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I'm generally aware that there have been attempts to describe things like magnetism and the other forces geometrically, like with gravity, and that QFTs have essentially supplanted them. But it's also my understanding that per GR, we don't simply treat spacetime curvature as a model for how gravity works but rather consider space actually to be curved by the presence of mass-energy - or at least gravity is fundamentally indistinguishable from actual curvature.

My question is why is gravity special in this regard? The only obvious reason is that it's (as far as we know) asymmetric, but imagine an extraordinarily powerful magnetically charged body attracting a smaller metallic object in space. In that setup, if we only look at the magnetic field in one direction, it seems like all of the postulates of GR (equivalence principle, etc.) would be equally applicable and would lead one to derive essentially the same equations, just with different constants.

So why do we say that gravity curves space but the magnetic field doesn't? Is it simply the case that we use the best models for each force just because they are the best models that fit observation and not becuase we are convinced the forces are fundamentally different?

Maybe a clearer way to ask this is whether we believe that the acceleration caused by gravity is of a fundamentally different character (namely the curvature of space) than the acceleration caused by quantum fields and virtual particles? Or am I just asking the $billion question underlying the search for a TOE?

(Note I'm not talking about the gravity caused by the mass-energy associated with the magnetic field).

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  • $\begingroup$ Related: physics.stackexchange.com/q/357488/2451 , physics.stackexchange.com/q/57228/2451 $\endgroup$
    – Qmechanic
    Mar 16, 2023 at 21:24
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    $\begingroup$ Different objects with different masses and types of alloy would have different accelerations in the same region of space. It is not a geometric property as for gravity. $\endgroup$ Mar 16, 2023 at 21:39
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    $\begingroup$ Of course GR is a model. No mathematical object is physical. $\endgroup$
    – John Doty
    Mar 17, 2023 at 1:42
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    $\begingroup$ Any question with "Why" up front should auto-generate a comment with this link: youtube.com/watch?v=36GT2zI8lVA :) $\endgroup$
    – AnoE
    Mar 17, 2023 at 9:05
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    $\begingroup$ @AnoE haha that is an awesome clip. Note I was very careful to ask why we SAY gravity curves space but not the other forces, rather than to ask why gravity curves space. In other words why do we use one very special model for gravity and no other forces even though it seems like we could use it with the others and this has indeed been tried - to which JEB provided a masterful answer. $\endgroup$ Mar 17, 2023 at 18:49

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One reason is that it just doesn't work, for example, C/S's comment: inertial mass and would-be magnetic mass are not the same. In Newtonian gravity, inertial mass in the 1st law and gravitational mass in his law of gravity are equal. In GR, they're the same thing.

Electromagnetism starts with Coulomb's law: electric fields tell charges how to move, and charges tell electric fields how to diverge (to paraphrase John Wheeler). It shouldn't be hard to convince yourself that Coulomb's law on it's own is not Lorentz Covariant (that is, it does't work in all reference frames on its own...in fact, it only works in the reference frame where charges are at rest). In frames moving w.r.t. the charges, the charges become currents which tell magnetic fields how to curl. Along with (free-space, Gaussian units):

$$ \vec{\nabla}\times \vec{E} = -\dot{\vec{B}} $$ $$ \vec{\nabla}\times \vec{B} = \frac 1 c \dot{\vec{E}} $$

and Lorentz contraction and time dilation, Lorentz covariance is restored. Note the $B$ is "weaker" than $E$ by a factor of $c$.

In four-vector notation, the source of the fields is the four-current (for a point charge):

$$ j_{\mu} = qu_{\mu} = q\gamma(c, \vec v) $$

So the source is a 4-vector, unrelated to mass, and E&M is a vector force.

Like Coulomb's $1/r^2$ law, Newton's Law of gravity is not covariant. If it is just sourced by mass, it would be a scalar field (which it is not). A covariant form is constructed via the stress-energy tensor:

$$ T_{\mu\nu} = mu_{\mu}u_{\nu} = m\gamma^2(c, \vec v)_{\mu} (c, \vec v)_{\nu}$$

(a more formal description with tensor densities is a bit more involved). In the rest frame of the particle, only the time-time component is non-zero:

$$ T_{00} = mc^2 $$

For objects moving at $v \ll c$, far from the Schwarzschild radius, the curvature of space-time is entirely described by gravitational time-dilation: masses move on inertial trajectories that maximize their proper time (a la the Twin Paradox).

At higher velocities, the other terms matter, so that a single gravitational field is insufficient to describe motion. In the weak regime, there is so-called gravioelectromagnetism in which a magnetic-like field circles around momentum density, and moving masses are deflected by a Lorentz-force like term sourced by $T_{i, 0}$.

Like the magnetic field, it is supressed by a factor of $1/c$.

Near compact objects, all terms matter, and gravitation is sourced by mass, momentum, energy flux, and the space-space stress tensor (suppress by $1/c^2$). That is just classical stress-tensor, with the trace being the pressure. It becomes too non-linear to describe with simple fields.

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    $\begingroup$ A very comprehensive and understandable answer thank you. My takeaway - and please correct me if I'm wrong - is that it is indeed then a question of using the best model to fit nature and not because we have some foundational reason to believe that the accelerations caused by each are of a different fundamental character. Would you agree? $\endgroup$ Mar 17, 2023 at 0:17
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    $\begingroup$ @PeterMoore The foundations of physics are experiments and observations of the phenomena. We have no oracle to give us "foundational reasons". $\endgroup$
    – John Doty
    Mar 17, 2023 at 1:45
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    $\begingroup$ @JEB What experiment do you propose to test these? Without testing, all you have is opinions, not science. $\endgroup$
    – John Doty
    Mar 17, 2023 at 1:53
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    $\begingroup$ Lol oh dear I didn't mean to open a can of worms. :). I disclaim anything that follows. Thanks again @jeb $\endgroup$ Mar 17, 2023 at 2:01
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    $\begingroup$ @JohnDoty yes, that is why I mentioned them. Plenty of granting agencies think they are science, in spite of the lack of observables (Penrose's CMB rings notwithstanding). $\endgroup$
    – JEB
    Mar 17, 2023 at 16:03
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Can I offer a super simplistic answer, but which might capture some of the same ideas as the much more comprehensive answer already given although with a potentially faster route to intuition?

As far as we can tell, Gravity deflects the path of everything ('deflects' meaning: causes a deviation from the motion it would have had if the gravity wasn't there.) Other forces don't do this.

It doesn't just deflect physical motion but also movement through time - again, other forces don't seem to do this.

So, we could either try to model gravity as some kind of force that affects more than all the other forces AND then throw in an extra, pretty complicated effect on time, which only affects the time-behavior of the other forces OR we could model this all as a curvature of space-time.

You might ask: "But, is spacetime truly curved?" In answer, an under-appreciated quote from Poincaré comes to mind: "One geometry cannot be more true than another; it can only be more convenient."


(long side note below, re: comments on above answer - basically just a harsher version of a Karl Popper idea)

There are different types of 'foundational reasons.'

The type that strict, empirical physicists usually like to think of are actual observations that affect how you set up your rules, simply because the predictions of your rules need to match with experiment. This is where the "I don't care about the philosophy of your physics or how pretty its mathematics is" opinion makes sense and where things like String Theory and most modern-particle physics run into a lot of problems.

Then there are intermediary meta-rules, such as 'Conservation of Energy.' Every modern theory has its own version of this, which is of course not a meta-rule in that case but a very concrete rule, but the idea that any model should have a constructible quantity, typically called Energy, that is conserved over the evolution of another variable, typically called time, is a meta-rule about how collections of rules in Models should act.

Finally, another type that are actually meta-meta-rules. So, for example, if you notice that all 'good' rules of physics so far in history have a certain property, you're probably going to give the new one(s) you're trying to design that property as well. An example of this would be the "The Maths should look pretty" rule, which is definitely a 'foundational reason' for other Laws/Models in physics and was a huge favorite of Dirac, or the "This theory has many more predictions than the currently accepted one - most of which we can't test right now - but where there is overlap, their predictions are miraculously close considering how differently we came up with this other one" rule. Two things to say about this second kind -

  1. MANY scientists would get upset at the suggestion that they adhere to such rules, and will argue vigorously against the idea, but if you just look at their behavior it's clear to see how closely they follow them, and
  2. These are not necessarily bad meta-rules: it is true that certain theories that are considered overly speculative by some, such as String Theory, fall strongly into this category, but then again so did others at some point in time, which we now love and cherish - for example Special (for a short time) and G (for a longer time) relativity.
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  • $\begingroup$ I like the answer but I can't say I follow your attempt to distinguish gravity from, say, electromagnetism. Acceleration from EM forces can cause time dilation too. As for deflection, all the other forces can do what gravity does (attract) but also can repel and thus cancel out which gravity can't. EM follows the same 1/r^2 law. And all the particles that comprise normal matter have charge. So I'd say that EM, due to the asymmetry, is certainly more complicated than gravity but that doesn't intuitively explian why a geometric model doesn't fit it as well. $\endgroup$ Mar 17, 2023 at 19:13
  • $\begingroup$ No, the other forces CAN'T do what gravity does - at least not in terms of attraction. Think about the fact that an extremely strong magnet will not deflect the path of light at all (except for the deflection caused by the mass of the magnet), but a gravitationally massive object will. (Interesting side note: a strong magnet can't bend the travel path of gravitational waves either, so there isn't some sort of symmetry there.) $\endgroup$
    – M Jones
    Mar 18, 2023 at 13:06
  • $\begingroup$ Ok I see your point about how gravity literally deflects everything such that we cannot in principle distinguish it from curved space, whereas other forces might mimic curved space for some objects under some conditions but not for others. My takeaway is that if we ever did find something that could repel or ignore gravity then the geometric model would have to be thrown out the window. $\endgroup$ Mar 18, 2023 at 23:29
  • $\begingroup$ @MJones "No, the other forces CAN'T do what gravity does - at least not in terms of attraction".The strength of a horse magnet can beat the gravity of a planet gravity is so weak... $\endgroup$ Mar 22, 2023 at 0:14
  • $\begingroup$ @appliedSciences I don't think you read my whole comment. I am not talking about the strength of Gravity, I am talking about what sorts of deviations it can induce. A horse magnet can't bend light - simple. $\endgroup$
    – M Jones
    Mar 22, 2023 at 18:55
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I think this is a philosophical and neuro-psychological question, rather than a physics one. Any physics answer will not be satisfying. It may feel satisfying if somebody gives you a sufficiently obfuscated answer that is hard enough to prevent you from figuring out it also isn’t satisfying because you still don’t know why that is the way it is. Case in point: The other answer here.

So only a meta answer may help:

Nature needs no reason to be a certain way. Only we humans do. Because our brains are literally bias-based pattern-detecting prediction machines. You can always ask “why”. And you will never be satisfied. You will only end up at the Münchhausen Trilemma: Either there is an infinite chain of “why”s, or there is a thing without a “why”, or it goes in a circle. Neither of which are satisfying.
This is why people make up creation myths. From gods to a big bang and whatnot. Not that it solves the problem, because you can still ask “why?”, but it sometimes lets them trick their minds into calming down if they enforce the rule of not thinking any further hard enough. So they can keep going.

Additionally, nature has no need to be symmetric. That’s just our brains trying to simplify several patterns into one, by trying to force a similarity where there is none. It is the same mechanism that lets us see things in clouds; academic researcher edition. ;)

An actually satisfying answer is:
That nature is exactly what we observe. Nothing more can be said of it, and nothing less. And that is what the other answer’s useful aspect boils down to: Because it’s what we see! (In the sense of “observe/measure”.) At least it’s the most likely choice that is not self-contradicting. And we humans obviously need it to be consistent, to be able to use it.
That’s how we get to “It just doesn’t work.”. Translation: There is no common pattern there, judging by what we see. You can spend your lifetime trying to find it, any may succeed, but may also end up having chased down windmills.

Anything beyond that… is literally just pseudo-science. You can make up elaborate string theories and multiverses and or even wave function “collapses” based on nonsense like a “consciousness” (a term from a time when philosophy and religion were still the same thing). But in the end, unless you can observe it, that’s “not even wrong”. In other words: It is of no use. You may as well state “$Deity did it.”

I hope this gives you a more more general and hence more long-term-satisfying answer. :)

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  • $\begingroup$ This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review $\endgroup$
    – Toffomat
    Mar 17, 2023 at 12:21
  • $\begingroup$ Indeed I intentionally asked why we SAY gravity curves space as I wished to understand better why the geometric model so successfully describes gravity but not the other forces. But I appreciate your perspective. $\endgroup$ Mar 17, 2023 at 18:55
  • $\begingroup$ If you did want to attribute a philosophical motive to my question though, you could read into it the argument that if we don't model anything other than gravity as resulting from curved spacetime, how do we know gravity really is the result of curved spacetime and that GR isn't just a convenient approximation of something deeper? But I already know the answer to that - we don't know that and won't know that until we have quantum gravity. And that is WAAAAY above my paygrade to opine on. $\endgroup$ Mar 17, 2023 at 22:09
  • $\begingroup$ @Toffomat: Yes it does. You seem to have missed the entire point of my answer. $\endgroup$
    – anon
    Mar 18, 2023 at 0:45
  • $\begingroup$ @PeterMoore: I’m sorry that I could not communicate the entire point of my answer so it was possible to follow it in its entirety. You are asking a question that is already based on invalid assumptions. Namely, that “gravity is special”, which it only is if you assume it is a force in the first place. From that can only come invalid and not useful answers, that will misguide you down a dead end. … If you want that, then yes, mine “is not an answer”. … But I had hoped to move the mindset out of that box. I’m sorry. $\endgroup$
    – anon
    Mar 18, 2023 at 0:54
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Gauge theories(QED,QCD) are written on a manifold(spacetime) while GR which descibes gravity, describes the manifold itself.That's why gravity is different than other forces.

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The electromagnetic field does curve spacetime, therefore it is incorporated in the equation that describes the spacetime and its solution. To wit, the Reissner–Nordström and Kerr-Newman metrics are well-known black hole spacetimes with electromagnetic field. There even exist solutions with only magnetic/electric field, such as the Melvin metric.

Mass is privileged in physics, as almost all particles, i.e. objects physical theories deal with, are massive, and, according to the equivalence principle and the energy-mass equations, energy is massive, too. This indicates that free-fall is ubiquitous and we can use geodesics as particle trajectories.

Note I'm not talking about the gravity caused by the mass-energy associated with the magnetic field

There is one kind of gravity, if we keep in mind the equivalence principle. Its source may be the inertial mass or the gravitational mass. As you exclude the inertial mass, I'm not certain what the question is about. If it is about interaction of gravity with quantum fields in general, this is outside our experimental and theoretical capabilities at the moment.

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  • $\begingroup$ When you say the EM field does curve spacetime are you just referring to standard General Relativity and the stress-energy tensor? If so that's what I mean when I say I'm not asking about the effect of the mass-energy associated with the magnetic field. The force caused by the EM field itself is obviously many orders of magnitude greater than that. I'm asking why we don't use spacetime curvature to describe the acceleration caused by other forces besides gravity. $\endgroup$ Mar 17, 2023 at 22:00
  • $\begingroup$ My answer is about the classical theory, yes. But we do use spacetime curvature to describe the acceleration caused by electromagnetism in the cases described above. You can forget that it is about the mass-energy of the field; still, the derived metric is curved and its geodesics are roughly the trajectories of test particles. $\endgroup$
    – auxsvr
    Mar 18, 2023 at 9:40

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