Construction of \((}_k,}_k)\)

This appendix discusses the existence and global properties of k-self-similar naked singularities, cf. Theorem 3. The original argument of [7] takes place in Bondi coordinates (u, r), with the area radius r treated as a coordinate. For the main estimates underlying Theorems 4 and 5, it is more convenient to consider the Einstein-scalar field system in double-null gauge. Thus, the main role of the appendix is to effect a translation of the results in [7] to a particular gauge, here termed renormalized double-null coordinates. The proof presented here is not self-contained, and relies crucially on the self-similar analysis of the spherically symmetric Einstein-scalar field system presented in [7].

Section A.1 motivates the class of k-self-similar solutions, and defines self-similar double-null coordinates adapted to the symmetry. Section A.2 reduces the full Einstein-scalar field system to an autonomous ODE system, which is shown to be identical with the system obtained in [7]. Applying the ODE analysis of the latter paper and transforming back to a double-null gauge, we discuss in Sects. A.2.1, A.3 the resulting interior and exterior regions of spacetime. The final step in the construction consists of an asymptotically flat truncation in the exterior region. Although technically straightforward, we discuss the truncation procedure explicitly and collect various bounds on the solution.

A key insight from the construction is the breakdown of self-similar double-null coordinates along the past and future similarity horizons, motivating the introduction of a renormalized double-null gauge (see Sect. A.2.1). The properties of the self-similar solutions in the renormalized gauge, including blowup rates of various double-null unknowns, directly motivate the assumptions for admissible spacetimes outlined in Sect. 3.1.

In the remainder of this appendix, we drop the overline and “k” subscript on metric and scalar field quantities associated to the k-self-similar spacetime.

1.1 Consequences of k-Self-Similarity

The point of departure for Christodoulou’s construction is the notion of k-self-similarity. Assume that the spacetime \((},}}_)\) admits a spherically symmetric, conformally Killing vector field S, which generates dilations about a central point \(}\). It follows that the metric and scalar field quantities satisfy

$$\begin }_S g_ = 2 g_, \ Sr = r, \ S\phi = -k, \end$$

(A.1)

where \(g_\) is the metric on the quotient spacetime \(}\). Explicitly, \(g = -\Omega ^2(u,v) dudv\). Moreover, we will work exclusively with \(k \in }_+\) satisfying \(k^2 \in (0, \frac)\). While [6, 7] consider a wider range of values, it is shown that such an extension does not lead to new naked singularity solutions. In particular, the so-called scale-invariant solutions with \(k\!=\!0\) can be written down explicitly [6], and do not model singularity formation.

The solutions will be constructed in self-similar double-null coordinates, adapted to the scaling symmetry. Wherever this system is regular, it will follow that S takes the simple form

$$\begin S = }\partial _}} + }\partial _}}. \end$$

(A.2)

The assumption (A.1) and coordinate condition (A.2) place strong restrictions on the form of the metric functions and the scalar field. Compute

$$\begin (}_S g)_ = S(g(\partial _, \partial _\nu )) - g([S,\partial _\mu ],\partial _\nu ) - g(\partial _\mu , [S,\partial _\nu ]), \end$$

with \(\mu ,\nu \in \},} \}.\) The non-vanishing commutators are given by

$$\begin S,\partial _}}] = - \partial _}}, \ \ [S,\partial _}}] = - \partial _}}, \end$$

implying

$$\begin (}_S g)_} }}&= (}_S g)_} }} = 0, \ (}_S g)_} }} = -S(\Omega ^2) - 2\Omega ^2. \end$$

It follows from (A.1) that

$$\begin S(\Omega ^2) = 0, \ Sr = r, \ S \phi = -k, \end$$

(A.3)

and it remains to analyze these reduced equations. Define the variable \(} = -\frac}}}}\), which is well-defined on \(}\), and introduce the coordinate system

$$\begin (},}) \doteq \left( }, -\frac}}}}\right) . \end$$

(A.4)

The coordinate derivatives are related byFootnote 12

$$\begin \partial _}} = \partial _}} - \frac}}}}\partial _}}, \ \partial _}} = -\frac}}\partial _}}. \end$$

(A.5)

The conformal Killing field is given in the new coordinates by

$$\begin S = }\partial _}}. \end$$

The equations (A.3) are equivalent to

$$\begin S(\Omega ^2) = S(\frac}}) = S(\phi + k\log (-})) = 0, \end$$

and thus there exist functions \(\mathring(}}), \mathring(}}), \mathring(}})\) such that

$$\begin \Omega ^2(},}}) = \mathring^2(}}), \ \ r(},}}) = -}\mathring(}}), \ \ \phi (},}}) = \mathring(}}) - k\log (-}). \end$$

(A.6)

We can in fact derive similar expressions for all the double-null unknowns appearing in the system (2.6)–(2.10). Applying (A.6) along with the coordinate derivative expression (A.5) gives

$$\begin \nu&= \partial _}} r = (\partial _}} - \frac}}}}}\partial _}})(-}\mathring(}})) = -\mathring(}}) + }}\partial _}} \mathring(}}) \doteq \mathring(}}) \end$$

(A.7)

$$\begin \lambda&= \partial _}}r = -\frac}}\partial _}} (-u\mathring(}})) = \partial _}} \mathring(}}) \doteq \mathring(}}), \end$$

(A.8)

$$\begin \mu&= 1 + \frac = 1 + \frac(}}) \mathring(}})}(}})^2} \doteq \mathring(}}), \end$$

(A.9)

$$\begin m&= \frac \doteq -} \mathring(}}), \end$$

(A.10)

$$\begin \partial _}} \phi&= (\partial _}} - \frac}}}}}\partial _}})(\mathring(}}) - k\log (-})) = -\frac}}}}}\partial _}} \mathring(}}) - \frac}}, \end$$

(A.11)

$$\begin \partial _}} \phi&= -\frac}}\partial _}} (\mathring(}}) - k\log (-})) = -\frac}}\partial _}} \mathring(}}), \end$$

(A.12)

where we have introduced functions \(\mathring(}}), \mathring(}}), \mathring(}}),\) and \(\mathring(}})\). Comparing (A.7) and (A.8) gives the useful algebraic relationship

$$\begin \mathring + \mathring = }}\mathring. \end$$

(A.13)

We conclude this section with an overview of the solution manifold. With respect to a self-similar double-null coordinate system \((},})\), the \((}_k,}_k)\) solutions will initially be defined on

$$\begin } = \},})\ | -\infty \le }< 0, \ } \le } < \infty \}. \end$$

In Sect. A.3.1 this spacetime will be truncated in the region \(} \gg 1\) to generate an asymptotically flat spacetime; however, the underlying self-similar solution is defined on the whole of \(}\).

The timelike curve \(\} = }, } < 0\}\) will be denoted \(\Gamma \), and coincides with the set \(\}, }) = 0\}\). An interesting question concerns the possible extensions to \(} > 0\), and the existence of a regular center and/or singularities in the extended spacetime. For details on extensions with a regular center with \(} > 0\), see [7]. We will not discuss extensions further here.

The axis is generated by the vector field \(\partial _}} + \partial _}}\), which will be useful for translating regularity conditions along \(\Gamma \) to conditions on the coordinate derivatives. In particular, for a suitably regular solution the relation \(\mathring + \mathring = 0\) holds along the axis. We will in fact have that the solution is smooth in a \(}}\)-neighborhood of \(\Gamma \), and thus the natural regularity conditions will be applicable.

We have not yet exhausted the gauge freedom inherent to self-similar double-null coordinates. A 1-parameter scaling freedom

$$\begin } \rightarrow a}, \ } \rightarrow a} \end$$

remains, for any \(a > 0\). Under this scaling, \(\Omega ^2(},})\) transforms as \(\Omega ^2 \rightarrow a^\Omega ^2\), and so a can be fixed in order to set \(\mathring^2(-1) = 1\). Make this choice, and thereby fix the choice of self-similar double-null gauge.

1.2 The Interior Solution

The previous section explored the consequences of the self-similar ansatz, and identified a natural self-similar coordinate system spanned by \((},}}).\) Using the expressions (A.6)–(A.12) to rewrite the scalar field system in terms of self-similar coordinates, we arrive at the following proposition:

Proposition A.1

Under the assumption of k-self-similarity, (2.6)–(2.10) formally implies the following system for the variables \((\mathring,\mathring,\mathring,\mathring,\mathring')\). We use the notation \(\frac}}} \mathring \doteq \mathring'\).

$$\begin \mathring}}\frac}}} \mathring&= -\mathring \mathring - \frac\mathring^2, \end$$

(A.14)

$$\begin \mathring\frac}}} \mathring&= -\mathring \mathring - \frac\mathring^2, \end$$

(A.15)

$$\begin 2 \mathring^ \mathring }} \frac}}}\mathring&= }}\frac}}}\mathring+\mathring(}}\mathring'+k)^2, \end$$

(A.16)

$$\begin 2\mathring^\mathring\frac}}}\mathring&= \frac}}}\mathring + \mathring (\mathring')^2, \end$$

(A.17)

$$\begin \mathring}}\frac}}}\mathring'&= -2\mathring}} \mathring' - k\mathring. \end$$

(A.18)

The self-similar reduction of the wave equation for \(\mathring\) is not included in the above set, and is not required for generating the autonomous system below. The strategy for constructing the interior solution will be to solve the ODE system (A.14)–(A.18) with “initial data” along \(\Gamma \), or equivalently, \(\}}=-1\}\). It will follow from the reduction below that \(\Gamma \) is a critical point for the ODE system, and therefore care will have to be taken when discussing local existence of solutions.

In addition to the condition \(\mathring(-1) = 0\), the gauge normalization implies \(\mathring(-1) = 1\). It therefore remains to identify \(\mathring(-1), \mathring(-1),\) and \(\mathring'(-1).\) Regularity for \(\mathring, \mathring\) at the axis yields

$$\begin \mathring(-1) = - \mathring(-1), \end$$

and

$$\begin 0 = \mathring(-1) = 1 + \frac(-1)\mathring(-1)}^2(-1)}. \end$$

Taken together, these equations imply \(\mathring(-1) = -\frac, \ \mathring(-1) = \frac. \) Finally, regularity of \(\frac}}}\mathring'\) at \(}}=-1\) implies the right hand side of (A.18) must vanish, i.e., \(\mathring'(-1) = \frac.\) In summary, the initial conditions are given by

$$\begin (\mathring, \mathring, \mathring, \mathring, \mathring' )(-1) = \left( 0,1,-\frac, \frac, \frac\right) . \end$$

(A.19)

1.2.1 Reduction to an Autonomous System

The full system (A.14)–(A.18) is difficult to analyze in its present form. Thankfully, one may reduce the system to a pair of autonomous equations. The starting point for this reduction is an algebraic identity for \(\mathring^2\), allowing its elimination from the system.

Lemma A.1

The following algebraic identity holds:

$$\begin \frac\mathring^2 = -\mathring\mathring + \mathring^2}}(\mathring')^2 + 2k }}\lambda \mathring \mathring'+ \mathring\mathring k^2. \end$$

(A.20)

Proof

Comparing (A.16) and (A.17) and eliminating \(2\mathring^\frac}}}\mathring\) gives the equality

$$\begin \frac}\frac}}}\mathring + \frac}}}\mathring}(}}\mathring'+k)^2 = \frac}\frac}}}\mathring + \frac}}(\mathring')^2. \end$$

Inserting (A.14), (A.15) and clearing denominators gives

$$\begin (\mathring\mathring+\frac\mathring^2)(}}\mathring-\mathring) = \mathring^2 \mathring(}}\mathring'+k)^2 - \mathring^2}}\mathring(\mathring')^2. \end$$

Using (A.13) and simplifying gives the result. \(\square \)

Define the functions

$$\begin \psi (}}) \doteq \frac(})}(})+\mathring(})}, \ \ \ \theta (}}) \doteq }} \psi (}}) \mathring'(}}). \end$$

(A.21)

The goal is to reduce the above ODE system to one for the quantities \(\psi (}}), \ \theta (}})\). Differentiating these quantities, inserting (A.14)–(A.18), (A.20), and simplifying yields the pair of equations

$$\begin \frac}}}\psi&=\frac}}}\big ((\theta +k)^2+(1-k^2)(1-\psi )\big ) \end$$

(A.22)

$$\begin \frac}}}\theta&= \frac}}\psi }\big (k\psi (k\theta -1) + \theta ( (\theta +k)^2-(1+k^2))\big ). \end$$

(A.23)

To make contact with the ODE system considered by Christodoulou in [7], we change variables from \(}\) to a parameter s. Assume a solution for \(\psi (}})\) is given on some interval \(}} \in [-1,}}_f)\), with \(}}\psi (}}) \ge 0\) holding for all \(}} \in (-1,}}_f)\). Choose any \((}}_0, s_0)\) with \(}}_0 \in (-1,}}_f)\), \(s_0 > -\infty \), and define the formal change of variables \(s(}})\) by

$$\begin s(}}) \doteq s_0- \int \limits _}}}^}}_0}\frac}}'\psi (}}')}d}}'. \end$$

(A.24)

Observing that \((\psi , \theta )(}=-1) = (0,0)\), it follows by (A.22) that in a regular solution to the ODE system we must have an expansion

$$\begin \psi (}}) = -|}}+1| + O(|}}+1|^2). \end$$

(A.25)

Inserting into (A.24) shows that in a neighborhood of \(}}=-1\),

$$\begin s(}}) = \log |}}+1| + O_}}\rightarrow -1}(1), \end$$

(A.26)

where \(O_}}\rightarrow -1}(1)\) denotes terms that are bounded as \(}} \rightarrow -1\). In particular, the axis corresponds to \(s = -\infty \).

The system for \((\psi ,\theta )\) takes an especially nice form in terms of the parameter s. The following autonomous system is identical to that in [7], with the functions \((\alpha ,\theta )\) of that paper replacing \((\psi , \theta )\) here.

$$\begin \frac\psi&= \psi \big ( (\theta +k)^2 + (1-k^2)(1-\psi )\big ), \end$$

(A.27)

$$\begin \frac\theta&= k\psi (k\theta -1) + \theta \big ( (\theta +k)^2-(1+k^2)\big ). \end$$

(A.28)

In the remainder of this section we use results of [7], in which a careful study of the existence and long time behavior of solutions to (A.27), (A.28) is undertaken. It remains to translate these results on the solution \((\psi , \theta )\) into control on the complete set of double-null unknowns. Two complications arise in directly using the analysis of [7]. One issue is that the parameter s may only be translated back to the original parameter \(}}\) through the coordinate transformation defined by (A.24), which itself depends on the function \(\psi \). Another difficulty is that the unknowns considered in [7], by virtue of the Bondi coordinate system used, are not directly comparable to the double-null unknowns we need to estimate here. Still, the bulk of the hard analysis in the construction lies in analyzing (A.27), (A.28), and this analysis may be used as a black box.

Before turning to the results of the ODE analysis, we discuss the initial conditions for \(\psi ,\theta \) as \(s \rightarrow -\infty \) (equivalently, as \(}} \rightarrow -1\)). The data for \(\mathring,\mathring, \mathring'\) implies

$$\begin \lim _ \psi (s) = 0, \ \ \ \lim _ \theta (s) = 0, \end$$

and therefore the axis data constitutes a critical point of the system (A.27), (A.28). The linearization analysis shows \((\psi ,\theta ) = (0,0)\) to be a saddle point with eigenvalues \(\pm 1\). Moreover, local existence and uniqueness follows given a choice of the limit

$$\begin \lim _ \psi e^. \end$$

(A.29)

Comparing (A.25), (A.26) gives

$$\begin \lim _ \psi e^ = -1. \end$$

(A.30)

The above computations are formal, and one may phrase the logic more rigorously as follows. Choose the boundary condition as in (A.30), and once the local existence theory gives the existence of a function \(\psi (s)\), defining the coordinate \(}}(s)\) and writing all functions in terms of \(}}\) will reproduce the desired asymptotics (A.25).

We are now in a position to state results concerning the local and global ODE analysis of (A.27), (A.28).

Proposition A.2

There exists an \(s_* < \infty \), and a unique solution to the system (A.27), (A.28) with asymptotic initial condition (A.30) on a parameter range \(s \in (-\infty , s_*)\) such that the following statements hold:

1.

\(\psi (s) < 0\), \(\theta (s) > 0 \) on \((-\infty , s_*).\)

2.

\(\lim _ \psi (s) = -\infty \), \(\lim _ \theta (s) = \frac\).

3.

\(\psi (s), \ \theta (s)\) are bounded, smooth functions on \( (-\infty ,s_0)\) for any fixed \(s_0 < s_*\).

4.

In a neighborhood of \(s = -\infty \), the solution admits the expansion

$$\begin & \psi (s) = - e^s + O(e^), \end$$

(A.31)

$$\begin & \theta (s) = \frace^ + O(e^). \end$$

(A.32)

5.

There exists a constant \(a_1\), which a priori may vanish, and a non-vanishing constant c(k) such that as \(s \rightarrow s_*^- \) the solution \((\psi , \theta )\) admits the expansion

$$\begin \frac&= (1-k^2)(s-s_*) + O((s-s_*)^2), \end$$

(A.33)

$$\begin \theta (s)&= \frac + c(k)\frac + a_1 (-(1-k^2)(s-s_*))^} + O((s-s_*)^}). \end$$

(A.34)

A first step in recovering \((\psi (}}),\theta (}}))\) is understanding the behavior of the change of coordinates \(}}(s)\).

Lemma A.2

The function \(}}(s): [-\infty , s_*) \rightarrow [-1, }}(s_*))\) is smooth and increasing. Moreover \(\lim _}}(s) = 0\), and there exists a constant \(c_1>0\) such that as \(s \rightarrow s_*^-\),

$$\begin |s-s_*| = c_1|}}|^ + O(|}|^). \end$$

(A.35)

Proof

By definition (A.24) of the coordinate transformation it follows that

$$\begin \frac}}} = }}\psi (}}(s)), \end$$

(A.36)

and therefore \(}}(s)\) is increasing wherever \(}}\psi \ge 0\). As \(\psi > 0\) for all \(s < s_*\), it suffices to check that \(}}(s) < 0\) holds. Suppose by way of contradiction that there exists an \(-\infty< s_1 < s_*\) with \(}}(s_1) = 0\). Fix any \(s_0 < s_1\) finite. Applying (A.24) between \((s_0, }}(s_0))\) and \((s_1,0)\) gives

$$\begin s_1 = s_0 + \int \limits _}}_0}^\frac}}'\psi (}}')}d}}', \end$$

Since we assume \(s_1 < s_*\), it follows that \(|\psi (}})| \lesssim 1\) for all \(}} \in [}}_0, }}(s_0)]\). The above integral then diverges, a contradiction.

It thus follows that \(}}(s) < 0\) on \((-\infty ,s_*)\), and that the map \(}}(s)\) is strictly increasing. Smoothness follows by differentiating (A.24) and using the smoothness of \(\psi \).

It remains to consider the behavior of \(}}(s)\) as \(s \rightarrow s_*\). Integrating (A.36) and applying the asymptotic behavior (A.33), we have

$$\begin \log \big |\frac}}}}}_0}\big |&= \int \limits _^}})}\psi (s')ds' \nonumber \\&= (1-k^2)^\log \big |\frac}})-s_*}\big | + O_}} \rightarrow 0}(1). \end$$

(A.37)

Re-arranging gives (A.35), which moreover implies \(\lim _}}(s) = 0\). \(\square \)

The breakdown of the solution at \(}}=0\) physically corresponds to the solution arriving at the past light cone of the singular point \((},}) = (0,0)\). The region \(-1 \le }} < 0\) on which the solution is defined is termed the interior of the naked singularity.

In the following sections the global properties of this interior solution are explored, with special attention paid to the consequences of (A.33), (A.34).

1.2.2 Regular Coordinates and Global Bounds

The goal is now to derive quantitative bounds on all the double-null unknowns. As mentioned above, the main difficulties lie in (1) understanding rates as a function of the more natural coordinate \(}}\), and (2) converting bounds on \((\psi , \theta )\) into bounds on the full set of double-null unknowns.

Away from \(\,\) or equivalently \(\}}=0\}\), Proposition A.2 asserts that \((\psi ,\theta )\) are smooth functions of s (and by Lemma A.2, as functions of \(}}\)). On any compact \(}}\) subinterval of \([-1,0)\) bounds will directly follow. It is therefore natural to start by considering the behavior as \(}} \rightarrow 0\).

The following lemma is a restatement of (A.33), (A.34) in terms of \(}}\), applying the change of variables (A.35):

Lemma A.3

As \(}} \rightarrow 0^-\), the following expansions hold for constants \(c_1, c_2\), with \(c_1 \ne 0\).

$$\begin & \frac}})} = c_1|}}|^ + O(|}}|^), \end$$

(A.38)

$$\begin & \theta (}}) = \frac - c_2|}}|^ + O(|}}|^). \end$$

(A.39)

An immediate consequence of these expansions is the following set of bounds for the double-null unknowns as \(}} \rightarrow 0\).

Lemma A.4

In a neighborhood of \(}}=0\), the following asymptotics hold:

$$\begin \mathring&\sim 1, \quad \mathring \sim |}}|^, \quad (-\mathring) \sim 1, \end$$

(A.40)

$$\begin |\mathring|&\lesssim 1, \quad |\mathring'| \sim |}}|^, \quad |\mathring''| \sim |}}|^, \end$$

(A.41)

$$\begin \mathring^2&\sim |}}|^, \quad 0< \mathring < 1. \end$$

(A.42)

Moreover, the following identities hold:

$$\begin & \mathring(0) = -\mathring(0), \end$$

(A.43)

$$\begin & \mathring(0) = \frac. \end$$

(A.44)

Proof

Begin with \(\mathring\), \(\mathring\), and \(\mathring\). By definition of \(\psi (}})\) and (A.38)–(A.39), it follows that

$$\begin \frac}})} = \frac}} \partial _}} \mathring}} \sim |}}|^, \end$$

and therefore

$$\begin \partial _}} \log \mathring \sim |}}|^, \end$$

(A.45)

implying \(\mathring \sim 1\) holds in a sufficiently small neighborhood of \(}=0\). Inserting this result into (A.45) gives the stated bound on \(\mathring = \partial _}} \mathring\). The algebraic relationship (A.13) combined with the rates for \(\mathring, \mathring\) gives the rate for \(\mathring\). Equation (A.43) also follows.

Now we turn to \(\mathring'\). Unpacking definitions of \(\theta , \psi \) yields

$$\begin \mathring' = \frac}}\psi } = \frac(c_1+o_}}\rightarrow 0 }(1))|}}|^. \end$$

Integrating this bound forward in \(}}\) from some reference \(}}_0 < 0\) gives a bound on \(\mathring\). To estimate \(\mathring''\), use the equation

$$\begin \mathring'' = -\frac}}}}(}}\mathring' + k). \end$$

The term in parentheses can be estimated as \(k + o_}}\rightarrow 0 }(1)\), and the preceding factor is asymptotically of size \(|}}|^\). Combining these statements gives the estimate for \(\mathring''\).

To estimate \(\mathring\), recall the algebraic identity

$$\begin \frac\mathring^2 = -\mathring\mathring + \mathring^2}}(\mathring')^2 + 2k }}\lambda \mathring \mathring'+ \mathring\mathring k^2. \end$$

The individual terms are of order \(|}}|^, |}}|^, |}}|^, |}}|^\) respectively. The \(|}}|^\) terms have the same sign, and therefore do not cancel. Moreover, \(k^2 < 1\) implies that the leading order behavior is determined by the \(|}}|^\) terms, which gives the first bound in (A.42).

Noting \(\mathring = -\mathring + O(|}}|^)\), we in fact have the more precise statement

$$\begin \frac\mathring^2 = -(1+k^2)\mathring\mathring + O(|}}|^). \end$$

We conclude

$$\begin \mathring = 1+ \frac\mathring}^2} = 1 - \frac + O(|}}|^) = \frac+O(|}}|^). \end$$

The range of k considered here implies \(1-3k^2 > 0\), giving the remaining statements of the lemma. \(\square \)

The rates proved in the previous lemma for \(\mathring, \mathring^2, \mathring',\) and \(\mathring''\) imply that these quantities blow up as \(}} \rightarrow 0\). Returning to the spacetime picture, it follows that \(\partial _}} r, \partial _}}\phi \) do not have regular limits as \(} \rightarrow 0\), even away from the singular point. This blowup should be distinguished from blowup near \((},}) = (0,0)\), corresponding to the presence of the singularity.

This discussion suggests the self-similar double-null coordinate system becomes irregular as \(} \rightarrow 0\). While it is possible to work with a restricted set of double-null unknowns that do have regular limits (e.g., quantities tangential to \(\}=0\}\), alongside quantities weighted by suitable powers of \(\mathring^2\)), it is technically easier here to change coordinates into a regular double-null gauge. The price to pay will be the symmetry of \(}\) and \(}\) built into the definition of the self-similar coordinate \(}}\).

Define a renormalized, non-self-similar double-null coordinate pair (u, v) by

$$\begin (u,v) = (}, -|}|^). \end$$

(A.46)

In moving between coordinate systems the u coordinate derivative is unchanged, whereas

$$\begin \frac = \frac}|^} \frac}}. \end$$

(A.47)

The axis \(\Gamma \) becomes the set \(\} \},\) with generator

$$\begin T = \partial _u + (1-k^2)|v|^}\partial _. \end$$

The conformal Killing field \(S = }\partial _}} + }\partial _}}\) becomes \(u\partial _u + (1-k^2)v \partial _.\)

The effect of defining v is to introduce additional factors of \(}^\) into the definition of the \(}\) derivative, and thereby compensate for blowup as \(} \rightarrow 0\). As mentioned above, however, one disadvantage of the formalism is that the natural self-similar coordinate \(}} = -\frac}}}}\) is replaced by the asymmetric \(}} = \frac},\) where \(p = (1-k^2)^.\)

The next lemma shows that in the renormalized coordinate system all quantities at the first derivative level of the solution have finite limits as \(v \rightarrow 0\). Moreover, the results apply not just in a neighborhood of \(\}} = 0\}\), but globally in the interior region. Recall the notation \(}^\) for the interior region of the spacetime.

Lemma A.5

The solution in \(}^\) has the following properties:

\(r(u,v) \ge 0\), and \(r(u,v) > 0\) in \(}^ \Gamma \).

The renormalized coordinate derivatives \(\nu \doteq \partial _u r, \ \lambda \doteq \partial _v r\) satisfy

$$\begin (-\nu ) \sim 1, \ \ \ \lambda \sim |u|^. \end$$

(A.48)

Define the set \(}_ \doteq }^\cap \} \le \frac\}\). Then

$$\begin r \lesssim |u| \ \text \ }^, \ \ \ r \sim |u| \ \text \ }_. \end$$

(A.49)

There exists a \(c_\mu < 1\) such that \(\mu \) satisfies \(0 \le \mu \le c_\mu < 1\). Moreover, the axis is regular in the following sense:

$$\begin \big |\frac\big | \lesssim \frac. \end$$

(A.50)

The scalar field satisfies modified self-similar bounds

$$\begin |\partial _u \phi | \lesssim \frac, \ \ \ |\partial _v \phi | \lesssim \frac}. \end$$

(A.51)

Along \(\\), we have the identity

$$\begin \partial _u \phi (u,0) = \frac. \end$$

(A.52)

Proof

Start with \(r = -} \mathring\), which is a coordinate independent quantity. Let \(\mathring = \partial _} r\) be associated to the self-similar coordinate system. To show non-negativity of r it suffices to show \(\mathring\) is always non-negative. In fact, suppose \(\mathring(}}_0) = 0\) for some \(}} \in (-1,0)\). The endpoints may be ruled out by considering initial conditions along \(}} = -1\), and the blowup \(\lim _}} \rightarrow 0}\mathring(}}) = \infty \).

Without loss of generality, choose \(}}_0\) to be the first point in \((-1,0)\) at which \(\mathring(}}_0)\) vanishes. Therefore, for all \(}} < }}_0\) we have \(\mathring > 0\). But then it follows that \(\mathring\) is a strictly increasing function on \([-1,}}_0),\) and \(\mathring(}}_0) > 0\). If \(\mathring(}}_0)=0\), then \(\psi (}}_0) = \frac(}}_0)}}}_0\mathring(0)} = -\infty \), contradicting the statement that \(|\psi (}})| < \infty \) for all \(}} < 0\).

This argument shows \(\mathring(}}) > 0\) for all \(}} \in [-1,0]\). Compactness of the interval implies \(\mathring\) has a positive lower bound. This in turn implies \(\mathring(}}) >0\) for \(}} > -1\), and \(r = -u\mathring > 0\) for