Principal Invariants of the Weyl Tensor

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Discussion Overview

The discussion revolves around the naming conventions for the principal invariants of the Weyl tensor in the context of differential geometry and general relativity. Participants explore whether there are established names for these invariants, which are often referred to as I1 and I2, and the implications for a computational relativity package being developed by the original poster.

Discussion Character

  • Exploratory, Debate/contested, Technical explanation

Main Points Raised

  • One participant inquires about standard naming conventions for the two principal invariants of the Weyl tensor, noting that the Riemann tensor has established names.
  • Another participant suggests checking a Wikipedia article on curvature invariants, implying it may provide the needed information.
  • A subsequent reply asserts that the Wikipedia article confirms the invariants are simply referred to as I1 and I2, arguing that not all invariants need to have names associated with individuals.
  • In response, a participant challenges this assertion, stating that the absence of information in a single source does not imply that alternate names do not exist, citing other related invariants as examples.
  • Another participant suggests that the original poster might find relevant information in relativity textbooks, although they are unaware of any alternate names for the Weyl tensor invariants in those texts.
  • One participant questions the necessity of having names for the invariants at all.

Areas of Agreement / Disagreement

Participants express differing views on the existence of established names for the Weyl tensor invariants, with some asserting that they are simply I1 and I2, while others argue that there may be additional names or conventions not covered in the sources mentioned.

Contextual Notes

The discussion highlights the limitations of available literature, as some participants reference specific articles and textbooks while acknowledging that these may not be comprehensive. There is also a noted dependence on the definitions and contexts in which the invariants are discussed.

getjonwithit
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TL;DR
Are there any standard names for the principal invariants of the Weyl tensor (akin to Kreschmann, Chern-Pontryagin, etc. for the Riemann tensor)?
It's possible that this may be a better fit for the Differential Geometry forum (in which case, please do let me know). However, I'm curious to know whether anyone is aware of any standard naming convention for the two principal invariants of the Weyl tensor. For the Riemann tensor, the names of the principal invariants (i.e. Kretschmann scalar, Chern-Pontryagin scalar and Euler scalar) are at least somewhat standardised, yet from the literature search that I've been able to perform thus far, I have yet to encounter any such naming convention for the Weyl tensor invariants, which are usually simply referred to as I1 and I2. Is anyone aware of any?

(For context, this is for a computational relativity package that I'm currently developing - I'd like to have some reasonably consistent, and not totally obscure, naming convention for the various quadratic curvature invariants.)
 
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getjonwithit said:
Yes, obviously. Why do you ask?
Because that answers your question: the two Weyl tensor invariants are indeed just called I1 and I2. There is no law that says every invariant needs to be named after a person.
 
PeterDonis said:
Because that answers your question: the two Weyl tensor invariants are indeed just called I1 and I2. There is no law that says every invariant needs to be named after a person.
No, it manifestly does not answer the question: the fact that a single (introductory and rather incomplete) Wikipedia article happens not to mention something does not imply that it doesn't exist.

That same article, for instance, makes reference to the Carminati-McLenaghan invariants, yet makes no mention of the deeply related real scalar invariants of Zakhary and McIntosh for Petrov and Segre-type metrics; this omission, however, does not mean that the latter do not exist, merely that Wikipedia is not omniscient.
 
getjonwithit said:
That same article, for instance, makes reference to the Carminati-McLenaghan invariants, yet makes no mention of the deeply related real scalar invariants of Zakhary and McIntosh for Petrov and Segre-type metrics
Whatever source you are getting the information from that you mention here would likely also contain the information you're looking for, if it exists. Also, the Wikipedia article gives two references.

You could also try relativity textbooks like Wald or Hawking & Ellis that are more oriented towards geometry and geometric invariants. I'm not aware of any alternate names for the Weyl tensor invariants in either of those books, but those would be good places to check.
 
Why do you need names?!
 

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