I have done relativistic calculations for multiple metrics at this point. I have worked with the exterior Schwarzschild metric, the Godel metric, the Morris-Thorne wormhole metric, and the Friedman Robertson Walker metric. For these metrics I have derived multiple tensors and tensor-like objects, such as: The metric tensors (both covariant and contravariant) Christoffel Symbols The Ricci tensor The Riemann tensor The curvature scalar The Einstein tensor The stress energy momentum tensor The Weyl tensor The Tidal tensor. Etc... Now, I take the time to derive these quantities, but whenever I have inquired about what these tensors/quantities tell me with regards to the creation of some of the special features of the metrics, I am always essentially told that these tensors tell you absolutely nothing about the special features. Allow me to explain what I mean: Firstly, by "special features of a metric", I mean special objects such as closed time like curves in the Godel metric, or wormholes in the Morris-Thorne metric (or other metrics that contain wormholes), or like black holes in the Schwarzschild metric. Now here are some examples of what I mean when I say that these tensors tell you nothing about the special features: 1. Take the Godel metric for example. We know that it contains within it closed time like curves. Do any of the tensors or quantities in the list above indicate the presence (or absence) of closed time like curves? No! Do any of these tensors/quantities tell you how a closed timed like curve is created, or how the matter and energy in the space - time have to move around and interact in order to generate a closed time like curve? No! Unless I have misunderstood some of people's replies to my questions, then it seems as though these tensors and quantities tell you absolutely nothing about closed time like curves (and those curves were actually my original reason for studying the Godel metric in the first place). 2. Take the Morris - Thorne wormhole metric as another example. The whole point of the metric is supposedly to explain the dynamics of a type of traversable wormhole, yet I've derived most of the tensors listed above for this metric, and still got no information on how the wormhole is actually generated, nor what factors determine where the wormhole leads to. 3. When I worked with the Schwarzschild metric, I got a bunch of 4 x 4 matrices out of it, but no information about black holes at all (despite the fact that the Schwarzschild metric contains black holes within it). This is what I mean when I say that these tensors/ quantities give me no info about the special features of the metrics. The fact that I have encountered this problem multiple times leads me to believe that I won't get any info about how to create any of these special features from a rank 2 or a rank 4 tensor. Of course this brings up the question, what special mathematical information do I need to learn in order to find out how special objects like wormholes or closed time like curves are generated? I think I might have to study some special type of vectors such as killing vectors and those basis vectors that are used in deriving the 4-velocity vector for matter in a metric. Additionally, I might have to study some more differential geometry. What do you guys think I should study in order to learn about these special features (because clearly higher ranked tensors alone aren't cutting it)?