Ok, I’m not touching the ECE thing; as noted, I’m not a theorist. I just measure stuff. I’ve taken classes in formal QFT, but I don’t use it day-to-day, so it’s a weak point for me. However, it seems a bit odd to describe things that can be produced in collisions and (at least in principle) fired at your enemies to kill them by radiation poisoning as ‘illusory’. If you bang two electrons together, measuring the cross-section as a function of the center-of-mass energy, you will observe a classic 1/s decline interrupted by equally classic resonance bumps. That is, at certain energies the electrons are much more likely to interact with each other; that’s because those are the energies that are just right for producing other particles. Increase the CM energy through 80 GeV or so, and you’ll find a Breit-Wigner shape like any other particle; that’s the W, and if it weren’t so short-lived you could make a beam of them to kill your enemies. (With asymmetric electron energies you can produce a relativistic-speed W and get arbitrarily long lifetimes in the lab frame, but that gets on for being difficult engineering. In fact, just colliding two electrons at these energies is difficult, they’re too light; that’s why CERN used an electron and a proton in LEP.)
Now, returning to the math, my memory of this is that particles appear as creation and annihilation operators when field theories with particular gauge symmetries are quantized. If you want to call the virtual particles that appear in Feynmann diagrams illusory, I won’t necessarily argue with you; they are just a convenient way of expressing a huge path integral. But the math doesn’t spring fully-formed from Feynmann’s brow; the particular gauge symmetry that is quantised is chosen such that it describes particles or forces already known to exist. (Historically, forces, since the theory ran ahead of the experiments in the sixties—we saw beta decay long before we saw actual W bosons.) If the forces were different, the theorists would have chosen a different gauge symmetry and got out a different set of particles.
I’m not sure if I’m answering your question, here? My basic approach to QFT has always been “shut up and calculate”, not because of QM confusion but because I find it very confusing when someone says that a particular mathematical operation is “causing” something. I prefer to think of the causality as flowing from the observations, so that the sequence is thus:
We observe these forces / cross-sections / particles.
We know that by quantising field theories with gauge symmetries, we can get things that look very much like particles.
Searching through gauge-symmetry space, we find that this one gives us the particles and forces we observe.
I meant illusory in the same sense that “sure, the force of gravity can cause me to fall down and get ouchies… but by a bit of a coordinate change and so on, we can see that there really is no ‘force’, but instead that it’s all just geometry and curvature and such. Gravity is real, but the ‘force’ of gravity is an illusion. There’s a deeper physical principle that gives rise to the effect, and the regular ‘force’ more or less amounts to summing up all the curvature between here and there.”
My understanding was that gauge bosons are similar “we observe this forces/fields/etc… but actually, we don’t need to explicitly postulate those fields as existing. Instead, we can simply state that these other fields obey these symmetries, and that produces the same results. Obviously, to figure out which symmetries are the ones that actually are valid, we have to look at how the universe actually behaves”
ie, my understanding is that if you deleted from your mind the knowledge of the electromagnetic and nuclear forces and instead just knew about the quark and lepton fields and the symmetries they obeyed, then the forces of interaction would automatically “pop out”. One would then see behaviors that looks like photons, gluons, etc, but the total behavior can be described without explicitly adding them to the theory, but simply taking all the symmetries of the other stuff into account when doing the calculations.
That’s what I was asking about. Is this notion correct, or did I manage to critically fail to comprehend something?
And thanks for taking the time to explain this, btw. :) (I’m just trying to figure out if I’ve got a serious misconception here, and if so, to clear it up)
I guess you can think of it that way, but I don’t quite see what it gains you. Ultimately the math is the only description that matters. Whether you think of gravity as being a force or a curvature is just words. When you say “there is no force, falling is caused by the curvature of space-time” you haven’t explained either falling or forces, you’ve substituted different passwords, suitable for a more advanced classroom. The math doesn’t explain anything either, but at least it describes accurately. At some point—and in physics you can reach that point surprisingly fast—you’re going to have to press Ignore (being careful to avoid Worship, thanks), at least for the time being, and concentrate on description rather than explanation.
Well, my question could be viewed as about the math. ie: “does the math of the standard model have the property that if you removed any explicit mention of electromagnetism, strong force, or weak force and just kept the quark and lepton fields + the math of the symmetries for those, would that be sufficient for it to effectively already contain EM, strong, and weak forces?”
And as far as gravity being force or geometry, uh… there’s plenty of math associated with that. I mean, how would one even begin to talk about the meaning of the Einstein field equation without interpreting it in terms of geometry?
Perhaps there is a deeper underlying principle that gives rise to it, but the Einstein field equation is an equation about how matter shapes the geometry of spacetime. There’s no way really (that I know of) to reasonably interpret it as a force equation, although one can effectively solve it and eventually get behaviors that Newtonian gravity approximates (at low energies/etc...)
(EDIT: to clarify, I’m trying to figure out how to semivisualize this. ie, with gravity and curvature, I can sorta “see” and get the idea of everything’s just moving in geodesics and the behavior of stuff is due to how matter affects the geometry. (though I still can only semi “grasp” what precisely G is. I get the idea of curvature (the R tensor), I get the idea of metric, but the I currently only have a semigrasp on what G actually means. (Although I think I now have a bit of a better notion than I used to). Anyways, loosely similar, am trying to understand if the fundamental forces arise similarly, rather than being “forces”, they’re more an effect of what sorts of symmetries there are, what bits of configuration space count as equivalent to other bits, etc...)
I guess I’m not enough of a theorist to answer your question: I do not know whether the symmetries alone are sufficient to produce the observed particles. My intuition says not, for the following reason: First, SU(3) symmetry is broken in the quarks; second, the Standard Model contains parameters which must be hand-tuned, including the electromagnetic/weak separation phase that gives you the massless photon and the very massive weak-force carriers. Theories which spring purely from symmetry ought not to behave like that! But this is hand-waving.
As an aside, I seem to recall that GR does not produce our universe from symmetries alone, either; there are many solutions to the equations, and you have to figure out from observation which one you’re in.
If you like, I can quote our exchange and ask some local theorists if they’d like to comment?
But GR explains (or explains away, depending on how you look at it) the force of gravity in terms of geometry. I meant “does the standard model do something similar with the gauge bosons via symmetry?”
May still leave some tunable parameters, not sure. But does the basic structure of the interactions pop straight out of the symmetries?
And yeah, I’d like that, thanks. It’s nothing urgent, just am unclear if I have the basic idea or if I have severe misconceptions.
The basic fact about quantum field theory is field-particle duality. Quantum field states can be thought of either as a wavefunction over classical field configuration space, or as a wavefunction over a space of multi-particle states. You can build the particle states out of the field states (out of energy levels of the Fourier modes), so the field description is probably fundamental. But whenever there is a quantum field, there are also particles, the quanta of the field.
In classical general relativity, particles follow geodesics, they are guided by the local curvature of space. This geometry is actually an objective, coordinate-independent property of space, though the way you represent it (e.g. the metric you use) will depend on the coordinate system. Something similar applies to the gauge fields which produce all the other forces in the Standard Model. Geometrically, they are “connections” describing “parallel transport” properties, and these connections are not solely an artefact of a coordinate system. See first paragraph here.
You will see it said that the equations of motion in a gauge field theory are obtained by taking a global symmetry and making it local. These global symmetries apply to the matter particle (which is usually a complex-valued vector): if the value of the matter vector is transformed in the same way at every point (e.g. multiplied by a unit complex number), it makes no difference to the equation of motion of the “free field”, the field not yet interacting with anything. Introducing a connection field allows you to compare different transformations at different points (though the comparison is path-dependent, depending on the path between them that you take), so now you can leave one particle’s state vector unmodified, and transform a distant particle’s vector however you want, and so long as the intervening gauge connection transforms in a compensatory fashion, you will be talking about the same physical situation. However, as the link above states, the gauge connection is not solely a bookkeeping device; there are topologically distinct gauge field states which are not equivalent under some continuously varying transformation.
It all sounds horrifically abstract, but if you follow the link above you’ll see some simple examples which may help. Anyway, the bottom line is that classical gauge field configurations do contain a coordinate-independent geometric content just as gravity does, so you can’t completely do without them; and whenever you quantize a field, you have particles.
Gravity is real, but the ‘force’ of gravity is an illusion.
What is the difference between saying gravity is a force and saying it’s a curvature of spacetime?
What is your definition of “a force” that makes it inapplicable to gravity? Is electromagnetism a force, or is it a curvature in the universe’s phase space?
I don’t know much about physics, please enlighten me...
What is the difference between saying gravity is a force and saying it’s a curvature of spacetime?
To say that gravity is a curvature of spacetime means that gravity “falls out of” the geometry of spacetime. To say that gravity is something else (e.g., a force) means that, even after you have a complete description of the geometry of spacetime, you can’t yet explain the behavior of gravity.
Isn’t it equally valid to say that the geometry of spacetime falls out of gravity? I.e., given a complete description of any one of them, you get the other for free.
What is a force by your definition? Something fundamental which can’t be explained through something else? But it seems to me that “the curvature of spacetime” is the same thing as gravity, not a separate thing that is linked to gravity by causality or even by logical necessity. They’re different descriptions of the same thing. So we can still call gravity a fundamental force, it’s not being caused by something else that exists in its own right.
What I meant is that the notion of gravity as “something that pulls on matter” goes away.
There’re a couple of concepts that’re needed to see this. First is “locality is really important”
For instance, you’re in an elevator that’s freefalling toward the earth… or it’s just floating in space. Either way, the overall average net force you feel inside is zero. How do you tell the difference? “look outside and see if there’s a planet nearby that you’re moving toward”? Excuse me? what’s this business about talking about objects far away from you?
Alternately, you’re either on the surface of the earth, or in space accelerating at 9.8m/s^2
Which one? “look outside” is a disallowed operation. It appeals to nonlocal entities.
Once again, return to you being in the box and freefalling toward the ground. What can you say locally?
Well… I’m going to appeal to Newtonian gravity briefly just to illustrate a concept, but then we’ll sort of get rid of it:
Place two test particles in the elevator, one above the other. What do you see? You’d see them accelerating away from each other, right? ie, if one’s closer to the earth than the other, then you get tidal force pulling them away from each other.
Similarly, placing them side by side, well, the lines connecting each of them to the center of the earth make a bit of an angle to each other. So you’ll see them accelerate toward each other. Again, tidal force.
From the perspective of locality, tidal force is the fundamental thing, it’s the thing that’s “right here”, rather than far away, and regular gravity is just sort of the sum (well, integral) of tidal force.
Now, let’s do a bit of a perspective jump to geometry. I’ll get back to the above in a moment. To help illustrate this, I’ll just summarize the “Parable of the Apple” from Gravitation:
Imagine you see ants crawling on the side of the apple. You see them initially seem to move parallel, then as they crawl up, you see them moving toward each other.
“hrm… they attract each other perhaps?” you suppose.So you get out your knife and you cut a bit of the apple, a thin cut a millimeter to either side of the path of one of the ants. when you peel off the bit of the apple’s skin, you find that the path is… straight!
“huh! Well, maybe it’s the other one that’s curving its path...” you think to yourself. So you go and cut out the other one’s path in the same way… and lo and behold, that one’s straight too.
“augh! WTF? what sort of witchery is this?”
The answer? You look again and you see that it’s the shape of the apple that brings those paths together, even if they’re individually straight (ie, geodesics).
Also, you might note that as the ants crawl toward the stem, as they move on the indentation near the stem, their paths seem to change a bit more… Is the stem exerting some sort of mysterious force on them?
Having learned your lesson once, you look closer, and you see that it’s simply the different curvature of the apple over there.
The shape of the apple there affects what sorts of curvature there can be nearby, etc etc.
Hrm… the behavior of the ants sounds similar to tidal force. Perhaps then the apparent gravitational “forces” are really just geometric properties. Everything in freefall moves in straight lines, it’s just that curvature changes how geodesics relate to each other.
There is one other catch: in basic GR, it’s not space that’s curved, but spacetime that’s curved. Objects in freefall have geodesic paths through spacetime.
It’s not that the earth is pulling down on you, it’s that the earth is pushing up on you, and in a local inertial reference frame, you would be accelerating downwards. If you zoom in on any small part of spacetime, it’s locally flat. (I’m ignoring stuff like zooming in so close that stuff like quantum foam may or may not show up. I’m just talking classical GR here.) Curvature could be viewed as controlling how all those locally flat bits are “stitched together”
Does that make sense? So the idea of gravity “pulling” on you goes away completely. The “force” of gravity amounts to nothing more than the geometry of spacetime.
What’s left is the Einstein field equation which tells how matter shapes spacetime. (It doesn’t control the metric or the curvature directly, but rather it controls a certain “average” or sum of certain properties of the curvature.)
Edited: I understand what you’ve said (and thanks for taking the time to write all that out!). But I’m not sure why “the concept of gravity as something that pulls on matter goes away”. Is it the case that it’s mathematically impossible to define gravity as attraction between matter and still have a correct relativistic physics? Is it impossible to generalize Newton’s law that way?
Well, it goes away in the sense that “this particular theory of physics explains gravity without directly having a ‘force’ associated with it as such”
In GR, one doesn’t see any forces locally pulling on objects. One instead sees (if one zooms in closely) objects moving on straight (geodesic) paths through spacetime. It simply happens to be that spacetime is in some cases shaped in ways that alter the relationships between nearby geodesics.
I guess it’s an attraction, sort of, but once one starts taking locality seriously, that’s not that good, is it? “don’t tell me what’s going on way over there, tell me what’s happening right here!”
There may be alternate theories, but GR itself is a geometric theory and I wouldn’t even know how to interpret the central equations as force equations. Saying “there could be other explanations” or such is a separate issue. What I meant was “In GR, once one has the geometry, nothing more needs to be said, really.” (Well, I’m skipping subtleties, stuff gets tricky in that you have shape of space affecting motion of matter, and motion of matter affecting shape of space, but yeah...)
Actually, there’s really no way for Newtonian stuff to be reasonably extended to describe GR effects without going to geometry in some form. I mean, GR predicts stuff about measured distances not quite obeying the rules that they would in flat spacetime. Measured times too, for that matter. One would have to get really creatively messy to produce a theory that is more an extension of Newtonian gravity, isn’t at all based on geometry, curvature, etc… any more than regular Newtonian gravity, yet still produces the same predictions for experimental outcomes that GR does.
It would, at best, be rather complicated and messy, I’d expect. If it’s even possible. Actually, I don’t think it is. More or less no matter what, other stuff would have to be added on that doesn’t at all even resemble Newtonian stuff.
I think the Wikipedia page on Gravitomagnetism might be relevant; it seems to be an approximation to GR that looks an awful lot like classical electromagnetism.
Incidentally, what about electromagnetism and the other fundamental forces? Can they be described the same way as gravity? In classical mechanics they’re the same kind of thing as gravity, except they can be repulsive as well. And a lot of popsci versions of modern physics research seems to postulate the same kind of properties for gravity as we know from electromagnetism: like repulsive gravity, or gravitational shields, or effects due to gravitational waves propagating at speed of light, or artificial gravity. And all forces are related through inertial mass.
So is there a description of all these things, including gravity, in the same terms? Either all of them “forces” or fields with mediating particles, or all of them affecting some kind of geometry?
Scott Aaronson has a nice post about the differences between gravity and electromagnetism. It seems his thoughts were running along the same lines as yours when he wrote it; he asks almost all the same questions. http://www.scottaaronson.com/blog/?p=244
Gravity waves come straight out of GR. (Actually, weak gravity waves show up in the linearized theory (the linearized theory of GR being a certain approximation of it that’s easier to deal with, good for low energies and such))
And that was part of what I was asking about. Well, others have tried to find that sort of thing, but I was asking something like “in the standard model and such, are the forces really aspects of what would amount to the geometry (specifically the symmetries) of configuration space rather than additional dimensions in the config space?”
And, of course, one of the BIG questions for modern physics is how to get a quantum description of gravity or to otherwise find a model of reality which contains both QM and GR in a “natural” way.
So, basically, at this point, all I can say is “I don’t really know.” :)
(well, also, I guess depending on how you look at it, curvature either explains or explains away tidal force. It explains the effects/behaviors, but explains away any apparent “forces” being involved.)
...but forces fall out of something—electromagnetic interactions, for example. As an engineer, I am inclined to call something a force if it goes on the “force” side of the equation in the domain I’m modeling, and not worry about whether to call it “real”.
(Then again, as an engineer, I rarely need to exceed Newtonian mechanics.)
Ok, I’m not touching the ECE thing; as noted, I’m not a theorist. I just measure stuff. I’ve taken classes in formal QFT, but I don’t use it day-to-day, so it’s a weak point for me. However, it seems a bit odd to describe things that can be produced in collisions and (at least in principle) fired at your enemies to kill them by radiation poisoning as ‘illusory’. If you bang two electrons together, measuring the cross-section as a function of the center-of-mass energy, you will observe a classic 1/s decline interrupted by equally classic resonance bumps. That is, at certain energies the electrons are much more likely to interact with each other; that’s because those are the energies that are just right for producing other particles. Increase the CM energy through 80 GeV or so, and you’ll find a Breit-Wigner shape like any other particle; that’s the W, and if it weren’t so short-lived you could make a beam of them to kill your enemies. (With asymmetric electron energies you can produce a relativistic-speed W and get arbitrarily long lifetimes in the lab frame, but that gets on for being difficult engineering. In fact, just colliding two electrons at these energies is difficult, they’re too light; that’s why CERN used an electron and a proton in LEP.)
Now, returning to the math, my memory of this is that particles appear as creation and annihilation operators when field theories with particular gauge symmetries are quantized. If you want to call the virtual particles that appear in Feynmann diagrams illusory, I won’t necessarily argue with you; they are just a convenient way of expressing a huge path integral. But the math doesn’t spring fully-formed from Feynmann’s brow; the particular gauge symmetry that is quantised is chosen such that it describes particles or forces already known to exist. (Historically, forces, since the theory ran ahead of the experiments in the sixties—we saw beta decay long before we saw actual W bosons.) If the forces were different, the theorists would have chosen a different gauge symmetry and got out a different set of particles.
I’m not sure if I’m answering your question, here? My basic approach to QFT has always been “shut up and calculate”, not because of QM confusion but because I find it very confusing when someone says that a particular mathematical operation is “causing” something. I prefer to think of the causality as flowing from the observations, so that the sequence is thus:
We observe these forces / cross-sections / particles.
We know that by quantising field theories with gauge symmetries, we can get things that look very much like particles.
Searching through gauge-symmetry space, we find that this one gives us the particles and forces we observe.
I wasn’t bringing up the ECE thing.
I meant illusory in the same sense that “sure, the force of gravity can cause me to fall down and get ouchies… but by a bit of a coordinate change and so on, we can see that there really is no ‘force’, but instead that it’s all just geometry and curvature and such. Gravity is real, but the ‘force’ of gravity is an illusion. There’s a deeper physical principle that gives rise to the effect, and the regular ‘force’ more or less amounts to summing up all the curvature between here and there.”
My understanding was that gauge bosons are similar “we observe this forces/fields/etc… but actually, we don’t need to explicitly postulate those fields as existing. Instead, we can simply state that these other fields obey these symmetries, and that produces the same results. Obviously, to figure out which symmetries are the ones that actually are valid, we have to look at how the universe actually behaves”
ie, my understanding is that if you deleted from your mind the knowledge of the electromagnetic and nuclear forces and instead just knew about the quark and lepton fields and the symmetries they obeyed, then the forces of interaction would automatically “pop out”. One would then see behaviors that looks like photons, gluons, etc, but the total behavior can be described without explicitly adding them to the theory, but simply taking all the symmetries of the other stuff into account when doing the calculations.
That’s what I was asking about. Is this notion correct, or did I manage to critically fail to comprehend something?
And thanks for taking the time to explain this, btw. :) (I’m just trying to figure out if I’ve got a serious misconception here, and if so, to clear it up)
I guess you can think of it that way, but I don’t quite see what it gains you. Ultimately the math is the only description that matters. Whether you think of gravity as being a force or a curvature is just words. When you say “there is no force, falling is caused by the curvature of space-time” you haven’t explained either falling or forces, you’ve substituted different passwords, suitable for a more advanced classroom. The math doesn’t explain anything either, but at least it describes accurately. At some point—and in physics you can reach that point surprisingly fast—you’re going to have to press Ignore (being careful to avoid Worship, thanks), at least for the time being, and concentrate on description rather than explanation.
Well, my question could be viewed as about the math. ie: “does the math of the standard model have the property that if you removed any explicit mention of electromagnetism, strong force, or weak force and just kept the quark and lepton fields + the math of the symmetries for those, would that be sufficient for it to effectively already contain EM, strong, and weak forces?”
And as far as gravity being force or geometry, uh… there’s plenty of math associated with that. I mean, how would one even begin to talk about the meaning of the Einstein field equation without interpreting it in terms of geometry?
Perhaps there is a deeper underlying principle that gives rise to it, but the Einstein field equation is an equation about how matter shapes the geometry of spacetime. There’s no way really (that I know of) to reasonably interpret it as a force equation, although one can effectively solve it and eventually get behaviors that Newtonian gravity approximates (at low energies/etc...)
(EDIT: to clarify, I’m trying to figure out how to semivisualize this. ie, with gravity and curvature, I can sorta “see” and get the idea of everything’s just moving in geodesics and the behavior of stuff is due to how matter affects the geometry. (though I still can only semi “grasp” what precisely G is. I get the idea of curvature (the R tensor), I get the idea of metric, but the I currently only have a semigrasp on what G actually means. (Although I think I now have a bit of a better notion than I used to). Anyways, loosely similar, am trying to understand if the fundamental forces arise similarly, rather than being “forces”, they’re more an effect of what sorts of symmetries there are, what bits of configuration space count as equivalent to other bits, etc...)
I guess I’m not enough of a theorist to answer your question: I do not know whether the symmetries alone are sufficient to produce the observed particles. My intuition says not, for the following reason: First, SU(3) symmetry is broken in the quarks; second, the Standard Model contains parameters which must be hand-tuned, including the electromagnetic/weak separation phase that gives you the massless photon and the very massive weak-force carriers. Theories which spring purely from symmetry ought not to behave like that! But this is hand-waving.
As an aside, I seem to recall that GR does not produce our universe from symmetries alone, either; there are many solutions to the equations, and you have to figure out from observation which one you’re in.
If you like, I can quote our exchange and ask some local theorists if they’d like to comment?
But GR explains (or explains away, depending on how you look at it) the force of gravity in terms of geometry. I meant “does the standard model do something similar with the gauge bosons via symmetry?”
May still leave some tunable parameters, not sure. But does the basic structure of the interactions pop straight out of the symmetries?
And yeah, I’d like that, thanks. It’s nothing urgent, just am unclear if I have the basic idea or if I have severe misconceptions.
It is a while since I thought about this. But…
The basic fact about quantum field theory is field-particle duality. Quantum field states can be thought of either as a wavefunction over classical field configuration space, or as a wavefunction over a space of multi-particle states. You can build the particle states out of the field states (out of energy levels of the Fourier modes), so the field description is probably fundamental. But whenever there is a quantum field, there are also particles, the quanta of the field.
In classical general relativity, particles follow geodesics, they are guided by the local curvature of space. This geometry is actually an objective, coordinate-independent property of space, though the way you represent it (e.g. the metric you use) will depend on the coordinate system. Something similar applies to the gauge fields which produce all the other forces in the Standard Model. Geometrically, they are “connections” describing “parallel transport” properties, and these connections are not solely an artefact of a coordinate system. See first paragraph here.
You will see it said that the equations of motion in a gauge field theory are obtained by taking a global symmetry and making it local. These global symmetries apply to the matter particle (which is usually a complex-valued vector): if the value of the matter vector is transformed in the same way at every point (e.g. multiplied by a unit complex number), it makes no difference to the equation of motion of the “free field”, the field not yet interacting with anything. Introducing a connection field allows you to compare different transformations at different points (though the comparison is path-dependent, depending on the path between them that you take), so now you can leave one particle’s state vector unmodified, and transform a distant particle’s vector however you want, and so long as the intervening gauge connection transforms in a compensatory fashion, you will be talking about the same physical situation. However, as the link above states, the gauge connection is not solely a bookkeeping device; there are topologically distinct gauge field states which are not equivalent under some continuously varying transformation.
It all sounds horrifically abstract, but if you follow the link above you’ll see some simple examples which may help. Anyway, the bottom line is that classical gauge field configurations do contain a coordinate-independent geometric content just as gravity does, so you can’t completely do without them; and whenever you quantize a field, you have particles.
What is the difference between saying gravity is a force and saying it’s a curvature of spacetime?
What is your definition of “a force” that makes it inapplicable to gravity? Is electromagnetism a force, or is it a curvature in the universe’s phase space?
I don’t know much about physics, please enlighten me...
To say that gravity is a curvature of spacetime means that gravity “falls out of” the geometry of spacetime. To say that gravity is something else (e.g., a force) means that, even after you have a complete description of the geometry of spacetime, you can’t yet explain the behavior of gravity.
Isn’t it equally valid to say that the geometry of spacetime falls out of gravity? I.e., given a complete description of any one of them, you get the other for free.
What is a force by your definition? Something fundamental which can’t be explained through something else? But it seems to me that “the curvature of spacetime” is the same thing as gravity, not a separate thing that is linked to gravity by causality or even by logical necessity. They’re different descriptions of the same thing. So we can still call gravity a fundamental force, it’s not being caused by something else that exists in its own right.
What I meant is that the notion of gravity as “something that pulls on matter” goes away.
There’re a couple of concepts that’re needed to see this. First is “locality is really important”
For instance, you’re in an elevator that’s freefalling toward the earth… or it’s just floating in space. Either way, the overall average net force you feel inside is zero. How do you tell the difference? “look outside and see if there’s a planet nearby that you’re moving toward”? Excuse me? what’s this business about talking about objects far away from you?
Alternately, you’re either on the surface of the earth, or in space accelerating at 9.8m/s^2
Which one? “look outside” is a disallowed operation. It appeals to nonlocal entities.
Once again, return to you being in the box and freefalling toward the ground. What can you say locally?
Well… I’m going to appeal to Newtonian gravity briefly just to illustrate a concept, but then we’ll sort of get rid of it:
Place two test particles in the elevator, one above the other. What do you see? You’d see them accelerating away from each other, right? ie, if one’s closer to the earth than the other, then you get tidal force pulling them away from each other.
Similarly, placing them side by side, well, the lines connecting each of them to the center of the earth make a bit of an angle to each other. So you’ll see them accelerate toward each other. Again, tidal force.
From the perspective of locality, tidal force is the fundamental thing, it’s the thing that’s “right here”, rather than far away, and regular gravity is just sort of the sum (well, integral) of tidal force.
Now, let’s do a bit of a perspective jump to geometry. I’ll get back to the above in a moment. To help illustrate this, I’ll just summarize the “Parable of the Apple” from Gravitation:
Imagine you see ants crawling on the side of the apple. You see them initially seem to move parallel, then as they crawl up, you see them moving toward each other.
“hrm… they attract each other perhaps?” you suppose.So you get out your knife and you cut a bit of the apple, a thin cut a millimeter to either side of the path of one of the ants. when you peel off the bit of the apple’s skin, you find that the path is… straight!
“huh! Well, maybe it’s the other one that’s curving its path...” you think to yourself. So you go and cut out the other one’s path in the same way… and lo and behold, that one’s straight too.
“augh! WTF? what sort of witchery is this?”
The answer? You look again and you see that it’s the shape of the apple that brings those paths together, even if they’re individually straight (ie, geodesics).
Also, you might note that as the ants crawl toward the stem, as they move on the indentation near the stem, their paths seem to change a bit more… Is the stem exerting some sort of mysterious force on them?
Having learned your lesson once, you look closer, and you see that it’s simply the different curvature of the apple over there.
The shape of the apple there affects what sorts of curvature there can be nearby, etc etc.
Hrm… the behavior of the ants sounds similar to tidal force. Perhaps then the apparent gravitational “forces” are really just geometric properties. Everything in freefall moves in straight lines, it’s just that curvature changes how geodesics relate to each other.
There is one other catch: in basic GR, it’s not space that’s curved, but spacetime that’s curved. Objects in freefall have geodesic paths through spacetime.
It’s not that the earth is pulling down on you, it’s that the earth is pushing up on you, and in a local inertial reference frame, you would be accelerating downwards. If you zoom in on any small part of spacetime, it’s locally flat. (I’m ignoring stuff like zooming in so close that stuff like quantum foam may or may not show up. I’m just talking classical GR here.) Curvature could be viewed as controlling how all those locally flat bits are “stitched together”
Does that make sense? So the idea of gravity “pulling” on you goes away completely. The “force” of gravity amounts to nothing more than the geometry of spacetime.
What’s left is the Einstein field equation which tells how matter shapes spacetime. (It doesn’t control the metric or the curvature directly, but rather it controls a certain “average” or sum of certain properties of the curvature.)
Edited: I understand what you’ve said (and thanks for taking the time to write all that out!). But I’m not sure why “the concept of gravity as something that pulls on matter goes away”. Is it the case that it’s mathematically impossible to define gravity as attraction between matter and still have a correct relativistic physics? Is it impossible to generalize Newton’s law that way?
Well, it goes away in the sense that “this particular theory of physics explains gravity without directly having a ‘force’ associated with it as such”
In GR, one doesn’t see any forces locally pulling on objects. One instead sees (if one zooms in closely) objects moving on straight (geodesic) paths through spacetime. It simply happens to be that spacetime is in some cases shaped in ways that alter the relationships between nearby geodesics.
I guess it’s an attraction, sort of, but once one starts taking locality seriously, that’s not that good, is it? “don’t tell me what’s going on way over there, tell me what’s happening right here!”
There may be alternate theories, but GR itself is a geometric theory and I wouldn’t even know how to interpret the central equations as force equations. Saying “there could be other explanations” or such is a separate issue. What I meant was “In GR, once one has the geometry, nothing more needs to be said, really.” (Well, I’m skipping subtleties, stuff gets tricky in that you have shape of space affecting motion of matter, and motion of matter affecting shape of space, but yeah...)
Actually, there’s really no way for Newtonian stuff to be reasonably extended to describe GR effects without going to geometry in some form. I mean, GR predicts stuff about measured distances not quite obeying the rules that they would in flat spacetime. Measured times too, for that matter. One would have to get really creatively messy to produce a theory that is more an extension of Newtonian gravity, isn’t at all based on geometry, curvature, etc… any more than regular Newtonian gravity, yet still produces the same predictions for experimental outcomes that GR does.
It would, at best, be rather complicated and messy, I’d expect. If it’s even possible. Actually, I don’t think it is. More or less no matter what, other stuff would have to be added on that doesn’t at all even resemble Newtonian stuff.
I think the Wikipedia page on Gravitomagnetism might be relevant; it seems to be an approximation to GR that looks an awful lot like classical electromagnetism.
OK, now I understand better, thanks :-)
Incidentally, what about electromagnetism and the other fundamental forces? Can they be described the same way as gravity? In classical mechanics they’re the same kind of thing as gravity, except they can be repulsive as well. And a lot of popsci versions of modern physics research seems to postulate the same kind of properties for gravity as we know from electromagnetism: like repulsive gravity, or gravitational shields, or effects due to gravitational waves propagating at speed of light, or artificial gravity. And all forces are related through inertial mass.
So is there a description of all these things, including gravity, in the same terms? Either all of them “forces” or fields with mediating particles, or all of them affecting some kind of geometry?
Scott Aaronson has a nice post about the differences between gravity and electromagnetism. It seems his thoughts were running along the same lines as yours when he wrote it; he asks almost all the same questions. http://www.scottaaronson.com/blog/?p=244
That was very interesting and relevant. Thanks.
Gravity waves come straight out of GR. (Actually, weak gravity waves show up in the linearized theory (the linearized theory of GR being a certain approximation of it that’s easier to deal with, good for low energies and such))
And that was part of what I was asking about. Well, others have tried to find that sort of thing, but I was asking something like “in the standard model and such, are the forces really aspects of what would amount to the geometry (specifically the symmetries) of configuration space rather than additional dimensions in the config space?”
And, of course, one of the BIG questions for modern physics is how to get a quantum description of gravity or to otherwise find a model of reality which contains both QM and GR in a “natural” way.
So, basically, at this point, all I can say is “I don’t really know.” :)
(well, also, I guess depending on how you look at it, curvature either explains or explains away tidal force. It explains the effects/behaviors, but explains away any apparent “forces” being involved.)
...but forces fall out of something—electromagnetic interactions, for example. As an engineer, I am inclined to call something a force if it goes on the “force” side of the equation in the domain I’m modeling, and not worry about whether to call it “real”.
(Then again, as an engineer, I rarely need to exceed Newtonian mechanics.)