               | How long can that definition of a force have existed? 60
               | years? "Carriers" (carrying particles if I understand you
               | correctly) only makes sense in the context of the
               | standard model. It rubs me the wrong way when a field co-
               | opts layperson definitions of something and redefines
               | them to be something incompatible with the original
               | definition, but I can understand that maybe physicists
               | are not interested in helping people maintain what little
               | physical intuition can be gleaned from the world around
               | them.
               | 
               | But I digress, are gravitons not a carrier of gravity?
        
               | forgotpwd16 wrote:
               | There're no gravitons in the standard model and every
               | attempt to introduce them has failed. That's the problem.
               | Our current theoretical framework describes all
               | fundamental forces but gravity.
        
             | tsimionescu wrote:
             | Gravity is sort of special compared to the 3 forces in the
             | standard model because of its relation to inertia.
             | 
             | For all 3 SM forces, there is a distinction between the
             | charge of a particle and how easy it is to move that
             | particle.
             | 
             | For gravity no such distinction has ever been measured -
             | the "gravity charge" of an object seems to be the same
             | thing as its tendency to oppose movement (its inertial
             | mass) in any experiment ever conducted, which has led to
             | Einstein's observation that gravity seems to be a dual of
             | acceleration in a curved space time. No similar explanation
             | is required for the other forces, as they are not
             | intimately tied to inertia in the same way.
             | 
             | Now, unfortunately general relativity's curved space time
             | has never been successfully meshed with QM's Uncertainty
             | principle and/or wave function. Most physicists, at least
             | particle physicists, do tend to believe that it is in fact
             | GR that is limited in its description, since QM (in
             | particular QFT) is the most precisely confirmed
             | experimental theory ever devised, while GR is extremely
             | difficult to work with for making precise experimental
             | predictions (Einstein's equations are non-linear, and
             | almost all predictions are actually based on linear
             | approximations of the equations) - so there is much more
             | "room" for GR to be modified at very low scales without
             | contradicting the high scale results than the other way
             | around.
        
             | cygx wrote:
             | Acceleration being independent of the mass of the test
             | particle being acted upon only holds true for two types of
             | forces: Inertial forces, and the gravitational force.
             | 
             | According to General Relativity, this is no coincidence:
             | Free-fall spacetime trajectories ('worldlines') get
             | modelled as straight lines ('autoparallels'/'geodesics'),
             | indicating an absence of net forces. Apparent gravitational
             | forces then arise the same way any of the other inertial
             | forces do.
             | 
             | Such an interpretation of gravity is natural insofar that
             | it handily explains the measurements of accelerometers,
             | which only will measure nonzero values if motion deviates
             | from free-fall due to the presence of non-gravitational
             | forces.
             | 
             | But note that as is often the case, it boils down to a
             | semantic argument about the definition of 'force'. The
             | effects of 'fictitious' forces can certainly be as real as
             | the effects of 'real' forces (in case of gravity, eg tidal
             | heating, or the extreme example of spaghettification, ...);
             | personally, I have no issue with calling gravity a force.
        
           | criddell wrote:
           | > Mathematically all interactions can be described as fields.
           | 
           | Is there any other way a field can be described? What's the
           | physical reality of a field?
        
             | forgotpwd16 wrote:
             | Particles as shown in most introductory particle physics
             | courses. Because of particle/wave (alternatively
             | particle/field) duality the question of whether particles
             | or fields are the fundamental constituents remains open but
             | the current consensus is the later (fields).
        
               | criddell wrote:
               | Describing a field in terms of particles seems circular
               | if particles are excitations of a quantum field.
        
               | forgotpwd16 wrote:
               | Indeed. That's why it is a hard question to answer which
               | is fundamental.
        
         | grae_QED wrote:
         | This sums it up [1].
         | 
         | [1] https://www.youtube.com/embed/XRr1kaXKBsU
        
         | jerf wrote:
         | In _General Relativity_ , gravity is not a force.
         | 
         | Despite the fact we know that GR is broken and can't be a
         | description of reality, even educated physicists seem to
         | persist in speaking as if it is absolutely true, even though
         | they know better.
         | 
         | See also all the talk about "what happens in a black hole",
         | which should almost always be qualified with "in General
         | Relativity". When we have a unified theory a lot of the crazy
         | stuff, like rotating singularities leading to "other
         | universes", will probably disappear.
         | 
         | Is gravity going to be a force in the Unified Theory? Probably.
         | I believe it is in our current best candidates. But they aren't
         | right either yet, so I don't know.
         | 
         | Edit: Some modders seem to be confused and are probably reading
         | this as some sort of criticism of science itself or something.
         | It is not. It is literally true. It is not a force in GR, but
         | that doesn't make it "not a force", because GR is _known_ to be
         | false. It is an exceedingly good approximation to something,
         | but we know it is not the underlying truth, which is why we are
         | still seeking out a Unified Theory... precisely because GR isn
         | 't it. Thus, I find confident proclamations about whether or
         | not gravity is or is not a force to be premature. We don't
         | know. The thing it is _definitely_ not a force in is known to
         | be not the truth.
         | 
         | So to the extent you are confused about why it isn't a force...
         | well, stay tuned, because this whole area is ripe for
         | reconsideration in the next few decades. There are also
         | theories that have attempted to rewrite all of physics as "not
         | a force"s like gravity too, turning all forces into geometry.
         | While these are not currently favored, IIRC these are the
         | orginal physics theories that added rolled-up dimensions.
         | String theory built on those.
        
         | teachingassist wrote:
         | I say that 'weight is the force due to gravity' - that weight
         | is the force and gravity is the abstract concept (in college-
         | level physics terms).
         | 
         | Same for 'EM force' and 'electromagnetism'
         | 
         | Conflating the two is a type of 'metonym' that is common in
         | every-day and journalistic speech as here. If it's obvious what
         | aspect you mean, then it doesn't really matter.
        
         | cygx wrote:
         | Nope, that's not what a physicist means when they say gravity
         | isn't a force. What is meant is that apparent gravitational
         | acceleration can be understood as a consequence of Newton's
         | first instead of his second law.
         | 
         | According to Newton's first law, a body on its own (ie with no
         | net external forces acting) will continue to move uniformly in
         | a straight line.
         | 
         | Now, drag a marker uniformly and in a straight line (from your
         | perspective) across a spinning disc. It will trace out a curved
         | line, meaning an observer sitting on the disc will conclude
         | that a force must have been present as the line would have been
         | straight otherwise. We call this particular apparent force the
         | Coriolis force, one of the pseudo-forces (aka fictitious forces
         | or inertial forces) present in rotating reference frames. Being
         | accelerated by such a pseudo-force won't register on an
         | accelerometer, and the force vanishes if we analyze the motion
         | from an inertial reference frame.
         | 
         | According to General Relativity, gravity is like that, a
         | pseudo-force, except that there's generally no frame that can
         | make gravitational (pseudo-)forces vanish in an extended
         | region.
         | 
         | Taking a differential-geometric perspective, we note that
         | there's no inherent notion of "continuing on in the same
         | direction" on arbitrary manifolds. We need additional structure
         | such as a covariant derivative, which gives us a notion of
         | velocity change along a trajectory. Gravity hooks into that,
         | with bodies in free-fall moving in 'straight lines' according
         | to the Levi-Civita connection of spacetime.
        
           | dilippkumar wrote:
           | > ...Being accelerated by such a pseudo-force won't register
           | on an accelerometer...
           | 
           | Just checked the accelerometer on my phone. It reads around
           | 9.8m/s/s pointing downwards.
        
             | DarmokJalad1701 wrote:
             | If you were in free-fall, (preferably in a stable orbit and
             | not falling off a tall building) and checked your
             | accelerometer, you would read zero. Even though you are
             | still being accelerated by gravity.
        
             | m4r35n357 wrote:
             | _You_ are providing that acceleration. You need to drop it!
        
             | cygx wrote:
             | Assuming you're standing on the surface of the earth,
             | that's wrong: You're being accelerated upwards, not
             | downwards. Cf the introductory paragraph and "Physical
             | principles" section of
             | https://en.wikipedia.org/wiki/Accelerometer
        
               | dilippkumar wrote:
               | Sorry, I interpreted the - sign in the opposite
               | direction.
               | 
               | Either ways, the accelerometer is registering something.
               | If gravity was purely a 'pseudo force' as GGP said, I
               | should've expected a 0 reading.
               | 
               | Am I misunderstanding the point made about the
               | accelerometer?
        
               | roywiggins wrote:
               | The feeling you have of force is due to resting on
               | something that's holding you up. If you take that away,
               | you would be in the same gravitational field but feel
               | _no_ forces on you and your accelerometer would measure
               | 0m /s/s.
               | 
               | If you jump off a bridge, during your brief flight you
               | could look at your accelerometer and see zero. The
               | gravitational influence on you hasn't changed at all by
               | jumping off a bridge- the only change was removing the
               | upward force that stopped you from falling.
        
               | o-__-o wrote:
               | Or record your screen and drop your phone from a second
               | floor balcony to see what the accelerometer reads
        
               | cygx wrote:
               | You're at rest relative to Earth's surface. From the
               | perspective of that particular frame, the gravitational
               | (pseudo-)force gets balanced by the normal force. But the
               | gravitational force is 'actually' zero, leaving the
               | normal force accelerating you upwards to be measured by
               | your accelerometer.
        
               | zackees wrote:
               | The gravitational force is zero, but it balances out with
               | a real force which is not zero?
        
               | [deleted]
        
             | chronial wrote:
             | You misread. Your phone is accelerating upwards at
             | 9.8m/s^2. That is also what it displays. If you have an
             | accelerometer that displays a history (I recommend the app
             | phyphox), you will also see that acceleration while your
             | phone is falling.
        
               | wrycoder wrote:
               | The phone will measure zero acceleration while it's
               | falling.
        
               | zackees wrote:
               | If we replace gravity with magnetism can we make the same
               | conclusion? How about the forces experience from title
               | forces. The difference in gravitational pull on the moon
               | (tidal forces) caused the moon to be locked in orbit with
               | the earth. If gravity isn't a force, what stopped the
               | moon from rotating faster than its orbit?
        
               | whoopdedo wrote:
               | Any skydivers wish to confirm this? But the wind would
               | push on the phone. So any lunar skydivers wish to
               | confirm?
        
               | DarmokJalad1701 wrote:
               | A better analogy is something that is in a stable orbit.
        
               | chronial wrote:
               | Apparently I forgot to type the "go away" that was
               | supposed to go between the "acceleration" and "while" in
               | that sentence ^^.
        
             | jon_richards wrote:
             | That's because you aren't in free fall.
        
               | ctdonath wrote:
               | If I step off a bridge, why does the distance between me
               | and the very large nearby spherical mass begin decreasing
               | at an, umm, accelerating rate?
        
               | gumby wrote:
               | Compared to the radius of the earth the Bridge is on the
               | surface of the earth. In other words you don't travel far
               | enough / long enough to observe any difference in g.
        
               | babesh wrote:
               | The surface is accelerating up at you relative to the
               | gravitational field.
        
               | mrow84 wrote:
               | What happens if two people jump off bridges on opposite
               | sides of the earth? How does that work in that
               | interpretation?
        
         | vincent-toups wrote:
         | Whether it is pedantic depends on your goals and perspective,
         | but if you are interested in a history and philosophical
         | development of Force that culminates in the observation that
         | gravitational forces are fictitious, I can heartily recommend
         | "Concepts of Force" by Max Jammer.
        
         | TchoBeer wrote:
         | Isn't the same true of electromagnetism?
        
       | throwaway316943 wrote:
       | Someone needs to do a better visual explanation of these particle
       | interactions. If you can describe it mathematically then surely
       | you can create a computer simulation.
        
         | 317070 wrote:
         | People can and do make computer simulations of particle
         | interactions? It's just that 12 dimensional PDE's are not
         | exactly fast, but we can e.g. use them to compute the dipole
         | moment of fundamental particles like muons.
         | 
         | Think of it like knowing the rules, but the rules not being
         | efficient to compute with.
        
           | throwaway316943 wrote:
           | 3D rendering was unbelievably slow for a long time too, the
           | good thing about computers is they have storage so you can
           | render 1 frame an hour if you want and just record it and
           | play it back later.
        
         | hirako2000 wrote:
         | Exactly my thought while reading the prose. A picture is worth
         | a thousand words, a formula a thousand pictures. Yet having a
         | bit of the three kinds is often needed to easily comprehend the
         | concept for those who are neither particularly visual, logical,
         | or patient.
        
         | bfvjkfmrmrm wrote:
         | Quantum simulations ran on classical computers have exponential
         | complexity class.
         | 
         | So it's only doable for tiny N.
        
           | throwaway316943 wrote:
           | Even a visual demonstration with a few particles would help.
           | I'm sure the statistical calculations could be visually
           | represented as well.
        
         | roywiggins wrote:
         | Well, there's Feynman diagrams. The problem is that there's a
         | combinatorial explosion that severely limits certain
         | simulations. Apparently this double-copy procedure can be used
         | to cut down on the combinatorics, which is why physicists are
         | excited about it.
        
         | stared wrote:
         | Visualizing particles in an easy way is HARD.
         | 
         | To the point, I've founded a startup precisely for that:
         | https://quantumflytrap.com/
        
           | gambitown wrote:
           | What you have linked is just bog-standard entanglement
           | simulations. There are a dozen free software apps to do that.
           | It's basically just linear matrix algebra with complex
           | numbers.
           | 
           | What the parent meant was _real_ particle simulation, for
           | example simulating the collision between two electrons or
           | computing the dipole moment of a muon, from fundamental
           | standard model constants. It requires numerically solving 12
           | dimensional PDE 's. That's much more complicated than
           | entanglement simulations.
        
             | stared wrote:
             | > There are a dozen free software apps to do that.
             | 
             | Put your links when your mouth is. :) There is a handful of
             | interactive simulations for qubits. I don't know any other
             | interactive simulation for quantum many-particle systems.
             | 
             | > It's basically just linear matrix algebra with complex
             | numbers.
             | 
             | It always has been. For Quantum Field Theory it gets much
             | more complicated, and classical simulations are way to slow
             | to simulate most systems. Quantum computers (or even
             | quantum simulators) are likely to change the game. Anyway,
             | being able to simulate does not result in being able to
             | visualize in a meaningful way.
        
               | gambitown wrote:
               | > Put your links when your mouth is
               | 
               | List of available QC simulators grouped by programming
               | language: https://quantiki.org/wiki/list-qc-simulators
               | 
               | I count more than 40 QC simulator projects. All open
               | source. There is an abundance of quantum computer (QC)
               | simulators, as opposed to a paucity of quantum field
               | theory (QFT) simulators (most of them written in Fortran,
               | yuck).
        
               | stared wrote:
               | QC simulation (qubits only, no dedicated vis) is much
               | easier. You just make a vector, and multiply it by
               | matrices. Here for Python interface I like QuTiP even
               | more than Qiskit (though, I might be biased, as I
               | contributed to the former). For interactive, IBM Quantum
               | Computing Experience and Quirk
               | (https://algassert.com/quirk).
               | 
               | As "easier" I mean that 2^10 is still managable, as you
               | can use dense vectors and operators. For, say, this
               | quantum optics simulation you get roughly 1000 dimensions
               | per particle. For 3 particles it is 10^9. Don't even
               | think about dense vectors, let alot - dense operators.
               | Sparse operators are not enough, as even identity is big;
               | so for any real-time-ish simulation there are
               | considerably more tricks and methods.
               | 
               | Simulting QFT you need to deal with much more complexity
               | (including continuous dimensions). Instead of numerically
               | pretty much you need to solve some intergrals. (Side
               | note: I wanted at some point to write an interactive
               | editor of Feynman diagrams, turning it into formulae and
               | integrating.)
        
         | ddbb33 wrote:
         | "someone"
        
           | hungryforcodes wrote:
           | "somebody"
        
         | forgotpwd16 wrote:
         | Particle interactions are visualized with Feynman diagrams
         | which can be used for mathematical calculations (ref:
         | arXiv:1602.04182). Do you mean something better than them?
        
       | floatingatoll wrote:
       | I'd like to learn more about this symmetry. Does anyone recommend
       | a good article (or paper) about this?
       | 
       | > _Researchers note that electromagnetism, the weak force and the
       | strong force each follow directly from a specific kind of
       | symmetry -- a change that doesn't change anything overall (the
       | way rotating a square by 90 degrees gives us back the same
       | square)._
        
         | srl wrote:
         | I'm _pretty_ sure that this is referring to gauge symmetry, so
         | that's the term to search for.
         | 
         | I dunno of a good article as an introduction. As a grad
         | student, I found Terry Tao's explanation [1] rather helpful,
         | but of course it has a strongly mathematical flavor.
         | 
         | [1] https://terrytao.wordpress.com/2008/09/27/what-is-a-gauge/
        
           | floatingatoll wrote:
           | Thanks! It sounds like the next step is to look for curvature
           | transformations associated with terms used for each of those
           | three. (If someone knows more, always open to hear that too.)
           | 
           | Also, a textbook is revealed on what seems to be this exact
           | topic. Does this seem right?
           | https://physicstoday.scitation.org/doi/pdf/10.1063/PT.3.2421
        
       ___________________________________________________________________
       (page generated 2021-05-09 23:01 UTC)