Hi,

Below is my answer yesterday, and this text now is a correction: I am sorry I misread your question, @Rakshak . Somehow I missed the word "vacuum" and that by Metric search you meant you were using the **DifferentialGeometry:-Library:-MetricSearch applet**. Thanks to @Hullzie and @Jamie for somehow pointing out what I misread. Indeed, one would expect that if, on the applet, you click vacuum, the output should restrict the Petrov type II metrics to those with **EnergyMomentum = ****0****,** and so *Ricci = *0.

Now, the applet returns not just** [33, 8, 3]** but also** [20, 32, 1]** as "**Petrov type II and vacuum**", and both have non zero Ricci components, and have arbitrary functions that, if adjusted, result in **Ricci = 0**. To understand why the applet returns these two, load the **Physics:-Tetrads** package and check, for instance, this input/output

Physics:-Version(latest): # update your Physics

restart;

with(Physics):

g_[[20, 32, 1]];

with(Tetrads):

WeylScalars(TransformTetrad(canonicalform));

You will see that the scalars - now computed with the tetrad in canonical form - are those of Petrov II (see ?PetrovType for the expected scalars), and *they don't depend on the arbitrary functions* found in the metric. The same happens with your example, [33, 8, 3].

In conclusion: the spacetime is of type II *regardless of the arbitrary functions (or parameters) found in the metric*, but it is a vacuum solution only if you solve these parameters (functions or symbols) such that the components of Ricci are equal to 0. I will add a sentence clarifying this on the help page of **DifferentialGeometry:-Library:-MetricSearch**.

To the side, I note that it is also possible to specialize the arbitrary functions found in these two metrics in such a way that the resulting metric is not of Petrov type II anymore. How could that be?

The answer is that, as explained in the help page of **PetrovType**, the algorithm determines the type according to the multiplicity of the roots of the underlying principal polynomial. If that is done without specializing the parameters found in the metric (in these two examples: unknown functions), some roots may be different while becoming the same after specializing those parameters.

Edgardo S. Cheb-Terrab

Physics, Differential Equations and Mathematical Functions, Maplesoft

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**Original answer yesterday.**

Metrics of Petrov type II don't have the Ricci tensor equal to a matrix of zeros. The metric [33, 8, 3] is an example of that, Petrov type II, and the Ricci tensor has non-zero components.

I suggest you take a look at the help page ?Tetrads:-PetrovType, where you see the description of the types and how they are calculated, including different algorithms that you could invoke to perform that calculation. On that page, you will also find other examples of Petrov type II, with non-zero Ricci.

Answering your question: the Metric Search applet of DifferentialGeometry is correct, [33, 8, 3] is of Petrov type II, you are not misusing anything, nor is there a glitch in the programs. It is your statement (expecting Ricci to be equal to 0 for Petrov type II) that is not correct.

I think the page ?Tetrads:-PetrovType as well as ?Tetrads:-TransformTetrad present a clear exposition on the topic; still if you want literature outside Maple, you can take a look at The classification of the Ricci and Plebanski Tensors.

Edgardo S. Cheb-Terrab

Physics, Differential Equations and Mathematical Functions, Maplesoft