spacetime curvature for non-gravitational forces

[gib65]

GR explains gravity by the warping of spacetime near massive objects. I've always wondered if there was a way of explaining other forces in this way. For example, could the electromagnetic force be explained by spacetime distortions at the level of particles? Every thought experiment I've tried out doesn't seem to hold promise of this. For example, we could explain why a proton is attracted to an electron by assuming that time and space curve towards each of these particles, and the result would be much like an object falling towards a planet. But then all particles should be attracted to each other. Why would putting two electrons together result in their mutually repelling each other? Does spacetime curvature figure at all into any forces other than gravity?

[swansont]

I think spacetime curvature works because gravitational mass and inertial mass are (as far as we know) identical. That is, for a particular geometry, you have a certain acceleration. But that's not the case for e.g. a proton vs a positron in a given electric field. Same force, but different acceleration.

[gib65]

Really? So the proton is the heavier one, right? It accelerates more slowly than a positron in the same electric field?

[ajb]

Electromagnetism is explained in terms very similar to general relativity. The notions of a connection and curvature are more general that found in general relativity.

 

It is hard to explain here and requires many notions from modern differential geometry. But the bottom line is that EM (and Yang-Mills theories) is a geometric theory.

 

I have written a little on this subject as part of my PhD. You can find more details on my website.

[fredrik]

Like ajb mentions, generalized geometrical methods are very popular in physics, supposedly because it can sometimes be given intuitive geometric interpretations, and since it's well studied by mathematicians. So once you've got a geometric formulation, you've got an arsenal of theorems from mathematics to play with.

 

However I wouldn't say that alone "explains" anything in it's proper fundamental sense. It seems many physicists are guided by mathematical beauty and some like geometry and may think that a geometric model are more likely to described nature than anything else. I don't share that view. You can certainly often formulate the same information in several more or less equivalent ways. Pick the one of your choice, but either way it needs qualifying experimental support.

 

There are also geometric interpretations of probability theory, where probability distributions can be exploited to define distance in distribution space (information space). So there seems to be a few equivalent ways to described things. It would mean that the distance between two points, is associated with the probability that they are mixed up. So statistical models can actually map out geometries too if you prefer that interpretation.

 

But some of the arrived geometries are abstract one, and not something as universal as space.

 

/Fredrik

[swansont]

Really? So the proton is the heavier one, right? It accelerates more slowly than a positron in the same electric field?

 

Yes

 

a = qE/m

[gib65]

Electromagnetism is explained in terms very similar to general relativity. The notions of a connection and curvature are more general that found in general relativity.

 

It is hard to explain here and requires many notions from modern differential geometry. But the bottom line is that EM (and Yang-Mills theories) is a geometric theory.

 

I have written a little on this subject as part of my PhD. You can find more details on my website.

 

I would love to read your paper, ajb, and in fact I took a look at it. As I feared, however, its speaks at a level a little too indepth more me to understand (heavy math always screws me up). I really need to take a whole battery of math courses before I can understand this stuff. Until then, I'm stuck with conceptual models .

 

a = qE/m

 

m is mass, E is energy, right? What is q?

[timo]

Electric charge.

EDIT: D'oh, should read more than the last three words.

[fredrik]

I would love to read your paper, ajb, and in fact I took a look at it. As I feared, however, its speaks at a level a little too indepth more me to understand (heavy math always screws me up). I really need to take a whole battery of math courses before I can understand this stuff. Until then, I'm stuck with conceptual models .

 

When you talk to different people, of different background, or even similar backgrounds but different mindsets it's clear that everyone has their own way of preferred thinking. Some people are very mathematical and if you ask something philosophical they don't understand it. But there is also the opposite case.

 

Here is one of those classic engineering jokes...

 

"Engineers think that equations approximate the real world.

Scientists think that the real world approximates equations.

Mathematicians are unable to make the connection."

 

Gib, I've seen your great philosophical questions in various threads. Check this paper out and tell me if it makes any sense, or triggers any ideas in your mind? no?

 

"Change, Time and Information Geometry"

http://arxiv.org/PS_cache/math-ph/pdf/0008/0008018v1.pdf

 

It does contain a few basic equations, but there are also some text that I think is fairly readable. Information geometry is IMO yields a very intuitive understanding of geometry, even in the abstract case where visualisations stall.

 

The basic concept is the association of "shortest path" (straight line) with a sort of minimum transition probability or simply the "most probable path". The obvious question is of course what the measure is to define "most probable". But one can IMO find some very plausible argumentation for that. But I figure it's a matter of mindset if you like it or not. It talks about maximum entropy principles. And one obvious objection may be that entropy can be defined in different ways. But it can be shown that the exact definition/choice of this measure doesn't matter within limits. Also, there is seemingly a way to derive this principle without touching the concept of entropy in the first place. I am currenly working on that but I don't yet have anything readable. But it will come. I am also curious to see what Ariel Caticha will come with. It seems he is still working on deriving GR from these principles to make the intuitive connection explicit. Check out his other papers too. I was told he is currently writing a book on information physics.

 

If you got a basic idea of probability theory in the context of learning (which can be very intuitive in the first place) they might excite your imagination. You may not that alot of stuff is missing though, which is true. But it seems not many papers to find on this.

 

/Fredrik

[swansont]

m is mass, E is energy, right? What is q?

 

E is electric field. q is charge

[gib65]

Gib, I've seen your great philosophical questions in various threads. Check this paper out and tell me if it makes any sense, or triggers any ideas in your mind? no?

 

"Change, Time and Information Geometry"

http://arxiv.org/PS_cache/math-ph/pdf/0008/0008018v1.pdf

 

Thanks, fredrik. I like the brevity of that paper .

 

You know, this discussion (about the relation between math and the real world) has inspired me to start a new thread. I wrote a paper a looong time ago about this very issue in which I actually went through, step-by-step, the derivation of the formula for the standard deviation of the bell curve you often see in statistics and social sciences (or was it variance?). Starting with the elementary variables, I described what each step represented conceptually, going into great detail, until I got to the the overall formula (sometimes I wonder if I have way too much time on my hands ). If I find that paper, I'll probably start a new thread, not so much about the bell curve and what each step of the derivation represents, but about something I've always felt the discipline of mathematics needs: a branch that actually tries to tie each step in the derivation of the most common formuli in the sciences to easy-to-understand concepts. We already do this for the individual variables we start out with, and for what the formula calculates overall, but when you get half way through your caclulations, you're completely in math-land where all sight of the world of concepts is gone. Wouldn't it be nice if there was some kind of standard lookup table or reference book that explained exactly where you were conceptually at every step along the way? Not only would this lay the groundwork for potential new discoveries IMO, but it would bring a whole chunk of the educated public into the inner circles of the mathematical and scientific community. A lot more people would be enabled to understand what's going on.

 

Oh, and thanks for the kind words.

[Farsight]

I've got it all worked out. But I've decided not to share it for now. Sorry. An electric field is not at all like a gravitational field.

 

Swanson: gravitational mass and inertial mass are not identical. A photon has energy but it has no inertial mass. It's mass/energy that causes gravity, not mass.

[swansont]

Swanson: gravitational mass and inertial mass are not identical. A photon has energy but it has no inertial mass. It's mass/energy that causes gravity, not mass.

 

And what I wrote doesn't contradict this, as I was not talking about the source of the field, but rather the behavior of an object with mass in a gravitational field. But I think that that really has more to do with the equivalence principle than the geometry.