General Question on Schwarzschild Metric

[WannabeNewton]

Hi guys I have a quick question on the Schwarzschild Metric:
Since the metric is a solution to the EFEs does it intrinsically have the curvature of the gravitational field embedded in the metric? If so is it represented by the time and spatial components of the metric? If not could you please explain what those two components describe. Also, if the curvature isn't directly shown in the metric how exactly would one go about finding it?

 

[pervect]

Given the metric, you can calculate first the Christoffel symbols, and then the curvature tensor - to be precise, the Riemann curvature tensor. There are other curvature tensors used in GR , .e. the Ricci tensor and the Einstein tensor. The Ricci tensor can be expressed as the contraction of the Riemann, i.e R^ab = R^ab_ab. The Einstein tensor can be expressed in terms of the Ricci tensor, the metric, and a constant obtained by contracting the Ricci, i.e. R = R^a_a, G_ab = R_ab - (R/2) g_ab.

I'm not sure if the tensor equations will make sense to you, but offhand I'm not sure how to eliminate them without writing a much much longer post.

You might try http://math.ucr.edu/home/baez/einstein/

 

[WannabeNewton]

Sorry I phrased my question terribly. What I meant to ask was:
When people describe the Scwarzschild metric (or any space - time metric for that matter) they say it describes the geometry of the gravitational field for the given mass - energy distribution. So does the metric have the curvature of the field embedded in it such that when you are calculating space - time intervals in the field it automatically takes the curvature into account? And the actual value for the curvature would be calculated from the Riemann curvature tensor?

 

[haushofer]

So does the metric have the curvature of the field embedded in it such that when you are calculating space - time intervals in the field it automatically takes the curvature into account? And the actual value for the curvature would be calculated from the Riemann curvature tensor?

Yes, because in GR eventually you calculate the Riemann tensor from that particular metric.

In short, the Riemann tensor depends on the Christoffel symbols, and the Christoffel symbols depend on the metric.

 

[WannabeNewton]

So it would be safe to say that when calculating space - time intervals using the Schwarzschild metric it intrinsically accounts for the curvature of the gravitational field around it (for a static and spherically symmetric object)? I need not solve the curvature tensor to account for the proper time or proper distance resulting from the curved geometry?

 

[Russell E]

So it would be safe to say that when calculating space - time intervals using the Schwarzschild metric it intrinsically accounts for the curvature of the gravitational field around it (for a static and spherically symmetric object)? I need not solve the curvature tensor to account for the proper time or proper distance resulting from the curved geometry?

Yes, that's right.

 

[WannabeNewton]

Sorry just one last question:
Could you then describe the geometry of the space - time around the spherical mass by calculating a large number of proper distances for intervals around the given mass and culminating them together?

 

[Passionflower]

Since the metric is a solution to the EFEs does it intrinsically have the curvature of the gravitational field embedded in the metric? If so is it represented by the time and spatial components of the metric?

A little difficult to explain and a little controversial because traditionally most people like to view spacetime as three dimensions and time as a separate dimension.

However what is space and what is time is really observer dependent it is not a dimension of the manifold.

Thus it is better to think of a curved 4 dimensional manifold in which different observers have their own perspective of which 'mixture' of these 4 dimensions represent time and space to them.

 

[Dale]

Sorry just one last question:
Could you then describe the geometry of the space - time around the spherical mass by calculating a large number of proper distances for intervals around the given mass and culminating them together?

Distances and angles, yes. Consider the geometry on the surface of a sphere. If you make a triangle on the sphere you will wind up with three distances which could indeed represent the sides of a triangle in a flat space, but the sum of the interior angles will be greater than 180° which will let you know that the space is curved.