Electromagnetic Field & Space-Time: Relationship Explained

[crastinus]

What is the relationship between the electromagnetic field and space-time? I am basically assuming that space-time is one big gravitational field.

Is there a relationship between space-time and the field (I presume) created by the strong force (however negligible it may be at any significant distance)?

I am in particular wondering whether there is one big EM field that is everywhere in space or something else. I googled around and got only some papers that were way beyond me.

Thanks!

 

[ergospherical]

Don't assume things.

Spacetime is a manifold together with a metric, and the electromagnetic field is an antisymmetric tensor field $F_{ab}$ defined on the manifold. The stress tensor $T_{ab}$ depends on $F_{ab}$ and is related to the curvature of spacetime by Einstein's equation.

 

[crastinus]

Assumptions are so much fun, though! :)

 

[crastinus]

I can honestly say that I'm still trying to decipher what you said.

I summoned up Feynman's ghost to interpret for me. He says this:

"The guy is basically saying: Spacetime is the sheet of rubber and the EM field is the static on the sheet. Things move sometimes because the sheet is curved and sometimes because of the static and very often because of both!"

Snarky replies welcome, of course. (But they have to use LaTeX!)

 

[Ibix]

There have been attempts to make electromagnetism a part of spacetime geometry - Kaluza-Klein theory is one well known one. Unfortunately they've never worked - Kaluza-Klein successfully combines gravity and electromagnetism, but predicts an extra field that would be glaringly obvious if it existed.

Spacetime is the "background" on which the other fields do their thing. Don't take that Feynman quote too literally, but it's a decent enough attempt at explaining the distinction.

 

[Vanadium 50]

I can honestly say that I'm still trying to decipher what you said.

The intent of the B/I/A header is to try and avoid that, and to provide answers that don't need deciphering. As you saw, results are mixed.

I am basically assuming that space-time is one big gravitational field.

Don't think of space-time (or space, or time) as "made up" of anything. Any correct insight that this provides is more than compensated for by the number of wrong conclusions you can draw. It's better to think of it as a way of labeling where and when events happen (x,y,z and t). But the events matter more than the labeling.

I am in particular wondering whether there is one big EM field that is everywhere in space

A field is something that has a value at every point in space and time. That value may be zero. But the field is not the coordinate system used to label events.

 

[ergospherical]

@Vanadium 50 it's wrong to think of spacetime as a way of labelling where and when events happen, because there is not a unique way to cover $M$ with charts $(\mathscr{U}_{\alpha}, \phi_{\alpha} )$; the only concepts defined by the manifold structure are those independent of any coordinates.

 

[romsofia]

it's wrong to think of spacetime as a way of labelling where and when events happen

No it isn't, that's exactly WHAT spacetime is. It's literally a mathematical model that labels all events. If you think of spacetime as anything else, then I'd love to hear it.

 

[PeterDonis]

No it isn't, that's exactly WHAT spacetime is. It's literally a mathematical model that labels all events.

No, that's not spacetime, that's a coordinate chart (and as has already been pointed out, there is no one unique coordinate chart for any given spacetime geometry). Spacetime is the geometric object in which the events are points; but the events don't come with any intrinsic "labeling" by numbers. If something of interest happens at an event (e.g., "observer A received a light signal from observer B"), you can use that happening as a way of referencing the event, but not all events (points in spacetime) have something of interest happening at them.

 

[crastinus]

To help myself understand the difficulty here, since I'm the OP, I think the question discussed in the last coupled of posts is this:

"Is spacetime itself a map or is it the thing that gets mapped by us in some way?"

Is that the question you guys are discussing? Or something else?

What is the difference between being a manifold and being represented by a manifold? (I do have some notion of what a manifold is, by the way.)

 

[Ibix]

The way GR models gravity is to propose that all events (an event is a place at a particular time) form a manifold, one with a metric with a particular signature and whose curvature is directly or indirectly responsible for all the phenomena we call gravity. Typically we label all events in a systematic way, assigning four numbers we call coordinates to each one.

The argument seems to be whether this process of labelling events is part of the definition of spacetime or not. I'd go with no - spacetime is all events and the labelling is an extra system that we impose because it's useful.

To answer your last question, the difference between being a manifold and being represented by a manifold is philosophical. Formally, we build mathematical models and associate entities in those models with physical experience. So the mathematical object, the manifold, represents whatever spacetime is. I think this is the correct formalism, not least because someone may come along with a theory of quantum gravity in which spacetime is modeled by something else. Spacetime doesn't change when we revise our models, but the representation can do.

On the other hand, "spacetime is a manifold" is a lot shorter to type. You should just keep in the back of your mind the rather pedantic thought that any "x is a y" statement in science would be more precisely stated as "in our model x is represented by a y".

 

[PeterDonis]

I think the question discussed in the last coupled of posts is this:

"Is spacetime itself a map or is it the thing that gets mapped by us in some way?"

Yes. The answer is that spacetime is the thing that gets mapped; coordinate charts are the maps.

 

[romsofia]

Is that the question you guys are discussing? Or something else?

It's basically that, but the caveat is that they'd rather blast you with math which isn't physical. Coordinate charts aren't physical. Your question was a physical one in nature, and I took issue to how it was being approached. I, also, philosophically differ with them. This has been my stance since my first GR course, and it's a hill I die on, but that's for another thread.

Now, I realize it wasn't the time nor place for it, so in the spirt of PF I will answer your OP question to the best of my ability, and if you're more into the math-y answers than by all means tell me that, and I'll move on.

What is the relationship between the electromagnetic field and space-time?

If you view this from the lens of general relativity, and I'm assuming you are since this is the relativity forum, the relationship between the E+M field and space-time can be described in the concept known as "minimal coupling". It's *mainly* a math procedure, but the result of it is physical (if you go through the procedure for the given field, this will always satisfy the equivalence principle, and general covariance). If you take your well known E+M relationships (those on a minkowski (flat) space time), and try to generalize them (riemannian (potentially curved) spacetime), what you'll see is that the relationships between your electromagnetic fields and potential don't change, and neither do your conservation laws.

But what DOES change? The dynamics of your electromagnetic fields. In other words, your Maxwell Equations become dependent on the geometry of your spacetime (seems kind of obvious, eh?).

Is there a relationship between space-time and the field (I presume) created by the strong force (however negligible it may be at any significant distance)?

No, the relationship is simply that you take spacetime to be fundamental in general relativity. Spacetime exists, and things get coupled to the geometry. You can then talk about those relationships, but first you must ask "how does the (potential) geometry effect my dynamical equations?". As far as I'm aware, there is no physical system that doesn't care about the curvature/geometry of spacetime.

I am in particular wondering whether there is one big EM field that is everywhere in space or something else. I googled around and got only some papers that were way beyond me.

Yes Sir! That's how you should think about it. However, if you want to be a relativist, you first have to think of your spacetime, then your E+M field comes next.

I do apologize for the inconvenience I have caused by having your thread shut down for a few days, as your question is a fun one to ponder when starting out in GR!

 

[PeterDonis]

the caveat is that they'd rather blast you with math which isn't physical.

You were also doing that. See below.

(Note, btw, that the overly mathematical posts have been deleted from this thread.)

Coordinate charts aren't physical.

Exactly. Which means that you, in your previous posts, by talking about how events are labeled, were focusing on the non-physical thing--the coordinates--instead of the physical thing--spacetime itself, the set of events. That's what you got pushback about.

Your latest post does focus on the actual physical things instead of the coordinates, which is good.