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.)
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.)