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.
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
no wonder no one can follow any of this. they intentionally obfuscate everything.
Nobody is trying to intentionally obfuscate anything; the authors working in this area deserve tremendous credit for their sincere and sustained (decade+) effort in trying to elucidate the structure of gravitational interactions. Please don’t go around slandering them just because you don’t understand what’s going on.
Essentially, we've incentivised intellectual wankery.
Contrast this to any field of engineering: intelligibility is selected for by market forces. Obscure, hard to follow stuff is slowly (but surely) weeded out in favour of the clear and simple.
In the more theoretical sciences, that's just not a goal. Quite the contrary: The goal is to make the published papers look impressive. More jargon, more complex equations, fewer diagrams, all help achieve this goal.
I looked for a better entry into the subject and found this:
And a longer review article:
Saying God created the world or the universe is a simulation is as dissatisfying a response as saying humans were made on Earth. Sure why not but what made it, did it start, will it end, how does it work at the most fundamental level? How does God's processing of information actually works ? That s what we must understand, up to the end.
Here's an old one for example that explains what they are doing at Fermi wit the G-2 machine:
And just check the science sites.
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?
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.
force = vector under specific coordinate system with physical source
isn't a force = vector + properties in specific coordinate system produce some kind of force-like phenomenon
Just checked the accelerometer on my phone. It reads around 9.8m/s/s pointing downwards.
An accelerometer works by comparing the difference between some mass that is acted upon by an external force and a related mass that is loosely connected to it (e.g. by a spring-like linkage. whose strain we can measure) as not to be acted upon by that force directly.
If you move the casing of an accelerometer, the sprung mass inside it lags behind, and thereby the acceleration can be measured via strain gauges or whatever.
Gravity defies accelerometers because, unsurprisingly, gravity acts on all of the the masses contained in an object equally.
(At least any gravitational field you are likely to encounter in ordinary experience is going be almost perfectly uniform across an everyday object.)
It is when you are in free-fall, that is when you're accelerating due to gravity. And that's exactly when the accelerometer doesn't tell you anything, because every part of the object is in the same free-fall; all of you is accelerating. The accelerometer is differential. It doesn't see a difference, so it doesn't see acceleration.
You now that free-fall is acceleration because orbiting satellites are in free fall and they follow circular paths. Objects will not move in a circle unless a force accelerates them toward the center. Like when you spin a mass on the end of as string, it goes in a circle due to the centripetal force exerted by the tension carried in the string.
When you're standing on the planet, your accelerometer is lying to you. Gravity is acting on the sprung mass inside the accelerometer, and it's also acting on the case which holds that mass. But the case is constrained from moving, whereas the sprung mass isn't. So a differential strain develops, which "looks and feels" like acceleration.
In summary, an accelerometer is an instrument which measures the acceleration due to a force which is applied just to its casing, and not to some critical piece of mass held inside. Gravity causes equal acceleration of the casing and that piece of mass, and so gravitational acceleration is immeasurable by that accelerometer.
Acceleration can be measured with regard to some frame of reference via distance and time calculations.
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?
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.
Why is the soil not in free-fall? Because it feels normal force of all the earth below.
And why is not the whole earth in free fall towards its core? Because it has an electrostatic force (a real one!) which keeps all the atoms and molecules from collapsing on itself. If it would collapse on itself, it would first become a neutron star and then a black hole.
To be clear, my confusion is about how this interpretation is consistent with the facts, not the facts themselves (I think).
So the field is going radially inwards towards the center of the massive object.
"Gravity" is the curvature of spacetime. An inertial "straight line" follows that curvature, like going "straight" on the Earth's surface draws a spatial curve. Remove what's supporting me (dirt, bridge) and I'll continue on my inertial straight line, which actually orbits and settles down to the center. Even "at rest" at the center/bottom of the gravity well is not zero inertia per se, it is a continuing inertial line orbiting with still-constant velocity at zero radius. Ergo gravity isn't a force; gravitational attraction is just masses' inertial lines being distorted by the infinite-reaching distortions of other masses. There is no "gravitational force" to unify with other forces. OMG.
I've been trying to get my head wrapped around gravity for years. You just tied the knot.
Next questions: why does mass distort spacetime? As E=mc^2, does/can energy distort spacetime? Mass & energy being just different units of the same thing, what's the nature of that thing without thinking in terms of divided mass/energy colloquial thought? And then questions on to weak & strong nuclear forces, and reconsideration of electromagnetic force.
Do you know of any diagrams that might illustrate this idea to a layperson?
The electrostatic forces and bonds between atoms are exactly what prevents these atoms from following inertial paths. The atoms act on each other, and collectively produce a force radially outwards from the center of Earth supporting its mass.
Earth isn't pulling itself apart because nothing is pulling in the first place. All the mass is pushing against itself, refusing to be packed tighter, despite the flow of spacetime around it drawing it together over time.
When we're standing on Earth, our particles are following the same flows of spacetime toward the core of Earth. From our inertial frame, the ground below us is moving toward us, because as noted Earth resists being compacted further. So we meet the surface, we both resist being compacted by the other, and we experience a stable force.
That said, your description does kind of explain it for me, unless I am misunderstanding, though at the moment I only get it as a "negative" argument: we are being accelerated upwards/outwards with respect to the field's reference frame, because otherwise we would be moving along our natural inertial path.
It is still a bit puzzling that if the ground disappears we appear to accelerate, whereas my current reading of what you wrote seems to imply that we would immediately return to our "inertial path".
What I'm really wondering is if all of this "acceleration relative to the field" business is a misleading abstraction, and "relative reference frames" fix it (and indeed that was my very primitive understanding of relativity). However, on balance it seems much more likely that I'm simply misunderstanding things.
Fair, and I'm not sure I follow that, either. I'm not sure how an entire field can really have a reference frame in the first place.
> though at the moment I only get it as a "negative" argument: we are being accelerated upwards/outwards with respect to the field's reference frame, because otherwise we would be moving along our natural inertial path.
That pretty succinctly captures my understanding.
> It is still a bit puzzling that if the ground disappears we appear to accelerate, whereas my current reading of what you wrote seems to imply that we would immediately return to our "inertial path".
Yes, that's the sticky bit! I think "appear" is load-bearing here, because what are you measuring relative to? The surface of the Earth is not an inertial reference frame (according to general relativity, it's accelerating!).
It might be easier to think about if we use centrifugal force -- the fictitious force you feel when being swung in a circle from some central point. From an inertial reference frame, you have some velocity in the tangential direction, and a rope (say) is pulling on you in the radial direction. This force accumulates with the existing velocity over time to produce a circular path.
From your perspective, however, you feel a force pushing you outwards, which is resisted by the rope you're attached to. You would feel that, if the rope was cut, you'd be "pulled" out into space (radially away). But this is because you're not aware of -- you don't sense -- the tangential velocity you possess, because it's canceled by the choice of a frame that's moving with you.
The change to an inertial frame introduces an extra velocity term, and this in turn allows us to replace the unattributable outward pull of the centrifugal force with a force merely attributed to the tension of the rope you're attached to. (In fact, this might be a good intuition for what an "inertial frame" even is: it's a frame in which all forces are equal and opposite to other forces. Gravity and the centrifugal force are unbalanced in this sense.)
The gravitational field (curvature of spacetime) plays the same role that the hidden velocity does. In an Earth surface frame, the feeling of being pulled "down" is attributed vaguely to the fact that there's a bunch of mass "down there". In an inertial frame, the feeling of being pulled "down" is attributed to the ground pushing "up" at you.
In both scenarios, we can find a frame involving an extra term that lets us redescribe the phenomenon we observe in terms of forces attributable to physical entities.
The "Fictitious forces" Wikipedia page has some pretty good schematic animations for the centrifugal force, FWIW.
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.
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.)
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.
Also the other issue is that the word "force" in physics doesn't just refer to F=ma, it also refers to the general concept of an interaction mediator. That's why they're called the four fundamental forces -- meaning that saying "gravity is not a force, it's a field" is not only overly pedantic but is also arguably wrong if you take it literally.
Edit: This is what article is about and how double copy helps with this. The previous is described in the article's second section.
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.
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...
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 :)
Massive particles don't have that limitation but they only travel time-like geodesics (as far as we know).
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.
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.
When we learn, a force will be described in a way that adheres to the new/modified framework.
But I digress, are gravitons not a carrier of gravity?
Is there any other way a field can be described? What's the physical reality of a field?
Thank you for your explanation.
Small nit - I think it is actually called electrostatic force, at least when we are talking about charges at rest.
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.
The right kind of object in the right kind of force field experiences a force.
A mass experiences a force in a gravitational field; a charge experiences a force in an electric field.
A point mass is difficult to reason about because a point mass has infinite density, and a gravitational field that gets arbitrarily large the closer that the point is approached.
Non-point masses, like uniform spheres, have a gravitational field that increases up to their surface, and then decreases. A particle at the exact centre of a uniformly dense field experiences no net force.
Also, a particle floating inside a hollow sphere of uniform thickness experiences no net force; the field sums to zero everywhere inside. This is why the field gets weaker toward the centre of a solid sphere. Every point inside a solid sphere can be regarded as simultaneously being inside a hollow sphere (experiencing no net force from that), and being just on the surface of a smaller interior sphere: that part of the sphere which is deeper than the particle.
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.
Despite its central presence in physics 101 the notion of force is actually not all that clearly present in fundamental physics.
Newton's innovation might be said to be the representation of gravity as a force, but even in the Principia he doesn't commit to the idea because of an intuitive rejection of the non-locality implied by such a conception. So the idea that gravity is a force of the same sort as you pushing on a cart full of apples (whatever _that_ is) was never a particularly strongly held idea. I'm not an expert on the history of physics (I have read the Pricipia in translation, though) and I suspect Newton, if questioned appropriately, might have himself admitted there are issues with the conception that go beyond just non-locality.
Even at the level of Lagranian physics in the contemporary style (which despite that description is hundreds of years old) the idea of forces has an ad-hoc quality to it. The description of a completely closed system doesn't really contain forces unless you explicitly include them in the form of lagrange multipliers for the constraints, and then the forces that you calculate are, in a sense, provided "externally" by processes outside of the system itself.
For instance, the easiest way to represent the motion of two particles held together by a perfectly rigid rod doesn't mention the force exerted by the rod to maintain that constraint and yet simulates the motion perfectly. If we are interested in how much force the rod must exert to maintain the constraint, we can add in a largrange multiplier, but what that quantity tells us is in a sense a summary of the non-represented physics of the rod itself, which is still excluded from the system. A force then is a kind of interface between that which you are considering in detail and that which is assumed to be part of the external world. If everything in the world were simulated, then nothing would deviate from its expected motion, and thus there would be no forces at all, since all a force is is a description of how something deviates from the path it would ordinarily take if not for the intervention.
On this view Newton's first law is much more important and fundamental than his second: objects always move in straight lines but we must take great care in understanding what a straight line actually is for a given physical system. Really, the seed of general relativity and the geometric formulation of field theories is right there in the first law. This explains why the second law is of so much less importance in modern physics.
I'm not a professional, just an interested failed physicist, so standard equivocations apply to the above.
This video by an actual professional hits on some of these ideas:
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.
In an Einsteinian context, forces are not even discussed.
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.
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.
> 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).
Could you explain this? I'm not making the connection where "you should sum over this potential" implies "there exists a corresponding particle for the potential". If that implication is true, that feels like a pretty important generalizable principle that someone should have told me in undergrad...
Basically the thing you need to think about is the energy associated with the field itself. For the electromagnetic fields, the energy in the field is like (E^2 + B^2), or in terms of the electromagnetic field tensor F (see  if this isn't a familiar concept), it's F^2. You can write F in terms of a potential A, basically F = dA, where d means gradient . So the energy looks like (dA)^2, which is a kinetic energy for the field A, because it tells you that an oscillation in the field costs energy. Integrating over the possible values of the A field is a lot like the integral you would do for the matter fields , in particular the math looks basically the same as what you would do for electrons for example . And just like with electrons, the local propagating degrees of freedom are the ones we call particles.
So this is how you make dynamics happen in QFT--you integrate over all the possible field values, and the local propagating degrees of freedom are called particles. And for EM and other gauge theories, the math is basically the same as for electrons, and it should be interpreted the same.
The same story carries over for gravity, too, where the Einstein-Hilbert Lagrangian (R) is like the EM field strength (F^2), and the metric g enters into it in a similar way as A does for EM (basically, R = (dg)^2).
 This isn't quite right, you have to make sure everything is gauge invariant, but that's not important for understanding what's going on at a high level.
 See .
 If this isn't a story that you understand yet, take some time to study free scalar field theory, and make sure to understand both the path integral method and the canonical method. By seeing how these translate between each other, you can build intuition.
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?
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.
Sure, but what if there is no field for GR? The way I understand gravity to work (warping spacetime in such a way that proper time gets "crunched" towards mass, retarding the system's action, shifting the variationally stationary path towards the mass) does not require a force carrying field (as far as I can tell).
To the point, I've founded a startup precisely for that: https://quantumflytrap.com/
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.
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.
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).
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.)
Think of it like knowing the rules, but the rules not being efficient to compute with.
Squiggles. Little arrows. Hand-drawn scribbles.
I've looked into why there's a total vacuum of numerical simulation, and the reason is simply snobbery. There's a philosophy that numerical methods are what mere engineers do, and not as intellectually pure as symbolic solutions.
You regularly hear comments like: "Such and such cannot be solved", what is often meant is: "It can only be solved numerically, but not with closed form algebraic expression."
People that say things like the former statement have excluded numerical solutions as valid in their mind. Such solutions may as well not exist, as far as they're concerned.
I see this attitude everywhere in theoretical physics.
The other issue is that because of the seventeen layers of assumptions and abstractions, most physicists are now at the point where they're not really making theories about physics, but instead they're debating the properties of the abstractions themselves. It's like kids trading basketball cards and saying one player is better than the other because his card is printed better.
This can all be boiled down to an acid test: Can you render your equations? As in, full 3D numerical simulation with an image as an output? Not a graph, not a scalar value, but a picture?
For simple (non-diffracting) optics? Yes! That's literally every 3D computer renderer!
For classical electromagnetics the answer is: yes.
For plasma physics the answer is: yes.
Even for special and general relativity, the answer is: yes.
For QED? Err... maybe? I've seen some toy examples in 2D, and I think I've seen a 3D example once. In principle, it's doable.
For anything else? Nothing. Or at any rate, I've seen nothing despite years of searching.
E.g.: Can you show me any "rendering" that extends a QED simulation with the weak interaction?
Can you find any animated examples of any particle interaction? E.g.: a free electron being captured by a proton, or an electron changing orbitals?
Again: Images or animations please. Not graphs or scalars. Everyone I ever challenge gets confused and links me to a paper with a histogram in it. That's not what I mean!
This is basically false as far as I can see. Numerical methods are basically the only way to solve either general relativity or quantum equations, but even then it's extremely difficult to get accurate calculations with a practical amount of compute power. There's intense competition over who can get the best results here (in fact the leading group in lattice QCD keeps their code a secret, which is not great science).
I think the core reason you don't see much visualisations is the visualisations are not very useful. The little visualisations I've seen of extremely simplified versions quickly dissolve into complete mush as everything overlaps. It would not suprise me if the more complex interactions (like two real particles in 3d) were basically unintelligable and extremely difficult to make intelligable without becoming 'cartoons'.
There's no visualisation of what goes on at the interaction point. Just an idealisation of what goes on far from it.
Where in that diagram can I see diffraction? Interference? Entanglement? The particles interacting?
Or can I not see any of that because it's a cartoon representation that shows the classical measurements only?
Are you asking for a picture of 2 spheres and then another sphere? What are asking for?
Pick any one.
I find even questions like this revealing: Why are you asking me such incredibly basic questions instead of having a dozen renderings ready to hyperlink?
If you ask a NASA guy to link some airflow simulations, he won't retort with "what criteria would satisfy you?". He'll just straight up link a bunch of gorgeous renderings of physically accurate airflow simulations.
I guess my confusion (which may be from ignorance of physics) is about how to handle depicting superpositions of whether or not the electron has been absorbed by the proton.
I don’t mean that that isn’t a problem with a good (known?) solution, just I don’t know one.
This is the "measurement problem", and nobody can give you a straight answer.
This is why I give this acid test: it reveals this gap in the theory that is otherwise easy to gloss over.
It's the difference between pseudocode on a whiteboard, and something that compiles, runs, and gives the correct answer.
You can hand-wave one of those and claim correctness. The other will give you a compiler error.
This is why I asked you to specify what you were asking for.
So it's only doable for tiny N.
> 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.
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?
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.
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
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
> 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).
Physics From Symmetry by Jakob Schwichtenberg
Also, (though I've not read it) this appears to be a longer, more detailed text in the same vein:
Quantum Theory, Groups and Representations: An Introduction by Peter Woit
This one gets to the EM field on page 573 :-/
I dunno of a good article as an introduction. As a grad student, I found Terry Tao's explanation  rather helpful, but of course it has a strongly mathematical flavor.
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
(Which I realize is the least helpful possible reply.)
The textbook Peskin&Schroeder is standard in this field. "If" you can find a PDF online, there's a chapter introducing gauge theories (the chapter on yang-mills) that has a reasonably clear explanation near the beginning, I think.
> “Gauge theory” is a term which has connotations of being a fearsomely complicated part of mathematics – for instance, playing an important role in quantum field theory, general relativity, geometric PDE, and so forth. But the underlying concept is really quite simple:
Then goes on to write an incomprehensible novella length explanation.
I like Tao, but this is a problem in 'pure' maths. I feel the issue is exacerbated by the fact that if you can follow along with Terry then you likely will be unable to recognise the issue.
A good counter point is this series by Timothy Gowers developing a proof of "Pingala's Determinant".
When Gowers finds the proof he states:
> Since this clearly has determinant 1 and this clearly has determinant 1, so does their products, ie. Pascal's[sic] determinant is 1. Could have done that, but that would have been uh, well depends on what you find interesting, but that would have been really sort of rabbit out of the hat, look what a billiant clever mathematician I am. I can just sort of produce this fabulous identity out of nowhere and it's got a nice simple proof that also came out of nowhere. What I want to emphasize is these sort of proofs don't come out of nowhere.
Timestamped to quotation: https://youtu.be/m8R9rVb0M5o?t=886
The series is great and will show anyone outside the field of pure maths that even pure mathematicians use numerical reasoning while trying to reason about a subject.
Not all people will agree that Tao's exposition is unclear. I linked it precisely because I found it more clear than other, "more physical" explanations (as a graduate student in physics, not math). When I was trying to understand gauge symmetry, it was that article that finally made things click for me. And, of course, many people I know would disagree, and point to other expositions as superior. That's the point: different people will not agree on what is well-written and what is incomprehensible.
You state "even pure mathematicians use numerical reasoning while trying to reason about a subject". Yes, some pure mathematicians do. Others use graphical reasoning. Others tend to lean purely on algebraic manipulations. Others tend to "reason" by analogy. And I'm sure there are other modes of thinking with which I'm too poorly acquainted to name. The result is that it's perfectly reasonable for people to disagree, quite strongly, about what is comprehensible and what is not, particularly people coming from different intellectual traditions.
It's a tangential nitpick, but I can't resist:
> [...] novella length explanation
As an undergrad I wanted to learn differential geometry, so I went to the library to pick a diff geo textbook. Well, I didn't want a difficult one, so I decided to pick the shortest book I could find --- surely that would be the easiest textbook to read? Haha.
As far as I'm concerned, the length of Tao's explanation is a strong merit.
> I feel the issue is exacerbated by the fact that if you can follow along with Terry then you likely will be unable to recognise the issue.
On the topic of "heterogeneity" it would seem to follow from my saying "If you can follow along with Terry" that I recognise some people are capable to find his explanation clear.
My concern is providing that link as educational stead insightful.
On the heterogeneity of pure mathematicians it is very frustrating to see someone try to claim, in a public forum where people unfamiliar with the subject can read, that a Mathematician uses anything other than "numerical reasoning while trying to reason about a subject", and I feel it causes great harm to people outside of mathematics.
You say, "some use graphical reasoning", well where did their graphical intuitions come from? Are they born with it? Or perhaps they were taught graphical reasoning by way of numerical reasoning and now are able to just use the graphical reasoning on new problems.
This is part of what I was trying to express by referencing Gowers. Using identities to reason about a problem is fine, and arguably necessary, but introducing identities when teaching without explanation is befuddling and I argue deters learning.
The op said "I'd like to learn more about symmetry", you used the word "introduction" and Tao used the word "simple".
If the op said, "I went through a physics program in undergrad and grad school and I feel like I have all the mathematical pieces but I'm still struggling to put them all together... can anyone recommend a resource that avoids getting bogged down in prerequisite explanations and just focuses on the relations."
Then I would have scrolled past thinking, yeah that Tao article is perfect.
If you would have said, "After years of higher education this article made it click for me in ways multiple professors failed to but you'll need an equivalent amount of the numerical examples from my education of the various topics used to understand this."
Then I would have scrolled past thinking, yeah Tao does a great job of illuminating concepts.
If Tao would have used any other word than "simple".
Then I would have left out the "novella" jab. Though now I'm tempted to ask what about the short textbook was more difficult, because in my similar library experiences the short books were the ones that just had equations and identities and left out any of the numerical reasoning. Is that what you discovered when you Haha'd at that little textbook?
I worry for Mathematics when someone says, "Hey I think I might be interested in that thing..." and the response they recieve is a bludgeon of latin characters, and the dubious unqualified follow up "well other people get it".
It is now doubly hard for that person to explore their interest because they need to both find their own resources that actually teach the material, as well as decide whether its even worth it if this is the community they are working to become part of.
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.
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.
> In this document, we discuss rotations, including simple rotations in the plane but also including compound rotations around multiple axes in three or more dimensions. We briefly survey four ways of pictorially representing rotations: two vectors in the plane of rotation, triad before and after rotation, axis plus amount of rotation, and yaw/pitch/roll. These can (respectively) be formalized in terms of (respectively) Clifford algebra i.e. quaternions, matrices, Rodrigues vectors, and Euler angles.
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.
I wanted to say they are both emergent phenomenons... or something in that direction.
Newtonian mechanics is a useful approximation at certain scales. At other scales, they are problematic.
"Is Infinity Real?" Sabine Hossenfelder, about 6:25 in.
Perhaps this is what OP was referring to?
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
This hints to me that you might be thinking of the concept of invariant vs relativistic mass
> 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?
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.
Maybe, gravity is like that in some more complex (or abstract?) way.
I am curious though, as to where you got such a strong notion that shadows can move faster than light...
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.
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.
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.
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.
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).
For example, imagine we live in a simulation, how does it affect your example?
You now seem to be arguing that all science must be done by proving things from first principles, which is of course absurd.
Just look at various explanations for "dark matter" phenomena
With regard to dark matter: yes there are many competing / mutually exclusive theories surrounding it, but that does not then mean in order to invalidate one of those theories you need to have picked a different one that you like better.
If I would try to find and explain alternative hypotheses for your "human intuition/everyday experience " examples they would sound contrived this invalidating themselves. Imagine trying to explain quantum theory/general relativity to a Victorian and their counter-examples all use objects from their life.