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

## Main Question or Discussion Point

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.

Related Special and General Relativity News on Phys.org
Chris Hillman
Lambda should be R/24

Hi, cosmicstring1,

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.

pervect
Staff Emeritus
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.

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.

Chris Hillman
Low tech verbatim

Hi, cosmicstring1, I don't understand--- is the mystery solved or not? Are you saying you still get $Lambda \neq R/24$? 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.

Chris Hillman
Oh, NOW you tell us--- you are looking at a euclidean signature! 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.