# Counting solutions to the EFE?

• Chris Hillman
In summary: Chris HillmanIn summary, the conversation discusses the relationship between the number of physical situations in spacetime and the number of solutions to Einstein's field equations. While it may seem intuitive that knowing the physical situations would give insight into the solutions, the highly complex and nonlinear nature of the equations makes it difficult to determine the exact number of solutions. Some researchers have proposed methods, such as constraint counting, to estimate the number of solutions, but these methods have limitations. Further research and advancements in mathematical and computational techniques are needed to fully understand the solutions to these equations.
Chris Hillman
Way back in July 2004, kurious asked:

kurious said:
So if we know how many physical situations there are in spacetime we should be able to say how many solutions Einstein's field equations have?

Just thought I'd mention that Einstein himself proposed an interesting method for "counting" the solutions of a PDE which may have been indirectly inspired by the landmark work by his colleague David Hilbert on what is now called "the hilbert polynomial" (both ideas involve analyzing the asymptotics of dimension counting in a series of finite dimensional vector spaces). See http://en.wikipedia.org/w/index.php?title=Constraint_counting&oldid=39565830 (notice that I cited a specific version, namely the one listed at http://en.wikipedia.org/wiki/User:Hillman/Archive, bearing in mind that Wikipedia articles can change drastically in seconds and that I haven't checked the current version, just the last version I worked on myself.) For example, by counting alone we can conclude that a solution to the standard one dimensional wave equation should be specified by two freely chosen real functions of one real variable each, and of course just such a representation is known.

His charming idea has been taken up in the context of gtr by Siklos and (independently) Schutz, and others; see the 1996 paper by Siklos cited in the above cited Wikipedia article, which is one of my favorites. Siklos shows that from counting alone, we should expect a vacuum solution to be given by four real functions of three real variables, plus six real functions of two real variables, and in fact such a representation is known.

See also the more recent paper by Llosa and Soler, CQG 22 (2005).

Chris Hillman

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Dear kurious,

Thank you for bringing up this interesting question. I can confirm that knowing the number of physical situations in spacetime does not necessarily equate to knowing the number of solutions to Einstein's field equations.

Einstein's field equations are a set of ten partial differential equations that describe the relationship between the curvature of spacetime and the distribution of matter and energy. These equations are highly complex and nonlinear, making it difficult to determine the exact number of solutions.

While it may seem intuitive that knowing the physical situations in spacetime would give us a better understanding of the solutions to these equations, it is not a direct correlation. In fact, many solutions to these equations have been discovered through mathematical techniques and numerical simulations rather than through physical observations.

As you mentioned, there have been attempts to use methods such as constraint counting to estimate the number of solutions to these equations. However, these methods are still limited and do not provide a definitive answer.

I am familiar with the work of Siklos and Schutz, and I agree that their approach is intriguing. However, it is important to note that their results are based on certain assumptions and approximations, and may not be applicable to all situations.

In conclusion, the number of physical situations in spacetime does not necessarily determine the number of solutions to Einstein's field equations. Further research and advancements in mathematical and computational techniques are needed to fully understand the solutions to these equations. Thank you again for bringing up this thought-provoking topic.

Thank you for sharing this information about Einstein's approach to "counting" solutions to the EFE. It's fascinating to see the connections between his work and that of his colleagues, and how they have contributed to our understanding of the solutions to this important equation. It's also interesting to see how this approach has been applied in the context of gtr and how it continues to be relevant in current research. I will definitely look into the papers you mentioned for further reading. Thanks again for your insights!

## 1. What is the EFE?

The EFE stands for the Einstein Field Equations, which are a set of equations developed by Albert Einstein to describe the relationship between matter and energy and the curvature of space-time in the theory of general relativity.

## 2. What does it mean to count solutions to the EFE?

Counting solutions to the EFE involves finding all possible mathematical solutions that satisfy the equations. These solutions can represent different physical scenarios, such as the behavior of particles in a gravitational field or the evolution of the universe.

## 3. How do scientists count solutions to the EFE?

Scientists use mathematical techniques and computer simulations to solve the EFE and determine the solutions. This involves solving complex equations and analyzing the results to understand the behavior of matter and energy in different scenarios.

## 4. Why is it important to count solutions to the EFE?

Counting solutions to the EFE allows scientists to understand the behavior of matter and energy in the universe and make predictions about the evolution of the universe. It also helps to test the accuracy of the theory of general relativity and potentially uncover new insights into the nature of space-time.

## 5. What are some current challenges in counting solutions to the EFE?

One of the main challenges in counting solutions to the EFE is the complexity of the equations and the difficulty in finding exact solutions. This requires advanced mathematical and computational techniques, and there may be some scenarios where solutions cannot be found. Additionally, the EFE may need to be modified or extended to account for phenomena such as dark matter and dark energy, which adds further complexity to the problem.

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