In this section we compute in detail the expansions used in Sect. 2.4.3. The goal is to study the autonomous system of ODEs:

$$\begin \left\ \frac=\alpha \big [(\theta +k)^2+(1-k^2)(1-\alpha )\big ] \\ \frac=k\alpha \big (k\theta -1\big )+\theta \big [(\theta +k)^2-(1+k^2)\big ] \end\right. \end$$

(28)

in a neighborhood of the critical point \((\alpha ,\theta )=(0,\pm \sqrt).\) We set \(x_1=\alpha ,\ x_2=\theta \mp \sqrt,\) to obtain the system:

$$\begin \left\ \frac=2x_1+x_1\bigg [x_2^2\pm \frac}x_2-\fracx_1\bigg ] \\ \frac=-2kx_1+4x_2+\fracx_1x_2\pm \frac}x_2^2+x_2^3 \end\right. \end$$

(29)

We diagonalize the linear part of this system by setting \(y_1=x_1,\ y_2=-kx_1+x_2\). We obtain the system:

$$\begin \left\ \frac=2y_1-2y_1^2\pm \frac}y_1y_2+U_1^(y_1,y_2) \\ \frac=4y_2\pm \frac}y_2^2-3y_1y_2+U_2^(y_1,y_2) \end\right. \end$$

(30)

where \(U_1^(y_1,y_2),\ U_2^(y_1,y_2)\) are homogeneous polynomials of degree 3. The critical point (0, 0) is stable to the past. The eigenvalues \(\lambda _1=2,\ \lambda _2=4\) are resonant, since \((2,0)\cdot (\lambda _1,\lambda _2)=\lambda _2.\) Using the Poincaré–Dulac theorem, we obtain that the solutions have the form:

$$\begin \left\ y_1=z_1+Q_1^(z_1,z_2) \\ y_2=z_2+Q_2^(z_1,z_2) \end\right. \end$$

(31)

where \(Q_1^(z_1,z_2),\ Q_2^(z_1,z_2)\) are analytic function that vanish to order 2 at the origin, and \(z_1,\ z_2\) are solutions to the system:

$$\begin \left\ \frac=2z_1\\ \frac=4z_2+C_0z_1^2 \end\right. \end$$

(32)

We compute \(z_1=a_1e^,\ z_2=a_2e^+C_0a_1^2se^.\) We introduce the equivalence relation \((a_1,a_2)\sim (Aa_1,A^2a_2)\), for some \(A>0,\) and we denote by \(}\) the set of equivalence classes. There is a one to one correspondence between \(}\) and the orbits of (30) near the origin. We write:

$$\beginQ_1^(z_1,z_2)=b^z_iz_j+Q_1^(z_1,z_2),\ Q_2^(z_1,z_2)=d^z_iz_j+Q_1^(z_1,z_2),\end$$

and we obtain that \(C_0=0\). This corresponds to the fact that even though the eigenvalues are resonant, the resonant monomial \(\begin 0&y_1^2 \end\) does not appear on the right-hand side of (30). We also obtain \(b^=-1.\) Thus:

$$\beginy_1=a_1e^-a_1^2e^+O(e^),\ y_2=\big (a_2+a_1^2d^\big )e^+O(e^).\end$$

Finally, in a neighborhood of the negatively stable critical point \((0,\pm \sqrt)\), the orbits of (28) satisfy the expansion:

$$\begin & \alpha =a_1e^-a_1^2e^+O(e^),\\ & \theta =\pm \sqrt\mp \frac}e^+\bigg (\pm \frac}+a_1^2d^+a_2\bigg )e^+O(e^),\end$$

for some \((a_1,a_2)\in }.\)

In this section, we compute the sectional curvatures of \(S^2\) and \(}^\) with respect to the metric:

$$\begin\tilde=-\frace^du^2-\frace^dudr+\fracr^2d\sigma _2^2+f^2d\sigma _1^2.\end$$

We express these using the corresponding 4-dimensional metric:

$$\beging=-e^du^2-2e^dudr+r^2d\sigma _2^2,\end$$

and the quantities defined in Sect. .

In general, we compute for \(a,b,c,d\ne \psi :\)

$$\begin}_ ^d= & R_ ^d-\delta ^d _\nabla _\nabla _c\log f+g_\nabla _\nabla ^d\log f+\frac\big (\nabla _\log f\big )\delta ^d _\nabla _c\log f\\ & -\frac\big (\nabla _\log f\big )g_\nabla ^d\log f-\fracg_\delta ^d _g(\nabla \log f,\nabla \log f).\end$$

We apply this formula to compute:

$$\begin }_= & \frac}_ ^=\frac\bigg [R_ ^+\nabla _\nabla _\log f-\fracr^2g(\nabla \log f,\nabla \log f)\bigg ]\\= & \frac\bigg [\frac-k(1-\mu )(\theta +\zeta )-k^2r^2g(\nabla \phi ,\nabla \phi )\bigg ]\\= & \frac\bigg [1-(1-\mu )(k\theta +1)(k\zeta +1)\bigg ].\end$$

Similarly, we also compute:

$$\begin }_}^}= & \frac}(n,l,n,l)=f^2K_}^}+\frac\nabla _\nabla _\log f\\ & +\frac\nabla _\log f\nabla _\log f+\fracg(\nabla \log f,\nabla \log f)\\= & \frac\bigg [1-(1-\mu )(k\theta +1)(k\zeta +1)\bigg ]+(1+k^2)\frac(1-\mu )\theta \zeta .\end$$

In this section, we write the Einstein-Scalar field equations in spherical symmetry for a self-similar solution in double null coordinates as a system of autonomous ODEs. We consider the metric:

$$\beging=-\Omega ^2dudv+r^2d\sigma _2^2.\end$$

The Einstein-Scalar field equations are:

$$\begin & \partial _u(\Omega ^\partial _ur)=-r\Omega ^(\partial _u\phi )^2 \end$$

(33)

$$\begin & \partial _v(\Omega ^\partial _vr)=-r\Omega ^(\partial _v\phi )^2 \end$$

(34)

$$\begin & \partial _u\partial _v\log \Omega =-\frac\partial _u\partial _v r-4\partial _u\phi \partial _v\phi \end$$

(35)

$$\begin & \partial _u\partial _vr^2=-\frac\Omega ^2 \end$$

(36)

$$\begin & r\partial _u\partial _v\phi +\partial _v\phi \partial _ur+\partial _u\phi \partial _vr=0 \end$$

(37)

We notice that equation (35) is implied by the other equations. Moreover, we can rewrite the Raychaudhuri’s equations (33), (34) as:

$$\begin \partial _um= & -2r^2\Omega ^\partial _vr(\partial _u\phi )^2 \end$$

(38)

$$\begin \partial _vm= & -2r^2\Omega ^\partial _ur(\partial _v\phi )^2 \end$$

(39)

where m is the Hawking mass:

$$\begin m=\frac(1+4\Omega ^\partial _ur\partial _vr). \end$$

(40)

Following Sect. 3.3.1, we assume we are in self-similar double null gauge, so \(S=u\partial _u+v\partial _v\) and:

$$\beginS\Omega =0,\ Sr=r,\ S\phi =-k.\end$$

As before, we consider the following quantities:

$$\begin\theta =\frac,\ \zeta =\frac,\ \beta =\frac.\end$$

We introduce the self-similar coordinate:

The change of coordinates is given by:

$$\begin\partial _v=-\frac\partial _y,\ \partial _u=\frac\partial _y+\partial _u.\end$$

Thus, in (u, y) coordinates we have \(S=u\partial _u.\) We recall the notation:

$$\begin\chi =\phi +k\log (-u),\ R=-\frac.\end$$

The self-similar condition is equivalent to \(\Omega =\Omega (y),\ R=R(y),\ \chi =\chi (y).\) We compute:

$$\begin\beta =\frac,\ \zeta =\frac,\ \Omega ^2=-\frac\partial _yR\big (y\partial _yR-R\big ).\end$$

We can rewrite the wave equation (37) as:

$$\beginy\partial _y\theta +\theta \bigg (1+\frac\bigg )+k=0.\end$$

Also, we can rewrite equation (36) as:

$$\beginy\partial _y\beta +\bigg (2-\frac\bigg )\beta (\beta -1)=0.\end$$

Using this in equations (38),(39) eventually gives:

$$\begin\frac=\beta \theta ^2-\frac+1.\end$$

Finally, we conclude that the Einstein-Scalar field system is equivalent to:

$$\begin \left\ y\partial _y\theta =k\big (k\theta -1\big )+\beta \theta \big [(\theta +k)^2-(1+k^2)\big ] \\ y\partial _y\beta =\beta (1-k^2)-\beta ^2\big [(\theta +k)^2+(1-k^2)\big ] \end\right. \end$$