Rotation, Embedding and Topology for the Szekeres Geometry

The Lemaître-Tolman (LT) spacetime [22, 27] represents a spherically symmetric cloud of dust particles that is inhomogeneous in the radial direction; both the density and the rate of expansion or contraction can vary with radius. Its metric is

$\displaystyle{\rm d}s^{2}=-{\rm d}t^{2}+\frac{R^{\prime 2}\,{\rm d}r^{2}}{1+f}+R^{2}\big{(}{\rm d}\theta^{2}+\sin^{2}\theta{\rm d}\phi^{2}\big{)}~{},$

(1)

where $R=R(t,r)$ is the areal radius, and $f=f(r)$ is a geometry-energy factor. From the Einstein field equations (EFEs), the evolution equation and the density are

	$\displaystyle\dot{R}^{2}$	$\displaystyle=\frac{2M}{R}+f+\frac{\Lambda R^{2}}{3}~{},$		(2)
	$\displaystyle\kappa\rho$	$\displaystyle=\frac{2M^{\prime}}{R^{2}R^{\prime}}~{},$		(3)

where $M=M(r)$ is the total gravitational mass interior to each constant $r$ shell, and $\Lambda$ is the cosmological constant. When $\Lambda=0$, the evolution equation (2) has parametric solutions, for example when $f<0$,

$\displaystyle R=\frac{M}{(-f)}\big{(}1-\cos\eta\big{)}~{},~{}~{}~{}~{}~{}~{}t=a+\frac{M}{(-f)^{3/2}}\big{(}\eta-\sin\eta\big{)}~{}.$

(4)

and $a=a(r)$ is the local ‘bang time’; the time on each constant $r$ worldline at which $R(t,r)=0$ is $t=a$. The factors

$\displaystyle L=\frac{M}{(-f)}~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}T=\frac{M}{(-f)^{3/2}}$

(5)

can be thought of as a scale length and a scale time for the worldline at $r$; $2L$ is the maximum areal radius reached, and $2\pi T$ is the duration from bang to crunch. The derivative $R^{\prime}$ that appears in the metric can be expressed parametrically as

	$\displaystyle R^{\prime}$	$\displaystyle=\frac{M^{\prime}}{(-f)}(1-\phi_{1})(1-\cos\eta)-\frac{f^{\prime}M}{f^{2}}\left(\frac{3}{2}\phi_{1}-1\right)(1-\cos\eta)$
		$\displaystyle~{}~{}~{}~{}-(-f)^{1/2}a^{\prime}\phi_{2}(1-\cos\eta)~{},$		(6)
	where	$\displaystyle\phi_{1}=\frac{\sin\eta(\eta-\sin\eta)}{(1-\cos\eta)^{2}}~{},~{}~{}~{}~{}~{}~{}\phi_{2}=\frac{\sin\eta}{(1-\cos\eta)^{2}}~{}.$		(7)

The Ellis [9] metrics are equivalents of the LT case, having planar and hyperboloidal (pseudo-spherical) symmetry.

The Szekeres (S) spacetimes [25, 26] can be thought of as distortions of the LT and Ellis ones. In addition to the free functions $f(r)$, $M(r)$ and $a(r)$ they have 3 more functions $S(r)$, $P(r)$ and $Q(r)$ that specify the deviation from symmetry — spherical, planar, or hyperboloidal. The metric is

$\displaystyle{\rm d}s^{2}=-{\rm d}t^{2}+\frac{\left(R^{\prime}-\dfrac{RE^{\prime}}{E}\right)^{2}{\rm d}r^{2}}{\epsilon+f}+\frac{R^{2}}{E^{2}}\big{(}{\rm d}p^{2}+{\rm d}q^{2}\big{)}~{},$

(8)

where $\epsilon=+1,0,-1$, and

$\displaystyle E=\frac{S}{2}\left(\frac{(p-P)^{2}}{S^{2}}+\frac{(q-Q)^{2}}{S^{2}}+\epsilon\right)~{}.$

(9)

Since $S=0$ is not possible for a regular metric, we assume $S>0$. In fact the last term in (8) is $R^{2}$ times the 2-metric for a unit sphere, pseudo-sphere, or plane, depending on whether $\epsilon$ is $+1$, $-1$, or $0$. Thus the 3-spaces are foliated by a collection of symmetric 2-spaces, but they are not arranged symmetrically, as we shall see. By the EFEs, $R(t,r)$ obeys exactly the same evolution equation (2), while the density $\rho$ is more complicated,

$\displaystyle\kappa\rho$

$\displaystyle=\frac{2\left(M^{\prime}-\dfrac{3ME^{\prime}}{E}\right)}{R^{2}\left(R^{\prime}-\dfrac{RE^{\prime}}{E}\right)}~{}.$

(10)

For more information about the Szekeres metric, see for example [20, 16, 17, 14, 28, 6, 7].

The standard stereographic mapping, for the $\epsilon=+1$ case,

$\displaystyle p=P+S\cot\left(\frac{\theta}{2}\right)\cos\phi~{},~{}~{}~{}~{}~{}~{}q=Q+S\cot\left(\frac{\theta}{2}\right)\sin\phi~{},$

(11)

transforms the metric into a more complicated, non-diagonal form,

	$\displaystyle{\rm d}s^{2}$	$\displaystyle=\Bigg{[}\frac{1}{\epsilon+f}\left(R^{\prime}+\frac{R}{S}\big{\{}S^{\prime}\cos\theta+\sin\theta\big{(}P^{\prime}\cos\phi+Q^{\prime}\sin\phi\big{)}\big{\}}\right)^{2}$
		$\displaystyle~{}~{}~{}~{}+\frac{R^{2}}{S^{2}}\big{\{}S^{\prime}\sin\theta+(1-\cos\theta)\big{(}P^{\prime}\cos\phi+Q^{\prime}\sin\phi\big{)}\big{\}}$
		$\displaystyle~{}~{}~{}~{}+\frac{R^{2}}{S^{2}}\big{\{}(1-\cos\theta)^{2}\big{(}P^{\prime}\sin\phi-Q^{\prime}\cos\phi\big{)}\big{\}}\Bigg{]}{\rm d}r^{2}$
		$\displaystyle~{}~{}~{}~{}-\frac{R^{2}}{S}\big{\{}S^{\prime}\sin\theta+(1-\cos\theta)\big{(}P^{\prime}\cos\phi+Q^{\prime}\sin\phi\big{)}\big{\}}\,{\rm d}r\,{\rm d}\theta$
		$\displaystyle~{}~{}~{}~{}-\frac{R^{2}\sin\theta}{S}\big{\{}(1-\cos\theta)^{2}\big{(}P^{\prime}\sin\phi-Q^{\prime}\cos\phi\big{)}\big{\}}\,{\rm d}r\,{\rm d}\phi$
		$\displaystyle~{}~{}~{}~{}+R^{2}({\rm d}\theta^{2}+\sin^{2}\theta\,{\rm d}\phi^{2})~{}.$		(12)

For each $(t,r)$ 2-sphere, it is sometimes convenient to define a local cartesian frame by

$\displaystyle\begin{aligned} x&=R\sin\theta\cos\phi\\ y&=R\sin\theta\sin\phi\\ z&=R\cos\theta~{}.\end{aligned}$

(13)

The factor ${\mathcal{D}}=E^{\prime}/E$ that appears in both the metric (8) and the density (10) controls the deviation from spherical, hyperboloidal or planar symmetry, and for $\epsilon\neq 0$ it behaves like a dipole. The dipole has maximum value and orientation

$\displaystyle\begin{aligned} {\mathcal{D}}_{m}=\left.\frac{E^{\prime}}{E}\right|_{m}&=\frac{J}{S}~{},~{}~{}~{}~{}~{}~{}p_{m}-P=\frac{P^{\prime}S\big{\{}\epsilon S^{\prime}-J\big{\}}}{H^{2}}~{},~{}~{}~{}~{}~{}~{}q_{m}-Q=\frac{Q^{\prime}S\big{\{}\epsilon S^{\prime}-J\big{\}}}{H^{2}}~{},\\ \mbox{where}&~{}~{}~{}~{}~{}~{}~{}~{}J=\sqrt{\epsilon^{2}S^{\prime 2}+\epsilon\big{(}P^{\prime 2}+Q^{\prime 2}\big{)}}\;~{},~{}~{}~{}~{}~{}~{}~{}H=\sqrt{P^{\prime 2}+Q^{\prime 2}}\;~{}.\end{aligned}$

(14)

The locus $E^{\prime}=0$ lies on the $(p,q)$ circle

$\displaystyle\left(\frac{(p-P)S^{\prime}}{S}+P^{\prime}\right)+\left(\frac{(q-Q)S^{\prime}}{S}+Q^{\prime}\right)=P^{\prime 2}+Q^{\prime 2}+\epsilon S^{\prime 2}~{}.$

(15)

For the quasi-spherical case, $\epsilon=+1$, the dipole function can be written as

$\displaystyle{\mathcal{D}}=\frac{E^{\prime}}{E}=-\frac{S^{\prime}\cos\theta+\sin\theta\big{(}P^{\prime}\cos\phi+Q^{\prime}\sin\phi\big{)}}{S}~{},$

(16)

and it is evident that $E^{\prime}/E$ ranges between opposite extremes, passing through zero on an ‘equatorial’ circle. Note that here $E^{\prime}$ still represents the $r$ derivative at constant $p$ & $q$. The angular position of the dipole maximum is found from

$\displaystyle\sin\theta_{m}$

$\displaystyle=\frac{-H}{J}~{},~{}~{}~{}~{}~{}~{}\cos\theta_{m}=\frac{-S^{\prime}}{J}~{},~{}~{}~{}~{}~{}~{}\cos\phi_{m}=\frac{P^{\prime}}{H}~{},~{}~{}~{}~{}~{}~{}\sin\phi_{m}=\frac{Q^{\prime}}{H}~{},$

(17)

or, expressed in local Cartesian coordinates, the position of the maximum on the $(x,y,z)$ unit sphere is

$\displaystyle x_{m}$

$\displaystyle=\sin\theta_{m}\cos\phi_{m}=\frac{-P^{\prime}}{J}~{},~{}~{}~{}~{}~{}~{}y_{m}=\sin\theta_{m}\sin\phi_{m}=\frac{-Q^{\prime}}{J}~{},~{}~{}~{}~{}~{}~{}z_{m}=\cos\theta_{m}=\frac{-S^{\prime}}{J}~{},$

(18)

while the $E^{\prime}=0$ locus is in the plane

$\displaystyle P^{\prime}x+Q^{\prime}y+S^{\prime}z=0~{}.$

(19)

The dipole has two obvious effects — in the $g_{rr}$ component of (8) it creates a non-uniform separation between adjacent 2-spheres of constant $r$, and in (10), it creates a variation of the density distribution around each sphere. These effects are in addition to the $r$-dependent inhomogeneity of the underlying LT model.

Relative to the angular form of the Szekeres metric (12), there is another more subtle effect of $S$, $P$ & $Q$. The angular coordinates $\theta$ & $\phi$ of (12) do not in fact represent a constant orientation, and their cardinal directions do not parallel transport from one shell to the next. Adjacent 2-spheres have a relative rotation: the sphere at $r+\delta r$ is rotated

	by	$\displaystyle\frac{Q^{\prime}}{S}\delta r~{}~{}~{}~{}\mbox{about the $x$ axis}$		(20a)
	and by	$\displaystyle\frac{-P^{\prime}}{S}\delta r~{}~{}~{}~{}\mbox{about the $y$ axis}~{},$		(20b)

relative to the one at $r$ [6]. Four justifications for this were given in [7]. Firstly there is an argument about nearest points on the two spheres, explained in PG, section V.B and fig 4. Secondly, it was shown that geodesics look much straighter once these rotations are incorporated into plots. Thirdly was the calculation in PG appendix C, perhaps not entirely rigorous, that added displacements and rotations to the LT metric, ending up with the S metric. Fourthly, an embedding of any given constant $t$ 3-space of the angular S metric for $\epsilon=+1$, into a 4-d space that is flat or has constant curvature, turned out to require the above-stated rotations.

Let $\mathbb{E}^{4}$ be a 4-d Euclidean space with Cartesian coordinates, $X,Y,Z,W$, and let the LT 3-spaces have positive curvature, $-1\leq f\leq 0$. At a fixed time $t$, $R$ becomes a function of $r$ only. We define a 3-surface $\Sigma$ by

	$\displaystyle X$	$\displaystyle=R(r)\sin\theta\cos\phi~{},$		(21a)
	$\displaystyle Y$	$\displaystyle=R(r)\sin\theta\sin\phi~{},$		(21b)
	$\displaystyle Z$	$\displaystyle=R(r)\cos\theta~{},$		(21c)
	$\displaystyle W$	$\displaystyle=\int_{0}^{r}R^{\prime}(r)\alpha(r)\,{\rm d}r~{},$		(21d)

so then

	$\displaystyle{\rm d}X$	$\displaystyle=R^{\prime}\sin\theta\cos\phi\,{\rm d}r+R\cos\theta\cos\phi\,{\rm d}\theta-R\sin\theta\sin\phi\,{\rm d}\phi~{},$		(22a)
	$\displaystyle{\rm d}Y$	$\displaystyle=R^{\prime}\sin\theta\sin\phi\,{\rm d}r+R\cos\theta\sin\phi\,{\rm d}\theta+R\sin\theta\cos\phi\,{\rm d}\phi~{},$		(22b)
	$\displaystyle{\rm d}Z$	$\displaystyle=R^{\prime}\cos\theta\,{\rm d}r-R\sin\theta\,{\rm d}\theta~{},$		(22c)
	$\displaystyle{\rm d}W$	$\displaystyle=R^{\prime}\alpha(r)~{}.$		(22d)

Here $R(r)$ is the LT areal radius $R(t,r)$ at a particular time, and $\alpha$ is

$\displaystyle\alpha$

$\displaystyle=\pm\sqrt{\frac{-f(r)}{1+f(r)}}\;~{}.$

(23)

Now in [13] the function $f$ was interpreted as $f=-\cos^{2}\psi$ where $\psi$ is the angle of the tangent cone to the embedded surface at $r$, and in this notation, then, $\alpha=\cot\psi$:

	$\displaystyle\tan\psi=\frac{{\rm d}R}{{\rm d}W}=\frac{R^{\prime}}{\left|R^{\prime}\sqrt{\dfrac{-f}{1+f}}\;\right|}=\pm\sqrt{\dfrac{1+f}{-f}}\;=\frac{1}{\alpha}$		(24)
	$\displaystyle\to~{}~{}~{}~{}~{}~{}~{}~{}\cos\psi=\sqrt{-f}\;~{},~{}~{}~{}~{}\sin\psi=\sqrt{1+f}\;~{},~{}~{}~{}~{}\cot\psi=\alpha~{}.$		(25)

Since $f\leq 0$, closed models are quite likely, in which case there will be at least one point $r_{m}$ that is a maximum (or minimum) in $R$ where $R^{\prime}=0$, $f=-1$, but $R^{\prime}/\sqrt{1+f}\;$ is finite [18, 13]. As a spatial extremum is approached and traversed, $R^{\prime}$, $\psi$ and $\alpha$ change sign, $R^{\prime}$ & $\psi$ passing through zero and $\alpha$ diverging. This ensures ${\rm d}W/{\rm d}r$ retains a constant sign¹¹1This is not essential, and other choices could lead to different valid embeddings..

Using this embedding, (21) and (22) show that the metric of the 3-surface becomes

$\displaystyle{\rm d}X^{2}$

$\displaystyle+{\rm d}Y^{2}+{\rm d}Z^{2}+{\rm d}W^{2}=\frac{R^{\prime 2}\,{\rm d}r^{2}}{1+f}+R^{2}\big{(}{\rm d}\theta^{2}+\sin^{2}\theta\,{\rm d}\phi^{2}\big{)}$

(26)

which is the spatial part of the LT metric, ${\rm d}t=0$.

In the case that

	$\displaystyle R=K\sin r~{},~{}~{}~{}~{}R^{\prime}=K\cos r$	$\displaystyle~{},~{}~{}~{}~{}f=-\sin^{2}r~{},~{}~{}~{}~{}K~{}\mbox{constant,}$		(27)
we find
	$\displaystyle\frac{R^{\prime 2}\,{\rm d}r^{2}}{1+f}+R^{2}\big{(}{\rm d}\theta^{2}+\sin^{2}\theta\,{\rm d}\phi^{2}\big{)}$	$\displaystyle=K^{2}\Big{(}{\rm d}r^{2}+\sin^{2}r\big{(}{\rm d}\theta^{2}+\sin^{2}\theta\,{\rm d}\phi^{2}\big{)}\Big{)}$		(28)

and $\Sigma$ becomes the 3-sphere.

For quasi-spherical S models with $f<0$, their constant $t$ 3-spaces can be embedded in a 4-d flat space, but for general $f$, one must embed in a 4-space of constant curvature [7]. We here consider the former case. Let $\mathbb{E}^{4}$ be a 4-d Euclidean space with Cartesian coordinates, $X,Y,Z,W$. In this 4-space, a 3-surface is constructed from a sequence of 2-spheres, by expanding, shifting and rotating a unit sphere, as a function of parameter $r$. Suitable choices of the expansion, shift and rotation functions ensure the intrinsic metric of the 3-surface is identical with the positively curved, quasi-spherical Szekeres 3-spaces of constant $t$. While an embedding is primarily a visualisation tool, in this case it also provides a clear confirmation of Buckley and Schlegel’s rotations, given in equation (20).

It is also convenient to define local Cartesian coordinates, $x,y,z,w$ near each constant $r$ shell. In these coordinates, the 2-spherical shell lies in the $w=0$ 3-space, according to (13), so the accumulated displacements and rotations are ignored, and the focus is on the local rate of displacement and rotation.

Buckley & Schlegel’s embedding equation is

$\displaystyle V(r,\theta,\phi)=R(r)A^{T}(r)U(\theta,\phi)+\Delta(r)~{},$

(29)

where $V$ is a point in $\mathbb{E}^{4}$ and $U$ is a unit sphere,

$\displaystyle V=\begin{pmatrix}X\\ Y\\ Z\\ W\end{pmatrix}~{},~{}~{}~{}~{}~{}~{}U=\begin{pmatrix}\sin\theta\cos\phi\\ \sin\theta\sin\phi\\ \cos\theta\\ 0\end{pmatrix}~{},$

(30)

while the rotation matrix $A$, and the displacement vector $\Delta$, are functions of $r$ that satisfy the following differential equations (DEs)

	$\displaystyle A^{\prime}(r)$	$\displaystyle=\Omega^{\prime}(r)A(r)=\begin{pmatrix}0&0&\dfrac{P^{\prime}}{S}&\dfrac{P^{\prime}}{S}\alpha\\[5.69054pt] 0&0&\dfrac{Q^{\prime}}{S}&\dfrac{Q^{\prime}}{S}\alpha\\[5.69054pt] -\dfrac{P^{\prime}}{S}&-\dfrac{Q^{\prime}}{S}&0&\dfrac{S^{\prime}}{S}\alpha\\[5.69054pt] -\dfrac{P^{\prime}}{S}\alpha&-\dfrac{Q^{\prime}}{S}\alpha&-\dfrac{S^{\prime}}{S}\alpha&0\end{pmatrix}A(r)~{},$		(31)
	$\displaystyle\Delta^{\prime}(r)$	$\displaystyle=A^{T}(r)D^{\prime}(r)=A^{T}(r)\begin{pmatrix}R\dfrac{P^{\prime}}{S}\\[5.69054pt] R\dfrac{Q^{\prime}}{S}\\[5.69054pt] R\dfrac{S^{\prime}}{S}\\[5.69054pt] R^{\prime}\alpha\end{pmatrix}~{},$		(32)

where $\alpha$ is given by (23), and here too one may specify that it have the same sign as $R^{\prime}$. The intrinsic 3-metric of the embedded 3-surface is derived from the differential of $V$, using (29)-(32),

	$\displaystyle{\rm d}s^{2}$	$\displaystyle=g({\rm d}V,{\rm d}V)={\rm d}V^{T}\,{\rm d}V$		(33)
	$\displaystyle{\rm d}V$	$\displaystyle=(R^{\prime}\,{\rm d}r)A^{T}U+R((A^{T})^{\prime}\,{\rm d}r)U+RA^{T}(U_{\theta}\,{\rm d}\theta+U_{\phi}\,{\rm d}\phi)+\Delta^{\prime}\,{\rm d}r$		(34)
		$\displaystyle=R^{\prime}A^{T}U\,{\rm d}r+R(A^{T}(\Omega^{\prime})^{T})U\,{\rm d}r+RA^{T}(U_{\theta}\,{\rm d}\theta+U_{\phi}\,{\rm d}\phi)+A^{T}D^{\prime}\,{\rm d}r$		(35)
		$\displaystyle=A^{T}\big{[}(R^{\prime}U+R(\Omega^{\prime})^{T}U+D^{\prime}){\rm d}r+RU_{\theta}\,{\rm d}\theta+RU_{\phi}\,{\rm d}\phi\big{]}$		(36)
	$\displaystyle{\rm d}s^{2}$	$\displaystyle=\big{[}(R^{\prime}U^{T}+RU^{T}\Omega^{\prime}+(D^{\prime})^{T}){\rm d}r+RU_{\theta}^{T}\,{\rm d}\theta+RU_{\phi}^{T}\,{\rm d}\phi\big{]}$
		$\displaystyle~{}~{}~{}~{}~{}~{}\big{[}(R^{\prime}U+R(\Omega^{\prime})^{T}U+D^{\prime}){\rm d}r+RU_{\theta}\,{\rm d}\theta+RU_{\phi}\,{\rm d}\phi\big{]}$		(37)
		$\displaystyle={\rm d}r^{2}\big{\{}R^{\prime 2}U^{T}U+RR^{\prime}\big{[}U^{T}(\Omega^{\prime})^{T}U+U^{T}\Omega^{\prime}U\big{]}+R^{\prime}\big{[}U^{T}D^{\prime}+(D^{\prime})^{T}U\big{]}$
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+R^{2}U^{T}\Omega^{\prime}(\Omega^{\prime})^{T}U+R\big{[}U^{T}\Omega^{\prime}D^{\prime}+(D^{\prime})^{T}(\Omega^{\prime})^{T}U\big{]}+(D^{\prime})^{T}D^{\prime}\big{\}}$
		$\displaystyle~{}~{}~{}+{\rm d}r{\rm d}\theta\big{\{}RR^{\prime}\big{[}U^{T}U_{\theta}+U_{\theta}^{T}U\big{]}+R^{2}\big{[}U^{T}\Omega^{\prime}U_{\theta}+U_{\theta}^{T}(\Omega^{\prime})^{T}U\big{]}$
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+R\big{[}(D^{\prime})^{T}U_{\theta}+U_{\theta}^{T}D^{\prime}\big{]}\big{\}}$
		$\displaystyle~{}~{}~{}+{\rm d}r{\rm d}\phi\big{\{}RR^{\prime}\big{[}U^{T}U_{\phi}+U_{\phi}^{T}U\big{]}+R^{2}\big{[}U^{T}\Omega^{\prime}U_{\phi}+U_{\phi}^{T}(\Omega^{\prime})^{T}U\big{]}$
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+R\big{[}(D^{\prime})^{T}U_{\phi}+U_{\phi}^{T}D^{\prime}\big{]}\big{\}}$
		$\displaystyle~{}~{}~{}+{\rm d}\theta^{2}\big{\{}R^{2}U_{\theta}^{T}U_{\theta}\big{\}}+{\rm d}\theta{\rm d}\phi\big{\{}R^{2}\big{[}U_{\theta}^{T}U_{\phi}+U_{\phi}^{T}U_{\theta}\big{]}\Big{\}}+{\rm d}\phi^{2}\big{\{}R^{2}U_{\phi}^{T}U_{\phi}\big{\}}$		(38)

where

$\displaystyle U_{\theta}=\begin{pmatrix}\cos\theta\cos\phi\\ \cos\theta\sin\phi\\ -\sin\theta\\ 0\end{pmatrix}~{},~{}~{}~{}~{}~{}~{}~{}~{}U_{\phi}=\begin{pmatrix}-\sin\theta\sin\phi\\ \sin\theta\cos\phi\\ 0\\ 0\end{pmatrix}$

(39)

Evaluating the various matrix products, we find

	$\displaystyle 0=U^{T}\Omega^{\prime}U=U^{T}(\Omega^{\prime})^{T}U=U^{T}U_{\theta}=U_{\theta}^{T}U=U^{T}U_{\phi}=U_{\phi}^{T}U=U_{\theta}^{T}U_{\phi}=U_{\phi}^{T}U_{\theta}$		(40a)
	$\displaystyle 1=U^{T}U=U_{\theta}^{T}U_{\theta}$		(40b)
	$\displaystyle U_{\phi}^{T}U_{\phi}=\sin^{2}\theta$		(40c)
	$\displaystyle(D^{\prime})^{T}U=U^{T}D^{\prime}=\frac{R}{S}\big{\{}\sin\theta\big{(}P^{\prime}\cos\phi+Q^{\prime}\sin\phi\big{)}+S^{\prime}\cos\theta\big{\}}$		(40d)
	$\displaystyle(D^{\prime})^{T}(\Omega^{\prime})^{T}U=U^{T}\Omega^{\prime}D^{\prime}=\frac{R}{S^{2}}\big{\{}S^{\prime}\sin\theta\big{(}P^{\prime}\cos\phi+Q^{\prime}\sin\phi\big{)}-\big{(}P^{\prime 2}+Q^{\prime 2}\big{)}\cos\theta\big{\}}$
	$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{R^{\prime}\alpha^{2}}{S}\big{\{}\sin\theta\big{(}P^{\prime}\cos\phi+Q^{\prime}\sin\phi\big{)}+\cos\theta S^{\prime}\big{\}}$		(40e)
	$\displaystyle(D^{\prime})^{T}D^{\prime}=\frac{R^{2}}{S^{2}}\big{\{}P^{\prime 2}+Q^{\prime 2}+S^{\prime 2}\big{\}}+R^{\prime 2}\alpha^{2}$		(40f)
	$\displaystyle U^{T}\Omega^{\prime}(\Omega^{\prime})^{T}U=\frac{\alpha^{2}}{S^{2}}\Big{(}\big{\{}S^{\prime}\cos\theta+\sin\theta\big{(}P^{\prime}\cos\phi+Q^{\prime}\sin\phi\big{)}\big{\}}^{2}$
	$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{\sin^{2}\theta}{S^{2}}\big{\{}P^{\prime}\cos\phi+Q^{\prime}\sin\phi\big{\}}^{2}+\cos^{2}\theta\big{\{}P^{\prime 2}+Q^{\prime 2}\big{\}}\Big{)}$		(40g)
	$\displaystyle U^{T}\Omega^{\prime}U_{\theta}=U_{\theta}^{T}(\Omega^{\prime})^{T}U=-\frac{\big{\{}P^{\prime}\cos\phi+Q^{\prime}\sin\phi\big{\}}}{S}$		(40h)
	$\displaystyle U^{T}\Omega^{\prime}U_{\phi}=U_{\phi}^{T}(\Omega^{\prime})^{T}U=\frac{\cos\theta\sin\theta\big{\{}P^{\prime}\sin\phi-Q^{\prime}\cos\phi\big{\}}}{S}$		(40i)
	$\displaystyle(D^{\prime})^{T}U_{\theta}=U_{\theta}^{T}D^{\prime}=\frac{R}{S}\big{\{}\cos\theta\big{(}P^{\prime}\cos\phi+Q^{\prime}\sin\phi\big{)}-S^{\prime}\sin\theta\big{\}}$		(40j)
	$\displaystyle(D^{\prime})^{T}U_{\phi}=U_{\phi}^{T}D^{\prime}=-\frac{R\sin\theta}{S}\big{\{}P^{\prime}\sin\phi-Q^{\prime}\cos\phi\big{\}}$		(40k)

and we recover the metric (12) of the angular form of the quasi-spherical Szekeres metric.

Although there is no physical significance to the embedding of a given spacetime into one of higher dimension, it can be very helpful for visualising the spacetime geometry, and that is what we explore here.

We note that the constant $w$ projection of $D^{\prime}$, (32), is anti-parallel to the $(x,y,z)$ direction of the dipole maximum, (18). For the local rate-of-rotation matrix $\Omega^{\prime}$, the fixed point locus is a 2-plane,

$\displaystyle\Omega^{\prime}\>V=0~{}~{}~{}~{}~{}~{}\to~{}~{}~{}~{}~{}~{}V_{fp}=\begin{pmatrix}S^{\prime}\lambda\\ S^{\prime}\mu\\[2.84526pt] -(P^{\prime}\lambda+Q^{\prime}\mu)\\[5.69054pt] \dfrac{(P^{\prime}\lambda+Q^{\prime}\mu)}{\alpha}\end{pmatrix}~{},~{}~{}~{}~{}\lambda,\mu~{}\mbox{independent parameters},$

(41)

which makes $\Omega^{\prime}$ the derivative of a simple rotation (not a double or isoclinic rotation). The constant $w$ projections of $D^{\prime}$ and $V_{fp}$ are orthogonal. This last fact ensures the maximum tilt-displacements occur along the dipole axis (where the max & min are located) — see fig 3.3 — and thus enables the following argument.

In the local $(x,y,z,w)$ frame, the constant $r$ 2-sphere lies in the $x$-$y$-$z$ 3-space, and the direction $o=(0,0,0,1)$ is orthogonal to that space. In going from the 2-sphere at $r$ to the 2-sphere at $r+\delta r$, the displacement of the sphere centres is $D^{\prime}\,\delta r$; the rate of perpendicular displacement is $R^{\prime}\alpha$, and the rates of sideways displacement are $RP^{\prime}/S$, $RQ^{\prime}/S$, & $RS^{\prime}/S$, towards the $+$ve $x$, $y$, & $z$ directions²²2We here assume that $P^{\prime}$, $Q^{\prime}$, & $S^{\prime}$ are positive. Where the opposite sign occurs, the relevant direction is changed in the obvious way.. The slant angle, between $o$ and the line of centres, is

$\displaystyle\cos\gamma=o\cdot\frac{D^{\prime}}{|D^{\prime}|}=\frac{R^{\prime}\alpha}{\sqrt{R^{2}{\mathcal{D}}_{m}^{2}+R^{\prime 2}\alpha^{2}}\;}$

(42)

which we could re-write as ‘component’ angles³³3These ‘components’ are the projections of $\gamma$ onto the $x$-$w$, $y$-$w$, and $z$-$w$ planes, such that $\tan^{2}\gamma=\tan^{2}\gamma_{x}+\tan^{2}\gamma_{y}+\tan^{2}\gamma_{z}$.

$\displaystyle\tan\gamma_{x}=\frac{RP^{\prime}}{R^{\prime}\alpha S}~{},~{}~{}~{}~{}~{}~{}\tan\gamma_{y}=\frac{RQ^{\prime}}{R^{\prime}\alpha S}~{},~{}~{}~{}~{}~{}~{}\tan\gamma_{z}=\frac{RS^{\prime}}{R^{\prime}\alpha S}~{}.$

(43)

Since the total rotation used in (29) is $A^{T}$, we see from (31) that the local rotation-rate matrix to examine is $(\Omega^{\prime})^{T}$. The components of the local rate-of-shell-rotation are thus

$\displaystyle\omega_{xz}^{\prime}\,\delta r=\frac{-P^{\prime}}{S}\,\delta r~{},~{}~{}~{}~{}~{}~{}\omega_{yz}^{\prime}\,\delta r=\frac{-Q^{\prime}}{S}\,\delta r~{},~{}~{}~{}~{}~{}~{}\omega_{xy}^{\prime}\,\delta r=0~{},$

(44)

with senses as noted above, and the components of rate-of-tilt are

$\displaystyle\zeta_{xw}^{\prime}\,\delta r=\frac{-P^{\prime}\alpha}{S}\,\delta r~{},~{}~{}~{}~{}~{}~{}\zeta_{yw}^{\prime}\,\delta r=\frac{-Q^{\prime}\alpha}{S}\,\delta r~{},~{}~{}~{}~{}~{}~{}\zeta_{zw}^{\prime}\,\delta r=\frac{-S^{\prime}\alpha}{S}\,\delta r~{}.$

(45)

Therefore the rates of tilt are $\alpha/R$ times the rates of sideways displacement, and a 3-plane parallel to $w=0$ is tilted down on the $-x$, $-y$, & $-z$ sides by $\zeta_{xw}$, $\zeta_{yw}$, & $\zeta_{zw}$, respectively.

Both these effects, the centre displacement and the tilt, decrease the shell separation where ${\mathcal{D}}$ has the same sign as $R^{\prime}$ (the ’closing edge’) and increase it where ${\mathcal{D}}$ has the opposite sign (the ’opening edge’). The no-shell-crossing conditions ensure the separation stays positive all round each sphere. Fig 3.3 illustrates the arrangement for the case when $R^{\prime}>0$. Although finite displacements are shown, one should think of $\delta r$ as infinitesimal, so that only first order effects are relevant. The four displacements shown are:

	the $w$-separation of shell centres	$\displaystyle=\delta w_{0}=R^{\prime}\alpha\,\delta r~{},$		(46a)
	the increase in shell radius	$\displaystyle=\delta d_{0}=R^{\prime}\alpha\tan\psi\,\delta r=R^{\prime}\,\delta r~{},$		(46b)
	the tilt down displacement where ${\mathcal{D}}$ is max	$\displaystyle=\delta w_{1}=R\zeta^{\prime}\,\delta r=R\alpha{\mathcal{D}}_{m}\,\delta r~{},$		(46c)
	the dipole displacement of the shell centre	$\displaystyle=\delta d_{1}=R{\mathcal{D}}_{m}\,\delta r~{}.$		(46d)

Interestingly, $\delta d_{0}$ & $\delta w_{0}$ make the same angle as $\delta d_{1}$ & $\delta w_{1}$. This angle coincidence is possibly the reason why the calculation in appendix C of PG actually works — because the displaced & tilted shell ar $r+\delta r$ more or less lies on the LT tangent cone to the 3-surface at $r$, to first order.

Fig 1.   Sketch of the displacement and tilt effects in the embedding, with one dimension suppressed. Because the sketch shows finite displacements, instead of infinitesimal ones, the distances shown are not exact. The green shows the embedding of the underlying LT model — the shells at $r$ (lower) and $r+\delta r$ (upper) and the local tangent cone. The blue vectors show the $w$ displacement between the two shells and the radial expansion of the second shell (assumed positive here). The cyan shows the displaced and tilted shell at $r+\delta r$ of the Szekeres model. The red vectors show the sideways displacement of the shell centre, and the down or up displacement due to the tilt. The dipole maximum is located at the left side, and the minimum at the right. The $E^{\prime}=0$ locus, the ‘widest’ part of the 2-sphere at $r+\delta r$ is shifted by $\delta d_{1}$; this is consistent — a tilted slice through the LT cone has a displaced ‘greatest width’. The tilt axis is the magenta dashed line, and it is perpendicular to the red displacement $\delta d_{1}$. The $\theta$-$\phi$ rotation is not shown.

Looking at the next shell pair, at $r+\delta r$ and $r+2\delta r$, the local LT tangent cone is now tilted by $\zeta$, and the $r+2\delta r$ shell is tilted further. The slant angle $\gamma$ is also tilted by $\zeta$.

A reflection in the bottom plane gives an idea of the $R^{\prime}<0$ case. Where $R^{\prime}<0$, $\alpha$ also flips sign and ${\mathcal{D}}_{m}$ goes to $-{\mathcal{D}}_{m}$ in (46).

Using the expressions in (46), we examine the limiting values of these embedding quantities near origins and spatial extrema. We assume there are no shell crossings. If there are, then $R^{\prime}$ & $\alpha$ do not always flip signs together. We also assume a well behaved $r$ coordinate, with $R$ finite & non-zero everywhere except at an origin, $r=r_{o}$, and $R^{\prime}$ finite & non-zero everywhere except at a spatial extremum, $r=r_{e}$. We further assume ‘generic’ arbitrary functions, so that, for example, $R^{\prime}$ & ${\mathcal{D}}$ go linearly through zero at an extremum⁴⁴4One may intentionally choose functions that give other behaviour at specific locations, such as $R^{\prime}$ or ${\mathcal{D}}$ going quadratically to zero and not changing sign. . The results are gathered in table 3.4.
flips sign with flips sign behaviour at behaviour ${\mathcal{D}}_{m}\to-{\mathcal{D}}_{m}$ with $R^{\prime}$ $R^{\prime}\to 0\leftarrow 1/\alpha$ at $R\to 0$ $R$ $R~{}\to~{}$const $\to~{}0$ as $(r-r_{o})$ $R^{\prime}$ $R^{\prime}~{}\to~{}0$ as $(r-r_{e})$ $\to~{}$const $\neq 0$ $\alpha$ $\alpha~{}\to~{}\infty$ as $(r-r_{e})^{-1}$ $\to~{}0$ as $(r-r_{o})$ ${\mathcal{D}}_{m}$ ${\mathcal{D}}_{m}~{}\to~{}0$ as $(r-r_{e})$ $\to~{}$const (or $0$) $\delta w_{0}$ No No $R^{\prime}\alpha~{}\to~{}$const $\to~{}0$ as $(r-r_{o})$ $\delta d_{0}$ No Yes $R^{\prime}~{}\to~{}0$ as $(r-r_{e})$ $\to~{}$const $\neq 0$ $\delta w_{1}$ Yes Yes $R\alpha{\mathcal{D}}_{m}~{}\to~{}$const $\to~{}0$ as $(r-r_{o})^{2}$ $\delta d_{1}$ Yes No $R{\mathcal{D}}_{m}~{}\to~{}0$ as $(r-r_{e})$ $\to~{}0$ as $(r-r_{o})$ dipole slant $\delta d_{1}/\delta w_{0}$ Yes No $(R{\mathcal{D}}_{m})/(R^{\prime}\alpha)~{}\to~{}0$ as $(r-r_{e})$ $\to~{}$const tilt rate $\delta w_{1}/R$ Yes Yes $\alpha{\mathcal{D}}_{m}~{}\to~{}$const $\to~{}0$ as $(r-r_{o})$
Table 1.   The behaviour of embedding displacements near origins and spatial extrema. See (46) and the illustration in fig. 3.3.