On the Bondi mass of Maxwell–Klein–Gordon spacetimes

Abstract

In this paper we calculate the Bondi mass of asymptotically flat spacetimes with interacting electromagnetic and scalar fields. The system of coupled Einstein–Maxwell–Klein–Gordon equations is investigated and corresponding field equations are written in the spinor form and in the Newman–Penrose formalism. Asymptotically flat solution of the resulting system is found near null infinity. Finally we use the asymptotic twistor equation to find the Bondi mass of the spacetime and derive the Bondi mass-loss formula. We compare the results with our previous work (Bičák et al. in Class Quantum Gravity 27(17):175011, 2010) and show that, unlike the conformal scalar field, the (Maxwell–)Klein–Gordon field has negatively semi-definite mass-loss formula.

Appendix 1: Field equations in the Newman–Penrose formalism

Four-potential $A_a = A_{AA'}$ is a real vector field and its components with respect to the spin basis will be denoted by

$$\begin{aligned} A_0&= A_{XX'}o^X\overline{o}^{X'},&A_1&= A_{XX'}o^X\overline{\iota }^{X'}, \end{aligned}$$

(71a)

$$\begin{aligned} A_{\overline{1}}&= A_{XX'} \iota ^X \overline{o}^{X'},&A_2&= A_{XX'}\iota ^X\overline{\iota }^{X'}. \end{aligned}$$

(71b)

Similarly we introduce the Newman–Penrose components of electromagnetic spinor $\phi _{AB}$ by

$$\begin{aligned} \phi _0 = \phi _{AB}\,o^A\,o^B, \qquad \phi _1 = \phi _{AB}\,o^A\,\iota ^B, \qquad \phi _2 = \phi _{AB}\,\iota ^A\,\iota ^B. \end{aligned}$$

(72)

Potential $A_a$ is governed by Eq. (8),

$$\begin{aligned} \nabla ^{A'}_A A_{BA'} = -\,\phi _{AB}. \end{aligned}$$

Projections of this equation onto the spin basis are

$$\begin{aligned} DA_1-\delta A_0&= (\overline{\pi }-\overline{\alpha }-\beta )A_0 + (\varepsilon -\overline{\varepsilon }+\overline{\rho })A_1+\sigma A_{\overline{1}} - \kappa A_2+\phi _0, \end{aligned}$$

(73a)

$$\begin{aligned} DA_2-\delta A_{\overline{1}}&= -\mu A_0 +\pi A_1 +(\overline{\pi }-\overline{\alpha }+\beta )A_{\overline{1}}+(\overline{\rho }-\varepsilon -\overline{\varepsilon })A_2+\phi _1, \end{aligned}$$

(73b)

$$\begin{aligned} \varDelta A_0-\overline{\delta }A_1&= (\gamma +\overline{\gamma }-\overline{\mu })A_0 + (\overline{\beta }-\alpha -\overline{\tau })A_1 - \tau A_{\overline{1}} + \rho A_2 - \phi _1, \end{aligned}$$

(73c)

$$\begin{aligned} \varDelta A_{\overline{1}}-\overline{\delta }A_2&= \nu A_0 - \lambda A_1 + (\overline{\gamma }-\gamma -\overline{\mu })A_{\overline{1}} + (\alpha +\overline{\beta }-\overline{\tau })A_2-\phi _2. \end{aligned}$$

(73d)

The Lorenz condition $\nabla ^a A_a = 0$ in the Newman–Penrose formalism acquires the form

$$\begin{aligned} DA_2 - \varDelta A_0 - \delta A_{\overline{1}} - \overline{\delta }A_1&= (\gamma +\overline{\gamma }-\mu -\overline{\mu })A_0 + (\pi -\alpha +\overline{\beta }-\overline{\tau })A_1 \nonumber \\&\quad +(\overline{\pi }-\overline{\alpha }+\beta -\tau )A_{\overline{1}} + (\rho +\overline{\rho }-\varepsilon -\overline{\varepsilon })A_2 = 0. \end{aligned}$$

(74)

Projections of the gradient $\varphi _{AA'}=\nabla _{AA'}\phi$ will be denoted by

$$\begin{aligned} \begin{aligned} \varphi _0 = D\phi , \qquad \varphi _2 = \varDelta \phi , \qquad \varphi _1 = \delta \phi , \qquad \varphi _{\overline{1}} = \overline{\delta }\phi , \\ \overline{\varphi }_0 = D\overline{\phi }, \qquad \overline{\varphi }_2 = \varDelta \overline{\phi }, \qquad \overline{\varphi }_1 = \delta \overline{\phi },\qquad \overline{\varphi }_{\overline{1}} = \overline{\delta }\,\overline{\phi } \end{aligned} \end{aligned}$$

(75)

Now we can complete equations for electromagnetic field. Equation (9),

$$\begin{aligned} \nabla ^A_{B'}\phi _{AB} = \frac{ie}{2}\left( \overline{\phi }\,\varphi _b - \phi \,\overline{\varphi }_b \right) - e^2\,\phi \,\overline{\phi }\,A_b, \end{aligned}$$

is the spinor version of Maxwell’s equations with four-current $j^a$ on the right hand side. Projections of this equation onto the spin basis follow:

$$\begin{aligned} D\phi _1-\overline{\delta }\phi _0&= (\pi -2\alpha )\phi _0+2\rho \phi _1-\kappa \phi _2+\frac{ie}{2} \left( \phi \overline{\varphi }_0-\overline{\phi }\varphi _0\right) +e^2\phi \overline{\phi }A_0, \end{aligned}$$

(76a)

$$\begin{aligned} D\phi _2-\overline{\delta }\phi _1&= -\lambda \phi _0+2\pi \phi _1+(\rho -2\varepsilon )\phi _2+\frac{ie}{2}\left( \phi \overline{\varphi }_{\overline{1}}-\overline{\phi }\varphi _{\overline{1}}\right) +e^2\phi \overline{\phi }A_{\overline{1}}, \end{aligned}$$

(76b)

$$\begin{aligned} \varDelta \phi _0-\delta \phi _1&= (2\gamma -\mu )\phi _0 - 2\tau \phi _1+\sigma \phi _2 +\frac{ie}{2}\left( \overline{\phi }\varphi _1-\phi \overline{\varphi }_1\right) -e^2\phi \overline{\phi }A_1, \end{aligned}$$

(76c)

$$\begin{aligned} \varDelta \phi _1-\delta \phi _2&= \nu \phi _0 - 2\mu \phi _1 +(2\beta -\tau )\phi _2+\frac{ie}{2}\left( \overline{\phi }\varphi _2-\phi \overline{\varphi }_2\right) -e^2\phi \overline{\phi }A_2. \end{aligned}$$

(76d)

Dynamical equation for the gradient $\varphi _{AA'}$ is provided by Eq. (12)

$$\begin{aligned} \nabla ^A_{A'}\varphi _{AB'}&= \,i\,e\,A^c\varphi _c\,\varepsilon _{A'B'} + \frac{1}{2}\left( m^2 - e^2\,A^c A_c\right) \phi \,\varepsilon _{A'B'}. \end{aligned}$$

(77)

Projected on the spin basis, this equation is equivalent to any of the following four scalar equations:

$$\begin{aligned} D\varphi _{\overline{1}}-\overline{\delta }\varphi _0&= (\pi -\alpha -\overline{\beta })\varphi _0+\overline{\sigma }\varphi _1+(\rho +\overline{\varepsilon }-\varepsilon )\varphi _{\overline{1}}-\overline{\kappa }\varphi _2, \end{aligned}$$

(78a)

$$\begin{aligned} D\varphi _2-\overline{\delta }\varphi _1&= -\overline{\mu }\varphi _0+(\pi -\alpha +\overline{\beta })\varphi _1+\overline{\pi }\varphi _{\overline{1}}+(\rho -\varepsilon -\overline{\varepsilon })\varphi _2 -\phi \,m^2/2 \nonumber \\&\quad +e^2\phi \left( A_0 A_2-A_1 A_{\overline{1}}\right) + ie\left( A_1\varphi _{\overline{1}}+A_{\overline{1}}\varphi _1-A_0\varphi _2-A_2\varphi _0 \right) , \end{aligned}$$

(78b)

$$\begin{aligned} \varDelta \varphi _0-\delta \varphi _{\overline{1}}&= (\gamma +\overline{\gamma }-\mu )\varphi _0 -\overline{\tau }\varphi _1+(\beta -\overline{\alpha }-\tau )\varphi _{\overline{1}}+\overline{\rho }\varphi _2 -\phi \, m^2/2\nonumber \\&\quad +e^2\phi \left( A_0 A_2-A_1 A_{\overline{1}}\right) + ie\left( A_1\varphi _{\overline{1}}+A_{\overline{1}}\varphi _1-A_0\varphi _2-A_2\varphi _0 \right) , \end{aligned}$$

(78c)

$$\begin{aligned} \varDelta \varphi _1-\delta \varphi _2&= \overline{\nu }\varphi _0+(\gamma -\overline{\gamma }-\mu )\varphi _1-\overline{\lambda }\varphi _{\overline{1}}+(\overline{\alpha }+\beta -\tau )\varphi _2. \end{aligned}$$

(78d)

The Ricci spinor is related to the electro-scalar fields by Einstein’s equations (16) and is given by formula (18). The Newman–Penrose components of the Ricci spinor read:

$$\begin{aligned} \varPhi _{00}&= \phi _0\,\overline{\phi }_0 + \left( \fancyscript{D}_0 \phi \right) \left( \fancyscript{D}_0\overline{\phi }\right) = \phi _0\overline{\phi }_0 + \varphi _0\overline{\varphi }_0 + e^2 A_0^2 \phi \overline{\phi }+ieA_0\left( \phi \overline{\varphi }_0-\overline{\phi }\varphi _0\right) , \end{aligned}$$

(79a)

$$\begin{aligned} \varPhi _{01}&= \phi _0\,\overline{\phi }_1 + \left( \fancyscript{D}_{(0} \phi \right) \left( \fancyscript{D}_{1)}\overline{\phi }\right) =\phi _0\overline{\phi }_1 + \varphi _{(0}\overline{\varphi }_{1)} + e^2 \phi \overline{\phi } A_0 A_1 \nonumber \\&\quad + ie\phi A_{(0}\overline{\varphi }_{1)} - ie\overline{\phi } A_{(0}\varphi _{1)},\end{aligned}$$

(79b)

$$\begin{aligned} \varPhi _{11}&= \phi _1\,\overline{\phi }_1 + \frac{1}{2}\,\left[ \left( \fancyscript{D}_{(0}\phi \right) \left( \fancyscript{D}_{2)}\overline{\phi }\right) + \left( \fancyscript{D}_{(1}\phi \right) \left( \fancyscript{D}_{\overline{1})}\overline{\phi }\right) \right] \end{aligned}$$

(79c)

$$\begin{aligned}&= \phi _1\,\overline{\phi }_1 + \frac{1}{2}\left[ \varphi _{(0}\overline{\varphi }_{2)} + \varphi _{(1}\overline{\varphi }_{\overline{1})} + ie\phi \left( A_{(0}\overline{\varphi }_{2)} + A_{(1}\overline{\varphi }_{\overline{1})} \right) \right. \nonumber \\&\quad \left. - ie\overline{\phi } \left( A_{(0}{\varphi }_{2)} + A_{(1}{\varphi }_{\overline{1})} \right) + e^2 \phi \overline{\phi } \left( A_0 A_2 - A_1 A_{\overline{1}} \right) \right] , \end{aligned}$$

(79d)

$$\begin{aligned} \varPhi _{02}&= \phi _0 \overline{\phi }_2 + \left( \fancyscript{D}_1\phi \right) \left( \fancyscript{D}_{1}\overline{\phi }\right) = \phi _0 \overline{\phi }_2 + \varphi _{1}\overline{\varphi }_{1} + e^2 \phi \overline{\phi }A_1^2 +ie\left( \phi A_1 \overline{\varphi }_1 - \overline{\phi }A_1\varphi _1\right) , \end{aligned}$$

(79e)

$$\begin{aligned} \varPhi _{12}&= \phi _1 \overline{\phi }_2 + \left( \fancyscript{D}_{(1}\phi \right) \left( \fancyscript{D}_{2)}\overline{\phi }\right) = \phi _1\overline{\phi }_2 + \varphi _{(1}\overline{\varphi }_{2)} + e^2 \phi \overline{\phi } A_1 A_2\nonumber \\&\quad + ie\left( \phi A_{(2}\overline{\varphi }_{1)}-\overline{\phi }A_{(2}\varphi _{1)}\right) , \end{aligned}$$

(79f)

$$\begin{aligned} \varPhi _{22}&= \phi _2\overline{\phi }_2 + \left( \fancyscript{D}_{2}\phi \right) \left( \fancyscript{D}_{2}\overline{\phi }\right) =\phi _2\overline{\phi }_2 + \varphi _2\,\overline{\varphi }_2 + e^2\phi \overline{\phi } A_2^2 +ieA_2\left( \phi \overline{\varphi }_2-\overline{\phi }\varphi _2\right) . \end{aligned}$$

(79g)

$$\begin{aligned} 6 \varLambda&= \varphi _{(1}\overline{\varphi }_{\overline{1})} - \varphi _{(0}\overline{\varphi }_{2)} + i e \overline{\phi }\left( A_{(0}\varphi _{2)} - A_{(1}\varphi _{\overline{1})}\right) + i e \phi \left( A_{(1}\overline{\varphi }_{\overline{1})} - A_{(0}\overline{\varphi }_{2)}\right) \nonumber \\&\quad +e^2\phi \overline{\phi }\left( A_{\overline{1}}A_1-A_0 A_2 \right) + m^2\phi \overline{\phi }. \end{aligned}$$

(80)

The Ricci identities in the tetrad introduced in Sect. 3 simplify to the following set of equations.

$$\begin{aligned} D \rho&= \rho ^2 + \sigma \,\overline{\sigma } + \varPhi _{00}, \end{aligned}$$

(81a)

$$\begin{aligned} D\sigma&= 2\,\rho \,\sigma + \varPsi _0, \end{aligned}$$

(81b)

$$\begin{aligned} D\alpha&= \rho \,\alpha + \beta \,\overline{\sigma } + \rho \,\pi + \varPhi _{10}, \end{aligned}$$

(81c)

$$\begin{aligned} D\beta&= (\alpha +\pi )\,\sigma + \rho \,\beta + \varPsi _1, \end{aligned}$$

(81d)

$$\begin{aligned} D\gamma&= 2\,\overline{\pi }\,\alpha + 2\,\pi \,\beta + \pi \,\overline{\pi } + \varPsi _2 - \varLambda + \varPhi _{11}, \end{aligned}$$

(81e)

$$\begin{aligned} D\lambda - \overline{\delta }\pi&= \rho \,\lambda + \mu \,\overline{\sigma } + 2\,\alpha \,\pi + \varPhi _{20}, \end{aligned}$$

(81f)

$$\begin{aligned} D\mu - \delta \pi&= \rho \,\mu + \sigma \,\lambda + 2\,\beta \,\pi + \varPsi _2 + 2\,\varLambda , \end{aligned}$$

(81g)

$$\begin{aligned} D\nu - \varDelta \pi&= 2\,\pi \,\mu + 2\,\overline{\pi }\,\lambda + (\gamma -\overline{\gamma })\pi + \varPsi _3 + \varPhi _{21}, \end{aligned}$$

(81h)

$$\begin{aligned} D\tau&= 2\,\overline{\pi }\,\rho + 2\,\pi \,\sigma + \varPsi _1 + \varPhi _{01}, \end{aligned}$$

(81i)

$$\begin{aligned} \varDelta \rho - \overline{\delta }\tau&= (\gamma +\overline{\gamma }-\mu )\rho - \sigma \,\lambda - 2\,\alpha \,\tau - \varPsi _2 - 2\,\varLambda , \end{aligned}$$

(81j)

$$\begin{aligned} \varDelta \sigma - \delta \tau&= - (\mu -3\,\gamma +\overline{\gamma })\,\sigma - \overline{\lambda }\,\rho - 2\,\beta \,\tau - \varPhi _{02}, \end{aligned}$$

(81k)

$$\begin{aligned} \varDelta \lambda - \overline{\delta }\nu&= -(2\,\mu + 3\,\gamma - \overline{\gamma })\,\lambda - (3\,\alpha +\beta )\,\nu , \end{aligned}$$

(81l)

$$\begin{aligned} \varDelta \alpha - \overline{\delta }\gamma&= \rho \,\nu - (\beta +\tau )\,\lambda + (\overline{\gamma }-\mu )\,\alpha + (\overline{\beta }-\overline{\tau })\,\gamma - \varPsi _3, \end{aligned}$$

(81m)

$$\begin{aligned} \varDelta \beta - \delta \gamma&= -\,\mu \,\tau + \sigma \,\nu + (\gamma -\overline{\gamma }-\mu )\,\beta - \alpha \,\overline{\lambda } - \varPhi _{12}, \end{aligned}$$

(81n)

$$\begin{aligned} \varDelta \mu - \delta \nu&= - (\mu +\gamma +\overline{\gamma })\,\mu - \lambda \,\overline{\lambda }+ \overline{\nu }\,\pi + 2\,\beta \,\nu - \varPhi _{22}, \end{aligned}$$

(81o)

$$\begin{aligned} \delta \alpha - \overline{\delta }\beta&= \mu \,\rho - \lambda \,\sigma + \alpha \,\overline{\alpha } + \beta \,\overline{\beta } - 2\,\alpha \,\beta - \varPsi _2 + \varLambda + \varPhi _{11}, \end{aligned}$$

(81p)

$$\begin{aligned} \delta \lambda - \overline{\delta }\mu&= \pi \,\mu + (\overline{\alpha }-3\,\beta )\,\lambda - \varPsi _3 + \varPhi _{21}, \end{aligned}$$

(81q)

$$\begin{aligned} \delta \rho -\overline{\delta }\sigma&= (\overline{\alpha } + \beta )\,\rho - (3\,\alpha -\overline{\beta })\,\sigma - \varPsi _1 + \varPhi _{01}. \end{aligned}$$

(81r)