# Distinguish a spacetime from flat space

How to distinguish a generic spacetime from flat space?

This (apparently silly) question puzzled me after reading something on Vanishing Scalar Invariant (VSI) spacetime (for instance PP-waves, spaces with all the polynomial curvature invariants vanishing) and Kundt spaces. See for instance wikipedia.

Why we need to look at non polynomial invariants or Cartan–Karlhede algorithm to discriminate between VSI and flat space if naively we could simply look at the Riemann tensor and see that it is not identically zero?

Am I missing something trivial here?

You are right that if you get identically zero for all components of the Riemann tensor all invariants, polynomial or not, will also be zero. You can distinguish flat spacetimes in various ways, those are two. Clearly another one is if you can find a coordinate transformation to a Minkowski spacetime. You can use any of those.

What is more important about VSI spacetimes and Kundt spacetimes is that they have some interesting and useful physical properties. They can be classified and have served as a laboratory for studying various vacuum and even non-vacuum solutions of special types. That includes gravitational and also electromagnetic) wave exact solutions.

A paper from 2002 at https://arxiv.org/pdf/gr-qc/0209024v1.pdf finds all the spacetimes with vanishing curvature invariants, meaning all polynomial vanishing invariants or scalars. They discuss various reasons why they are important. All or most of those are spacetimes with special geometric algebraic properties, and have some physical significance.

The algebraically special spacetimes refers to Petrov type III, N or O spacetimes, which are classified that way based on how many null eigenvectors the Weyl tensor has. For the Petrov classifications see https://en.m.wikipedia.org/wiki/Petrov_classification, and for the Weyl tensor see https://en.m.wikipedia.org/wiki/Weyl_tensor.

The Weyl tensor is the only part of the Riemann tensor that is nonzero in vacuum spacetimes. It conveys information about how the spacetimes is distorted. The Ricci tensor, the trace of the Riemann tensor, is determined by the Einstein field equations which relate it to the stress-energy tensor, gives information on the volume changes of spacetime, and is zero in vacuum. A non flat vacuum has a nonzero Weyl tensor. For those Petrov types mentioned above there are VSI solutions. The 2002 paper cited above identifies all the VSI solutions (finds expressions for all the metrics and the functions with equations to be solved that denotes the different solutions).

The best known such solution is the pp-wave solution, parallel plane waves filling spacetime. They are physically interesting on their own right, like plane waves are in Quantum Field Theory, and have the interesting property that two such solutions they can be added if they share the direction of motion, so they show a kind of linearity. Those and some of the other special VSI solutions have unique properties of value for quantum gravity. Pp-waves, for instance, are also solutions of string theory, and generalized versions to superstring theory, and are valid to all perturbation orders in the string tension. Other solutions have vanishing corrections up to two loops, and then all corrections to all loop orders vanish.

So, those are of relevance whether you use non-polynomial invariants or not.

• Minor comment to the post (v1): In the future please use non-mobile-phone links, and link to abstract pages rather than pdf files. Sep 3, 2016 at 5:55
• Thanks. But not sure what you mean by non mobile cell links. Am using my iPad. You mean https link as reference at bottom with keyword in text, or do you mean something else? Sep 3, 2016 at 18:59
• ttp://en.wikipedia. instead of ttp://en.m.wikipedia. Sep 3, 2016 at 19:01
• No problem. Didn't even know that's what m stood for. It seems to me the server should automatically adapt, and not have different url's Sep 3, 2016 at 19:38