On the Continuity of Geometrized Newtonian Gravitation and General Relativity

Abstract

Pessimistic meta-induction is a powerful argument against scientific realism, so one of the major roles for advocates of scientific realism will be trying their best to give a sustained response to this argument. On the other hand, it is also alleged that structural realism is the most plausible form of scientific realism; therefore, the plausibility of scientific realism is threatened unless one is given the explicit form of a structural continuity and minimal structural preservation for all our current theories. This essay aims to present what we call expansive structures, which are the structures that can be reconstructed from geometrized Newtonian gravitation and are capable of expanding into general relativity, explicitly. In this way, pessimistic meta-induction will be undermined.

Appendix A: Decompositions and Expansions

First we note that according to the proposition (1.10.1) of Malament [47], there is an isomorphism between the tensor algebra of $\Sigma$-tensors at a point p of the metric submanifold $\Sigma$ of the manifold M and the tensor algebra of M-tensors that are tangent to $\Sigma$. Now let $\Sigma$ be a three-dimensional metric submanifold of M with induced metric field ${\mathbf {h}}_{ab}$, and let $h_{ab}$ be its corresponding symmetric tensor field that is tangent to $\Sigma$. Then $h_{ab}$ is completely determined by its action on the tangent and normal vectors to $\Sigma$ at each point p [47, p. 100]. Similar remarks hold in the case of $\Lambda$, a one-dimensional metric submanifold of M with induced metric filed ${\mathbf {t}}_{ab}$ for which $t_{ab}$ is its corresponding symmetric tensor field that is tangent to $\Sigma$.

Having pointed out the above remarks, here we introduce the decompositions and the expansions given in the Sect. 5. Before we see the decompositions and the expansions, let us define some useful notations.

• (a) $\sigma _i\equiv {g}^i_{ab_{|_p}}\xi ^a \eta ^b= g^{ab}_{ i_{|_p}} \xi _a\eta _b.$

• (b) $\beta _i\equiv h^{ab}_{ i_{|_p}} \xi _a\eta _b.$

• (c) $\gamma _{i}\equiv \lambda ^a\lambda ^b_ { |_{_p}}\xi _a\eta _b$.

• (d) $\delta _i\equiv {t}^i_{ab_{|_p}}\xi ^a \eta ^b$.

• (e) $\theta _i\equiv {{{\hat{h}}}^i}_{ab_{|_p}} \xi ^a\eta ^b.$

Note that in the relativistic space-time with the Minkowski metric $h^{ab}\xi _a\mu _a={{{\hat{h}}}}_{ab}\xi ^a\mu ^a$ since

$$\begin{aligned}&\eta ^{ab}\xi _a\mu _a=(h^{ab}-\lambda ^a\lambda ^b)\xi _a\mu _b \\&\quad = {\left\{ \begin{array}{ll} h^{ab}\xi _a\mu _b &{} \xi _a\in {(id_p)}^*{(T_p\Sigma )_a} \ {\text{and}} \ \mu _b\in {(id_p)}^*{(T_p\Sigma )_b}\ \\ \\ -\lambda ^a\lambda ^b\xi _a\mu _b \ &{} \ \xi _a\in {(id_p)}^*{(T_p\Lambda )_a}\ {\text{and}} \ \mu _b \ \in {(id_p)}^*{(T_p\Lambda )_b}\\ \end{array}\right. } \end{aligned}$$

(14)

also it should be clear³² that

$$\begin{aligned}&\eta _{ab}\xi ^a\mu ^a=({{\hat{h}}}_{ab}+t_{ab})\xi ^a\mu ^b \\&\quad ={\left\{ \begin{array}{ll} \hat{h}_{ab}\xi ^a\mu ^b &{} \quad \xi _a\in {(id_p)}_*{(T_p\Sigma )_a} \ {\text{and}} \ \mu _b\in {(id_p)}_*{(T_p\Sigma )_b}\ \\ \\ t_{ab}\xi _a\mu _b \ &{} \quad \xi _a\in {(id_p)}_*{(T_p\Lambda )_a}\ {\text{and}} \ \mu _b \in {(id_p)}_*{(T_p\Lambda )_b}.\\ \end{array}\right. } \end{aligned}$$

(15)

In addition, we know that the following equations hold:

• (A). $h^{ab}\xi _a\mu _b=0 \ {\text{if and only if}}\ \xi _a\in {(id_p)}_*{(T_p\Lambda )_a}\ {\text {or}} \ \mu _a\in {(id_p)}_*{(T_p\Lambda )_b}.$

• (B). $\lambda ^a\lambda ^b\xi _a\mu _b=0 \ {\text{if and only if}}\ \xi _a\in {(id_p)}_*{(T_p\Sigma )_a}\ {\text{or}} \ \mu _a\in {(id_p)}_*{(T_p\Sigma )_b}.$

• (C). $t_{ab}\xi ^a\mu ^b=0\ {\text{if and only if}}\ \xi ^a\in {(id_p)}_*{(T_p\Sigma )_a} \ {\text{or}} \ \mu ^a\in {(id_p)}_*{(T_p\Sigma )_b}.$

• (D). ${{\hat{h}}}_{ab}\xi ^a\mu ^b=0\ {\text{if and only if}}\ \xi ^a\in {(id_p)}_*{(T_p\Lambda )_a} \ {\text{or}} \ \mu ^a\in {(id_p)}_*{(T_p\Lambda )_b}.$

1.1 A.1 A Decomposition of $g_{ab}$

$$\begin{aligned}&{g}^1_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, \sigma _1):\xi ^a\in {(id_p)}_*(T_p\Sigma )^a \ {\text{and}} \ \eta ^b\in (id_p)_*(T_p\Sigma )^b \Big \}.\\&{g}^2_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, \sigma _2):\xi ^a\in (id_p)_*(T_p\Lambda )^a \ {\text{and}} \ \eta ^b\in (id_p)_*(T_p\Lambda )^b \Big \}. \\&{g}^3_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, 0):\xi ^a\in (id_p)_*(T_p\Sigma )^a \ {\text{and}} \ \eta ^b\in (id_p)_*(T_p\Lambda )^b \Big \}. \\&{g}^4_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, 0):\xi ^a \in (id_p)_*(T_p\Lambda )^a \ {\text{and}} \ \eta ^b\in (id_p)_*(T_p\Sigma )^b \Big \}.\\&{g}^5_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, \sigma _5):\xi ^a={\mathfrak{a}}_5\zeta ^a+{\mathfrak{b}}_5\mu ^a, \zeta ^a\in (id_p)_*(T_p\Sigma )^a \ \\&\quad {\text{and}} \ \mu ^a\in (id_p)_*(T_p\Lambda )^a, {\mathfrak{a}}_5\ne 0, {\mathfrak{b}}_5\ne 0 \ {\text{and}} \ \eta ^b\in (id_p)_*(T_p\Sigma )^b\Big \} .\\&{g}^6_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, \sigma _6):\xi ^a\in (id_p)_*(T_p\Sigma )^a, \eta ^b={\mathfrak{a}}_6\zeta ^b+{\mathfrak{b}}_6\mu ^b, \\&\quad \zeta ^b\in (id_p)_*(T_p\Sigma )^b, \ \mu ^b\in (id_p)_*(T_p\Lambda )^b,\ {\mathfrak{a}}_6\ne 0\ {\text{and}}\ {\mathfrak{b}}_6\ne 0 \Big \}.\\&{g}^7_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, \sigma _7):\xi ^a={\mathfrak{a}}_7\zeta ^a+{\mathfrak{b}}_7\mu ^a, \zeta ^a\in (id_p)_*(T_p\Sigma )^a, \\&\quad \mu ^a\in (id_p)_*(T_p\Lambda )^a,\ {\mathfrak{a}}_7\ne 0, {\mathfrak{b}}_7\ne 0 {\text{and}} \ \eta ^b\in (id_p)_*(T_p\Lambda )^b\Big \}.\\&{g}^8_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, \sigma _8):\xi ^a\in (id_p)_*(T_p\Lambda )^a, \eta ^b={\mathfrak{a}}_8\zeta ^b+{\mathfrak{b}}_8\mu ^b, \\&\quad \zeta ^b\in (id_p)_*(T_p\Sigma )^b, \ \mu ^b\in (id_p)_*(T_p\Lambda )^b,\ {\mathfrak{a}}_8\ne 0\ {\text{and}}\ {\mathfrak{b}}_8\ne 0 \Big \}.\\&{g}^9_{ab_{|_p}}=\Big \{ (\xi ^a, \eta ^b, \sigma _9):\xi ^a={\mathfrak{a}}_9\zeta ^a+{\mathfrak{b}}_9\mu ^a, \ {\mathfrak{a}}_9\ne 0, {\mathfrak{b}}_9\ne 0 ,\\&\quad \zeta ^a\in (id_p)_*(T_p\Sigma )^a, \ \mu ^a\in (id_p)_*(T_p\Lambda )^a,\ \xi ^{\prime b}\\&\quad ={\mathfrak{a}}^\prime _9\zeta ^{\prime b}+{\mathfrak{b}}^\prime _9\mu ^{\prime b}, {\mathfrak{a}}^\prime _9\ne 0, {\mathfrak{b}}^\prime _9\ne 0, \ \zeta ^{\prime b}\in (id_p)_*(T_p\Sigma )^b \ \\&\quad {\text{and}} \ \mu ^{\prime a}\in (id_p)_*(T_p\Lambda )^b\Big \}. \end{aligned}$$

1.2 A.2 A Decomposition of $g^{ab}$

$$\begin{aligned}&g^{ab}_{ 1_{|_p}}=\Big \{ (\xi _a, \eta _b, \beta _1e^{-\Omega (p)}):\xi _a\in {(id_p)}^*{(T_p\Sigma )_a} \\&\quad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Sigma )_b} \Big \}.\\&g^{ab}_{ 2_{|_p}}=\Big \{(\xi _a, \eta _b, \gamma _2e^{-\Omega (p)}):\xi _a\in (id_p)^*(T_p\Lambda )_a \\&\quad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Lambda )_b} \Big \}. \\&g^{ab}_{ 3_{|_p}}=\Big \{ (\xi _a, \eta _b, 0):\xi _a\in (id_p)^*{(T_p\Sigma )_a}\ \ \\&\quad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Lambda )_b} \Big \}. \\&g^{ab}_{ 4_{|_p}}=\Big \{(\xi _a, \eta _b, 0):\xi _a \in (id_p)^*{(T_p\Lambda )_a} \\&\quad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Sigma )_b} \Big \}.\\&g^{ab}_{ 5_{|_p}}=\Big \{ (\xi _a, \eta _b,{\mathfrak{a}}_5\beta _5e^{-\Omega (p)}):\xi _a={\mathfrak{a}}_5\zeta _a+{\mathfrak{b}}\mu _a, \zeta _a\in (id_p)^*{(T_p\Sigma )_a}, \\&\quad \mu _a\in (id_p)^*{(T_p\Lambda )_a}, {\mathfrak{a}}\ne 0 \ {\text{and}}\ {\mathfrak{b}}\ne 0\ \eta _b\in (id_p)^*(T_p\Sigma )_b\Big \} .\\&g^{ab}_{ 6_{|_p}}=\Big \{ (\xi _a, \eta _b, \mathfrak{a_6}\beta _6e^{-\Omega (p)}):\xi _a\in (id_p)^*{(T_p\Sigma )_a} , \eta _b \\&\quad ={\mathfrak{a}}_6\zeta _b+{\mathfrak{b}}_6\mu _b, \ \zeta _b\in (id_p)^*{(T_p\Sigma )_b}, \mu _b\in (id_p)^*{(T_p\Lambda )_b},\ {\mathfrak{a}}_6\ne 0\ {\text{and}} \ {\mathfrak{b}}_6\ne 0\Big \}.\\&g^{ab}_{ 7_{|_p}}=\Big \{ (\xi _a, \eta _b, {\mathfrak{a}}_7\gamma _7e^{-\Omega (p)}):\xi _a={\mathfrak{a}}_7\zeta _a+{\mathfrak{b}}_7\mu _a, \\&\quad \zeta _a\in (id_p)^*{(T_p\Sigma )_a}, \ \mu _a\in (id_p)^*{(T_p\Lambda )_a},\ {\mathfrak{a}}_7\ne 0, {\mathfrak{b}}_7\ne 0 \ {\text{and}} \ \eta _b\in (id_p)^*{(T_p\Lambda )_b}\Big \}.\\&g^{ab}_{ 8_{|_p}}=\Big \{ (\xi _a, \eta _b, {\mathfrak{a}}_8\gamma _8e^{-\Omega (p)}):\xi _a\in (id_p)^*{(T_p\Lambda )_a}, \eta _b \\&\quad ={\mathfrak{a}}_8\zeta _b+{\mathfrak{b}}_8\mu _b, \ \zeta _b\in (id_p)^*{(T_p\Sigma )_b} \ {\text{and}} \ \mu _b\in (id_p)^*{(T_p\Lambda )_b},\ {\mathfrak{a}}_8\ne 0, {\mathfrak{b}}_8\ne 0\Big \}.\\&g^{ab}_{ 9_{|_p}}=\Big \{ (\xi _a, \eta _b, ({\mathfrak{a}}_9{\mathfrak{a}}^\prime _9\beta _9+{\mathfrak{b}}_9{\mathfrak{b}}^\prime _9\gamma _9)e^{-\Omega (p)}):\xi _a={\mathfrak{a}}_9\zeta _a+{\mathfrak{b}}_9\mu _a, \\&\quad {\mathfrak{a}}_9\ne 0, {\mathfrak{b}}_9\ne 0 ,\ \zeta _a\in (id_p)^*{(T_p\Sigma )_a},\ \mu ^a\in (id_p)^*{(T_p\Lambda )_a}, \eta _b\\&\quad ={\mathfrak{a}}^\prime _9\zeta ^\prime _b+{\mathfrak{b}}^\prime _9\mu ^\prime _b, {\mathfrak{a}}^\prime _9\ne 0, {\mathfrak{b}}^\prime _9\ne 0, \ \zeta ^\prime _ b\in (id_p)^*{(T_p\Sigma )_b} \\&\quad {\text{and}} \ \mu ^\prime _ b\in (id_p)^*{(T_p\Lambda )_b}\Big \}. \end{aligned}$$

1.3 A.3 A Decomposition of $t_{ab}$

$$\begin{aligned}&{t}^1_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, 0):\xi ^a\in {(id_p)}_*(T_p\Sigma )^a \ {\text{and}} \ \eta ^b\in (id_p)_*(T_p\Sigma )^b \Big \}.\\&{t}^2_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, \delta _2):\xi ^a\in (id_p)_*(T_p\Lambda )^a \ {\text{and}} \ \eta ^b\in (id_p)_*(T_p\Lambda )^b \Big \}. \\&{t}^3_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, 0):\xi ^a\in (id_p)_*(T_p\Sigma )^a \ {\text{and}} \ \eta ^b\in (id_p)_*(T_p\Lambda )^b \Big \}. \\&{t}^4_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, 0):\xi ^a \in (id_p)_*(T_p\Lambda )^a \ {\text{and}} \ \eta ^b\in (id_p)_*(T_p\Sigma )^b \Big \}.\\&{t}^5_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, 0):\xi ^a={\mathfrak{a}}_5\zeta ^a+{\mathfrak{b}}_5\mu ^a, \zeta ^a\in (id_p)_*(T_p\Sigma )^a \\&\quad {\text{and}}\ \mu ^a\in (id_p)_*(T_p\Lambda )^a, {\mathfrak{a}}_5\ne 0, {\mathfrak{b}}_5\ne 0 \ {\text{and}} \ \eta ^b\in (id_p)_*(T_p\Sigma )^b\Big \} .\\&{t}^6_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, 0):\xi ^a\in (id_p)_*(T_p\Sigma )^a, \eta ^b={\mathfrak{a}}_6\zeta ^b+{\mathfrak{b}}_6\mu ^b, \\&\quad \zeta ^b\in (id_p)_*(T_p\Sigma )^b, \ \mu ^b\in (id_p)_*(T_p\Lambda )^b,\ {\mathfrak{a}}_6\ne 0\ {\text{and}}\ {\mathfrak{b}}_6\ne 0 \Big \}.\\&{t}^7_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, {\mathfrak{b}}_7\delta _7):\xi ^a={\mathfrak{a}}_7\zeta ^a+{\mathfrak{b}}_7\mu ^a, \zeta ^a\in (id_p)_*(T_p\Sigma )^a, \\&\quad \mu ^a\in (id_p)_*(T_p\Lambda )^a,\ {\mathfrak{a}}_7\ne 0, {\mathfrak{b}}_7\ne 0 \ {\text{and}} \ \eta ^b\in (id_p)_*(T_p\Lambda )^b\Big \}.\\&{t}^8_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, {\mathfrak{b}}_8\delta _8):\xi ^a\in (id_p)_*(T_p\Lambda )^a, \eta ^b={\mathfrak{a}}_8\zeta ^b+{\mathfrak{b}}_8\mu ^b, \ \zeta ^b\in (id_p)_*(T_p\Sigma )^b, \\&\quad \mu ^b\in (id_p)_*(T_p\Lambda )^b,\ {\mathfrak{a}}_8\ne 0\ {\text{and}}\ {\mathfrak{b}}_8\ne 0 \Big \}.\\&{t}^9_{ab_{|_p}}=\Big \{ (\xi ^a, \eta ^b, {\mathfrak{b}}_9{\mathfrak{b}}^\prime _9\delta _9):\xi ^a={\mathfrak{a}}_9\zeta ^a+{\mathfrak{b}}_9\mu ^a, \ {\mathfrak{a}}_9\ne 0, {\mathfrak{b}}_9\ne 0 ,\ \zeta ^a \\&\quad \in (id_p)_*(T_p\Sigma )^a, \ \mu ^a\in (id_p)_*(T_p\Lambda )^a,\ \eta ^b={\mathfrak{a}}^\prime _9\zeta ^{\prime b}+{\mathfrak{b}}^\prime _9\mu ^{\prime b}, {\mathfrak{a}}^\prime _9\ne 0, {\mathfrak{b}}^\prime _9\ne 0, \ \zeta ^{\prime b} \\&\quad \in (id_p)_*(T_p\Sigma )^b \ {\text{and}} \ \mu ^{\prime b}\in (id_p)_*(T_p\Lambda )^b\Big \}. \end{aligned}$$

It worth noting that in the case under consideration $e^{\Omega (p)}({\hat{h}}_{ab}+t_{ab})=(\gamma _{ab}-n_an_b) {t}^2_{ab_{|_p}}\xi ^a\eta ^b=\delta _2$.

$$\begin{aligned}&{t}^7_{ab_{|_p}}({\mathfrak{a}}_9\zeta ^a+{\mathfrak{b}}_9\mu ^a)\eta ^b={\mathfrak{b}}_7{t}^7_{ab_{|p}}\mu ^a\eta ^b={\mathfrak{b}}_7\delta _7.\\&{t}^8_{ab_{|_p}}\xi ^a({\mathfrak{a}}_8\zeta ^b+{\mathfrak{b}}_8\mu ^b)={\mathfrak{b}}_8{t}^8_{ab_{|p}}\xi ^a\eta ^b={\mathfrak{b}}_8\delta _8.\\&{t}^9_{ab_{|_p}}({\mathfrak{a}}_9\zeta ^a+{\mathfrak{b}}_9\mu ^a)({\mathfrak{a}}^\prime _9{\zeta ^\prime }^b+{\mathfrak{b}}^\prime _9{\mu ^\prime }^b) ={\mathfrak{b}}_9{\mathfrak{b}}^\prime _9{t}^9_{ab_{|_p}}\mu ^a{\mu ^\prime }^b={\mathfrak{b}}_9{\mathfrak{b}}^\prime _9\delta _9. \end{aligned}$$

1.4 A.4 An Expansion of $t_{ab}$

$$\begin{aligned}&{{\mathfrak{t}}}^1_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, \theta _1):\xi ^a\in {(id_p)}_*(T_p\Sigma )^a \ \\&\quad {\text{and}} \ \eta ^b\in (id_p)_*(T_p\Sigma )^b \Big \}.\\&{{\mathfrak{t}}}^2_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, \delta _2):\xi ^a\in (id_p)_*(T_p\Lambda )^a \\&\quad {\text{and}} \ \eta ^b\in (id_p)_*(T_p\Lambda )^b \Big \}. \\&{{\mathfrak{t}}}^3_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, 0):\xi ^a\in (id_p)_*(T_p\Sigma )^a \ \\&\quad {\text{and}} \ \eta ^b\in (id_p)_*(T_p\Lambda )^b \Big \}. \\&{{\mathfrak{t}}}^4_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, 0):\xi ^a \in (id_p)_*(T_p\Lambda )^a \ \\&\quad {\text{and}} \ \eta ^b\in (id_p)_*(T_p\Sigma )^b \Big \}.\\&{{\mathfrak{t}}}^5_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, {\mathfrak{a}}_5\theta _5):\xi ^a={\mathfrak{a}}_5\zeta ^a+{\mathfrak{b}}_5\mu ^a, \zeta ^a\in (id_p)_*(T_p\Sigma )^a \\&\quad {\text{and}} \ \mu ^a\in (id_p)_*(T_p\Lambda )^a, {\mathfrak{a}}_5\ne 0, {\mathfrak{b}}_5\ne 0 \ \\&\quad {\text{and}} \ \eta ^b\in (id_p)_*(T_p\Sigma )^b\Big \} .\\&{{\mathfrak{t}}}^6_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, {\mathfrak{a}}_6\theta _6):\xi ^a\in (id_p)_*(T_p\Sigma )^a, \eta ^b={\mathfrak{a}}_6\\&\quad \zeta ^b+{\mathfrak{b}}_6\mu ^b, \ \zeta ^b\in (id_p)_*(T_p\Sigma )^b, \ \mu ^b\in (id_p)_*(T_p\Lambda )^b,\ {\mathfrak{a}}_6\ne 0\ {\text{and}}\ {\mathfrak{b}}_6\ne 0 \Big \}.\\&{{\mathfrak{t}}}^7_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, \delta _7):\xi ^a={\mathfrak{a}}_7\zeta ^a+{\mathfrak{b}}_7\mu ^a, \\&\quad \zeta ^a\in (id_p)_*(T_p\Sigma )^a, \ \mu ^a\in (id_p)_*(T_p\Lambda )^a,\ {\mathfrak{a}}_7\ne 0, {\mathfrak{b}}_7\ne 0 {\text{and}} \ \eta ^b\in (id_p)_*(T_p\Lambda )^b\Big \}.\\&{{\mathfrak{t}}}^8_{ab_{|_p}}=\Big \{(\xi ^a, \eta ^b, \delta _8):\xi ^a\in (id_p)_*(T_p\Lambda )^a, \eta ^b={\mathfrak{a}}_8\zeta ^b+{\mathfrak{b}}_8\mu ^b, \\&\quad \ \zeta ^b\in (id_p)_*(T_p\Sigma )^b, \ \mu ^b\in (id_p)_*(T_p\Lambda )^b,\ {\mathfrak{a}}_8\ne 0\ {\text{and}}\ {\mathfrak{b}}_8\ne 0 \Big \}.\\&{{\mathfrak{t}}}^9_{ab_{|_p}}=\Big \{ (\xi ^a, \eta ^b, {\mathfrak{a}}_9{\mathfrak{a}}^\prime _9\theta _9+{\mathfrak{b}}_9{\mathfrak{b}}^\prime _9\delta _9):\xi ^a={\mathfrak{a}}_9\zeta ^a+{\mathfrak{b}}_9\mu ^a, \ {\mathfrak{a}}_9\ne 0, {\mathfrak{b}}_9\ne 0 ,\\&\quad \zeta ^a\in (id_p)_*(T_p\Sigma )^a, \ \mu ^a\in (id_p)_*(T_p\Lambda )^a,\ \eta ^b={\mathfrak{a}}^\prime _9\zeta ^{\prime b}+{\mathfrak{b}}^\prime _9\mu ^{\prime b}, {\mathfrak{a}}^\prime _9\ne 0, {\mathfrak{b}}^\prime _9\ne 0, \ \zeta ^{\prime b} \\&\quad \in (id_p)_*(T_p\Sigma )^b \ {\text{and}} \ \mu ^{\prime b}\in (id_p)_*(T_p\Lambda )^b\Big \}. \end{aligned}$$

1.5 A.5 A Decomposition of $h^{ab}$

$$\begin{aligned}&h^{ab}_{ 1_{|_p}}=\Big \{ (\xi _a, \eta _b, \beta _1):\xi _a\in {(id_p)}^*{(T_p\Sigma )_a} \ \\&\quad {\text{and}} \ \eta _b\in (id_p)^*{(T_p\Sigma )_b} \Big \}.\\&h^{ab}_{ 2_{|_p}}=\Big \{(\xi _a, \eta _b, 0):\xi _a\in (id_p)^*(T_p\Lambda )_a \\&\quad {\text{and}} \ \eta _b\in (id_p)^*{(T_p\Lambda )_b} \Big \}. \\&h^{ab}_{ 3_{|_p}}=\Big \{ (\xi _a, \eta _b, 0):\xi _a\in (id_p)^*{(T_p\Sigma )_a}\ \ \\&\quad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Lambda )_b} \Big \}. \\&h^{ab}_{ 4_{|_p}}=\Big \{(\xi _a, \eta _b, 0):\xi _a \in (id_p)^*{(T_p\Lambda )_a} \ \ \\&\quad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Sigma )_b} \Big \}.\\&h^{ab}_{ 5_{|_p}}=\Big \{ (\xi _a, \eta _b,{\mathfrak{a}}_5\beta _5):\xi _a={\mathfrak{a}}_5\zeta _a+{\mathfrak{b}}\mu _a, \zeta _a\in (id_p)^*{(T_p\Sigma )_a} \ \\&\quad {\text{and}} \mu _a\in (id_p)^*{(T_p\Lambda )_a}\ {\mathfrak{a}}\ne 0, {\mathfrak{b}}\ne 0 \ \eta _b\in (id_p)^*(T_p\Sigma )_b\Big \} .\\&h^{ab}_{ 6_{|_p}}=\Big \{ (\xi _a, \eta _b, {\mathfrak{a}}_6\beta _6):\xi _a\in (id_p)^*{(T_p\Sigma )_a} , \eta _b={\mathfrak{a}}_6\zeta _b+{\mathfrak{b}}_6\mu _b, \\&\quad \zeta _b\in (id_p)^*{(T_p\Sigma )_b}, \mu _b\in (id_p)^*{(T_p\Lambda )_b},\ {\mathfrak{a}}_6\ne 0\ {\text{and}} \ {\mathfrak{b}}_6\ne 0\Big \}.\\&h^{ab}_{ 7_{|_p}}=\Big \{ (\xi _a, \eta _b, 0):\xi _a={\mathfrak{a}}_7\zeta _a+{\mathfrak{b}}_7\mu _a, \zeta _a\in (id_p)^*{(T_p\Sigma )_a},\\&\quad \ \mu _a\in (id_p)^*{(T_p\Lambda )_a},\ {\mathfrak{a}}_7\ne 0, {\mathfrak{b}}_7\ne 0 \ {\text{and}} \ \eta _b\in (id_p)^*{(T_p\Lambda )_b}\Big \}.\\&h^{ab}_{ 8_{|_p}}=\Big \{ (\xi _a, \eta _b, 0):\xi _a\in (id_p)^*{(T_p\Lambda )_a}, \eta _b={\mathfrak{a}}_8\zeta _b\\&\quad +{\mathfrak{b}}_8\mu _b, \ \zeta _b\in (id_p)^*{(T_p\Sigma )_b} \ {\text{and}} \ \mu _b\in (id_p)^*{(T_p\Lambda )_b},\ {\mathfrak{a}}_8\ne 0, {\mathfrak{b}}_8\ne 0\Big \}.\\&h^{ab}_{ 9_{|_p}}=\Big \{(\xi _a, \eta _b, {\mathfrak{a}}_9{\mathfrak{a}}^\prime _9\beta _9):\xi _a={\mathfrak{a}}_9\zeta _a+{\mathfrak{b}}_9\mu _a, \ {\mathfrak{a}}_9\ne 0, {\mathfrak{b}}_9\ne 0 ,\ \zeta _a\in (id_p)^*{(T_p\Sigma )_a},\\&\quad \mu _a\in (id_p)^*{(T_p\Lambda )_a},\eta _b={\mathfrak{a}}^\prime _9\zeta ^\prime _b+{\mathfrak{b}}^\prime _9\mu ^\prime _b, {\mathfrak{a}}^\prime _9\ne 0, {\mathfrak{b}}^\prime _9\ne 0, \ \zeta ^\prime _ b\in (id_p)^*{(T_p\Sigma )_b} \ {\text{and}} \ \mu ^\prime _ b\in (id_p)^*{(T_p\Lambda )_b}\Big \}. \end{aligned}$$

1.6 A.6 An Expansion of $h^{ab}$

$$\begin{aligned}&{{\mathfrak{h}} }^{ab}_{1|_p}=\Big \{ (\xi _a, \eta _b, \beta _1):\xi _a\in {(id_p)}^*{(T_p\Sigma )_a} \\&\quad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Sigma )_b} \Big \}.\\&{{\mathfrak{h}} }^{ab}_{2|_p}=\Big \{(\xi _a, \eta _b, \gamma _2):\xi _a\in (id_p)^*(T_p\Lambda )_a \\&\quad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Lambda )_b} \Big \}.\\&{{\mathfrak{h}} }^{ab}_{3|_p}=\Big \{ (\xi _a, \eta _b, 0):\xi _a\in (id_p)^*{(T_p\Sigma )_a}\\&\quad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Lambda )_b} \Big \}. \\&{{\mathfrak{h}} }^{ab}_{4|_p}=\Big \{(\xi _a, \eta _b, 0):\xi _a \in (id_p)^*{(T_p\Lambda )_a} \\&\quad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Sigma )_b} \Big \}.\\&{{\mathfrak{h}} }^{ab}_{5|_p}=\Big \{ (\xi _a, \eta _b,{\mathfrak{a}}_5\beta _5):\xi _a={\mathfrak{a}}_5\zeta _a+{\mathfrak{b}}\mu _a, \zeta _a\in (id_p)^*{(T_p\Sigma )_a} \\&\quad {\text{and}}\ \mu _a\in (id_p)^*{(T_p\Lambda )_a}\ {\mathfrak{a}}\ne 0, {\mathfrak{b}}\ne 0 \\&\quad \eta _b\in (id_p)^*(T_p\Sigma )_b\Big \} .\\&{{\mathfrak{h}} }^{ab}_{6|_p}=\Big \{ (\xi _a, \eta _b, {\mathfrak{a}}_6\beta _6):\xi _a\in (id_p)^*{(T_p\Sigma )_a} , \eta _b={\mathfrak{a}}_6\zeta _b+{\mathfrak{b}}_6\mu _b, \ \zeta _b\in (id_p)^*{(T_p\Sigma )_b},\\&\quad \mu _b\in (id_p)^*{(T_p\Lambda )_b},{\mathfrak{a}}_6\ne 0\ {\text{and}} \ {\mathfrak{b}}_6\ne 0\Big \}.\\&{{\mathfrak{h}} }^{ab}_{7|_p}=\Big \{ (\xi _a, \eta _b, {\mathfrak{a}}_7\gamma _7):\xi _a={\mathfrak{a}}_7\zeta _a+{\mathfrak{b}}_7\mu _a, \zeta _a\in (id_p)^*{(T_p\Sigma )_a}, \\&\quad \mu _a\in (id_p)^*{(T_p\Lambda )_a},\ {\mathfrak{a}}_7\ne 0, {\mathfrak{b}}_7\ne 0 \ {\text{and}} \ \eta _b\in (id_p)^*{(T_p\Lambda )_b}\Big \}.\\&{{\mathfrak{h}} }^{ab}_{8|_p}=\Big \{ (\xi _a, \eta _b, {\mathfrak{a}}_8\gamma _8):\xi _a\in (id_p)^*{(T_p\Lambda )_a}, \eta _b={\mathfrak{a}}_8\zeta _b+{\mathfrak{b}}_8\mu _b, \ \zeta _b\in (id_p)^*{(T_p\Sigma )_b} \ {\text{and}} \\&\quad \mu _b\in (id_p)^*{(T_p\Lambda )_b}, \ {\mathfrak{a}}_8\ne 0, {\mathfrak{b}}_8\ne 0\Big \}.\\&{{\mathfrak{h}} }^{ab}_{9|_p}=\Big \{(\xi _a, \eta _b, ({\mathfrak{a}}_9{\mathfrak{a}}^\prime _9\beta _9+{\mathfrak{b}}_9{\mathfrak{b}}^\prime _9\gamma _9)):\xi _a={\mathfrak{a}}_9\zeta _a\\&\quad +{\mathfrak{b}}_9\mu _a, \ {\mathfrak{a}}_9\ne 0, {\mathfrak{b}}_9\ne 0 ,\ \zeta _a\in (id_p)^*{(T_p\Sigma )_a},\\&\quad \mu _a\in (id_p)^*{(T_p\Lambda )_a},\eta _b={\mathfrak{a}}^\prime _9\zeta ^\prime _b+{\mathfrak{b}}^\prime _9\mu ^\prime _b, {\mathfrak{a}}^\prime _9\ne 0, {\mathfrak{b}}^\prime _9\ne 0, \ \zeta ^\prime _ b\in (id_p)^*{(T_p\Sigma )_b} \ {\text{and}} \ \mu ^\prime _ b\in (id_p)^*{(T_p\Lambda )_b}\Big \}. \end{aligned}$$

1.7 A.7 A Decomposition of $R^N_{ab}$

By equation (4.3.1) we can decompose $R^N_{ab}$ into the following sets.

$$\begin{aligned}&{R}^{N1}_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, 0):\xi _a\in {(id_p)}^*{(T_p\Sigma )_a} \\&\qquad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Sigma )_b} \Big \}.\\&{R}^{N2}_{ab_{|_p}}=\Big \{(\xi _a, \eta _b, 4\pi G \rho \delta _2):\xi _a\in (id_p)^*(T_p\Lambda )_a \ \ {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Lambda )_b} \Big \}. \\&{R}^{N3}_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, 0):\xi _a\in (id_p)^*{(T_p\Sigma )_a}\\&\qquad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Lambda )_b} \Big \}. \\&{R}^{N4}_{ab_{|_p}}=\Big \{(\xi _a, \eta _b, 0):\xi _a \in (id_p)^*{(T_p\Lambda )_a} \\&\qquad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Sigma )_b} \Big \}.\\&{R}^{N5}_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, 0):\xi _a={\mathfrak{a}}_5\zeta _a+{\mathfrak{b}}\mu _a, \zeta _a\in (id_p)^*{(T_p\Sigma )_a}, \\&\qquad \mu _a\in (id_p)^*{(T_p\Lambda )_a}\ {\mathfrak{a}}\ne 0 \ {\text{and}}\ {\mathfrak{b}}\ne 0 \ \eta _b\in (id_p)^*(T_p\Sigma )_b\Big \} .\\&{R}^{N6}_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, 0):\xi _a\in (id_p)^*{(T_p\Sigma )_a} , \eta _b\\&\qquad ={\mathfrak{a}}_6\zeta _b+{\mathfrak{b}}_6\mu _b, \ \zeta _b\in (id_p)^*{(T_p\Sigma )_b}, \mu _b\in (id_p)^*{(T_p\Lambda )_b},\ {\mathfrak{a}}_6\ne 0\ {\text{and}} \ {\mathfrak{b}}_6\ne 0\Big \}.\\&{R}^{N7}_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, 4\pi G \rho {\mathfrak{b}}_7 \delta _7):\xi _a={\mathfrak{a}}_7\zeta _a+{\mathfrak{b}}_7\mu _a, \zeta _a\in (id_p)^*{(T_p\Sigma )_a}, \\&\qquad \mu _a\in (id_p)^*{(T_p\Lambda )_a},\ {\mathfrak{a}}_7\ne 0, {\mathfrak{b}}_7\ne 0 \ {\text{and}} \ \eta _b\in (id_p)^*{(T_p\Lambda )_b}\Big \}.\\&{R}^{N8}_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, 4\pi G \rho {\mathfrak{a}}_8 \delta _8):\xi _a\in (id_p)^*{(T_p\Lambda )_a}, \eta _b\\&\quad ={\mathfrak{a}}_8\zeta _b+{\mathfrak{b}}_8\mu _b, \ \zeta _b\in (id_p)^*{(T_p\Sigma )_b} \ {\text{and}} \ \mu _b\in (id_p)^*{(T_p\Lambda )_b},\ {\mathfrak{a}}_8\ne 0, {\mathfrak{b}}_8\ne 0\Big \}.\\&{R}^{N9}_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, 4\pi G \rho {\mathfrak{b}}_9{\mathfrak{b}}^\prime _9 \delta _9):\xi _a={\mathfrak{a}}_9\zeta _a+{\mathfrak{b}}_9\mu _a, \ {\mathfrak{a}}_9\ne 0, {\mathfrak{b}}_9\ne 0 ,\ \\&\qquad \zeta _a\in (id_p)^*{(T_p\Sigma )_a},\\&\quad \mu _a\in (id_p)^*{(T_p\Lambda )_a},\eta _b={\mathfrak{a}}^\prime _9\zeta ^\prime _b+{\mathfrak{b}}^\prime _9\mu ^\prime _b, {\mathfrak{a}}^\prime _9\ne 0, {\mathfrak{b}}^\prime _9\ne 0, \ \zeta ^\prime _ b\in (id_p)^*{(T_p\Sigma )_b} \ {\text{and}} \ \mu ^\prime _ b\in (id_p)^*{(T_p\Lambda )_b}\Big \}. \end{aligned}$$

1.8 A.8 A Decomposition of $R_{ab}$

From [48], it follows that³³

$$\begin{aligned}&{R}^{1}_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, d_1):\xi _a\in {(id_p)}^*{(T_p\Sigma )_a} \\&\qquad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Sigma )_b} \Big \}.\\&{R}^{2}_{ab_{|_p}}=\Big \{(\xi _a, \eta _b, d_2+4\pi G \rho \delta _2):\xi _a\in (id_p)^*(T_p\Lambda )_a \\&\qquad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Lambda )_b} \Big \}. \\&{R}^{3}_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, d_3):\xi _a\in (id_p)^*{(T_p\Sigma )_a}\\&\qquad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Lambda )_b} \Big \}. \\&{R}^{4}_{ab_{|_p}}=\Big \{(\xi _a, \eta _b, d_4):\xi _a \in (id_p)^*{(T_p\Lambda )_a} \\&\qquad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Sigma )_b} \Big \}.\\&{R}^{5}_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, d_5):\xi _a={\mathfrak{a}}_5\zeta _a+{\mathfrak{b}}\mu _a, \zeta _a\in (id_p)^*{(T_p\Sigma )_a}, \\&\qquad \mu _a\in (id_p)^*{(T_p\Lambda )_a}\ {\mathfrak{a}}\ne 0 \ {\text{and}}\ {\mathfrak{b}}\ne 0 \ \eta _b\in (id_p)^*(T_p\Sigma )_b\Big \} .\\&{R}^{6}_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, d_6):\xi _a\in (id_p)^*{(T_p\Sigma )_a} , \eta _b={\mathfrak{a}}_6\zeta _b+{\mathfrak{b}}_6\mu _b, \ \zeta _b\in (id_p)^*{(T_p\Sigma )_b},\\&\qquad \mu _b\in (id_p)^*{(T_p\Lambda )_b},\ {\mathfrak{a}}_6\ne 0\ {\text{and}} \ {\mathfrak{b}}_6\ne 0\Big \}.\\&{R}^{7}_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, d_7+4\pi G \rho {\mathfrak{b}}_7 \delta _8):\xi _a={\mathfrak{a}}_7\zeta _a+{\mathfrak{b}}_7\mu _a, \zeta _a\in (id_p)^*{(T_p\Sigma )_a}, \\&\qquad \mu _a\in (id_p)^*{(T_p\Lambda )_a},\ {\mathfrak{a}}_7\ne 0, {\mathfrak{b}}_7\ne 0 \ \\&\qquad {\text{and}} \ \eta _b\in (id_p)^*{(T_p\Lambda )_b}\Big \}.\\&{R}^{8}_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, d_8+4\pi G \rho {\mathfrak{a}}_8 \delta _8):\xi _a\in (id_p)^*{(T_p\Lambda )_a}, \eta _b\\&\quad ={\mathfrak{a}}_8\zeta _b+{\mathfrak{b}}_8\mu _b, \ \zeta _b\in (id_p)^*{(T_p\Sigma )_b} \ {\text{and}} \ \mu _b\in (id_p)^*{(T_p\Lambda )_b},\ {\mathfrak{a}}_8\ne 0, {\mathfrak{b}}_8\ne 0\Big \}.\\&{R}^{9}_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, d_9+4\pi G \rho {\mathfrak{b}}_9{\mathfrak{b}}^\prime _9 \delta _9):\xi _a\\&\quad ={\mathfrak{a}}_9\zeta _a+{\mathfrak{b}}_9\mu _a, \ {\mathfrak{a}}_9\ne 0, {\mathfrak{b}}_9\ne 0 ,\ \zeta _a\in (id_p)^*{(T_p\Sigma )_a},\ \quad \mu _a\in (id_p)^*{(T_p\Lambda )_a},\eta _b={\mathfrak{a}}^\prime _9\zeta ^\prime _b+{\mathfrak{b}}^\prime _9\mu ^\prime _b, {\mathfrak{a}}^\prime _9\ne 0, {\mathfrak{b}}^\prime _9\ne 0, \ \zeta ^\prime _ b\in (id_p)^*{(T_p\Sigma )_b} \ {\text{and}} \ \mu ^\prime _ b\in (id_p)^*{(T_p\Lambda )_b}\Big \}. \end{aligned}$$

1.9 A.9 An Expansion of $R^N_{ab}$

$$\begin{aligned}&{{\mathfrak{R}}}^1_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, d_1):\xi _a\in {(id_p)}^*{(T_p\Sigma )_a} \\&\quad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Sigma )_b} \Big \}.\\&{{\mathfrak{R}}}^2_{ab_{|_p}}=\Big \{(\xi _a, \eta _b, d_2+4\pi G \rho \delta _2):\xi _a\in (id_p)^*(T_p\Lambda )_a \\&\quad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Lambda )_b} \Big \}. \\&{{\mathfrak{R}}}^3_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, d_3):\xi _a\in (id_p)^*{(T_p\Sigma )_a}\\&\quad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Lambda )_b} \Big \}. \\&{{\mathfrak{R}}}^4_{ab_{|_p}}=\Big \{(\xi _a, \eta _b, d_4):\xi _a \in (id_p)^*{(T_p\Lambda )_a} \\&\quad {\text{and}} \ \ \eta _b\in (id_p)^*{(T_p\Sigma )_b} \Big \}.\\&{{\mathfrak{R}}}^5_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, d_5):\xi _a={\mathfrak{a}}_5\zeta _a+{\mathfrak{b}}\\&\quad \mu _a,\zeta _a\in (id_p)^*{(T_p\Sigma )_a}, \ \mu _a\in (id_p)^*{(T_p\Lambda )_a}\ {\mathfrak{a}}\ne 0 \ {\text{and}}\ {\mathfrak{b}}\ne 0 \ \eta _b\in (id_p)^*(T_p\Sigma )_b\Big \} .\\&{{\mathfrak{R}}}^6_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, d_6):\xi _a\in (id_p)^*{(T_p\Sigma )_a} , \eta _b={\mathfrak{a}}_6\zeta _b+{\mathfrak{b}}_6\mu _b, \ \zeta _b\in (id_p)^*{(T_p\Sigma )_b}, \\&\quad \mu _b\in (id_p)^*{(T_p\Lambda )_b},\ {\mathfrak{a}}_6\ne 0\ {\text{and}} \ {\mathfrak{b}}_6\ne 0\Big \}.\\&{{\mathfrak{R}}}^7_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, d_7+4\pi G \rho {\mathfrak{b}}_7 \delta _7):\xi _a={\mathfrak{a}}_7\zeta _a\\&\quad +{\mathfrak{b}}_7\mu _a, \zeta _a\in (id_p)^*{(T_p\Sigma )_a}, \ \mu _a\in (id_p)^*{(T_p\Lambda )_a},\ {\mathfrak{a}}_7\ne 0, {\mathfrak{b}}_7\ne 0 \ {\text{and}} \ \eta _b\in (id_p)^*{(T_p\Lambda )_b}\Big \}.\\&{{\mathfrak{R}}}^8_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, d_8+4\pi G \rho {\mathfrak{a}}_8 \delta _8):\xi _a\in (id_p)^*{(T_p\Lambda )_a}, \eta _b={\mathfrak{a}}_8\zeta _b\\&\quad +{\mathfrak{b}}_8\mu _b, \ \zeta _b\in (id_p)^*{(T_p\Sigma )_b} \ {\text{and}} \ \mu _b\in (id_p)^*{(T_p\Lambda )_b},\ {\mathfrak{a}}_8\ne 0, {\mathfrak{b}}_8\ne 0\Big \}.\\&{{\mathfrak{R}}}^9_{ab_{|_p}}=\Big \{ (\xi _a, \eta _b, d_9+4\pi G \rho {\mathfrak{b}}_9{\mathfrak{b}}^\prime _9 \delta _9):\xi _a={\mathfrak{a}}_9\zeta _a+{\mathfrak{b}}_9\mu _a, \ {\mathfrak{a}}_9\ne 0, {\mathfrak{b}}_9\ne 0 ,\\&\quad \zeta _a\in (id_p)^*{(T_p\Sigma )_a}, \quad \mu _a\in (id_p)^*{(T_p\Lambda )_a},\eta _b={\mathfrak{a}}^\prime _9\zeta ^\prime _b+{\mathfrak{b}}^\prime _9\mu ^\prime _b, {\mathfrak{a}}^\prime _9\ne 0, {\mathfrak{b}}^\prime _9\ne 0, \ \zeta ^\prime _ b\in (id_p)^*{(T_p\Sigma )_b} \ {\text{and}} \ \mu ^\prime _ b\in (id_p)^*{(T_p\Lambda )_b}\Big \}. \end{aligned}$$

1.10 A.10 A Decomposition of $g_{ab}$

$$ \begin{aligned} & \sigma_1=g^1_{ab_{|p}}\xi^a\eta^b=-(N^2-\gamma_{ij}N^iN^j)d_at^1d_bt^1\xi^a\eta^b+\gamma^1_{ij}d_ax^id_bx^j\xi^a\eta^b+\gamma^1_{ij}N^jd_ax^id_bt\xi^a\eta^b\ +\gamma^1_{ij}N^id_atd_bx^j\xi^a\eta^b=\iota_1.\\ & \alpha_2=g^2_{ab_{|p}}\xi^a\eta^b=-(N^2-\gamma_{ij}N^iN^j)d_at^2d_bt^2_{|p}\xi^a\eta^b+\gamma^2_{ij}d_ax^id_bx^j_{|p}\xi^a\eta^b+\gamma^2_{ij}N^jd_ax^id_bt_{|p}\xi^a\eta^b+\gamma^2_{ij}N^id_atd_bx^j_{|p}\xi^a\eta^b=-(N^2-\gamma_{ij}N^iN^j)_{|p}\delta_2.\\ & \sigma_3=g^3_{ab_{|p}}\xi^a\eta^b=-(N^2-\gamma_{ij}N^iN^j)d_at^3d_bt^3\xi^a\eta^b+\gamma^3_{ij}d_ax^id_bx^j\xi^a\eta^b+\gamma^3_{ij}N^jd_ax^id_bt\xi^a\eta^b+\gamma^3_{ij}N^id_atd_bx^j\xi^a\eta^b=\epsilon_3.\\ & \sigma_4=g^4_{ab_{|p}}\xi^a\eta^b=-(N^2-\gamma_{ij}N^iN^j)d_at^4d_bt^4\xi^a\eta^b+\gamma^4_{ij}d_ax^id_bx^j\xi^a\eta^b+\gamma^4_{ij}N^jd_ax^id_bt\xi^a\eta^b+\gamma^4_{ij}N^id_atd_bx^j\xi^a\eta^b=\rho_4.\\ & \sigma_5=g^5_{ab_{|p}}\xi^a\eta^b=-(N^2-\gamma_{ij}N^iN^j)d_at^5d_bt^5_{|p}({\mathfrak{a}}_5\zeta^a+{\mathfrak{b}}_5\mu^a)\eta^b+\gamma^5_{ij}d_ax^id_bx^j_{|p}({\mathfrak{a}}_5\zeta^a+{\mathfrak{b}}_5\mu^a)\eta^b+\gamma^5_{ij}N^jd_ax^id_bt_{|p}({\mathfrak{a}}_5\zeta^a+{\mathfrak{b}}_5\mu^a)\eta^b+\gamma^5_{ij}N^id_atd_bx^j_{|p}({\mathfrak{a}}_5\zeta^a+{\mathfrak{b}}_5\mu^a)\eta^b={\mathfrak{a}}_5\gamma^5_{ij}d_ax^id_bx^j_{|p}\zeta^a\eta^b+{\mathfrak{b}}_5\gamma^5_{ij}N^id_atd_bx^j_{|p}\mu^a\eta^b={\mathfrak{a}}_5\iota_5+{\mathfrak{b}}_5\rho_5.\\ & \sigma_6=g^6_{ab_{|p}}\xi^a\eta^b=-(N^2-\gamma_{ij}N^iN^j)d_at^6d_bt^6_{|p}\xi^a({\mathfrak{a}}_6\zeta^b+{\mathfrak{b}}_6\mu^b)+\gamma^6_{ij}d_ax^id_bx^j_{|p}\xi^a({\mathfrak{a}}_6\zeta^b+{\mathfrak{b}}_6\mu^b)+\gamma^6_{ij}N^jd_ax^id_bt_{|p}({\mathfrak{a}}_6\zeta^a+{\mathfrak{b}}_6\mu^a)\eta^b+\gamma^6_{ij}N^id_atd_bx^j_{|p}\xi^a({\mathfrak{a}}_6\zeta^b+{\mathfrak{b}}_6\mu^b)={\mathfrak{a}}_6\gamma^6_{ij}d_ax^id_bx^j_{|p}\xi^a\zeta^b+{\mathfrak{b}}_6\gamma^6_{ij}N^jd_ax^id_bt_{|p}\xi^a\mu^b={\mathfrak{a}}_6\iota_6+{\mathfrak{b}}_6\epsilon_6.\\ & \sigma_7=g^7_{ab_{|p}}\xi^a\eta^b=-(N^2-\gamma_{ij}N^iN^j)d_at^7d_bt^7_{|p}({\mathfrak{a}}_7\zeta^a+{\mathfrak{b}}_7\mu^a)\eta^b+\gamma^7_{ij}d_ax^id_bx^j_{|p}({\mathfrak{a}}_5\zeta^a+{\mathfrak{b}}_7\mu^a)\eta^b+\gamma^7_{ij}N^jd_ax^id_bt_{|p}({\mathfrak{a}}_7\zeta^a+{\mathfrak{b}}_7\mu^a)\eta^b+\gamma^7_{ij}N^id_atd_bx^j_{|p}({\mathfrak{a}}_7\zeta^a+{\mathfrak{b}}_7\mu^a)\eta^b=-{\mathfrak{b}}_7(N^2-\gamma_{ij}N^iN^j)d_at^7d_bt^7_{|p}\mu^a\eta^b+{\mathfrak{a}}_7\gamma^7_{ij}N^jd_ax^id_bt_{|p}\zeta^a\eta^b={\mathfrak{a}}_7\epsilon_7-{\mathfrak{b}}_7(N^2-\gamma_{ij}N^iN^j)_{|p}\delta_7.\\ & \sigma_8=g^8_{ab_{|p}}\xi^a\eta^b=-(N^2-\gamma_{ij}N^iN^j)d_at^8d_bt^8_{|p}\xi^a({\mathfrak{a}}_8\zeta^b+{\mathfrak{b}}_8\mu^b)+\gamma^8_{ij}d_ax^id_bx^j_{|p}\xi^a({\mathfrak{a}}_8\zeta^b+{\mathfrak{b}}_8\mu^b)+\gamma^8_{ij}N^jd_ax^id_bt_{|p}({\mathfrak{a}}_8\zeta^a+{\mathfrak{b}}_8\mu^a)\eta^b+\gamma^8_{ij}N^id_atd_bx^j_{|p}\xi^a({\mathfrak{a}}_8\zeta^b+{\mathfrak{b}}_8\mu^b)=-(N^2-\gamma_{ij}N^iN^j)d_at^8d_bt^8_{|p}\xi^a\mu^b+{\mathfrak{b}}_8\gamma^8_{ij}N^jd_ax^id_bt_{|p}\xi^a\mu^b=-{\mathfrak{b}}_8(N^2-\gamma_{ij}N^iN^j)\delta_8+{\mathfrak{b}}_8\epsilon_8.\\ & \sigma_9=g^9_{ab_{|p}}\xi^a\eta^b=-(N^2-\gamma_{ij}N^iN^j)d_at^9d_bt^9_{|p}({\mathfrak{a}}_9\zeta^a+{\mathfrak{b}}_9\mu^a)({\mathfrak{a^\prime_9}}\zeta^{\prime b}+{\mathfrak{b^\prime_9}}\mu^{\prime b})+\gamma^9_{ij}d_ax^id_bx^j_{|p}({\mathfrak{a}}_9\zeta^a+{\mathfrak{b}}_9\mu^a)({\mathfrak{a^\prime_9}}\zeta^{\prime b}+{\mathfrak{b^\prime_9}}\mu^{\prime b})+\gamma^9_{ij}N^jd_ax^id_bt_{|p}({\mathfrak{a}}_9\zeta^a+{\mathfrak{b}}_9\mu^a)({\mathfrak{a^\prime_9}}\zeta^{\prime b}+{\mathfrak{b^\prime_9}}\mu^{\prime b})+\gamma^9_{ij}N^id_atd_bx^j_{|p}({\mathfrak{a}}_9\zeta^a+{\mathfrak{b}}_9\mu^a)({\mathfrak{a^\prime_9}}\zeta^{\prime b}+{\mathfrak{b^\prime_9}}\mu^{\prime b})=-(N^2-\gamma_{ij}N^iN^j){\mathfrak{b}}_9{\mathfrak{b^\prime_9}}+{\mathfrak{a}}_9{\mathfrak{a^\prime_9}}\iota_9+{\mathfrak{a}}_9{\mathfrak{b^\prime_9}}\epsilon_9+{\mathfrak{b}}_9{\mathfrak{a^\prime_9}}\rho_9. \end{aligned} $$