How come GR doesn't describe other forces with curvature?

[nashed]

Let me start by saying that I'm a physics student but I have no experience with GR other than some pop-sci and some clarifications on that pop-sci from various sources, so basically I'm saying that some math is fine but please bare in mind that I have no real experience with the relevant math.

With that out out of the way, here's my question:

As far as I understand what GR did, aside from dealing with frames of reference correctly, was that it descried the force of gravity as a curvature in space-time (basically a massive object will change the metric of space-time, at least locally), the thing is that up to that point both gravity and electromagnetism where thought to be inverse square distance laws, so how come gravity got to be described by space-rime curvature while electromagnetism ended up with Maxwell's laws and whatever qunatum theory does with it?
More generally what makes gravity so special that it's described by space-time curvature while other forces (not necessarily fundamental ones) get to move objects from their natural geodesics, why aren't they generalized the same way gravity is?

Again my experience with GR is quite limited, I just had a thought and figured I'd share, so please go easy on me as I'm not up to speed with the rest of you GR people.

 

[Lucas SV]

What you said is correct, in usual GR, gravity is not described as a 'force' but rather the curvature of spacetime, and all other forces are described differently. Gravity is indeed described by the metric, an analogue of the Newtonian gravitational potential. Eintein Field Equations are the analogue of maxwell's equations. They say that given a matter configuration, given by the energy-momentum tensor which should be thought as the source, you can solve the EFE to find the metric tensor, which tell spacetime how to curve.

Then the geodesic equation (and its extension to an arbitrary force field) gives an analogue of Newton's second law, where gravity does not appear as a force anymore, but it is part of what acceleration means in GR. Usually we take the force field to be the electromagnetic field, since it is prety much our only classical fundamental field.

However there is a way to incorporate electromagnetism into curvature of a 5 dimensional spacetime, and this is called Kaluza-Klein theory. I do not know of any other theory that treats the other forces as purely geometric in a similar sense to GR, besides we have to remember that GR is a classical theory. Things get more complicated when we move to the quantum world.

 

[PeterDonis]

what makes gravity so special that it's described by space-time curvature while other forces (not necessarily fundamental ones) get to move objects from their natural geodesics, why aren't they generalized the same way gravity is?

Gravity has two key features that no other force has:

(1) In Newtonian terms, it accelerates all objects the same, regardless of their mass, composition, or any other factor. Other forces don't work that way: electromagnetism, for example, acts differently on objects with different charges.

(2) Objects moving solely under the influence of gravity feel no force; they are weightless, in free fall. Other forces don't work that way: objects acted on by them feel a force and are not weightless.

These key features of gravity are what allow it to be modeled in GR as spacetime curvature, rather than as a force. Since other forces don't share those features, modeling them as spacetime curvature doesn't work the way it works for gravity.

 

[PeterDonis]

gravity does not appear as a force anymore, but it is part of what acceleration means in GR

This is misstated. A better statement is that, in GR, objects moving solely under the influence of gravity have zero acceleration, because the term "acceleration" in GR is best used to mean proper acceleration, i.e., acceleration that is actually felt (as weight--so objects moving solely under gravity are weightless). Objects acted on by other forces have nonzero proper acceleration.

there is a way to incorporate electromagnetism into curvature of a 5 dimensional spacetime, and this is called Kaluza-Klein theory.

This is true--and in fact higher-dimensional analogues of this technique can be used to model interactions similar to the Standard Model weak and strong interactions.

However, none of these methods change the fact that only gravity has the two key features I described in my previous post.

 

[haushofer]

Because of the equivalence principle.

 

[nashed]

Gravity has two key features that no other force has:

(1) In Newtonian terms, it accelerates all objects the same, regardless of their mass, composition, or any other factor. Other forces don't work that way: electromagnetism, for example, acts differently on objects with different charges.

(2) Objects moving solely under the influence of gravity feel no force; they are weightless, in free fall. Other forces don't work that way: objects acted on by them feel a force and are not weightless.

These key features of gravity are what allow it to be modeled in GR as spacetime curvature, rather than as a force. Since other forces don't share those features, modeling them as spacetime curvature doesn't work the way it works for gravity.

Thanks, this is exactly the kind of answer that I was looking for! Can you elaborate a little bit more though? specifically I've got two follow-up questions:

(1) While gravity does accelerate all objects equally, it doesn't apply the same moment to all objects equally, mainly due to the moment of inertia being a function of the shape and mass distribution of the object, how does that enter GR? I assume it would be something along the lines of the metric being some sort of a function of space-time (basically I'd guess at each point the metric would provide a different geometry with some sort of continuity requirement but this seems kinda weird and probably wrong)

(2) I tried to think of this in terms of a frame of reference attached to the object, but in such a frame of reference wouldn't the object always be weightless? I think I've confused some concepts in my head

 

[PeterDonis]

While gravity does accelerate all objects equally, it doesn't apply the same moment to all objects equally, mainly due to the moment of inertia being a function of the shape and mass distribution of the object, how does that enter GR?

Bear in mind that when we say objects acted on solely by gravity follow geodesics, we are referring to the center of mass of the object. Individual parts of the object that are not at the center of mass can feel a force and not be moving on geodesics. But the CoM will be (at least, if we ignore the small effects described below). So any difference in moments will not affect the geodesic motion of the object as a whole.

In GR there is a small effect predicted due to spacetime curvature varying from one part of the object to another. When this effect is taken into account it is no longer always the case that objects acted on solely by gravity follow geodesics. But this effect is extremely small; AFAIK it has never been measured because it is too small for us to detect with current technology.

 

[Ben Niehoff]

This is true--and in fact higher-dimensional analogues of this technique can be used to model interactions similar to the Standard Model weak and strong interactions.

KK reduction can give you classical gauge theories (and presumably, KK reduction of quantum gravity should give quantum gauge theories, provided that quantum gravity works in higher dimensions). However, certain details of the Standard Model are difficult, if not impossible, to achieve by KK reduction alone. Two examples are:

1. The Higgs mechanism. Standard KK reduction always gives massless gauge bosons, so it cannot give the weak force, whose gauge symmetry is broken. However, (in string theory anyway) the Higgs mechanism can be modeled by a stack of D-branes with open strings attached. Each open string represents a gauge boson, and the endpoints represent particles in the fundamental rep of the gauge group. The Higgs mass of the gauge bosons is then related to the transverse separation of the branes, which forces the strings to stretch. This is not a KK reduction, though; this is a completely different model.

2. Chiral matter. In the Standard Model, left-handed particles couple differently than right-handed ones. This cannot be achieved with KK reduction, because the spacetime fabric is agnostic about the handedness of particles. I'm not sure if any satisfactory models of chiral matter have been devised, it's a very hard thing to do apparently (and not my expertise, so maybe there is something I don't know about).

 

[haushofer]

In string theory you also obtain gauge bosons in the string spectrum, so compactification seems curious enoug to be a 'double strike': you don't need to put the vector dof's in the metric, as in KK-compact.

 

[stevendaryl]

I think that Kaluza Klein is a beautiful idea, but in some sense, it is going the wrong direction for unification of the forces. It explains something we already have a good quantum theory for, electrodynamics, in terms of something we don't have a good quantum theory for, gravity.

 

[Ben Niehoff]

I think that Kaluza Klein is a beautiful idea, but in some sense, it is going the wrong direction for unification of the forces. It explains something we already have a good quantum theory for, electrodynamics, in terms of something we don't have a good quantum theory for, gravity.

I agree. Interestingly, it is possible to get the Standard Model gauge groups by compactifying 11-dimensional M-theory down to 4 dimensions. But due to the problems I mentioned above, this doesn't really give you the Standard Model; it just gives another model that has the same gauge groups. There is another problem as well, which is that KK reduction gives you more than what you asked for: in addition to giving you gauge fields, it also gives you a bunch of massless scalar fields, which don't exist in nature.

So at best KK reduction is an interesting toy model, but it does not at all represent a way to get real-world physics out of a higher-dimensional theory.