Studying Special Features in Metrics: Tensors & Differential Geom.

[space-time]

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)?

 

[aleazk]

Do any of the tensors or quantities in the list above indicate the presence (or absence) of closed time like curves? No!

False. The metric is enough, as I already told you before: https://www.physicsforums.com/threa...-interpret-these-tensors.811708/#post-5095361 (it's not a general method, but works for most of the well known examples)

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!

It depends on the metric, If your metric is in some sense stationary, then you are not going to obtain from it processes about formations and these things, since these can be very complex and dynamical processes, it's unlikely to find an exact solution describing them (in some cases there are, but are highly idealized and physically unsatisfying). Anyway, this is a complex topic, check the Kip Thorne paper I mentioned in that thread (you can check it online). The main point is that the CTCs in Gödel or van Stockum are eternal, while what you need is a compactly generated horizon (i.e., a time machine in a compact region of spacetime).

The whole point of the metric is supposedly to explain the dynamics of a type of traversable wormhole

Incorrect. That's just a supposition you are making. In fact, the metric describes an eternal and static wormhole. It does not say anything about the dynamics because the metric itself decribes a different physical situation. Again, the creation of a wormhole (if that makes any sense), involves complex dynamical processes of which we know very little.

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

Well, the metric describes a spherically symmetric black hole... so... Of course, it's not going to say "hey, here I am, I'm a black hole". You need to have the tools to explore that metric. The mere calculation of the metric components is just the beginning.

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My friend @space-time: what you need is a more systematic and methodical approach to your study. What you are doing (reading bits of this and that on the net) is terribly inefficient. What you need to do is to get some university level textbooks and to read them from the A to the Z. That's what I did, at least. I recommend "General Relativity", by Robert Wald. Save some money and buy it from @amazon.com.

If you are in high school and already can grasp some of these topics, please, take yourself seriously and start a more systematic approach in your learning. If you plan to go to a physics degree, it will put you in real advantage.

 

[Dale]

Also, it sounds like you may be more interested in the topological features of the manifold than the metric.

 

[bcrowell]

Amplifying on DaleSpam's #3, tensors are local things. A tensor exists in the tangent space at a particular point. That means that looking at a tensor at a particular point will never *directly* tell you any global information. There are links between the local and the global, but they are often subtle. A good example from Riemannian geometry is the Myers theorem: http://en.wikipedia.org/wiki/Myers_theorem . It links curvature to topology. Roger Penrose and Stephen Hawking built their careers on global methods in relativity. The classic book on this is Hawking and Ellis.

Even if you don't go ahead and learn global methods, I can't help agreeing with aleazk that your #1 seems unnecessarily overwrought. Even at a pretty elementary level, the information you're talking about can be extracted from the metric. For example, if you want to extract information about the Schwarzschild spacetime from the metric, see Taylor and Wheeler, Exploring Black Holes: Introduction to General Relativity.