Solutions for the null-surface formulation in $2+1$ dimensions leading to spacetimes of Petrov types I, II, and D

Abstract

The only nontrivial exact solutions reported to-date for the field equations of the null-surface formulation (NSF) of general relativity are for the ($2+1$)-dimensional version of the theory, where three such solutions are known. This work presents a new family of NSF solutions. The corresponding general relativistic spacetimes are shown to span three different Petrov types, depending upon the choices that are made for various parameters. All of the scalar invariants for the spacetimes are constant, as are all of the eigenvalues of the Cotton-York tensor. The physical nature of a possible source term is discussed in detail, and two of the previously known NSF solutions are presented as special cases. The new family of solutions was derived by assuming additive separability—meaning that the dependent variable in the field equations is represented as a sum. This effectively turns the main NSF field equation (which is a partial differential equation) into an ordinary differential equation that is exactly solvable. The possibility of adapting this approach to the ($3+1$)-dimensional version of the NSF is discussed.

Appendix: Christoffel symbols and curvature tensors

The Christoffel symbols are

$$\begin{aligned} \Gamma ^{u}_{\; uu}= & {} -a\Gamma ^{u}_{\; \omega \omega } = -a\Gamma ^{\omega }_{\; \omega \rho } = 2^{-1} ay^{-1}\, \partial _{\rho }y , \\ \Gamma ^{u}_{\; u\omega }= & {} 8^{-1} Ay^{-1/2} + ay^{-3/2} , \\ \Gamma ^{u}_{\; u\rho }= & {} \Gamma ^{u}_{\; \omega \rho } = \Gamma ^{u}_{\; \rho \rho } = \Gamma ^{\omega }_{\; \rho \rho } = 0 , \\ \Gamma ^{\omega }_{\; uu}= & {} 2^{-1}Aay^{-1/2} + 3a^{2}y^{-3/2} - 2^{-1} kay^{1/2} , \\ \Gamma ^{\omega }_{\; u\omega }= & {} -(16^{-1}A + ay^{-1})\, \partial _{\rho }y , \\ \Gamma ^{\omega }_{\; u\rho }= & {} 8^{-1} Ay^{-1/2} + ay^{-3/2} , \\ \Gamma ^{\omega }_{\; \omega \omega }= & {} -4^{-1} Ay^{-1/2} - ay^{-3/2} + 2^{-1}ky^{1/2} , \\ \Gamma ^{\rho }_{\; uu}= & {} -2^{-1} a\, (-4^{-1} A + 3ay^{-1} + 2^{-1} ky)\, \partial _{\rho }y , \\ \Gamma ^{\rho }_{\; u\omega }= & {} -(16^{-1}Aky^{3/2} + a^{2}y^{-3/2} + kay^{1/2}) , \\ \Gamma ^{\rho }_{\; u\rho }= & {} (16^{-1}A - ay^{-1})\, \partial _{\rho }y , \\ \Gamma ^{\rho }_{\; \omega \omega }= & {} (2^{-1}\, ay^{-1} + 4^{-1}\, ky)\, \partial _{\rho }y , \\ \Gamma ^{\rho }_{\; \omega \rho }= & {} 8^{-1} Ay^{-1/2} - 2^{-1}ky^{1/2} , \\ \Gamma ^{\rho }_{\; \rho \rho }= & {} -y^{-1}\, \partial _{\rho }y .\\ \end{aligned}$$

Using the abbreviation $W := A + 8a\, y^{-1}$, the Ricci tensor can be written as a matrix, $[R_{ij}]$:

$$\begin{aligned}{}[R_{ij} ] = \left( \begin{array}{ccc} \frac{1}{256}\, (A^{3} + W^{3}) + \frac{1}{8}\, A k a &{} -\frac{1}{64}\, A W\, y^{-1/2}\, \partial _{\rho }y &{} \frac{1}{32}\, W^{2}\, y^{-1} \\ -\frac{1}{64}\, A W\, y^{-1/2}\, \partial _{\rho }y &{} -\frac{1}{32}\, A^{2}\, y^{-1} + \frac{1}{8}\, A k &{} -\frac{1}{8}\, A\, y^{-3/2}\, \partial _{\rho }y \\ \frac{1}{32}\, W^{2}\, y^{-1} &{} -\frac{1}{8}\, A\, y^{-3/2}\, \partial _{\rho }y &{} \frac{1}{4}\, W\, y^{-2} \\ \end{array} \right) . \end{aligned}$$

The scalar curvature, R, is given in Eq. (24).

The components of the Einstein tensor, $G_{ij} = R_{ij} - \frac{1}{2} R g_{ij}$, are

$$\begin{aligned} G_{uu}= & {} \frac{1}{256} (A^{3} + W^{3}) + \frac{1}{8} R (A + 12ay^{-1}) + \frac{1}{8} Aka , \\ G_{u\omega }= & {} -\frac{1}{64} (AW + 16R) y^{-1/2} \, \partial _{\rho } y , \\ G_{u\rho }= & {} \frac{1}{32} y^{-1} (W^{2} + 16R) , \\ G_{\omega \omega }= & {} -\frac{1}{32} y^{-1} (A^{2} + 16R) + \frac{1}{8} Ak , \\ G_{\omega \rho }= & {} -\frac{1}{8} Ay^{-3/2} \, \partial _{\rho } y , \\ G_{\rho \rho }= & {} \frac{1}{4} W y^{-2} . \end{aligned}$$

The covariant derivatives of the velocity can be written as a matrix, $[ U_{i \, ; \, j} ]$:

$$\begin{aligned} \left( \begin{array}{ccc} -\frac{1}{8} a^{1/2} y^{-1/2}\, W\, \partial _{\rho }y &{} -\frac{1}{64} a^{-1/2}\, W^{2} &{} -\frac{1}{16} a^{-1/2} y^{-1/2}\, W\, \partial _{\rho }y \\ a^{-1/2} \left( \frac{1}{2} ka - \frac{1}{64} W^{2} + a^{2} y^{-2} \right) &{} \frac{1}{16} a^{-1/2} y^{-1/2}\, A\, \partial _{\rho }y &{} \frac{1}{2}a^{-1/2} y^{-1} \left( ky - \frac{1}{4} A \right) \\ -a^{1/2} y^{-3/2}\, \partial _{\rho }y &{} -\frac{1}{8} a^{-1/2} y^{-1}\, W &{} -\frac{1}{2} a^{-1/2} y^{-3/2}\, \partial _{\rho }y \\ \end{array} \right) . \end{aligned}$$

The acceleration vector, ${\dot{U}}_{i}$, is defined by ${\dot{U}}_{i} := U_{i\, ;\, j}\, U^{j}$, and its components are

$$\begin{aligned} {\dot{U}}^{u}= & {} \left( 1 - \frac{1}{16} Wa^{-1}y \right) \, \partial _{\rho }y , \\ {\dot{U}}^{\omega }= & {} \frac{1}{4} Wy^{1/2} \left( 1 - \frac{1}{16} Wa^{-1}y \right) - \frac{1}{2} Ra^{-1}y^{3/2} , \\= & {} a^{-1} y^{3/2} \left( -R + \frac{1}{2} ka + a^{2}y^{-2} \right) , \\ {\dot{U}}^{\rho }= & {} - a \left( 1 - \frac{1}{16} Wa^{-1}y \right) \, \partial _{\rho }y - \frac{1}{4} Ra^{-1}y^{2} \, \partial _{\rho }y . \end{aligned}$$

The components of the heat-flux vector $q_{i}$ are

$$\begin{aligned} q_{u}= & {} \frac{1}{64\kappa } a^{-3/2} y^{3/2} AWR , \\ q_{\omega }= & {} -\frac{1}{16\kappa } a^{-3/2} y AR \, \partial _{\rho }y , \\ q_{\rho }= & {} \frac{1}{8\kappa } a^{-3/2} y^{1/2} AR . \end{aligned}$$

The only nonzero components of the vorticity tensor, $\omega _{ij}$, are

$$\begin{aligned} \omega _{u \omega }= & {} -\omega _{\omega u} = -\frac{1}{16} WRa^{-3/2}y , \\ \omega _{\omega \rho }= & {} -\omega _{\rho \omega } = \frac{1}{2} Ra^{-3/2} . \end{aligned}$$

The components of the shear tensor, $\sigma _{ij}$, are

$$\begin{aligned} \sigma _{uu}= & {} -\frac{1}{1024} A W^{2}a^{-3/2}y^{3/2} \, \partial _{\rho } y , \\ \sigma _{u \omega }= & {} \sigma _{\omega u} = \frac{1}{128} W^{2}k a^{-3/2}y^{2} - \frac{1}{16} WRa^{-3/2}y , \\ \sigma _{u \rho }= & {} \sigma _{u \rho } = -\frac{1}{128} AWa^{-3/2} y^{1/2} \, \partial _{\rho } y , \\ \sigma _{\omega \omega }= & {} -\frac{1}{32} Aka^{-3/2}y^{3/2}\, \partial _{\rho } y , \\ \sigma _{\omega \rho }= & {} \sigma _{\rho \omega } = \frac{1}{16} Wka^{-3/2}y - \frac{1}{2} Ra^{-3/2} , \\ \sigma _{\rho \rho }= & {} -\frac{1}{16} Aa^{-3/2}y^{-1/2}\, \partial _{\rho } y . \end{aligned}$$

The components of the Cotton-York tensor, $C^{i}_{\, j}$, are defined in Eq. (27) and are as follows:

$$\begin{aligned} C^{u}_{\;\; u}= & {} -\frac{1}{4} W \left( R + \frac{1}{32} AW \right) , \\= & {} -\frac{1}{4} A \left( R + \frac{1}{32} W^{2} \right) - 2 R a y^{-1} , \\ C^{u}_{\;\; \omega }= & {} y^{-1/2} \left( \frac{1}{2} R + \frac{1}{32} A^{2} \right) \partial _{\rho } y , \\ C^{u}_{\;\; \rho }= & {} -R y^{-1} - \frac{1}{16} AW y^{-1} , \\ C^{\omega }_{\;\; u}= & {} ay^{-1/2} \left( \frac{1}{2} R + \frac{1}{32} AW \right) \partial _{\rho } y , \\ C^{\omega }_{\;\;\omega }= & {} \frac{1}{128} A^{2} W + R ay^{-1} , \\ C^{\omega }_{\;\; \rho }= & {} \frac{1}{4} A a y^{-3/2}\, \partial _{\rho } y , \\ C^{\rho }_{\;\; u}= & {} \frac{1}{128} A W^{2} a - 2 R a^{2} y^{-1} + \frac{1}{2} R W a , \\ C^{\rho }_{\;\; \omega }= & {} -ay^{-1/2} \left( \frac{1}{2} R + \frac{1}{32} A^{2} \right) \partial _{\rho } y , \\ C^{\rho }_{\;\; \rho }= & {} \frac{1}{2} A a^{2} y^{-2} - 2 k a^{2} y^{-1} + \frac{1}{4} AR + 3 R a y^{-1} , \\= & {} \frac{1}{16} AW ay^{-1} + \frac{1}{4} AR + R a y^{-1} . \end{aligned}$$