What is the measure of space-time curvature and how is it calculated?

[Gingermolloy]

Space-Time Curvature Question!

Hi Guys,

A question about the curvture of space-time by mass.

Where is the point of maximum curvature??

Is it at the centre of mass (i.e.. the middle of the body)

The reason I ask, is that when space-time curvature is shown visually it makes out like it is a flat bottomed "dent" in space-time. If this was the case it would mean surely that there is no curvature at the centre of mass.

Where am I going wrong with this thought process.

Thanks

Ginger

 

[A.T.]

1) The "dent" just shows space-curvature, not space-time curvature. Check out:
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html

2) The "dent" is not flat. The spatial-curvature is the same everywhere within a uniform massive sphere (the spherical cap at the bottom):

3) The local space-time curvature is connected to tidal forces rather than gravitational attraction. At the center of a massive sphere the gravitational acceleration is zero, but the curvature isn't (unless there is a spherical cavity at the center).

 

[nitsuj]

3) ... At the center of a massive sphere the gravitational acceleration is zero, but the curvature isn't (unless there is a spherical cavity at the center).

How can there be curvature but no coordinate acceleration? If the gravitational potential is net zero, then wouldn't that mean there is no curvature?

 

[A.T.]

How can there be curvature but no coordinate acceleration?

It's explained in the part you sniped: The local space-time curvature is connected to tidal forces rather than gravitational attraction

If the gravitational potential is net zero, then wouldn't that mean there is no curvature?

The value of the gravitational potential is arbitrary. Its gradient matters, and determines the direction of coordinate acceleration. In the center the gradient or 1st metric derivatives are zero, therefore no gravity pull exists. But curvature or 2nd metric derivatives are not zero, therefore tidal forces exist.

 

[Gingermolloy]

I was under the impression that space-time became more curved the closer to a massive object you get. Is this true??

The area of constant curvature? Is that the region of space within the body itself?

And is there such thing as truly flat space-time?

 

[A.T.]

I was under the impression that space-time became more curved the closer to a massive object you get. Is this true??

Yes, outside of it.

The area of constant curvature? Is that the region of space within the body itself?

Yes, the interior spatial metric is spherical. A sphere has constant curvature.

And is there such thing as truly flat space-time?

Not in reality. Only regions with negligible curvature.

 

[pervect]

Yes, outside of it.

Yes, the interior spatial metric is spherical. A sphere has constant curvature.

Not in reality. Only regions with negligible curvature.

A quick pair of questions on the technical side. Are you using the Ricci scalar as a measure of the magnitude of the curvature? And setting it equal to the minus the trace of T_ab (possibly multiplied by some constant)?