Einstein's equation certainly have wave-type solutions, these are usually called radiative spacetimes. Today there are many exact solutions with gravitational waves, e.g. plane waves, pp waves, sandwich waves, impulsive plane waves, impulsive expanding waves, Robinson-Trautman solutions, etc.

I didn't know the paper by Fiziev and Shirkov and now I have read it only superficially. I think what they do is not rewriting Einstein's equations in terms of the Klein-Gordon equation. I believe they want to construct renormalizable quantum field theory. 

As far as I understood it, they consider a hypersurface in the flat spacetime (Minkowski). This hypersurface is curved and they restrict the Klein-Gordon equation (as the toy model) to this hypersurface. This was done in the earlier paper.

In the paper you refer to, they want this hypersurface to be more realistic, so they construct it as a solution of Einstein's equations in lower dimension (equal to dimension of hypersurface) and they include also the cosmological constant. In this way they obtain non-static, dynamical spacetime of dimension

Thank you, Martin, for your answer. So is it possible to put Einstein's equations in terms of KFG (Klein-Fock-Gordon) equation? Thanks

Dear Victor, I don't think it's possible, but maybe there is some trick I don't know.

However, I am not sure what do you mean by putting Einstein's equation in terms of KG equation. I would say that both these equations are a special case of quasi-linear second order partial differential equation.

The common property of both equations is that they have well posed initial value problem, for example, which is an important property of any physically reasonable theory. There is a very general theorem by Leray that equations of special type can be formulated as the Cauchy problem. And Einstein's equations and the KG equation fall into this category. In this sense, Einstein's equation can be recast to the form similar to KG equation, but I don't think this is really what you are asking about.

Dear Martin, thanks for your answer. I mean if it is possible to cast Einstein's equations in terms of some kind of wave equation. Btw, i just find this paper discussing KFG equation in Schwarzhild metric: http://www.jetp.ac.ru/cgi-bin/dn/e_066_01_0001.pdf. Best wishes

Victor, Einstein's equations have the wave-like solutions. Although there has been some controversy about that in the past, today we know that there are wave-like solutions.

Notice, by the way, that KG equation has mass term, so the solution of KG equation represents particles with the mass. 

The easiest way how to see the wave character of Einstein's equations is the linearized theory. Take the Minkowski flat spacetime and add a small perturbation to it. This perturbation satisfies linearized Einstein equations which have the form

d'Alembert operator on metric tensor = energy-momentum tensor.

This is exactly the wave equation with the source given by energy-momentum tensor, but with zero mass. So, linearized Einstein's equations are identical to massless KG equation with the source.

The question of solving KG equation on given background is a different problem which is not related to Einstein's equations.

Thanks a lot, Martin. Btw, do you mean KG or KFG? I thought there is slight difference between Klein-Gordon and Klein-Fock-Gordon equation. But may be i am wrong. Best wishes

Dear Victor, I always thought that KG and KFG equation are two names for the same equation, but after your question I am not sure :) If you have some knowledge about the difference between the two, I would be happy if you share this knowledge with me. 

The story I know is that Schrodinger first discovered the Klein-Gordon equation, which is Lorentz-invariant but he found that the solution of this equation does not describe the hydrogen atom to desired degree of precision. So then he derived the non-relativistic version which now bears his name. When the Schrodinger equation turned out to be appropriate, people were looking for relativistic generalization. Klein and Gordon then re-discovered original Schordinger's equation and it is named after them. Independently, in Russia, it was Fock who found the same equation, so today it is named after Klein-Gordon or Klein-Gordon-Fock. As the citizen of country which was occupied by the armies of the Soviet Union, I have tendency to avoid mentioning Russian contributions, but Fock was certainly a great physicist and he deserves appreciation (Fock space, Hartree-Fock equation). 

But I admit I can be wrong, so please, tell me what you think the difference between KG and KGF is. Best regards :)

Dear Martin, basically Fock's history that you told appear correct, but i think KG equation is related to quantum mechanics, while KFG eq. is related to some sort of curved spacetime because of its use of tensorial notation. See for instance the JETP paper that I cited before. I will try to find exact information on difference between the two. Best wishes

Martin, i read several articles on history of KFG and KG equations, including wikipedia: http://en.m.wikipedia.org/wiki/Klein–Gordon_equation. Perhaps you are right that both are essentially the same, except that Fock generalized the wave equation for magnetic field, and he also provided gauge extension of KG equation. That is why he introduced the tensor notation. What do you think? Thanks

OK Victor, but I believe that's a different issue. Anyway, thank you for interesting comment.

 Klein-Gordon (or Klein-Gordon-Fock) equation has been originally written for the flat space-time. It's form is

(d'Alembert operator + m^2) phi = 0.

(the sign depends on the signature of metric, so sometimes it appears in the form d'Alembert - m^2). By d'Alembert operator I mean usual operator

second derivative w.r.t. time ^2 - Laplace.

And yes, you are right that originally it was formulated with the quantum mechanics in mind. 

The d'Alembert operator in the flat spacetime is invariant with respect to the Lorentz transformation (or the Poincare transformation). However, this operator can be written in the way which is invariant with respect to arbitrary coordinate transformation. To do this, you have to introduce the covariant derivative, which works in arbitrary coordinate system, but in the flat spacetime it reduces to the usual wave operator. The covariant derivative is a tensorial operation, because if it acts on a tensor field, it produces another tensor field. If you replace ordinary partial derivatives in the KG equation by the covariant ones, you are able to generalize the KG equation to arbitrary curved spacetime. 

And this general form of KG equation is often used in relativity, because it's the simplest relativistically invariant equation you can write down. But this general form is just a natural generalization of original KG equation. In this sense, it is still the same equation.

One thing is that it it is related to quantum mechanics, beacuse the KG equation describes the massive particles with zero spin. Another thing is that it is a useful toy model for realistic physical fields in general relativity.

One interesting generalization of the KG equation is the so called conformally invariant KG equation, which has the form

(d'Alembert + R/6)phi  = 0,

where R is the scalar curvature of the spacetime. This equation has been studied extensively by Penrose, and I have written one paper on the properties of this equation. 

My point is, that KG equation can be naturally generalized to curved spacetime, but it is still the same equation. 

Dear Martin, thank you for your answer. Yes i agree with you that KG equation can be naturally generalized to curved spacetime, but as far as i know Fock generalize it further in 1927 to include magnetic field effect. I spent sometime to learn on the properties of KG equation using quaternionic expression, but i am still digging on KFG equation which is nonlinear eq.

Anyway, back to our question above, do you think that Einstein's equations can be expressed in terms of (nonlinear) KFG equation? I mean not the linearized ones, but the exact Eiinstein's equations. Thanks

What Fiziev and Shirkov have studied is what happens in 2+1 dimensions, assuming axial symmetry,  i.e. a very special class of problems and that is crucial. For Einstein's equations  describe a tensor, the metric; the Klein-Gordon equation describes a scalar field. Not all metrics can be described by determining a scalar field alone. One such class of metrics are Kähler metrics, determined by the Kähler potential, cf. http://people.mpim-bonn.mpg.de/hwbllmnn/archiv/kaehler0609.pdf and the manifolds are called Kähler manifolds. Cf. also, http://www.math.harvard.edu/~siu/siu_reprints/dmv_book.pdf

Einstein's equations *do* have wave-like solutions and this has been known for a long time. Such gravitational waves were, indirectly, observed by monitoring the period of binary pulsars (Nobel prize in physics to Hulse and Taylor in 1993)-so there's no connection between this  general fact and the special case  of expressing the metric tensor in terms of a scalar function.

Thank you Stam, for your answer. Best wishes