High School Do EFEs Physically Describe Something?

[Megaton]

Hi I'm new to this so please don't butcher me. I am just a enthusiastic individual with a huge interest in theoretical physics.

In GR, the Einstein field equations relate energy and momentum (I think) to curvature, then to local flat geometry (I think) my question is do the EFEs physically describe something (ie: a rotating spherical object like Earth or a large nebula etc...) or are they a purely mathematical tool used to describe local geometry in relation to energy and momentum. I know the geodesic equation is needed to explain the "force" of gravity which results, I'm just wondering if they also to describe the topology as well.

I have always found this confusing, and can not seem to get a clear answer on it, thanks to anyone would cold help.

 

[PeterDonis]

the Einstein field equations relate energy and momentum (I think) to curvature

More precisely, they relate stress-energy, which includes energy and momentum (density) but also includes pressure and other stresses, to a portion of curvature, the Einstein tensor.

then to local flat geometry

I'm not sure what you mean by this. The fact that spacetime can be considered flat locally is not due to the EFE (although the EFE is certainly consistent with it).

do the EFEs physically describe something (ie: a rotating spherical object like Earth or a large nebula etc...)

Solutions to the EFE, i.e., metric tensors, physically describe spacetimes that can contain various kinds of gravitating objects, like planets, stars, black holes, etc.

I know the geodesic equation is needed to explain the "force" of gravity

The geodesic equation is not the same as the EFE.

i'm just wondering if they also to describe the topology as well.

The topology is something else again; topology is not the same as geometry.

 

[Ibix]

You can think of the Field Equations as like Newton's law of gravity (note: the EFEs aren't quite equivalent to the Newtonian force equation - but it'll do for this explanation). Both tell you how gravity works in a general sense, but you need to feed in specific details in order to describe a specific situation. So, for example, to describe the solar system in Newtonian gravity you would write the gravitational force on a small mass m as
$$\vec F=-Gm\left (
\frac {M_{sun}}{|\vec r- \vec r_{sun}|^3}(\vec r- \vec r_{sun})+
\frac {M_{mercury}}{|\vec r- \vec r_{mercury}|^3}(\vec r- \vec r_{mercury})+
\frac {M_{venus}}{|\vec r- \vec r_{venus}|^3}(\vec r- \vec r_{venus})+\ldots\right) $$
plus terms for all other planets, moons, etc. Which is rather more complicated than $F=GMm/r^2$.

The EFEs are (mathematically) worse than Newtonian gravity for a number of reasons. First, as you and Peter noted, there are more things than just mass that go into the "source" term. Secondly, the equations are non-linear, so the solution for mass 1 plus the solution for mass 2 is not the same as the solution for mass 1 plus mass 2. The combination of the two (and other factors, like an extra dimension) is why I can write the Newtonian solution for multiple point masses off the top of my head but you need computer support to do the same thing for the full GR solution.

As Peter says, topology and geometry are different things. A good example is the Asteroids computer game. The whole thing takes place in flat space described by Euclidean geometry. If you couldn't move your spaceship off the edge of the screen, that would be a finite bounded Euclidean space - the topology and geometry of a sheet of paper. But the game let's you move off one edge of the screen and on to the opposite edge. The geometry is the same (triangles have 180 degree interior angles, circles have $c=2\pi r$ etc.), but the topology is toroidal - like the surface of a donut.

 

[pervect]

Hi I'm new to this so please don't butcher me. I am just a enthusiastic individual with a huge interest in theoretical physics.

In GR, the Einstein field equations relate energy and momentum (I think) to curvature, then to local flat geometry (I think) my question is do the EFEs physically describe something (ie: a rotating spherical object like Earth or a large nebula etc...) or are they a purely mathematical tool used to describe local geometry in relation to energy and momentum. I know the geodesic equation is needed to explain the "force" of gravity which results, I'm just wondering if they also to describe the topology as well.

I have always found this confusing, and can not seem to get a clear answer on it, thanks to anyone would cold help.

GR as a theory definitely makes experimental predictions. I'd go so far as to call them "physical" predictions, though it's not quite clear what that term means in abstract or to you.

I believe I recall reading that one doesn't need any more assumptions than the EFE to get the complete theory of GR, but perhaps I'm missing some small seemingly innocuous assumptions. For instance, I think I recall it being said that one doesn't need to assume geodesic motion of test particles as a separate assumption, but I also recall that there may be some seemingly innocuous assumptions required to prove that, related to the positivity of the energy density. I'm not sure if such seemingly fine points are of interest to you, though I suspect you're just looking for the "big picture".

The big picture would be that if you have a ball of matter, like a rotating Earth, you can get experimental predictions out of the math, for instance the "frame dragging" results of GPB, and other, less subtle, predictions.

You might find some situations where you would require information about the properties of the matter itself which is separate from the EFE's to make those predictions, and there might be some situations where the information is not known for a certanity. The structure of neutron stars would, I think, fall into this category, there are some reasonable theories about them, but I believe there may be some uncertanities in their structure relating not to the EFE's, but to the "physical" properties of neutron degenerate matter itself.

Sorry if this isn't an exact answer, but the question is general, and a lot depends on whether your trying to get a razor-sharp answer or just a general "feel".

 

[Megaton]

Thanks for your prompt answers that helps a lot.

I am currently studying philosophy and have a weird obsession with finding a Theory of everything. I understand that GR and QM (plus many other factors) are required to explain a TOE, and am curious to see what real world life applications a TOE would bring.

Understanding the overall picture of Theoretical Physics and Physical Cosmology are essential to understanding a TOE, which is what I am trying to do.

Thanks for all your help