When does Metric contain curved spacetime?

[Mad Dog]

I’m aware that the metric contains information regarding both the coordinate system and the curvature of spacetime, and have been trying to understand how a glance at it could tell one if the spacetime it expressed is curved.
At this point, I suspect that:
If any of the metric components involve mixing time and space in any way more complicated than ax+bt you’ve got “gravity.”
Can anybody give me some help with this line of reasoning?

 

[atyy]

You can calculate the Riemann tensor. If it is zero everywhere, then the metric is flat. Roughly speaking, curvature is characterised by second derivatives, and these are present in the Riemann tensor.

http://mathworld.wolfram.com/RiemannTensor.html
http://mathworld.wolfram.com/MinkowskiMetric.html

 

[Mad Dog]

Thanks atyy. One could also get there calculating the Ricci and Weyl tensors, but... but... but... ain't there no easier way? The simple rule I gave worked in the 7 examples I've checked. Still wondering.

 

[shoehorn]

I’m aware that the metric contains information regarding both the coordinate system

It doesn't. The metric as such contains no information about coordinates, as can be evinced by the fact that one can provide an entirely coordinate-independent expression for the action of the metric on vectors.

and the curvature of spacetime, and have been trying to understand how a glance at it could tell one if the spacetime it expressed is curved.

Without a great deal of experience, you can't.

At this point, I suspect that:
If any of the metric components involve mixing time and space in any way more complicated than ax+bt you’ve got “gravity.”
Can anybody give me some help with this line of reasoning?

Besides pointing out that this line of reasoning is nonsense, no. If you want to determine whether or not a given manifold-metric pair is curved, you've got to determine the curvature tensors. By definition, there's no way around this.

 

[Mad Dog]

Thank you Shoehorn- Please understand that I'm trying to learn GR (down here in the bowels of Mexico) from books and without the aid of either a teacher or fellow students. This forum is my only contact with those who know more than I, so I appreciate every response from anyone who offers their help by responding to my questions.
Now: Considering the information contained in the Metric - Foster and Nightingale's book "A Short Course in General Relativity", 3rd Edition, Section 3.5, page 112 says:
"The metric tensor contains two separate pieces of information:
(i) the relatively unimportant information concerning the specific coordinate system used (e.g., spherical coordinates, Cartesian coordinates, etc.);
(ii) the important information regarding the existence of any gravitational potentials."
Can you explain the difference between your point of view and theirs?

With regard to "this line of reasoning is nonsense": It may be, and you could convince me easily if you could point me to an example where a mixing of only spatial coordinates can result in a spacetime where movement through time "spills over" into movement in space.

Thanks again, Mexican Mad Dog

 

[atyy]

Try George Jones post #10:
https://www.physicsforums.com/showthread.php?t=265648