# A proposal of quasi-local mass for $2$-surfaces of timelike mean curvature

Bowen Zhao  Beijing Institute of Mathematical Sciences and Applications,
Beijing, China  Shing-Tung Yau  Yau Mathmatical Sciences Center, Tsinghua
University, Beijing, China  Lars Andersson  Beijing Institute of Mathematical
Sciences and Applications, Beijing, China

###### Abstract

A quasi-local mass, typically defined as an integral over a spacelike
$2$-surface $\Sigma$, should encode information about the gravitational field
within a finite, extended region bounded by $\Sigma$. Therefore, in attempts
to quantize gravity, one may consider an infinite dimensional space of
$2$-surfaces instead of an infinite dimensional space of $4$-dimensional
Lorentzian spacetimes. However, existing definitions for quasilocal mass only
applies to surfaces outside an horizon whose mean curvature vector is
spacelike. In this paper, we propose an extension of the Wang-Yau quasi-local
energy/mass to surfaces with timelike mean curvature vector, including in
particular trapped surfaces. We adopt the same canonical gauge as in the Wang-
Yau quasi-local energy but allow the pulled back ”killing vector” to the
physical spacetime to be spacelike. We define the new quasi-local energy along
the Hamiltonian formulation of the Wang-Yau quasi-local energy. The new
definition yields a positive definite surface energy density and a new
divergence free current. Calculations for coordinate spheres in Kerr family
spacetime are shown. In the spherical symmetric case, our definition reduces
to a previous definition [LSYJ07]

.

##  1 Introduction

In the path integral approach to the quantization of gravity, one must include
spacetime metrics with black holes as well as metrics that can be continuously
deformed to the flat metric. To avoid dealing with spacetime singularities
that necessarily appear in a black hole metric, Gibbons and Hawking [GH77]
proposed to consider Euclidean action for gravity. An alternative approach may
be employing the notion of quasi-local mass which is defined over a spacelike
$2$-surface. In principle, the quasi-local mass of a $2$-surface $\Sigma$
encodes information about the gravitational field of a finite region bounded
by $\Sigma$ which satisfies the dominant energy condition. In the case of
Schwarzschild-like singularity, there clearly exists a family of topological
$2$-spheres that foliates the spacetime outside the singularity. For more
general types of singularity such as the ring singularity of Kerr, one can use
surfaces of more general topology (e.g. torus for ring singularity) to foliate
the spacetime outside the singularities. One may then consider an infinite
dimensional space of $2$-surfaces with its intrinsic and extrinsic geometric
data to quantize gravity, instead of an infinite dimensional space of
$4$-dimensional Lorentzian spacetimes.

The equivalence principle of Einstein gravity theory excludes the existence of
a point-wise gravitational energy density. On the other hand, there exist
well-defined notions of total gravitational energy for an asymptotically flat
spacetime, e.g. ADM mass at spatial infinity and Bondi mass at null infinity.
A quasi-local mass aims to measure gravitational field energy within a finite,
extended region whose definition remains a major unresolved problem in general
relativity [Pen82, Col17]. There exist several propositions for quasi-local
energy/mass, e.g. Hawing mass[Haw68], Bartnik mass[Bar89], Brown-York
mass[BYJ93], Liu-Yau mass[LY03, LY06] etc. However, they all possess some
undesirable features [Sza09]. For example, the Hawking mass could be negative
even in the Minkowski spacetime; the Bartnik mass has no closed-form
expression; the Brown-York mass depends on the choice of a spacelike
$3$-manifold $\Omega$ bounded by the $2$-surface under consideration.
Moreover, there exist surfaces in the Minkowski spacetime with strictly
positive Liu–Yau mass as well as Brown–York mass [MST04], violating the
rigidity property that surfaces in the Minkowski spacetime should have zero
mass. In order to overcome these difficulties, Wang and Yau proposed a
definition that incorporates both metric and momentum information about the
$2$-surface under consideration [WY09a, WY09c]. Their definition is is
intrinsic to the spacelike $2$-surface $\Sigma$ under consideration and most
importantly, the positivity was proved. Moreover, the Wang-Yau quasi-local
mass was shown to be consistent with the ADM mass at spatial infinity [WY10]
and Bondi mass at null infinity [CWY11] and also exhibit reasonable small
sphere limit at a limiting point [CWY18].

However, the Wang-Yau quasi-local energy/mass as well as most other quasi-
local energy/mass, is only defined for spacelike $2$-surfaces of spacelike
mean curvature vector, including in particular future untrapped surfaces. To
consider a moduli space of $2$-surfaces that encodes gravitational field
information, one certainly needs to consider trapped surfaces or rather
surfaces of timelike mean curvature. This paper explores one possible
extension of the original Wang-Yau definition to surfaces with timelike mean
curvature vector. The rest of this paper is organized as follows. We briefly
review the setup of Wang-Yau quasi-local mass in section 2. We present our new
quasi-local energy in section 3. The new definition is given in section 3.1. A
new mass surface density $\rho^{new}$ and a new divergence-free current
$j^{new}$ are introduced in section 3.2. Continuity cross the horizon between
the definition here and the original Wang-Yau definition is discussed in
section 3.3. Examples from well known black hole spacetime are shown in
section 3.4. Lastly, we conclude and discuss in section 4.

##  2 Review of Wang-Yau quasi-local mass

The Wang-Yau quasi-local energy is defined as a difference between a reference
term and a physical term. This ensures that for surfaces lying in the
reference spacetime, typically taken as $\mathbb{R}^{3,1}$, the quasi-local
energy vanishes. Let us consider the reference spacetime first. Let $\Sigma$
be a spacelike $2$-surface from a general spacetime $\mathcal{M}^{3,1}$ and
let $X:\Sigma\to\mathbb{R}^{3,1}$ be an isometric embedding. Denote the image
by $\Sigma_{0}=X(\Sigma)$. One also fixes a future-pointing, timelike, unit
vector $T_{0}$ in the reference spacetime $\mathbb{R}^{3,1}$. Then the vector
$T_{0}$ determines a unique timelike unit normal $u_{0}$ at each point of
$\Sigma_{0}$ through

$T_{0}=\sqrt{1+|\nabla\tau|^{2}}u_{0}-\nabla\tau$

where $\tau=-\langle X,T_{0}\rangle$ is the time function of $\Sigma_{0}$
along $T_{0}$. We denote covariant derivative on $\Sigma$ by $\nabla$ and
covariant derivative in $\mathcal{M}^{3,1}$ by $\overline{\nabla}$. This leads
to a canonical basis $\\{v_{0},u_{0}\\}$ for the normal bundle $N\Sigma_{0}$
with

$\langle v_{0},u_{0}\rangle=\langle v_{0},T_{0}\rangle=0$

Further, [WY09b] employed the frame independent mean curvature vector
$H_{0}=(\nabla_{a}e_{a})^{\perp}$ and its conjugate vector $J_{0}$ (reflection
along the ingoing light cone in $N\Sigma_{0}$) to define a reference basis for
the normal bundle,

$e_{H_{0}}=-\frac{H_{0}}{|H_{0}|},e_{J_{0}}=\frac{J_{0}}{|H_{0}|}$

where $|H_{0}|=|J_{0}|=\sqrt{|\langle H,H\rangle|}$. For future-untrapped
surface, $e_{H_{0}}$ is outward pointing while $e_{J_{0}}$ is future pointing.
The basis transformation between the reference basis and the canonical basis
is

$\displaystyle v_{0}$
$\displaystyle=\cosh\theta_{0}\,e_{H_{0}}-\sinh\theta_{0}\,e_{J_{0}}$
$\displaystyle u_{0}$
$\displaystyle=-\sinh\theta_{0}\,e_{H_{0}}+\cosh\theta_{0}\,e_{J_{0}}$

with

$\sinh\theta_{0}=\frac{-\Delta\tau}{|H_{0}|\sqrt{1+|\nabla\tau|^{2}}}$

Now consider $\Sigma$ in the physical spacetime $\mathcal{M}^{3,1}$ with
spacelike mean curvature vector $H$. The canonical gauge condition

$\langle T_{0},H_{0}\rangle=\langle T,H\rangle$ (1)

together with two other implicit conditions

$\displaystyle\langle T,T\rangle$ $\displaystyle=-1,$ $\displaystyle
T^{T}=T_{0}^{T}$ $\displaystyle=-\nabla\tau\quad(\text{projection to
$T\Sigma$})$

uniquely determines a timelike unit vector $T$ at everypoint of $\Sigma$. The
uniquely determined $T$ then picks a canonical basis $\\{u,v\\}$ for $N\Sigma$
satisfying

$\displaystyle T$ $\displaystyle=\sqrt{1+|\nabla\tau|^{2}}u-\nabla\tau,\quad$
$\displaystyle\langle v,u\rangle$ $\displaystyle=\langle v,T\rangle=0$

This solves the issue of choosing a frame for $2$-surface $\Sigma$ in
$4$-dimensional spacetime $\mathcal{M}^{3,1}$. Similarly, oen can define
$\\{e_{H}=-\frac{H}{|H|},e_{J}=\frac{J}{|J|}\\}$ as the reference basis for
$N\Sigma$. The basis transformation is

$\displaystyle v$ $\displaystyle=\cosh\theta\,e_{H}-\sinh\theta\,e_{J}$
$\displaystyle u$ $\displaystyle=-\sinh\theta\,e_{H}+\cosh\theta\,e_{J}$ (2)

with

$\sinh\theta=\frac{-\Delta\tau}{|H|\sqrt{1+|\nabla\tau|^{2}}}$ (3)

determined by the canonical gauge (1).

By a theorem by Pogorelov [Pog64], a positive definite metric on a topological
$2$-sphere $\Sigma$ with Gaussian curvature $K_{\Sigma}>-\kappa^{2}$ can be
embedded into any hyperboloid
$\eta_{\mu\nu}X^{\mu}X^{\nu}=C>-\frac{1}{\kappa^{2}}$ in $\mathbb{R}^{3,1}$.
So an isometric embedding $X:\Sigma\to\mathbb{R}^{3,1}$ is far from unique.
Wang and Yau defined the minimum of quasi-local energy among all possible
isometric embeddings as the quasi-local mass. The variational problem can be
solved through varying the time function $\tau=\langle X,T_{0}\rangle$. The
Euler-Lagrangian equation or the optimal embedding equation (OEE) is

$\displaystyle\nabla\cdot j$ $\displaystyle=0,$ (4) $\displaystyle j$
$\displaystyle=\rho\nabla\tau-\alpha_{v_{0}}+\alpha_{v}=\rho\nabla\tau-\alpha_{H_{0}}-\nabla\theta_{0}+\alpha_{H}+\nabla\theta$
(5)

where the connection 1-forms are defined as

$\displaystyle\alpha_{v_{0}}$
$\displaystyle=\langle\nabla^{\mathbb{R}^{3,1}}v_{0},u_{0}\rangle=\alpha_{H_{0}}+\nabla\theta_{0}$
$\displaystyle\alpha_{v}$
$\displaystyle=\langle\overline{\nabla}v,u\rangle=\alpha_{H}+\nabla\theta$
$\displaystyle\alpha_{H_{0}}$
$\displaystyle=\langle\nabla\frac{-H_{0}}{|H_{0}|},\frac{J_{0}}{|J_{0}|}\rangle,\quad\alpha_{H}=\langle\nabla\frac{-H}{|H|},\frac{J}{|J|},\rangle$

Lastly, we note that the canonical gauge condition (1) is equivalent to

$\langle u,H\rangle=\langle u_{0},H_{0}\rangle$ (6)

which matches the “time” component of mean curvature vectors. Difference in
the other component measures the gravitational energy through the mass energy
surface density $\rho$,

$\rho=\frac{\langle H,v\rangle-\langle
H_{0},v_{0}\rangle}{\sqrt{1+|\nabla\tau|^{2}}}=\frac{\sqrt{|H_{0}|^{2}+\frac{\Delta\tau}{1+|\nabla\tau|^{2}}}-\sqrt{|H|^{2}+\frac{\Delta\tau}{1+|\nabla\tau|^{2}}}}{\sqrt{1+|\nabla\tau|^{2}}}$

In terms of $\rho$ and $j$, the quasi-local energy is

$QLE=\frac{1}{8\pi}\int_{\Sigma}\rho+j\cdot\nabla\tau$

which reduces to the quasi-local mass after imposing OEE $\nabla\cdot j=0$

$QLM=\frac{1}{8\pi}\int_{\Sigma}\rho$

##  3 Quasi-local mass for surfaces with time-like mean curvature vector

Here we present one possible extension of the Wang-Yau quasi-local mass to
surfaces with time-like mean curvature vector, including in particular trapped
surfaces inside an apparent horizon. Direct calculation of Hawking mass,
Brown-York mass etc. for surfaces inside the horizon yields a blowup at the
singularity [GY20]. On the other hand, [LSYJ07] extended the Brown-York mass
inside the horizon for spherical symmetric spacetimes based on physical
considerations and yield finite results even at the singularity. The proposal
here reduces to [LSYJ07] in spherical symmetric cases.

We first observe that the canonical gauge condition (1) breaks down at an
horizon where the mean curvature vector $H$ becomes null. Consider a future-
trapped $2$-surface $\Sigma$ with outgoing null expansion $\Theta_{+}<0$ and
ingoing null expansion $\Theta_{-}<0$. A future-pointing timelike unit vector
$T$ can be decomposed as

$T=Nu+\vec{N}$

where $N>0$, $\vec{N}\in T\Sigma$ and $u$ is a future-pointing unit normal
vector. Let $v$ be the unit normal vector such that $\langle
v,u\rangle=\langle v,T\rangle=0$. Then $\\{u,v\\}$ form an orthonormal basis
for $N\Sigma$. In this basis, one has

$H=-kv+pu,\quad\Theta_{\pm}=p\pm k$

where $k=-\langle H,v\rangle$ and $p=-\langle
H,u\rangle=\frac{\Theta_{+}+\Theta_{-}}{2}<0$. Then

$\langle H,T\rangle=\langle-kv+pu,Nu+\vec{N}\rangle=-Np>0$

On the other hand,

$\langle T_{0},H_{0}\rangle=-\Delta\tau$

cannot be sign-definite over $\Sigma$. Therefore, the canonical gauge (1)
fails for surfaces with timelike mean curvature vector.

Then one obvious rescue is to allow $T$ to change signature, in particular,
one can set $T$ to be everywhere spacelike when $H$ is everywhere timelike. In
other words, we regard $T_{0}$ as the killing vector that generates the
conserved energy in $\mathbb{R}^{3,1}$ and we determine the pulled back
“killing” vector $T$ through imposing (1). This is similar to that the Brill-
Lindquist coordinate $\partial_{t}$ turns to spacelike at the ergosphere.

###  3.1 Proposed definition for surfaces with time-like mean curvature

As argued above, for surfaces with timelike mean curvature vector $H$, we
instead take $\langle T,T\rangle=1$. Note that in the reference basis for
$N\Sigma$, $e_{H}=\frac{-H}{|H|}$ is now future-pointing timelike while
$e_{J}=\frac{J}{|H|}$ is now outward-pointing spacelike. The notation is that
$|H|=|J|=\sqrt{|\langle H,H\rangle|}$. The vector $T$ now decomposes as

$\displaystyle T$ $\displaystyle=\sqrt{1-|\vec{N}|^{2}}\,v+\vec{N},\quad
1-|\vec{N}|^{2}>0$ (7)

The basis transformation then becomes

$\displaystyle v$ $\displaystyle=\cosh\theta\,e_{J}-\sinh\theta\,e_{H}$
$\displaystyle u$ $\displaystyle=-\sinh\theta\,e_{J}+\cosh\theta\,e_{H}$ (8)

with

$\sinh\theta=\frac{\Delta\tau}{|H|\sqrt{1-|\vec{N}|^{2}}}$ (9)

determined by the canonical gauge (1).

We now work out the quasi-local energy along the Hamiltonian argument. As
reviewed in [ZAY24a], the quasi-local energy can be defined through the
generalized mean curvature vectors

$\displaystyle\Xi$ $\displaystyle=ku+K(v,\cdot)-(trK)v\in T\mathcal{M}^{3,1}$
$\displaystyle\Xi_{0}$
$\displaystyle=k_{0}u_{0}+K_{0}(v_{0},\cdot)-(trK_{0})\,v_{0}\in\mathbb{R}^{3,1}$

where

$\displaystyle\quad k_{0}=-\langle H_{0},v_{0}\rangle,\quad$ $\displaystyle
K_{0}(v_{0},\cdot)=\langle\nabla^{\mathbb{R}^{3,1}}_{v_{0}}u_{0},\cdot\rangle$
$\displaystyle k=-\langle H,v\rangle,\quad$ $\displaystyle
K(v,\cdot)=\langle\overline{\nabla}_{v}u,\cdot\rangle$

That is,

$\displaystyle
8\pi\,QLE=\int_{\Sigma}-\langle\Xi_{0},T_{0}\rangle-\int-\langle\Xi,T\rangle$
$\displaystyle=\int-\sqrt{1+|\nabla\tau|^{2}}\,\langle
v_{0},H_{0}\rangle+\alpha_{v_{0}}(-\nabla\tau)$
$\displaystyle-\int-\sqrt{1-|\vec{N}|^{2}}\,\langle
u,H\rangle+\alpha_{v}(\vec{N})$
$\displaystyle=\int\sqrt{1+|\nabla\tau|^{2}}\cosh\theta_{0}|H_{0}|+\theta_{0}\,\Delta\tau-\alpha_{\hat{e}_{3}}(\nabla\tau)$
$\displaystyle-\int-\sqrt{1-|\vec{N}|^{2}}\cosh\theta|H|-\theta\,\nabla\cdot\vec{N}+\alpha_{e_{3}}(\vec{N})$

If we extremize the integrand of the second integral, as in [WY09b], one
reaches at

$\sinh\theta=\frac{-\Delta\cdot\vec{N}}{|H|\sqrt{1-|\vec{N}|^{2}}}$

which agrees with (9) when $\vec{N}=-\nabla\tau$. From now on, we will make
the choice of $\vec{N}=-\nabla\tau$. Use the following inequality proved in
[WY09b]

$\int_{\hat{\Sigma}}\hat{k}=\int_{\Sigma}\sqrt{1+|\nabla\tau|^{2}}\cosh\theta_{0}|H_{0}|+\theta_{0}\,\Delta\tau-\alpha_{\hat{e}_{3}}(\nabla\tau)$

One reaches the new quasi-local energy definition as follows.

######  Definition 3.1 (Quasi-local energy for $2$-surface with time-like mean
curvature).

$\displaystyle QLE^{new}$
$\displaystyle=\frac{1}{8\pi}\int_{\hat{\Sigma}}\hat{k}-\frac{1}{8\pi}\int_{\Sigma}-\sqrt{1-|\nabla\tau|^{2}}\langle
H,u\rangle-\alpha_{v}(\nabla\tau)$
$\displaystyle=\frac{1}{8\pi}\int_{\hat{\Sigma}}\hat{k}-\frac{1}{8\pi}\int-\sqrt{1-|\nabla\tau|^{2}}\cosh\theta|H|-\nabla\theta\cdot\nabla\tau-\alpha_{J}(\nabla\tau)$
(10)

where

$\sinh\theta=\frac{\Delta\tau}{|H|\sqrt{1-|\nabla\tau|^{2}}}$ (11)

and

$\alpha_{J}(\cdot)=\langle\overline{\nabla}_{(\cdot)}\frac{J}{|H|},\frac{-H}{|H|}\rangle$
(12)

###  3.2 Surface mass density and current

Similar to the Wang-Yau quasi local energy, one would like to look for a
critical point among various embeddings $X$ or different killing vector
$T_{0}$. The first variation of the new definition with respect to the time
function $\tau=-\langle X,T_{0}\rangle$ again yields a divergence free current

$\nabla\cdot j^{new}=0$ (13)

where

$\displaystyle j^{new}$
$\displaystyle=\frac{\nabla\tau}{\sqrt{1+|\nabla\tau|^{2}}}|H_{0}|\cosh\theta_{0}-\frac{\nabla\tau}{\sqrt{1-|\nabla\tau|^{2}}}|H|\cosh\theta-\nabla(\theta_{0}-\theta)-\alpha_{H_{0}}+\alpha_{J}$
$\displaystyle=\tilde{\rho}_{1}\nabla\tau+\nabla\sinh^{-1}\frac{\tilde{\rho}_{2}\,\Delta\tau}{|H||H_{0}|}-\alpha_{H_{0}}+\alpha_{J}$

with

$\tilde{\rho}_{1}=\frac{\sqrt{|H_{0}|^{2}+\frac{(\Delta\tau)^{2}}{1+|\nabla\tau|^{2}}}}{\sqrt{1+|\nabla\tau|^{2}}}-\frac{\sqrt{|H|^{2}+\frac{(\Delta\tau)^{2}}{1-|\nabla\tau|^{2}}}}{\sqrt{1-|\nabla\tau|^{2}}}$
$\tilde{\rho}_{2}=\frac{\sqrt{|H_{0}|^{2}+\frac{(\Delta\tau)^{2}}{1+|\nabla\tau|^{2}}}}{\sqrt{1-|\nabla\tau|^{2}}}+\frac{\sqrt{|H|^{2}+\frac{(\Delta\tau)^{2}}{1-|\nabla\tau|^{2}}}}{\sqrt{1+|\nabla\tau|^{2}}}$

Further define a new mass surface density

$\displaystyle\rho^{new}$
$\displaystyle=\frac{|H_{0}|\cosh\theta_{0}}{\sqrt{1+|\nabla\tau|^{2}}}+\frac{|H|\cosh\theta}{\sqrt{1-|\nabla\tau|^{2}}}$
(14)
$\displaystyle=\frac{\sqrt{|H_{0}|^{2}+\frac{(\Delta\tau)^{2}}{1+|\nabla\tau|^{2}}}}{\sqrt{1+|\nabla\tau|^{2}}}+\frac{\sqrt{|H|^{2}+\frac{(\Delta\tau)^{2}}{1-|\nabla\tau|^{2}}}}{\sqrt{1-|\nabla\tau|^{2}}}$
(15)

Then the new quasi-local energy can be written as

$QLE^{new}=\frac{1}{8\pi}\int_{\Sigma}\rho^{new}+j^{new}\cdot\nabla\tau$ (16)

which reduces to

$QLM^{new}=\frac{1}{8\pi}\int_{\Sigma}\rho^{new}$

if the Euler-Lagrangian equation $\nabla\cdot j^{new}=0$ is imposed. Note that
the new mass surface density is nonnegative. The positivity of new quasi-local
mass follows from this. We recall that the surface mass density $\rho$ in
Wang-Yau quasi-local mass is not positive definite. However, the new quasi-
local energy with non-critical $\tau$ is not positive definite. Calculation
with Schwarzschild coordinate spheres indicate that the critical value of
$\tau$ seems to corresponds to a local maximum in QLEnew. We will study the
critical point in a separate work.

###  3.3 Continuity across an horizon

Having defined a quasi-local energy/mass for both trapped and untrapped
surfaces, one would naturally ask whether the two definition is continuous
cross the horizon, where the mean curvature vector is null. A careful analysis
of the Wang-Yau quasi-local energy in the limit of $|H|\to 0^{+}$ is shown in
[ZAY24b]. The analysis there shows that the Wang-Yau quasi-local energy blows
up to $+\infty$ unless the horizon can be isometrically embedded into
$\mathbb{R}^{3}$.

A similar conclusion applies to the new definition presented here. One can
rewrite the new quasi-local energy as in [ZAY24b]:

$\displaystyle 8\pi\,QLE^{new}=$
$\displaystyle\int_{\hat{\Sigma}}\hat{k}+\int_{\Sigma}\sqrt{(\Delta\tau)^{2}+|H|^{2}(1-|\nabla\tau|^{2})}+\nabla\tau\cdot\nabla\ln|\Theta_{-}|+\frac{1}{2}\langle\overline{\nabla}_{\nabla\tau}\,l^{-},l^{+}\rangle+\tau\Delta
y$ (17)

where
$y=\ln\bigg{[}\frac{-\Delta\tau}{\sqrt{1-|\nabla\tau|^{2}}}+\sqrt{|H|^{2}+\frac{(\Delta\tau)^{2}}{1-|\nabla\tau|^{2}}}\bigg{]}$.
Meanwhile the new Eular-Lagrangian equation can be rewritten as

$\Delta
y=\nabla\cdot\big{(}\tilde{\rho}_{1}\nabla\tau-\alpha_{H_{0}}-\nabla\theta_{0}\big{)}$
(18)

A similar analysis would yield that $\int\tau\Delta y\to-\infty$ and hence
$QLE^{new}\to-\infty$ unless $\tau$ approaches to a constant function at the
horiozn. We summarize in the following theorem.

######  Theorem 3.2.

Assume that an isometric embedding of $\Sigma$ remains smooth such that
$|\nabla\tau|$, $\Delta\tau$ all remain bounded as $\Sigma$ approaches an
horizon from both sides, i.e. $|H|\to 0$. Further assume that
$|\Theta_{-}|=-\Theta_{-}$ is bounded away from zero for the horizon under
consideration. If the horizon cannot be embeded into $\mathbb{R}^{3}$, then

$QLE\to+\infty,\textbf{ as }|H|\to 0^{+}$

while

$QLE^{new}\to-\infty,\textbf{ as }|H|\to 0^{-}$

and moreover, both Euler-Lagrangian equations

$\nabla\cdot j=0,\quad\nabla\cdot j^{new}=0$

are not solvable near the horiozn.

######  Remark 3.3.

If instead, the horizon can be isometrically embedded into $\mathbb{R}^{3}$
and moreover,

$x:=\lim_{|H|\to 0}\,\frac{\Delta\tau}{|H|\sqrt{1+|\nabla\tau|^{2}}}<\infty$

i.e. $\tau$ approaches to a constant function in a faster or comparable rate
as $|H|\to 0$, then

$lim_{|H|\to 0}QLE=\frac{1}{8\pi}\int_{{\Sigma}}|H_{0}|$

Therefore, unless the horizon can be isometrically embedded into $R^{3}$, the
Wang-Yau quasi-local energy blows up while the new quasi-local energy blows
down as $\Sigma$ approaches the horizon from outside or inside and hence there
is no continuity cross the horizon.

###  3.4 Examples of black hole spacetime

Recall that for coordinate spheres in an stationary and axisymmetric
spacetime, i.e. a Kerr family black hole, constant $\tau$ solves the optimal
embedding equation, both inside and outside the horizon since
$\nabla\cdot\alpha_{H}=-\nabla\cdot\alpha_{J}\equiv 0$ while
$\theta=\pm\sinh^{-1}\frac{\Delta\tau}{|H|\sqrt{1+|\nabla\tau|^{2}}}=0$ and
$\alpha_{v_{0}}=\langle\nabla^{\mathbb{R}^{3,1}}v_{0},T_{0}\rangle=0$ when
$\tau$ is constant and $|H|\neq 0$. Here we show calculations for coordinate
spheres from slowly rotating Kerr family for which one can simply take
constant $\tau$. The quasi-local energy with constant $\tau$ reduces to

$QLM=\begin{cases}\frac{1}{8\pi}\int|H_{0}|+|H|,&\text{ inside the horizon}\\\
\frac{1}{8\pi}\int|H_{0}|-|H|,&\text{ outside the horizon}\end{cases}$

We show in Figure 1-4 calculations for coordinate spheres of Schwarzschild
blackhole, Reissner-Nordstörm black hole with $Q=M/2$, extreme Reissner-
Nordstörm black hole with $Q=M$ and Kerr black hole with $a=M/2$,
respectively. Note that in these examples, the horizon can be embedded into
$\mathbb{R}^{3}$ and hence the QLE attains a finite limit as $|H|\to 0$

$\lim_{|H|\to 0}\,QLE=\frac{1}{8\pi}\int_{\Sigma}|H_{0}|$

The infinite slope at the horizon only manifests the fact that the horizon for
these black hole spacetime happen to be minimal surfaces, whose first
variation of area vanishes. Since the $x-axis$ is the area radius $r$, one has
$\delta r\propto\delta|\Sigma|=0$ and hence $\frac{\delta QLM}{\delta
r}\to\infty$ at the horizon.

For Schwarzschild black hole of mass $M$,

$QLM=\begin{cases}r[1-\sqrt{1-\frac{2M}{r}}],&r\geq 2M\\\
r[1+\sqrt{\frac{2M}{r}-1}],&r\leq 2M\end{cases}$

with the maximum at $r=(1+\frac{1}{\sqrt{2}})M$. For Reissner-Nordstörm of
charge $Q$ and mass $M$,

$QLM=\begin{cases}r[1-\sqrt{1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}}]&r\geq
r_{+}\\\ r[1+\sqrt{-1+\frac{2M}{r}-\frac{Q^{2}}{r^{2}}}]&r_{-}\leq r\geq
r_{+}\\\ r[1-\sqrt{1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}}]&r\leq
r_{-}\end{cases}$

with the zero at $r_{0}=\frac{Q^{2}}{2M}<r_{-}=M-\sqrt{M^{2}-Q^{2}}$. So the
negative mass for Reissner-Nordstörm happens inside the inner horizon, where
mean curvature vector of $\Sigma$ is spacelike. However, this does not
contradict previous proofs on positivity of Brown-York or Wang-Yau mass
because the singularity is essentially “naked” to surfaces inside the outer
horizon, violating assumptions in those proofs. The radius where quasi-local
energy turns negative is also when the textbook mass function

$M-\frac{Q^{2}}{2r}$

turns negative. The value at the singularity is $-|Q|$. These two spherical
symmetric examples recover the results of [LSYJ07] and reveal interesting
features that worths further exploration. For the Kerr black hole, when
rotation $a<\sqrt{3}/2M$, the horizon has positive Gaussian curvature, so we
can still take $\tau=0$ for surfaces inside but near the horizon. Further
inside, the calculation breaks down since those coordinate spheres cannot be
embedded into $\mathbb{R}^{3}$, i.e. $\tau=0$ is not a valid solution to the
Euler-Lagrangian equations.

##  4 Concluding remarks

As revealed in calculations for the Kerr family black hole, the new definition
of quasi-local energy exhibits some interesting features that may worth
further investigation. For example, the new quasi-local mass seems to attain
the maximum for a $2$-surface $\Sigma$ in the trapped region. Also, the
calculation for Reissner-Nordstörm black hole raises the question whether the
Wang-Yau quasi-local energy for $\Sigma$ with spacelike mean curvature vector
is bounded from below even when the spacetime singularity is not shielded from
$\Sigma$.

The definition here is based on the setup of Wang-Yau quasi-local energy and
hence reveals a similar pattern as one takes the limit of $\Sigma$ approaching
to an horizon. The basic assumption here is that the canonical gauge condition
of Wang and Yau [WY09b] still holds formally. It certainly needs further study
to investigate what the optimal gauge condition to impose for trapped surfaces
in order to pick a orthonormal basis $\\{u,v\\}$ for the normal bundle. This
study is a first attempt in trying to define a quasi-local energy for surfaces
inside the horizon.

We can now formulate the alternative approach to the path integral
quantization of gravity using the concept of quasi-local mass. Consider an
infinite dimensional space of topological $2$-spheres with geometric data
$(\sigma,\alpha,|H|)$ where $\sigma$ is a positive-definite metric, $\alpha$
is a one-form and $|H|$ is a scalar function on $\Sigma$. The proposal is to
rewrite the formal path integral over spacetime metrics $g$ as a formal path
integral over geometric data of topological $2$-sphere $\Sigma$:

$\displaystyle\int D[g]e^{iI[g]}=\int
D[\sigma]D[\alpha]D[|H|]D[\tau]D[T_{0}]e^{iQLM}$

In principle, one may consider another integral over the topology of $\Sigma$.
We note that this is different string theory, which quantizes the world sheet
geometry instead of the spacetime geometry.

Figure 1: Quasi-local energy with $\tau=0$ for coordinate spheres in the
Schwarzschild spacetime. The vertical black line indicates where the horizon
locates.

Figure 2: Quasi-local energy with $\tau=0$ for coordinate spheres in the
Reissner-Nordstörm with $Q=M/2$. The red and green vertical line indicates the
outer and inner horizon. Note that the mean curvature vector turns back to
spacelike inside the inner horizon. The negative value at $r=0$ is $-|Q|$.

Figure 3: Quasi-local energy with $\tau=0$ for coordinate spheres in the
Extremal Reissner-Nordstörm with $Q=M$. The negative value at $r=0$ is $-|Q|$.

Figure 4: Quasi-local energy with $\tau=0$ for coordinate spheres in the Kerr
spacetime with $a=M/2$. Calculation in done numerically with Mathematica.
Ingoing Kruskal coordinates is used. The calculation stops when $\tau=const$
yields complex values of QLE, indicating $\Sigma_{r}$ can not be embedded into
$\mathbb{R}^{3}$. Calculation for Kerr-Newmann black hole with $Q\neq 0$ is
similar.

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# Using child-woman ratios to infer demographic rates in historical
populations with limited data

John Bryant  Tahu Kukutai

###### Abstract

Data on historical populations often extends no further than numbers of people
by broad age-sex group, with nothing on numbers of births or deaths.
Demographers studying these populations have experimented with methods that
use the data on numbers of people to infer birth and death rates. These
methods have, however, received little attention since they were first
developed in the 1960s. We revisit the problem of inferring demographic rates
from population structure, spelling out the assumptions needed, and
specialising the methods to the case where only child-woman ratios are
available. We apply the methods to the case of Māori populations in
nineteenth-century Aotearoa New Zealand. We find that, in this particular
case, the methods reveal as much about the nature of the data as they do about
historical demographic conditions.

##  1 Introduction

In the era of big data, it is easy to lose sight of the fact that, for most
populations, for most of history, demographic data have been sparse at best.
Even in recent decades, many countries only have scattered survey and
administrative data, with a very occasional census [United Nations Population
Division, 2022]. Data on some populations, such as Indigenous populations, are
scarcer still.

In sparse data settings, the data that do exist often take the form of counts
of people by sex and age. Such data are among the least difficult to collect,
and meet the most urgent needs of government administrators. In twentieth
century censuses, the counts might be disaggregated by single-year or five-
year age groups. In older data sources, the counts might only use broad age
groups, such as counts of “children”, “men”, and “women”.

During the 1960s, Coale and Demeny, writing in a United Nations demographic
methods manual [UNDESA, 1967, ch. 1], described how census data on population
age structure could be used to infer fertility and mortality rates, especially
if data on population growth were also available. The basic intuition behind
these methods is that, since fertility and mortality rates determine age
structure and growth rates, data on age structure and growth rates should
allow some sort of inferences back to the underlying fertility and mortality
rates. The simplest version of the Coale-Demeny methods makes the assumption
that fertility and mortality rates have been constant, though Coale and Demeny
show how the methods can be extended to deal with changing rates. The methods
do not require data on births or deaths.

Since the 1960s, interest in these methods has waned among demographers
studying contemporary populations, as more direct evidence on fertility and
mortality levels, such as data from Demographic and Health Surveys, has become
available. The IUSSP-UNFPA compendium of demographic methods, for instance,
does not mention these methods [Moultrie et al., 2013].

The data available to historical demographers has grown much more slowly since
the 1960s. In historical demography, methods that can shed light on
demographic rates without the need for direct data on fertility and mortality
remain useful. Indeed, in Aotearoa New Zealand, data on age structure – and
particularly on the ratio between children and adult women – have played a
central role in the foundational research on Māori demographic history by Ian
Pool [Pool, 1973, 2013, 2015].

In this paper, we revisit methods for inferring demographic rates from data on
age structure and population growth. Following essentially the same logic as
the early methods of Coale and Demeny, we show how the relationship between
fertility, mortality, child-woman ratios, and population growth in stable
populations can be used to make inferences about fertility and mortality
rates. Whereas Coale and Demeny assume that census data on full age
distribution can be obtained, however, we focus on the case where only counts
of children versus adults are available, which allows us to develop a more
direct, graphical approach than their original methods. We investigate whether
these methods could shed light on demographic conditions experienced by Māori
in nineteenth-century Aotearoa New Zealand. We find that the ability to draw
out the implications of data on child-woman ratios and growth rates reveals as
much about the limits of the data as it does about nineteenth century
fertility and mortality.

##  2 The relationship between fertility, mortality and child-woman ratios

Throughout this paper, we define the child-woman ratio as the ratio of
children aged 0–14 to women aged 15 and over. Alternative definitions, such as
ones that restrict the denominator to women of reproductive age, are possible.
Use of alternative definitions would not, however, change the main arguments
of the paper.

The child-woman ratio can be used as an index of the age structure of a
population. The higher the child-woman ratio, the wider the base of the
population pyramid. The child-woman ratio increases when fertility increases,
since higher fertility implies that more recent cohorts are larger than
earlier ones. In high-mortality populations, the child-woman ratio also
increases when mortality falls. The reason is that, in high-mortality
populations, reductions in mortality entail larger absolute increases in the
probability of survival for infants and young children than for other groups.
Appendix A presents an illustration of this phenomenon. A large absolute
increase in the probability of survival for infants and young children has
essentially the same effect on population structure as a large increase in
fertility, in that it raises the number of ‘effective’ births, or births of
children who survive their first few years. The small size of the changes in
survival probabilities in other age groups mean that the changes in these
groups’ probabilities have relatively little effect on overall population
structure, so that the child-survival effect dominates.

To make these qualitative statements more precise, and to build a model of the
relationship between fertility, mortality, and child-woman ratios, we need to
make assumptions about the ways in which fertility and mortality rates vary.
Following Coale and Demeny [UNDESA, 1967], we build on elementary results from
stable population theory.

Figure 1 summarises our model. The inputs to the model are indicators for
overall levels of fertility and mortality. Combining these indicators with
assumptions about the age-sex-profiles of fertility and mortality yields age-
sex-specific fertility and mortality rates. If we assume, in addition, that
demographic rates have been constant for a long time, and that there is no
migration, then the resulting population must be stable – that is, the
relative shares of each age-sex group are not changing, even if total
population size is. Calculating the relative shares of each age-sex group is
straightforward [Keyfitz and Caswell, 2005]. Once we know the relative shares,
we can calculate the child-woman ratio.

Figure 1: Deriving child-woman ratios from values for fertility and mortality.
The inputs are the overall fertility and mortality rates, assumptions are
shown in grey, the output is child-woman ratios, and all other boxes represent
intermediate quantities.

Our model uses the total fertility rate (TFR) as the summary indicator for
overall fertility levels. The TFR is the average number of children that a
woman would bear, under current fertility rates, if she survived to the end of
the reproductive ages. For our initial calculations, we assume that age-
specific fertility rates follow the ‘Booth standard’ age pattern. The Booth
standard underlies some widely-used methods for fertility estimation [Booth,
1984, Moultrie et al., 2013]. Although not shown in Figure 1, we also assume
that there are 105 male births for every 100 female births.

Figure 1 uses life expectancy as the summary indicator for overall mortality
levels. In our initial calculations, we assume that age-specific mortality
patterns follow the shape predicted by “West” model life tables [Coale et al.,
1983, Jones, 2007]. Our choice of a West model life table is based on the fact
that it is a common default in studies of population dynamics, and that,
according to Pool [2013], it is likely to be a reasonable approximation to
nineteenth century Māori mortality patterns. In some of our analyses we work
with life expectancies that are lower than those in published West model life
tables. We derive life tables for these life expectancies that are consistent
with West model life table age patterns by applying the Brass logit model to
West model life tables [Preston et al., 2001, pp119-201]. (Specifically, we
use the published life tables with the lowest life expectancies as our
standard and vary the level parameter $\alpha$ until the implied life table
has the target life expectancy.)

The realism of the assumption of unchanging demographic rates varies from
population to population. It is, however, a convenient starting point, which
we revisit later in the paper. The assumption of zero migration is, similarly,
a convenient starting point.

Figure 2(a) displays the output from multiple applications of the model of
Figure 1. Each number in Figure 2(a) is a child-woman ratio obtained from a
particular value for TFR, which varies along the horizontal axis, and for
female life expectancy, which varies along the vertical axis. The number in
bold towards the left of the figure shows, for instance, that a child-woman
ratio of 1.00 is obtained from a TFR of 5.0 and a life expectancy of 27.5.

(a) All child-woman ratios.

(b) Child-woman ratios in the range 0.95–1.05.

Figure 2: Child-woman ratios in stable populations, and associated values for
fertility and mortality

Comparing the values on right of Figure 2(a) with the values on the left
illustrates how higher fertility is associated with larger child-woman ratios.
Comparing the values on the top of the figure with the values on the bottom
illustrates how lower mortality (and hence higher life expectancy) is also
associated with larger child-woman ratios.

##  3 Inferring fertility and mortality from child-woman ratios

When the assumptions behind Figure 2(a) are met, we can use the results from
the figure to infer fertility and mortality rates from child-woman ratios.
Given an observed value for child-woman ratio, we identify all combinations of
TFR and life expectancy that could have given rise to this ratio. If, for
instance, we observe a child-woman ratio of 1, we search through Figure 2(a)
to find all the 1s (including the case noted above) and record the associated
values for TFR and life expectancy.

In practice, it is prudent to work with ranges of child-woman ratios rather
than single values, to accommodate the fact that Figure 2(a) only shows ratios
at selected points, and as a way of acknowledging that the measurements of
child-woman ratios are imprecise, and that the assumptions behind the model
are unlikely to be fully met.

Figure 2(b) shows values of TFR and life expectancy that are associated with a
child-woman ratios between 0.95 and 1.05. It shows, for instance, that a
child-woman ratio of 0.95–1.05 can be produced by a TFR of 4–5 and a life
expectancy of about 35, but also from a TFR of 6–7 and a life expectancy of
about 15. It is impossible, using only information on child-woman ratios, to
say which of these possible combinations, or the many intermediate
combinations, is more likely.

##  4 Adding information on population growth rates

To narrow the range of possible values for fertility and mortality, we need to
bring in more information besides child-woman ratios. One possibility is to
bring in information on population growth rates. Estimates of population
growth rates can be calculated from the same census data that are used to
calculate child-woman ratios, provided that there are at least two censuses.
Every stable population has an associated population growth rate, so the model
of Figure 1 can easily be extended to include the population growth rate as a
second outcome alongside the child-woman ratio.

(a) All child-woman ratios and population growth rates.

(b) Child-woman ratios in the range 0.95–1.05 and population grow rates in the
range -0.4% to -0.2%.

Figure 3: Child-woman ratios and population growth rates in stable
populations, and associated values for fertility and mortality. Population
growth rates are shown in italics.

Figure 3 shows results from applying the extended model. Figure 3 is identical
to Figure 2(a) except that, in addition to child-woman ratios, it also shows
the annual population growth rates associated with each combination of TFR and
life expectancy. The figure shows, for instance, that a TFR of 5.0 and a life
expectancy of 27.5 leads to a child-woman ratio of 1.0 and an annual
population growth rate of 0.21%.

The benefits from supplementing information on child-woman ratios with
information on population growth rates are illustrated in Figure 3(b).
Combinations of TFR and life expectancy that are consistent with child-woman
ratios between 0.95 and 1.05 are shown in blue. Combinations that are
consistent with with annual population growth rates between -0.4% and -0.2%
are shown in red. Only two pairs of values fulfil both criteria: a TFR of 5.8
combined with a life expectancy of 20, and a TFR of 6.0 combined with a life
expectancy of 20. We can conclude that in the hypothetical population in
question, TFR is around 6 and life expectancy around 20.

##  5 Sensitivity to model assumptions

The assumptions behind our model of the relationship between fertility,
mortality, child-woman ratio, and population growth rate will never be exactly
met in any real population. In this section, we examine the effect of
violations of the assumptions.

###  5.1 Alternative age patterns for fertility and mortality

We begin by investigating sensitivity to assumptions about age patterns of
fertility and mortality. We examine how the relationship between fertility,
mortality, and child-woman ratios, and between fertility, mortality, and
population growth, changes when we depart from our original assumptions that
age-specific fertility rates follow the Booth standard and that age-specific
mortality rates follow the West model life table. Our alternative age pattern
for fertility is the age pattern experienced by Māori in 1962 [Statistics New
Zealand, 2020]. Our alternative age pattern for mortality is the South model
life table [Coale et al., 1983, Jones, 2007]. The alternative age patterns are
graphed in Figure B.1 in Appendix B. The 1962 Māori pattern has a sharper peak
than the Booth standard, and South model life table has higher rates at the
lowest and highest ages than the West model life table.

Figure 4: Child-woman ratios under alternative assumptions about age patterns
of fertility and mortality. Each panel shows contour lines for child-woman
ratios, that is, combinations of TFR and life expectancy that yield the same
value for the child-woman ratio. The dashed lines represent the baseline
combination of the Booth standard and West model life table.

Figure 4 shows how child-woman ratios vary with different combinations of age-
patterns for fertility and mortality. Rather than showing numeric values for
child-woman ratios, the figure shows contours lines. Each contour line in
Figure 4 depicts the combinations of TFR and life expectancy that are
associated with a given child-woman ratio. To help with comparisons, the
dashed lines in Figure 4 show child-woman ratios under the baseline
combination of the Booth standard and West model life table.

Judging by the results in Figure 4, the relationship between TFR, life
expectancy, and child-woman ratios is not at all sensitive to changes in the
age pattern for fertility. The relationship is, however, sensitive to changes
in the age pattern for mortality. For TFRs of 4.5–5.5, for instance, life
expectancy needs to be 1 or 2 years higher under a South model life table to
produce the same child-woman ratio that it does under a West model life table.

As discussed in Appendix B, similar conclusions apply to the relationship
between TFR, life expectancy, and population growth, with changes in the age
pattern for fertility rates having a much smaller impact on population growth
rates than changes in the age pattern for mortality.

###  5.2 Fertility and mortality rates changing over time

Our baseline model assumes that fertility and mortality rates have been fixed
for a long period, and that the age-sex structure of the population has
converged to its stable distribution. What happens if this assumption is
violated? When fertility and mortality change, how quickly do child-woman
ratios and population growth rates respond?

To assess the relationship between fertility and mortality rates, child-woman
ratios, and population growth rates, we conduct simulation experiments. We
look first at changes in fertility, examining two scenarios. In the first
scenario, at time 0 the TFR increases instantaneously from 5 births to 5.25
births, and then remains at 5.25 births forever. In the second scenario, the
TFR again starts at 5 births, but instead of just one increment of 0.25
births, there are five increments of 0.25 births, each five years apart.

Figure 5: Changes in child-woman ratios and population growth in response to
changes in fertility.

Results from these two scenarios are shown in Figure 5. The first scenario is
depicted in the two left-hand panels. In this scenario, child-woman ratios and
population growth rates both take around 50 years to attain their new long-run
values. The child-woman ratio gets most of the way towards its long-run value
within the first 15 years. The population growth rate follows a more
complicated path, with a rapid rise and fall followed by a slower rise.

The second scenario is depicted in the two right-hand panels. In this
scenario, the adjustment periods for each increment in fertility overlap,
producing a somewhat smoother transition. Once again, some of the adjustment
happens fairly quickly, but it takes several decades for all the changes to
work through.

Results from similar experiments with the effects of mortality changes are
shown in Appendix C. The results are similar to those for fertility, though
population growth adapts more quickly to a change in mortality than it does to
a change in fertility.

The results from the simulation experiments suggest that when fertility and
mortality rates are changing, the child-woman ratios and population growth
rates can no longer be interpreted as providing information on demographic
conditions at the time of the censuses. Instead, they should be interpreted as
a lagged indicator of average conditions over the decades preceding the
censuses. Changes in fertility and mortality rates have long-lasting effects
on child-woman ratios and population growth rates, so current child-woman
ratios and population growth rates as combined measures of fertility and
mortality rates over the past generation or two.

###  5.3 Migration

Migration into or out of an area might not be expected to affect child-woman
ratios in that area, provided that mothers and their children migrate
together. For migration to have absolutely no effect, however, various other
conditions need to be met: for instance, the probability that a woman migrates
needs to be independent of the number of children she has given birth to.

In addition, unless inflows perfectly match outflows, migration will almost
certainly affect population growth rates, even if mothers and children migrate
together. Substantial migration can therefore be expected to push populations
away from the combinations of indicators suggested by models based on
fertility and mortality only.

###  5.4 Measurement errors

The effect of under-coverage or over-coverage on the accuracy child-woman
ratios and population growth rates is potentially complex. For instance,
coverage errors only bias child-woman ratios to the extent that they vary
between children and women, and only bias growth rates to the extent that they
vary over time. In general, however, we would expect child-woman ratios to be
more robust than population growth rates to variation in coverage errors. A
10% difference in coverage rates between children and women will bias
estimates of the child-woman ratio by about 10%, but a 10% difference in
coverage rates between censuses can easily be larger than the true change in
population size.

##  6 Case study: Fertility and mortality of Māori in nineteenth-century
Aotearoa New Zealand

Quantitative evidence on the demographic effects on Māori of colonisation
during the nineteenth century is patchy and difficult to interpret [Chapple,
2017, Anderson, 2017]. We do not attempt assess the range of evidence, but
simply to illustrate the contribution that analysis of on child-woman ratios
and population growth can make.

Our input data are estimates of child-woman ratios and population growth from
censuses in the years 1874, 1878, 1881, 1886, 1891, 1896, and 1901. We use
data for all Aotearoa New Zealand. Rather than raw census counts, we use
counts assembled by Pool [2013, Tables 5.2, 5.3] that adjust for under-
coverage, as estimated by Pool and other historical investigators.

The child-woman ratios and population growth rates have different time
references, with child-woman ratios referring to census years, while
population growth rates refer to inter-censal periods. To apply our methods,
we need to bring the time references into alignment. We do this by aligning
everything to the mid-point of each period, with the child-woman ratio
equaling the average of the ratios at the two end points, and the population
growth rate equaling the average growth rate for that period.

Figure 6: Estimates of child-woman ratios and annual population growth rates
for Māori. Constructed from estimates presented in Table 5.2 and Table 5.3 of
Pool [2013].

The resulting values are shown in Figure 6. As can be seen in the figure, over
the 1870s and the 1890s the child-woman ratio increased from about 1.1 to
about 1.25. Between the 1870s, the population growth rate appears to have
increased from about -1% per year to about +1% per year.

Figure 7 shows the fertility and mortality rates implied by the estimated
child-woman ratio and population growth rate for all Māori in Aotearoa New
Zealand from the 1874–1878 censuses. TFR-life expectancy combinations
consistent with the estimated child-woman ratio are shown in blue, and TFR-
life expectancy combinations consistent with the estimated growth rate are
shown in red. The two sets of combinations meet at the bottom right of the
table, with a TFR of 7–8 and a life expectancy of 10–15.

The lower panel of Figure 7 shows the equivalent results derived from the
1896–1901 censuses. In this case, the two sets of combinations meet at the
upper-middle part of the table, with a TFR of 6-7 and a life expectancy in the
mid-to-high-20s.

(a) 1874–1878

(b) 1896–1901

Figure 7: Life expectancy and TFRs implied by child-woman ratios and annual
growth for Māori. The black numbers in the bottom right are derived with from
the 1874 and 1878 censuses, and the black numbers near the top are derived
from the 1896 and 1901 censuses.

Although the later combination of TFR and life expectancy is plausible, the
earlier combination is not. Rallu [1992, p. 182], for instance, obtains life
expectancies of 17–19 for the Marquesas in the early 20th century from good-
quality birth and death registration data, but notes that these values were
lower than they had been in the Marquesas during the late 19th century, and
lower than those in India around 1900 during a plague epidemic. It is unlikely
that Māori mortality rates in 19th century Aotearoa New Zealand were
significantly worse than those in the Marquesas. Moreover, given that
conception and child birth both require some minimum level of health and
security, it is extremely unlikely that very low life expectancies would have
been paired with a TFR as high as 7–8.

It appears, then that the child-woman ratio from the earlier period, or the
population growth rate, or both are wrong. Higher values for child-woman ratio
and population growth rate would yield more plausible values for fertility and
mortality. Finding some more plausible combination of child-woman ratio,
population growth rate, TFR and life expectancy would, however, require
additional information.

The later combination of TFR and life expectancy is much more plausible.
However, given the problems with the child-woman ratio and population growth
rate in the earlier census data, we need to treat the child-woman ratio and
population growth rate in the later census data with some suspicion. The
estimates of a TFR of 6-7 and life expectancy in the mid-to-high-20s need to
be treated as tentative.

##  7 Discussion

The original Coale-Demeny methods for analysing data on age-sex structure and
population growth rates, and the more specialised methods for child-woman
ratios that we have developed in this paper, are based on the relationship
between fertility and mortality and population age-sex structure and growth
rates. When data on age-sex structure and growth rates are sufficiently good,
and when assumptions about matters such as age-patterns of mortality are
approximately met, the data can be used to learn about fertility and
mortality. However, as our case study shows, inference can also work in the
opposition direction. If turns out that the population data implies fertility
and mortality rates that are implausible, then doubt is cast on the data
itself.

The population data that we use in our case study have played a central role
in analyses of the demographic impacts of colonisation in Aotearoa New Zealand
[Chapple, 2017, Pool, 2015]. Evidence that the population data, particularly
for early periods, may have substantial errors is therefore important. Much
work is still needed to reconstitute the population history of 19th century
Aotearoa New Zealand.

One possible avenue for investigation is to distinguish more carefully between
different areas, some of which are likely to have better data than others.
Much of the available historical data for Aotearoa New Zealand can be
disaggregated geographically. Enumerators’ reports provide at least some
information on how reliability varies across areas and across periods
[Statistics New Zealand, 2022]. A combination of demographic analysis,
involving measures such as child-woman ratios and growth rates, with
investigation of historical sources might allow researchers to distinguish
between areas where demgographic reconstructions were feasible and areas where
it was not. This would yield new, more detailed information on the demographic
impacts of colonisation, and give a clearer picture of the strengths and
weaknesses of the available evidence.

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##  Appendix A How improvements in life expectancy affect survival
probabilities of different age groups

Figure A.1: How survival probabilities at selected ages vary with life
expectancy, under West model life tables for females.

Figure A.1 shows how the probability of surviving from the start to the end of
the age groups 0–4, 5–9, 20–24, and 60–64 change as the level of overall
mortality falls from very high to moderate. The probabilities come from West
model life tables for females [Coale et al., 1983, Jones, 2007], which capture
experiences across a wide range of countries. The absolute increases in
survival probabilities for 0–4 year olds are much larger than the absolute
increases for 5–9 year olds and 20–24 year olds. The absolute increases in
survival probabilities for 60–64 year olds are somewhat larger, but older
people constitute a small proportion of the population in high-mortality
populations, so changes in the size of this group has only a small effect on
child-woman ratios.

##  Appendix B Alternative assumptions about age patterns of fertility and
mortality

Figure B.1 shows our baseline assumptions about age patterns for fertility and
mortality, versus the alternative assumptions that we use for sensitivity
testing in Section 5. Compared with the Booth standard, the Māori 1962 has a
sharper peak at young ages. Compared with the West age pattern, the South
pattern has higher childhood and old-age mortality, and lower middle-age
mortality.

Figure B.1: Original and alternative age patterns for fertility and mortality.
The left panel compares our original age-pattern for fertility, based on the
Booth standard, with an assumption based Māori fertility rates in 1962. The
right panel compares our original age-pattern for mortality, based on the West
model life table, with the age-pattern from the South model life table.

Figure B.2 is identical to Figure 4 , except that it shows population growth
rates rather than child-woman ratios. The results are essentially the same as
those for child-woman ratios. Switching to a new assumption about the age-
pattern of fertility has a negligible effect on the relationship between
fertility, mortality, and the population growth rate, while switching to a new
assumption about the age-pattern of mortality as a much more noticeable
effect.

Figure B.2: Annual population growth rates (percent) under alternative
assumptions about age patterns of fertility and mortality. Each panel shows
contour lines for population growth rates, that is, combinations of TFR and
life expectancy that yield the same value for the growth rate. The dashed
lines represent the baseline combination of the Booth standard and West model
life table.

##  Appendix C Allowing for changes in mortality rates

Figure C.1: _Changes in child-woman ratios and population growth in response
to changes in mortality._

Figure C.1 is identical to Figure 5, except that it uses life expectancy in
place of the TFR. In the two left-hand panels of Figure C.1, life expectancy
starts at 30 and then increases to 32.5. In the two right-hand panels, life
expectancy starts at 30 and then increases by 2.5 years ever 5 years. The
results are similar to those for the TFR, except that the adjustment process
is somewhat faster.