Scalar Curvature, R, for Dummies

[Farina]

Is it possible to explain, in one or two paragraphs, what the scalar curvature, R, is as it applies to General Relativity (the Einstein Field Equation, specifically?).

This needs to be understandable to a high school AP-C physics student.

Signed,
Me - the high school AP-C physics student.

This isn't a homework question. I started trying to picking apart the EFE a few weeks ago during my open study hour.

 

[bcrowell]

Interesting question. Although the Riemann curvature tensor is mathematically more complicated (a rank-2 tensor), I actually find it easier to interpret than the scalar curvature. If any other PF members have a good, succinct interpretation of the scalar curvature, I'd be interested to hear it.

 

[Naty1]

Looking for a dummy?? Here I am!...While the question is still above my pay grade, so maybe I'm "sub dummy", I'm listening to Susskind's online lectures for GR so oddly it's an interesting one...

There seem to be some really good (read that as "even I can understand them") descriptions here: http://en.wikipedia.org/wiki/Scalar_curvature

...the scalar curvature represents the amount by which the volume of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space

When the scalar curvature is positive at a point, the volume of a small ball about the point has smaller volume than a ball of the same radius in Euclidean space. On the other hand, when the scalar curvature is negative at a point, the volume of a small ball is instead larger than it would be in Euclidean space.

What do you experts think??

 

[bcrowell]

There seem to be some really good (read that as "even I can understand them") descriptions here: http://en.wikipedia.org/wiki/Scalar_curvature

I think that interpretation only works in a space with a Euclidean signature.

 

[Naty1]

picky,picky,picky!

 

[Naty1]

Ben, on a more serious note, I just stumbled across this in some notes...and wondered if it might add to the reply to the poster...none of this is mentioned in the source I posted above:

"The Ricci tensor Rab only keeps track of the change of volume of this ball. Namely, the second time derivative of the volume of the ball is -Rabva vb times the ball's original volume. The Weyl tensor tells the REST of the story about what happens to the ball. More precisely, it describes how the ball changes shape, into an ellipsoid.

(I don't have the source, but will Check...maybe it was Wikipedia. Sounds like this must be limited to euclidean coordinates also,right??) )

In the comments previously posted, the change in volume is NOT related to any changes in coordinates from point to point, right??..Euclidean coordinates maintain the same shape and separation everywhere, so a volume change is a physical change not due to any changes in coordinates... But if we used curvilinear coordinates with changing relationships over a manifold, then we get coordinate changes mixed up with manifold changes...so the "volume" change has a mix of causes?? Is that the idea...

 

[bcrowell]

Hi, Naty,

The issue isn't Cartesian versus non-Cartesian coordinates, it's Euclidean signature (+++) versus Lorentzian signature (+---). Signature refers to the set of plus and minus signs in the metric.

In a space with a Euclidean signature (known as a Riemannian space), the scalar curvature has a nice, simple geometric interpretation, as given in the WP article. In a space with a mixture of + and - in its signature (known as a semi-Riemannian space), I don't know of any simple geometric interpretation.

-Ben