# I List of discovered exact stationary spacetimes

#### Creedence

Summary
Is there a complete list of exact solution of the Einstein field equation?
Hi!
Is there a complete list of exact stationary solution of the Einstein field equation?
I started to solve it for an interesting electrovacuum case. I'd like to check my results.
I have found Thomas Müller's catalog of spacetimes and Hans Stephani's Exact Solutions of Einstein's Field Equations, but none of them contains a complete enumeration.
Thanks for the answer(s) - Robert

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#### martinbn

How could there be such a list! Any metric is a solution, how could you list all!

#### Creedence

How could there be such a list! Any metric is a solution, how could you list all!
By categorizing them according to their symmetries, asymptotical behavior and fields.

#### Heikki Tuuri

If you choose any metric g, then you can from the Einstein field equations calculate the stress-energy tensor T which would create that metric.

The problem in this is that it probably is impossible to find real, non-exotic, matter which produces that stress-energy tensor T. Furthermore, there has to exist a realistic history from the Big Bang to that stress-energy tensor T.

For example, if you choose the Miguel Alcubierre metric g, you find that to realize that metric you need very exotic matter.

As far as I know, it is an open problem if general relativity has any dynamic solutions with realistic matter. There exist static solutions, though. The best known static solution is the Schwarzschild internal and external metric, but it requires exotic matter (incompressible fluid), and cannot result from the Big Bang because it is an eternal solution which lives in the asymptotic Minkowski space.

I am currently studying if realistic dynamic solutions can exist at all.

Thus, the list of known realistic solutions of general relativity is very short: none.

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#### Creedence

If you choose any metric g, then you can from the Einstein field equations calculate the stress-energy tensor T which would create that metric.

The problem in this is that it probably is impossible to find real, non-exotic, matter which produces that stress-energy tensor T. Furthermore, there has to exist a realistic history from the Big Bang to that stress-energy tensor T.

For example, if you choose the Miguel Alcubierre metric g, you find that to realize that metric you need very exotic matter.

As far as I know, it is an open problem if general relativity has any dynamic solutions with realistic matter. There exist static solutions, though.
I'm searching for the list of exact stationary solutions.

#### martinbn

I'm searching for the list of exact stationary solutions.
This is not in your first post. May be you can list all the assumptions that you have.

#### George Jones

Staff Emeritus
Gold Member
The two standard compilations are "Exact Space-Times in Einstein's General Relativity" by Griffiths (Jerry, not David) and Podolsky, and the second edition of "Exact Solutions of Einstein's Field Equations" by Stephani, Kramer, MacCullum, Hoenselaers, and Herlt.

The second book is more comprehensive than first, but I find the first to be more interesting,

#### George Jones

Staff Emeritus
Gold Member
The two standard compilations are "Exact Space-Times in Einstein's General Relativity" by Griffiths (Jerry, not David) and Podolsky, and the second edition of "Exact Solutions of Einstein's Field Equations" by Stephani, Kramer, MacCullum, Hoenselaers, and Herlt.

The second book is more comprehensive than first, but I find the first to be more interesting,
When I made this post, I was rushing in order to finish in time to catch my bus. Now, I have the books at hand, and I can expand somewhat on this post.

Is there a complete list of exact stationary solution of the Einstein field equation?
I started to solve it for an interesting electrovacuum case. I'd like to check my results.
I have found Thomas Müller's catalog of spacetimes and Hans Stephani's Exact Solutions of Einstein's Field Equations, but none of them contains a complete enumeration.
I was unsure what you meant by "Hans Stephani's Exact Solutions of Einstein's Field Equations". It is important to give somewhat precise references. Did you mean:

1) the second reference the I gave above;
2) the first edition of the the second reference';
3) a possible review article solely authored by Stephani that evolved into a book with multiple authors?

The main article for this is
which contains the interesting reference (to me; I am not sure about the OP)

This reference compares the two editions of the second reference that I give in my post above: "The second edition of the exact solutions book contains about 400 pages of new material, covering hundreds of new solutions and references. In its preparation the authors read about 4000 new papers (as well as the 3000 read for the first edition). " Astonishing!

My second reference has a number of chapters on stationary solutions. At the start of these chapters, the authors write "Only in a few cases are exact stationary solutions without an additional symmetry known. They are given in §§18.5, 18.7 and 17.3. The stationary fields admitting a second Killing vector describing axial symmetry will be treated in the subsequent chapters."

@Creedence , have you checked your spacetime for additional symmetries (i.e., Killig vectors)? These are not always obvious, and symbolic computation packages (e.g., Mathematica or Maple) can be useful for this.

Several years ago, I used Maple to find the Killing vectors for a particular spacetime from a paper. Maple found Killing vectors for the symmetries of which the authors were aware, but it also found a Killing vector (field) of which the authors clearly were unaware. I use Maple for the pragmatic reason that my employer has a site license.

"List of discovered exact stationary spacetimes"

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