From the scalar of curvature (Newman-Penrose formalism) to the Ricci scalar

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Discussion Overview

The discussion revolves around the calculation of the Ricci scalar from trace-free Ricci scalars and the scalar of curvature within the context of the Newman-Penrose formalism, utilizing the GRTensorII computer package. Participants explore the relationships between these quantities and the implications of different tetrad choices, particularly in relation to spacetime metrics.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant calculates trace-free Ricci scalars and the scalar of curvature but questions how to derive the Ricci scalar, noting a discrepancy where Lambda is non-zero despite an expected zero Ricci scalar.
  • Another participant confirms that Lambda should equal R/24 and suggests a potential mistake in the tetrad used in GRTensorII, indicating that not all commands check assumptions.
  • A participant asserts that their tetrad satisfies necessary relations and considers the possibility that the Ricci scalar could be a combination of NP Ricci scalars, while also noting that both answers satisfy the Bianchi identities, implying non-uniqueness.
  • Some participants propose calculating the Ricci scalar directly from GRTensorII or computing the Ricci tensor and taking the trace, though uncertainty remains about the applicability of these methods with the null-tetrad formalism.
  • One participant confirms that they calculated the Ricci scalar and tensor directly from GRTensorII, obtaining zero for both.
  • Another participant expresses confusion about whether the issue has been resolved and questions the tetrad's derivation, emphasizing the importance of matching the expected metric tensor.
  • A participant resolves the mystery by confirming their tetrad gives the correct metric and acknowledges that their choice may not be suitable for GRTensorII's NP solver.
  • Another participant highlights the significance of the Euclidean signature in the discussion, suggesting it was an important detail that was initially overlooked.
  • One participant notes that changing the metric file for the Euclidean signature resolved some issues but that the computation of the Ricci scalars remained problematic.

Areas of Agreement / Disagreement

Participants express differing views on the validity of the tetrad used and the implications of the Euclidean signature. While some progress has been made in resolving specific issues, the overall discussion remains unresolved regarding the relationship between the calculated quantities.

Contextual Notes

Participants mention the need for careful consideration of tetrad choices and metric signatures, indicating that assumptions about these elements may significantly affect the calculations and results. There is also mention of the Bianchi identities, suggesting that certain mathematical properties hold, but the implications for the Ricci scalar remain unclear.

cosmicstring1
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I calculate trace-free Ricci scalars (Phi00, Phi01,Phi02, etc) and scalar of curvature (Lambda=R/24) in Newman-Penrose formalism using a computer package. How can I find the Ricci scalar out of them? I though R was the Ricci scalar but Lambda comes non-zero for a spacetime whose Ricci scalar is exactly zero.
 
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Lambda should be R/24

Hi, cosmicstring1,

cosmicstring1 said:
I calculate trace-free Ricci scalars (Phi00, Phi01,Phi02, etc) and scalar of curvature (Lambda=R/24) in Newman-Penrose formalism using a computer package. How can I find the Ricci scalar out of them? I though R was the Ricci scalar but Lambda comes non-zero for a spacetime whose Ricci scalar is exactly zero.

So are you referring to GRTensorII? A useful package indeed! Lambda is indeed just what you think it is, so you probably made a mistake. My first guessthat you entered what you thought was an NP tetrad, which is not in fact a tetrad. Not all GRTensorII built-in commands check that all relevant assumptions are satisfied, in order to save computation time.
 
Yes, I am referring to the great GRTensorII. I checked the tetrad from two articles and it satisfies the relations needed (I also checked by hand). I was thinking the Ricci scalar might be a combination of some NP Ricci scalars after having nonzero values. By the way, the both answers satisfy the Bianchi identities! So probably the solution is not unique in this perspective. But as you verified, I must take the zero Lambda value, not the nonzero one for a R=0 metric.
 
This might not be relevant but why not:

1) Calculate and display ricciscalar directly from grtensor?

I was also thinking you could compute the Ricci tensor and take the trace, but I'm not quite sure if that would work with the null-tetrad formalism, which I've only used indirectly.
 
pervect said:
This might not be relevant but why not:

1) Calculate and display ricciscalar directly from grtensor?

I was also thinking you could compute the Ricci tensor and take the trace, but I'm not quite sure if that would work with the null-tetrad formalism, which I've only used indirectly.

I calculated Ricci scalar and Ricci tensor directly from grtensor and it also gives zero for the both.
 
Low tech verbatim

Hi, cosmicstring1, I don't understand--- is the mystery solved or not? Are you saying you still get [itex]Lambda \neq R/24[/itex]? If so,how exactly did you obtain your NP tetrad? It sounds like you found a tetrad in a paper and then entered that by hand to make your GRTensorII "spacetime definition"--- that's perfectly fine, as long as you use the correct inner product for an NP tetrad!

A quick check: compute the metric tensor. Does it match the line element you expect?

(As you know, by convention, NP tetrads are usually reported using -+++ signature, and IIRC GRTensorII assumes you are using the standard sign conventions when you work with NP tetrads. In fact, if you use the built-in command which constructs an NP tetrad from a given real frame field (ONB), it automatically converts the signature. I usually find it preferable to construct a tetrad myself--- as you know, there are many possibilities for a given spacetime--- so I don't often use that command.)

Why not post your GRTensorII definition file so we can help you figure out what went wrong? To obtain verbatim text in a PF post, write [COMMAND]verbatim [/COMMAND] but with COMMAND replaced by "CODE".
 
Yes, the mystery is solved. I have already checked my tetrad and it gives the metric right. I have changed the signature by hand and I did not use the nptetrad command (it really changes the signature which is not valid for my Euclidean case). I think my tetrad is not the right choice for GRTensorII's NP solver.
 
Groan

Oh, NOW you tell us--- you are looking at a euclidean signature! :rolleyes: That's important to mention. I admit that the literature tends to encourage the notion that it doesn't matter, but that's a sin of omission over which I have no control.
 
Everything has seemed all right because I have changed the eta in my metric file for the Euclidean sign and it gave the Weyl scalars and spin coeff.s right. The Ricci scalars' computation was the problem.
 

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