Solving the Einstein field equations

TL;DR

This paper presents a perturbative method to solve Einstein's vacuum field equations, providing a graphical diagrammatic representation of the solutions to facilitate understanding and computation.

Contribution

It introduces a novel perturbation-based approach with a diagrammatic visualization for solving Einstein's equations.

Findings

Successful construction of solutions using perturbation theory

Diagrammatic representation simplifies understanding of solutions

Provides a new computational framework for Einstein's equations

Abstract

A solution of the Einstein vacuum field equations is constructed within the contex of perturbation theory. The solution possesses a graphical representation in terms of diagrams.

Figures

Equations

$$G_{\mu \nu }=0,$$

$$G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R,$$

$$R_{\mu \nu }={\Gamma }_{\mu \nu ,\beta }^{\beta }-{\Gamma }_{\mu \beta ,\nu }^{\beta }+{\Gamma }_{\mu \nu }^{\beta }{\Gamma }_{\beta \alpha }^{\alpha }-{\Gamma }_{\mu \beta }^{\alpha }{\Gamma }_{\nu \alpha }^{\beta },$$

$${\Gamma }_{\alpha \beta }^{\gamma }=\frac{1}{2}{g}^{\gamma \rho }({\partial }_{\beta }{g}_{\rho \alpha }+{\partial }_{\alpha }{g}_{\rho \beta }-{\partial }_{\rho }{g}_{\alpha \beta }).$$

$$\sum _{n=1}^{\infty }{G}_{\mu \nu }^n(h)=0.$$

$$G_{\mu \nu }^n(h)={\frac{1}{n!}\frac{d^n{G}_{\mu \nu }(\eta +\xi h)}{d^n\xi }|}_{\xi =0}.$$

$${\eta }^{\alpha \beta }{\partial }_{\alpha }{\phi }_{\beta \nu }=0,$$

$${\phi }_{\mu \nu }=h_{\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }{h}_{\alpha }^{\alpha },h_{\alpha }^{\alpha }={\eta }^{\alpha \beta }{h}_{\alpha \beta }.$$

$$h_{\mu \nu }={\phi }_{\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }{\phi }_{\alpha }^{\alpha },{\phi }_{\alpha }^{\alpha }={\eta }^{\alpha \beta }{\phi }_{\alpha \beta }.$$

$$\square \phi -2\sum _{n=2}^{\infty }{\tilde{G}}^n(\phi )=0,$$

$$\square {\phi }_{\mu \nu }=0$$

$${\phi }_{0\mu \nu }=\frac{1}{4\pi }M(v_{\mu \nu })+\frac{1}{4\pi }\frac{\partial }{\partial t}(tM(u_{\mu \nu })),$$

$$M(w_{\mu \nu })={\int }_S{w}_{\mu \nu }(x^1+t{\xi }^1,x^2+t{\xi }^2,x^3+t{\xi }^3)d{\sigma }_{\xi },$$

$${\phi }_{0\mu \nu }(0,x^1,x^2,x^3)=u_{\mu \nu }(x^1,x^2,x^3),$$

$${\frac{\partial {\phi }_{0\mu \nu }(t,x^1,x^2,x^3)}{\partial t}|}_{t=0}=v_{\mu \nu }(x^1,x^2,x^3),$$

$$v_{0\nu }-\sum _{i=1}^3{\partial }_i{u}_{i\nu }=0.$$

$$\square {\phi }_{\mu \nu }=f_{\mu \nu }$$

$$\mathcal{E}(f_{\mu \nu })=\frac{1}{4\pi }{\int }_0^t\tau d\tau {\int }_S{f}_{\mu \nu }(t-\tau ,x^1+\tau {\xi }^1,x^2+\tau {\xi }^2,x^3+\tau {\xi }^3)d{\sigma }_{\xi }.$$

$$\phi =\kappa {\phi }_0+\sum _{n=2}^{\infty }{p}_n(\phi ),$$

$$⟨\dots {⟩}_n:V^n\to V,n=2,3,\dots ,$$

$$⟨{\gamma }_1,\dots ,{\gamma }_n{⟩}_n=\frac{\partial }{\partial a_1}\dots \frac{\partial }{\partial a_n}{p}_n(a_1{\gamma }_1+\dots +a_n{\gamma }_n).$$

$$⟨\gamma ,\dots ,\gamma {⟩}_n=n!p_n(\gamma ).$$

$$\phi =\kappa {\phi }_0+\sum _{n=2}^{\infty }\frac{1}{n!}⟨\phi ,\dots ,\phi {⟩}_n.$$

$$P_{i_1\dots i_n}^m:V^m\to V^{m-n+1}$$

$$P_{i_1\dots i_n}^m({\gamma }_1,\dots ,{\gamma }_m)=(⟨{\gamma }_{i_1},\dots ,{\gamma }_{i_n}{⟩}_n,{\gamma }_1,\dots ,{\hat{\gamma }}_{i_1},\dots ,{\hat{\gamma }}_{i_n},\dots ,{\gamma }_m),$$

$$\varphi =P_{I_s}^{n_s}\dots P_{I_2}^{m-n_1+1}{P}_{I_1}^m({\gamma }_1,\dots ,{\gamma }_m)$$

$$({\gamma }_1,\dots ,{\gamma }_n)\to ⟨{\gamma }_1,\dots ,{\gamma }_n{⟩}_n$$

$$P_{ijk}^m({\gamma }_1,\dots ,{\gamma }_m)=(⟨{\gamma }_i,{\gamma }_j,{\gamma }_k{⟩}_3,{\gamma }_1,\dots ,{\hat{\gamma }}_i,\dots ,{\hat{\gamma }}_j,\dots ,{\hat{\gamma }}_k,\dots ,{\gamma }_m).$$

$$⟨\dots ⟩:V^m\to V,m=2,3,\dots ,$$

$$⟨{\gamma }_1,{\gamma }_2⟩=⟨{\gamma }_1,{\gamma }_2{⟩}_2,$$

Full text

In a recent article [1] for a wide class of nonlinear equations a perturbative solution was constructed.This class includes equations of motion of field theories. Solutions of the pure Yang-Mills field equations and equations of motions in spinor electrodynamics [2] were obtained. The solutions can be described by diagrams. The aim of the present paper is to construct a solution of the Einstein field equations in the vacuum.

Let $x=(x^{0},x^{1},x^{2},x^{3})$ be space-time coordinates, and $g(x)=g_{\mu\nu}(x)dx^{\mu}dx^{\nu}$ a space-time metric. The vacuum Einstein equations read

[TABLE]

where

[TABLE]

$R_{\mu\nu}$ is the Ricci tensor

[TABLE]

$R=R^{\mu}_{\mu},$ and $\Gamma_{\alpha\beta}^{\gamma}$ is the Cristofel symbol

[TABLE]

For $\eta=\eta_{\mu\nu}dx^{\mu}dx^{\nu},\eta_{\mu\nu}=diag(1,-1,-1,-1),$ we can expand equations (1) in powers of $h=g-\eta$

[TABLE]

Here

[TABLE]

We impose a gauge condition

[TABLE]

where

[TABLE]

From equation (4) it follows

[TABLE]

Let $\tilde{G}^{n}(\varphi)$ be defined by $\tilde{G}^{n}(\varphi)=G^{n}(h)_{\mu\nu}dx^{\mu}dx^{\nu},$ $\varphi=\varphi_{\mu\nu}dx^{\mu}dx^{\nu}.$ Then equation (2) takes the form [3]

[TABLE]

where $\Box=\eta^{\alpha\beta}\partial_{\alpha}\partial_{\beta}.$

The general solution $\varphi_{0\mu\nu}$ of the free equation

[TABLE]

is given by [4]

[TABLE]

where

[TABLE]

$\xi^{1},\xi^{2},\xi^{3}$ are coordinates on the unit sphere $S$, $\sigma_{\xi}$ is the area element on $S,$ $t=x^{0},$

[TABLE]

[TABLE]

and $u_{\mu\nu},v_{\mu\nu}$ are some functions. Equation (3) implies

[TABLE]

A specific solution ${\cal E}(f_{\mu\nu})$ of the inhomogeneous equation

[TABLE]

reads

[TABLE]

Then equation (6) can be written in the form

[TABLE]

where $\varphi_{0}=\varphi_{0\mu\nu}dx^{\mu}dx^{\nu},$ and $p_{n}=2{\cal E}(\tilde{G}^{n}_{\mu\nu})dx^{\mu}dx^{\nu}.$ Here we introduced an auxiliary parameter $\kappa$. Eventually it is set to be $1.$

Let $V$ be the space of symmetric 2-forms. We shall need the polylinear symmetric functions

[TABLE]

defined for $\gamma_{1},\ldots,\gamma_{n}\in V$ by

[TABLE]

One can check that

[TABLE]

Then equation (7) becomes

[TABLE]

For $m\geq 2,1\leq i_{1}<\ldots<i_{n}\leq m$ let

[TABLE]

be defined by

[TABLE]

where $\widehat{\gamma}$ means that $\gamma$ is omitted. If $\phi\in V$ is given by

[TABLE]

for some $I_{1}=(i_{1}^{1}\ldots i^{1}_{n_{1}}),I_{2}=(i^{2}_{1}\ldots i^{2}_{n_{2}}),\ldots,I_{s}=(i^{s}_{1}\ldots i^{s}_{n_{s}}),$ we say that $\phi$ is a descendant of $(\gamma_{1},\ldots,\gamma_{m}).$

Each descendant can be represented by a diagram. In this diagram an element of $V$ is represented by the line segment    . A product

[TABLE]

is represented by the vertex joining the line segments for $\gamma_{1},\ldots,\gamma_{n},$ $\langle\gamma_{1},\ldots,\gamma_{n}\rangle_{n}.$ The general rule for graphical representation of $P^{m}_{I}(\gamma_{1},\ldots,\gamma_{m})$ should be clear from Figure 1.Here we show the diagram for

[TABLE]

The points labeled by $1,\ldots,m$ represent the ends of the lines for $\gamma_{1},\ldots,\gamma_{m}.$ Using the representation for $P^{m}_{i_{1}\ldots i_{n}}(\gamma_{1},\ldots,\gamma_{m})$ one can consecutively draw the diagrams for $P^{n_{1}}_{I_{1}}(\gamma_{1},\ldots,\gamma_{m}),P^{n_{2}}_{I_{2}}P^{n_{1}}_{I_{1}}(\gamma_{1},\ldots,\gamma_{m}),\ldots,\phi$ (10).

Let us introduce a family of functions

[TABLE]

such that for $\gamma_{1},\ldots,\gamma_{m}\in V$ $\langle\gamma_{1},\ldots,\gamma_{m}\rangle$ is defined as the sum of all the descendants of its arguments. For example, for $m=2$ and $m=3$ we have

[TABLE]

[TABLE]

A solution of equation (9) is given by [1]

[TABLE]

where

[TABLE]

$\langle\varphi_{0}^{n}\rangle$ is defined as [math] if $n=0,$ and otherwise

[TABLE]

For example, the $O(\kappa^{3})$ contribution in $\varphi$ is given by

[TABLE]

The diagram for $\langle\langle\varphi_{0},\varphi_{0}\rangle_{2},\varphi_{0}\rangle_{2}$ is depicted in Figure 2.

The tensor $h_{\mu\nu}$ can be found by using (5).