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# Exact solutions to Einstein's equations

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Exact solutions to Einstein’s equations” is one of those books that are difficult even to imagine: the authors reviewed more than 4,000 papers containing solutions to Einstein’s equations in the general relativity literature, collecting, classifying, discarding repetitions in disguise, and organizing the whole material into chapters according to the physical properties of these solutions. The book is already in its second edition and it is a monumental piece of work.

As good as it is, however, the project resulted only in printed material, a textbook constituted of paper and ink. In 2006, when the DifferentialGeometry package was rewritten to enter the Maple library, one of the first things that passed through our minds was to bring the whole of “Exact solutions to Einstein’s equations” into Maple.

It took some time to start but in 2010, for Maple 14, we featured the first 26 solutions from this book. In Maple 15 this number jumped to 61. For Maple 17 we decided to emphasize the general relativity functionality of the DifferentialGeometry package, and Maple 18 added 50 more, featuring in total 225 of these solutions - great! but still far from the whole thing …

And this is when we decided to “step on the gas” - go for it, the whole book. One year later, working in collaboration with Denitsa Staicova from Bulgarian Academy of Sciences, Maple 2015 appeared with 330 solutions to Einstein’s equations. Today we have already implemented 492 solutions, and for the first time we can see the end of the tunnel: we are targeting finishing the whole book by the end of this year.

Wow2! This is a terrific result. First, because these solutions are key in the area of general relativity, and at this point what we have in Maple is already the most thorough digitized database of solutions to Einstein’s equations in the world. Second, and not any less important, because within Maple this knowledge comes alive. The solutions are fully searcheable and are set by a simple call to the Physics:-g_  spacetime metric command, and that automatically sets the related coordinates, Christoffel symbols , Ricci  and Riemann  tensors, orthonormal and null tetrads , etc. All of this happens on the fly, and all the mathematics within the Maple library are ready to work with these solutions. Having everything come alive completely changes the game. The ability to search the database according to the physical properties of the solutions, their classification, or just by parts of keywords also makes the whole book concretely more useful.

And, not only are these solutions to Einstein’s equations brought to life in a full-featured way through the Physics  package: they can also be reached through the DifferentialGeometry:-Library:-MetricSearch  applet. Almost all of the mathematical operations one can perform on them are also implemented as commands in DifferentialGeometry .

Finally, in the Maple PDEtools package , we already have all the mathematical tools to start resolving the equivalence problem around these solutions. That is: to answer whether a new solution is or not new, or whether it can be obtained from an existing solution by transformations of coordinates of different kinds. And we are going for it.

What follows is a basic illustration of what has already been implemented. As usual, in order to reproduce these results, you need to update your Physics library from the Maplesoft R&D Physics webpage.

Load Physics , set the metric to  (and everything else automatically) in one go

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 (1)

And that is all we do :) Although the strength in Physics  is to compute with tensors using indicial notation, all of the tensor components and related properties of this metric are also derived on the fly (and no, they are not in any database). For instance these are the definition in terms of Christoffel symbols , and the covariant components of the Ricci tensor

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 (2)
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These are the 16 Riemann invariants  for Schwarzschild solution, using the formulas by Carminati and McLenaghan

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 (4)

The related Weyl scalars  in the context of the Newman-Penrose formalism

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 (5)

These are the 2x2 matrix components of the Christoffel symbols of the second kind (that describe, in coordinates, the effects of parallel transport in curved surfaces), when the first of its three indices is equal to 1

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 (6)

In Physics, the Christoffel symbols of the first kind are represented by the same object (not two commands) just by taking the first index covariant, as we do when computing with paper and pencil

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 (7)

One could query the database, directly from the spacetime metrics, about the solutions (metrics) to Einstein's equations related to Levi-Civita, the Italian mathematician

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These solutions can be set in one go from the metrics command, just by indicating the number with which it appears in "Exact Solutions to Einstein's Equations"

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Automatically, everything gets set accordingly; these are the contravariant components of the related Ricci tensor

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 (10)

One works with the Newman-Penrose formalism frequently using tetrads (local system of references); the Physics subpackage for this is Tetrads

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 (11)

This is the tetrad related to the book's metric with number 12.16.1

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 (12)

One can check these directly; for instance this is the definition of the tetrad, where the right-hand side is the tetrad metric

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 (13)

This shows that, for the components given by (12), the definition holds

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 (14)

One frequently works with a different signature and null tetrads; set that, and everything gets automatically recomputed for the metric 12.16.1 accordingly

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 (16)
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The related 16 Riemann invariant

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The ability to query rapidly, set things in one go, change everything again etc. are at this point fantastic. For instance, these are the metrics by Kaigorodov; next are those published in 1962

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The search can be done visually, by properties; this is the only solution in the database that is a Pure Ratiation solution, of Petrov Type "D", Plebanski-Petrov Type "O" and that has Isometry Dimension equal to 1:

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Set the solution, and everything related to work with it, in one go

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 (22)

The related Riemann invariants:

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To conclude, how many solutions from the book have we already implemented?

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:)