Spherical Vacuum Model — Examples

Let $(M,g,N)$ be the Schwarzschild exterior datum with Schwarzschild radius $s>0$, $f(r)=1-s\mathrm{/}r$ and $g_{11}=1\mathrm{/}f$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation, The Schwarzschild exterior datum is a smooth time-oriented Lorentzian four-manifold). Let

$R(\rho ):=\rho (1+\frac{s}{4\rho }{)}^2,\rho >\frac{s}{4},$

be the isotropic-to-areal radius map of The isotropic coordinate form of the Schwarzschild exterior metric and the isothermal radius shift and let ${\rho }_0:(s,\mathrm{\infty })\to (s\mathrm{/}4,\mathrm{\infty })$ be its inverse, so that the isotropic coefficient of the spatial metric is the function $\rho \mapsto (1+s\mathrm{/}(4\rho ){)}^4$ and the areal coefficient is $r\mapsto (1-s\mathrm{/}r{)}^{-1}$. The two displayed functions are coefficients in different radial bases. The following scalar-function comparisons do not identify their tensor components after a change of basis; that transformation is included in clause 5.

1. Exact form and positivity. $R(\rho )=\rho +\frac{s}{2}+\frac{s^2}{16\rho }$ and $R^{\mathrm{\prime }}(\rho )=1-\frac{s^2}{16{\rho }^2}>0$ on $(s\mathrm{/}4,\mathrm{\infty })$; the inverse is ${\rho }_0(r)=\frac{(r-\frac{s}{2})+\sqrt{r(r-s)}}{2}$.

2. Conversion and areal expansion. With $\rho ={\rho }_0(r)$, ${\rho }_0(r)=r-\frac{s}{2}-\frac{s^2}{16r}+O\text{\hspace{0.17em}⁣}\left(\frac{s^3}{r^2}\right),\frac{1}{{\rho }_0(r)}=\frac{1}{r}+\frac{s}{2r^2}+O\text{\hspace{0.17em}⁣}\left(\frac{s^2}{r^3}\right)\mathrm{.}$ These are expansions as $r\mathrm{/}s\to \mathrm{\infty }$ with fixed $s>0$. The areal coefficient satisfies, exactly for $r>s$, $(1-s\mathrm{/}r{)}^{-1}=1+\frac{s}{r}+\frac{(s\mathrm{/}r{)}^2}{1-s\mathrm{/}r},$ so its expansion is $1+\frac{s}{r}+\frac{s^2}{r^2}+O\text{\hspace{0.17em}⁣}\left(\frac{s^3}{r^3}\right)$.

3. Isotropic coefficient after argument substitution. With $\rho ={\rho }_0(r)$ one has $\frac{s}{4{\rho }_0(r)}=\frac{s}{4r}+\frac{s^2}{8r^2}+O(s^3\mathrm{/}{r}^3)$ and $(1+\frac{s}{4{\rho }_0(r)}{)}^4=1+\frac{s}{r}+\frac{7}{8}\frac{s^2}{r^2}+O\text{\hspace{0.17em}⁣}\left(\frac{s^3}{r^3}\right)\mathrm{.}$ Hence these scalar functions agree through first order and differ at second order by $\frac{1}{8}\frac{s^2}{r^2}$.

4. Failure of the unconverted comparison. Evaluating the isotropic coefficient at $\rho =r$ instead of at $\rho ={\rho }_0(r)$ gives $(1+\frac{s}{4r}{)}^4=1+\frac{s}{r}+\frac{3}{8}\frac{s^2}{r^2}+O\text{\hspace{0.17em}⁣}\left(\frac{s^3}{r^3}\right),$ which differs from the areal coefficient of clause 2 by $\frac{5}{8}\frac{s^2}{r^2}$ at second order: the consistency of the leading Newtonian-type term is unaffected, but the comparison of coefficients in the unchanged chart is already wrong at relative order $(s\mathrm{/}r{)}^2$.

5. Radial tensor transformation. Put $a(r)=s\mathrm{/}(4{\rho }_0(r))$. Differentiating $R({\rho }_0(r))=r$ gives ${\rho }_0^{\mathrm{\prime }}(r)=(1-a(r{)}^2{)}^{-1}$. Consequently the radial coefficient in the areal basis is exactly $g_{rr}=(1+a(r){)}^4({\rho }_0^{\mathrm{\prime }}(r){)}^2=(1-s\mathrm{/}r{)}^{-1}\mathrm{.}$ Moreover $({\rho }_0^{\mathrm{\prime }}{)}^2=1+s^2\mathrm{/}(8r^2)+O(s^3\mathrm{/}{r}^3)$, so this coefficient is $1+s\mathrm{/}r+s^2\mathrm{/}{r}^2+O(s^3\mathrm{/}{r}^3)$: the apparent $1\mathrm{/}8$ discrepancy of clause 3 disappears when the basis is transformed.

• What the example checks. The computation is exact finite algebra with remainder bounds as $r\mathrm{/}s\to \mathrm{\infty }$: after argument substitution alone, the second-order isotropic coefficient is $1\mathrm{/}2+3\mathrm{/}8=7\mathrm{/}8$, where $1\mathrm{/}2$ comes from the coordinate shift in $1\mathrm{/}{\rho }_0$ and $3\mathrm{/}8$ from the fourth-power expansion. The areal coefficient has second-order coefficient $1$, so their difference is $1\mathrm{/}8$; the $O(s^2\mathrm{/}{r}^3)$ remainder of $1\mathrm{/}{\rho }_0$ contributes only at third order. The radial tensor coefficient additionally includes the squared Jacobian of clause 5 and agrees exactly with the areal coefficient. No numerical value of $s$, no observed system and no measurement occurs.

Let $(M,g,N)$ be the Schwarzschild exterior datum with Schwarzschild radius $s>0$, $f(r)=1-s\mathrm{/}r$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation) and static observer congruence $n=f^{-1\mathrm{/}2}{\mathrm{\partial }}_0$ (The static observer congruence of the Schwarzschild exterior model: unit field, integral curves, acceleration and clock-rate ratios clause (a), Observer congruences, their observer curves and adapted charts). By The static observer congruence of the Schwarzschild exterior model: unit field, integral curves, acceleration and clock-rate ratios clause (c), a static observer worldline at radius $r_0\in (s,\mathrm{\infty })$ has proper-time increment

$\mathrm{\Delta }\tau (r_0)=\sqrt{f(r_0)}\mathrm{\Delta }t=\sqrt{1-\frac{s}{r_0}}\mathrm{\Delta }t$

over a coordinate-time increment $\mathrm{\Delta }t=\mathrm{\Delta }{x}^0\mathrm{/}c>0$. Then:

1. Exact values at the two radii. For $r_1=2s$ and $r_2=4s$ one has $f(r_1)=\frac{1}{2}$, $f(r_2)=\frac{3}{4}$, so $\mathrm{\Delta }\tau (r_1)=\frac{\mathrm{\Delta }t}{\sqrt{2}},\mathrm{\Delta }\tau (r_2)=\frac{\sqrt{3}}{2}\mathrm{\Delta }t,\frac{\mathrm{\Delta }\tau (r_1)}{\mathrm{\Delta }\tau (r_2)}=\sqrt{\frac{2}{3}}=\frac{\sqrt{6}}{3}<1.$
2. Monotonicity in the radius. The function $r\mapsto \sqrt{f(r)}=\sqrt{1-s\mathrm{/}r}$ is smooth and strictly increasing on $(s,\mathrm{\infty })$ with derivative $\frac{d}{dr}\sqrt{f(r)}=\frac{s}{2r^2\sqrt{f(r)}}>0$; consequently a static observer at the larger radius accumulates strictly more proper time over the same coordinate interval: $\mathrm{\Delta }\tau (r_2)>\mathrm{\Delta }\tau (r_1)$ for $s<r_1<r_2$, and the ratio of clause 1 is the exact instance at $(r_1,r_2)=(2s,4s)$.
3. Large-radius limit. $\sqrt{f(r)}\to 1$ as $r\to +\mathrm{\infty }$; the ratios $\mathrm{\Delta }\tau (r_1)\mathrm{/}\mathrm{\Delta }\tau (r_2)$ are invariant under the common scaling of $\mathrm{\Delta }t$ and tend to $1$ as $\min (r_1,r_2)\to +\mathrm{\infty }$ with $r_2>r_1$.
4. The static observers are accelerated. At $r_1=2s$ the acceleration of the congruence is $a=\frac{s}{2r^2}{\mathrm{\partial }}_r$ (The static observer congruence of the Schwarzschild exterior model: unit field, integral curves, acceleration and clock-rate ratios clause (e)), whose coefficient at $r_1=2s$ is $\frac{s}{2(2s{)}^2}=\frac{1}{8s}>0$: the observer held at $r_1=2s$ has nonzero acceleration pointing in the direction of increasing $r$, so it is not freely falling.

• What the example computes. All four clauses are exact evaluations of the clock identity and the acceleration formula of the cited lemma at explicit radii; the coordinate-time increment $\mathrm{\Delta }t$ is an arbitrary positive coordinate difference, not a measurement, and no value of $c$, $G_N$, $m_{\ast }$ or $s$ in any unit system is asserted. In particular clause 1 does not report a clock comparison or a redshift observation; it is the conditional identity of the model evaluated at two radii.

Let $(M,g,N)$ be the Schwarzschild exterior datum with Schwarzschild radius $s>0$, $f(r)=1-s\mathrm{/}r$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation), with the circular-orbit data of Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2 and the radial potential $W_{\mathrm{\ell }}(r)=f(r)(c^2+{\mathrm{\ell }}^2\mathrm{/}{r}^2)$ of Conserved quantities and equatorial radial reduction of timelike geodesics of the Schwarzschild exterior model clause (c). Then the following exact evaluations hold.

In every $V_{\mathrm{\ell }}^{\mathrm{\prime }\mathrm{\prime }}$ evaluation below, differentiate with $\mathrm{\ell }$ fixed as a constant and substitute the circular-orbit value only after differentiation.

1. The innermost stable circular orbit at $r_0=3s$. With ${\mathrm{\ell }}^2={\displaystyle \frac{c^{2}s{r}_0^2}{2r_0-3s}}$ one has ${\mathrm{\ell }}^2=3c^2{s}^2$, $e^2=\frac{8}{9}{c}^2$ and $e=\frac{2\sqrt{2}}{3}c$, and the second derivative of $V_{\mathrm{\ell }}=\frac{1}{2}{W}_{\mathrm{\ell }}$ vanishes: $V_{\mathrm{\ell }}^{\mathrm{\prime }\mathrm{\prime }}(3s)={\displaystyle \frac{c^{2}s(3s-3s)}{(3s{)}^3(6s-3s)}}=0$. The circular angular velocity at this radius satisfies $(\frac{d\phi }{dt}{)}^2={\displaystyle \frac{G_N{m}_{\ast }}{(3s{)}^3}}={\displaystyle \frac{c^2}{54s^2}}$.
2. The photon sphere at $r_0=\frac{3s}{2}$. For $\mathrm{\ell }\mathrm{\ne }0$, with $e_0={\displaystyle \frac{\mathrm{\mid }\mathrm{\ell }\mathrm{\mid }\sqrt{f(r_0)}}{r_0}}={\displaystyle \frac{2\mathrm{\mid }\mathrm{\ell }\mathrm{\mid }}{3\sqrt{3}s}}$ one has $f\left(\frac{3s}{2}\right)=\frac{1}{3}$ and $(\frac{d\phi }{dt}{)}^2={\displaystyle \frac{c^{2}f(r_0)}{r_0^2}}={\displaystyle \frac{4c^2}{27s^2}}$; the null potential $W_{\mathrm{\ell }}^{\mathrm{n}}(r)=f(r){\mathrm{\ell }}^2\mathrm{/}{r}^2$ has $(W_{\mathrm{\ell }}^{\mathrm{n}}{)}^{\mathrm{\prime }}(\frac{3s}{2})=0$ and $(W_{\mathrm{\ell }}^{\mathrm{n}}{)}^{\mathrm{\prime }\mathrm{\prime }}(\frac{3s}{2})=-\frac{32{\mathrm{\ell }}^2}{81s^4}<0$.
3. A stable comparison radius. At $r_0=4s$ one has ${\mathrm{\ell }}^2={\displaystyle \frac{c^{2}s\cdot 16s^2}{8s-3s}}={\displaystyle \frac{16}{5}}{c}^2{s}^2$ and $V_{\mathrm{\ell }}^{\mathrm{\prime }\mathrm{\prime }}(4s)={\displaystyle \frac{c^{2}s(4s-3s)}{(4s{)}^3(8s-3s)}}={\displaystyle \frac{c^2}{320s^2}}>0$, a strict local minimum: the two threshold values $3s\mathrm{/}2$ (existence) and $3s$ (stability) separate the unstable circular orbits $\frac{3s}{2}<r_0<3s$ from the stable ones $r_0>3s$.
4. Blow-up at the existence threshold. As $r_0\downarrow \frac{3s}{2}$ the required angular momentum diverges, ${\mathrm{\ell }}^2={\displaystyle \frac{c^{2}s{r}_0^2}{2r_0-3s}}\to +\mathrm{\infty }$, so there is no finite angular momentum at the limiting radius; for $r_0\le \frac{3s}{2}$ no circular timelike geodesic exists (Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2 clause (b)).