The static observer congruence of the Schwarzschild exterior model: unit field, integral curves, acceleration and clock-rate ratios

Let $(M,g,N)$ be the Schwarzschild exterior datum of mass parameter $m_{\ast }$ with Schwarzschild radius $s>0$ and $f(r)=1-s\mathrm{/}r$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation), with metric-selected connection $\mathrm{\nabla }$ (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields). Define the smooth vector field

$n:=f^{-1\mathrm{/}2}{\mathrm{\partial }}_0=(1-\frac{s}{r}{)}^{-1\mathrm{/}2}{\mathrm{\partial }}_0$

on $M$. Then:

(a) Observer congruence. $n$ is a smooth observer congruence on $M$ in the sense of Observer congruences, their observer curves and adapted charts: $g(n,n)=-1$ and $g(n,N)<0$ at every point, so $n$ is a future-directed unit timelike field.

(b) Integral curves. For any $p=(c_0,r_0,{\omega }_0)\in M$, including sphere poles, the maximal integral curve of $n$ through $p$, with length-valued flow parameter $\lambda \in \mathbb{R}$, is $\lambda \mapsto (c_0+f(r_0{)}^{-1\mathrm{/}2}\lambda ,r_0,{\omega }_0)\mathrm{.}$ It is defined on all of $\mathbb{R}$ and is the static observer worldline at radius $r_0$. In a spherical-polar chart it has constant angular coordinates $({\vartheta }_0,{\phi }_0)$; in any angular chart the sphere point is constant, and the global coordinate $x^0$ increases monotonically.

(c) Proper time and clock rate. Along each static observer worldline the proper time $\tau$ is an affine function of the flow parameter $\lambda$, with $\frac{d\tau }{d\lambda }=\frac{1}{c}$, and

$\mathrm{\Delta }\tau =\frac{\sqrt{f(r_0)}}{c}\mathrm{\Delta }{x}^0=\sqrt{f(r_0)}\mathrm{\Delta }t,$

where $\mathrm{\Delta }{x}^0$ is the coordinate difference in metres between two events of the worldline and $\mathrm{\Delta }t=\mathrm{\Delta }{x}^0\mathrm{/}c$ is the same difference in seconds. In particular the proper-time increment is strictly positive for $\mathrm{\Delta }{x}^0>0$, and it is smaller than $\mathrm{\Delta }t$ for $\mathrm{\Delta }{x}^0>0$ at every finite radius because $0<f(r_0)<1$.

(d) Clock-rate ratio. For two static observers at radii $r_1,r_2\in (s,\mathrm{\infty })$ compared over the same coordinate interval $\mathrm{\Delta }{x}^0\mathrm{\ne }0$, the proper-time increments satisfy

$\frac{\mathrm{\Delta }{\tau }_1}{\mathrm{\Delta }{\tau }_2}=\sqrt{\frac{f(r_1)}{f(r_2)}}=\sqrt{\frac{1-s\mathrm{/}{r}_1}{1-s\mathrm{/}{r}_2}}\mathrm{.}$

(e) Acceleration. The acceleration field of the congruence is $a:={\mathrm{\nabla }}_{n}n=\frac{f^{\mathrm{\prime }}(r)}{2}{\mathrm{\partial }}_1=\frac{s}{2r^2}{\mathrm{\partial }}_r$; it is spatial, nowhere zero, and points in the direction of increasing $r$, so the static observers are not geodesics.

(f) Scope. These are identities of the geometric datum. The reading of $\tau$ as the time measured by an ideal clock and of the congruence as a family of static observers belongs to the framework's postulates and is not asserted here; no measurement and no comparison with any observed rate is made.

• Static does not mean geodesic. The acceleration $a=\frac{s}{2r^2}{\mathrm{\partial }}_1$ is nonzero at every finite radius: the static observers are held at fixed $r$ by an external constraint and are not free-fall worldlines. The magnitude $\mid a{\mid }_g=\frac{f^{\mathrm{\prime }}}{2\sqrt{f}}=\frac{s}{2r^2\sqrt{1-s\mathrm{/}r}}$ diverges as $r\downarrow s$; this is a statement about the congruence, not about the curvature.

• Relation to the redshift. The ratio of clause (d) is a ratio of proper times of static observers over the same coordinate interval; the related but distinct statement about the observer energies of a null geodesic between two static observers is proved in the physics theorem of this page, and it is the latter that involves the light-ray postulate.