Schwarzschild spacetime

In units $G=c=1$, the Schwarzschild exterior metric is

A spherical surface with areal radius $R>3m$, where $m=GM/c^2$, has apparent angular radius $\psi$ for a static exterior observer at $r_0>R$. Its grazing null geodesic has impact parameter $b=R/\sqrt{1-2m/R}$. Projecting the photon tangent into the observer's local rest frame gives $\sin \psi =b\sqrt{1-2m/r_0}/r_0$, yielding the displayed result. The condition $R>3m$ ensures the grazing ray escapes, since the surface is outside the photon sphere. For $R=4m$, the distant image radius is $\sqrt{2}$ times the flat-space prediction.

For an unbound equatorial Schwarzschild spacetime geodesic, let $E_{\infty }$ and $L_{\infty }$ be its conserved energy and angular momentum measured relative to infinity. The ratio $J=c^2{L}_{\infty }/E_{\infty }$ has dimensions of area divided by time. For a massive test particle, $E_{\infty }=m{c}^2{\gamma }_{\infty }$ and $L_{\infty }=m\ell$, so $J=\ell /{\gamma }_{\infty }=b{v}_{\infty }$, where $\ell$ is specific angular momentum and $b$ the impact parameter. For a photon, $E_{\infty }=c{p}_{\infty }$ and $L_{\infty }=b{p}_{\infty }$, giving $J=bc$ without assigning a finite Lorentz factor or a proper time to the photon. This ratio removes the arbitrary normalization of a null affine parameter.

A stationary time-coordinate change $t^{\prime }=t+F(r)$ leaves both the areal radius and the time-translation Killing vector field unchanged. The conserved Killing energy and angular momentum of a null geodesic therefore yield the same radial equation in either chart, ${\dot{r}}^2+h^2(1-2\mu /r)/r^2=c^2{k}^2$. The effective potential and radial turning points agree, even when the coordinate time velocities and horizon regularity differ.

An observer at fixed exterior Schwarzschild spatial coordinates. Its proper time is $dT=\sqrt{f}dt$, with $f=1-2GM/(r{c}^2)>0$. It is accelerated, not freely falling; no such static timelike worldline exists on or inside the event horizon.

For radial motion in Schwarzschild spacetime, $a^r=d{u}^r/d\tau +GM/r^2$. At a prospective radial turning point $u^r=0$, bounded outward proper acceleration $g$ permits $d{u}^r/d\tau \le g\sqrt{1-2GM/(c^{2}r)}-GM/r^2$. The equality $g=g_{\mathrm{hover}}$ defines a unique critical radius outside the horizon. A ship already falling inward cannot turn at or below it. The Newtonian proposed radius $\sqrt{GM/g}$ is strictly smaller and is therefore too late for reversal. A radius beyond the critical one is necessary, but braking must also remove the pre-existing inward velocity.

For a radial timelike geodesic of specific Killing energy $k=fdt/d\tau$, the static observer measures $v_{\mathrm{loc}}=(dr/dt)/f=\dot{r}/k$. Its relative Lorentz factor is $k/\sqrt{f}$, giving $v_{\mathrm{loc}}^2=c^2(1-f/k^2)$.

For asymptotic speed $0\le u<c$, $k=(1-u^2/c^2{)}^{-1/2}$ and the local outward speed obeys $v_{\mathrm{loc}}^2=u^2+(2GM/r)(1-u^2/c^2)$. It tends to $c$ as the horizon is approached from outside, though a static observer cannot actually remain on the horizon.

The outgoing radial timelike geodesic with zero speed at infinity has $k=1$ and $dr/d\tau =\sqrt{2GM/r}$. Thus $r(\tau )=[r_0^{3/2}+(3/2)\sqrt{2GM}\tau {]}^{2/3}$ and the local static-observer speed is $\sqrt{2GM/r}$.

For the Schwarzschild metric with $\mu =GM/c^2$, let $k=(1-2\mu /r)\dot{t}$ and $h=r^2\dot{\varphi }$ on an equatorial timelike geodesic. The normalization of the four-velocity gives ${\dot{r}}^2+V(r)=c^2(k^2-1)$, where $V(r)=(1-2\mu /r)h^2/r^2-2\mu c^2/r$. A circular geodesic obeys $V^{\prime }(R)=0$, giving the displayed constants and requiring $R>3\mu$. Its coordinate angular velocity is ${\Omega }^2=GM/R^3$, and its proper clock rate is $d\tau /dt=\sqrt{1-3\mu /R}$. Differentiating $V$ twice at the circular solution gives stability for $R>6\mu$.

The circular timelike orbit with conserved energy equal to rest energy at infinity. Substituting $E/(m{c}^2)=(1-2\mu /r)/\sqrt{1-3\mu /r}$ and solving $E=m{c}^2$ gives $r=4\mu$, with $\mu =GM/c^2$. It is unstable. Strictly bound circular orbits have $r>4\mu$ and $E<m{c}^2$; radial stability requires $r>6\mu$. This limiting orbit is not the smallest radius ever reached by a bound-energy plunging trajectory.

A Schwarzschild static observer at a circular orbit measures speed $v_{\mathrm{loc}}=r(d\varphi /dt)/\sqrt{1-2\mu /r}$. Combining the circular geodesic frequency with this clock rate gives the displayed formula. The observer-relative Lorentz factor is $\gamma =\sqrt{(r-2\mu )/(r-3\mu )}$, and $rd\varphi /d\tau =\gamma v_{\mathrm{loc}}$. Thus the local speed is $c/2$ at the Schwarzschild innermost stable circular orbit. Coordinate and locally measured speeds are distinct.

A source on a timelike circular Schwarzschild geodesic at radius $r>3GM/c^2$ emits light to a static observer at infinity on the orbital symmetry axis. Such a ray has zero conserved axial angular momentum. The locally measured photon energy is the contraction of its four-momentum with the observer's four-velocity. Comparing the emitter with the observer at infinity therefore gives ${\nu }_{\mathrm{em}}=U^t{\nu }_{\infty }$, where $U^t=[1-3GM/(c^{2}r){]}^{-1/2}$, proving the formula.

Equivalently the gravitational redshift factor $\sqrt{1-2GM/(c^{2}r)}$ multiplies the transverse Doppler effect factor $1/\gamma$ measured by a local static observer. For $R_S=2GM/c^2\ll r$,The coefficient contains both gravitational and orbital time dilation. It assumes an observer at infinity on the symmetry axis; other viewing directions generally introduce a first-order longitudinal Doppler effect.

A clock following a timelike circular Schwarzschild geodesic of radius $R_s$ advances at $\sqrt{1-3\mu /R_s}$ per unit Schwarzschild coordinate time. A static clock at radius $R_E>2\mu$ advances at $\sqrt{1-2\mu /R_E}$. Their ratio over the same coordinate-time interval gives the displayed result. For periodic signal comparisons, successive corresponding signals must also have the same propagation delay, as happens for repeated overhead passages in a fixed circular orbit observed from a static pole. To first order in $\mu /R$, the ratio is $1+\mu /R_E-3\mu /(2R_s)$.

For the receding edge of a circular emitter ring, the direct-projection model gives $1+z=(1-3r_g/r{)}^{-1/2}[1+\sqrt{r_g/(r-2r_g)}\sin i]$, with $r_g=GM/c^2$. Gravitational and transverse shifts combine with the line-of-sight Doppler factor. Mapping the distant inclination directly to a local photon direction neglects gravitational ray bending.

At $r=6r_g$, the local circular speed is $c/2$, so the receding direct-projection shift is $\sqrt{2}(1+\sin i/2)$. The upper envelope over inclinations is $3/\sqrt{2}$. It is not the shift for a face-on or arbitrary azimuthal emitter, and it excludes plunging-region emission and ray-tracing corrections.

For radial timelike geodesic fall released from rest at $r_0>2M$, the Killing energy is $E=\sqrt{1-2M/r_0}$ and the proper time radial equation is $(dr/ds{)}^2=2M/r-2M/r_0$. The displayed parameterization integrates its inward branch, with $s=0$ at release. The Schwarzschild event horizon is reached at ${\eta }_H=\arccos (4M/r_0-1)$ in finite proper time, even though Schwarzschild time diverges.

// Target: general-relativity.bigb

A freely falling source emits optical pulses at proper times $\delta +nP$, for integers $n\ge 0$. If its horizon-crossing proper time is finite, only pulses beginning before $s_H$ can escape. Their number is the displayed ceiling, for $0\le \delta <P<s_H$. Infinite Schwarzschild time at crossing does not give infinitely many emitted pulses. The pulse phase $\delta$ fixes both the count and the redshift of the last pulse. If a finite pulse straddles the horizon, its exterior portion has an arbitrarily late, increasingly redshifted received tail.

Inside the future black-hole region, the radius decreases along every future-directed timelike path. Its unit tangent obeys $|dr/ds|\ge \sqrt{2M/r-1}$ from the metric normalization, independently of radial acceleration or angular motion. Integrating from $r_0<2M$ to zero gives the displayed finite upper bound. The statement concerns future continuation within the black-hole region; reversing time in the white-hole region reverses the radial direction.

For smooth radial infall through a future Schwarzschild event horizon, outgoing retarded time obeys $u\sim -4M\log ({\tau }_H-\tau )$. A distant observer sees the redshift grow as $e^{t_{\mathrm{rec}}/(4M)}$. Its exponent in emission Schwarzschild time is instead $1/(2M)$, because $t_{\mathrm{rec}}\sim 2t_{\mathrm{em}}$. At finite fixed observer radius the wavelength ratio includes $\sqrt{1-2M/r_B}$.

Schwarzschild time $t$ is the exterior time coordinate normalized to the proper time of static observers at infinity. At fixed exterior radius, $d\tau =\sqrt{1-2M/r}dt$. A received signal and its emission occur at different values of this coordinate.

The Kruskal spacetime is the maximal analytic extension of the positive-mass vacuum Schwarzschild spacetime. It contains two exterior regions, a black hole, and a white hole. Kruskal–Szekeres coordinates remove both horizon coordinate singularities, while $r=0$ remains a curvature singularity.

In Kruskal–Szekeres coordinates, every future-directed radial causal curve has nondecreasing $U$ and $V$. The right exterior has $U<0,V>0$, while the left has $U>0,V<0$. No future-directed causal curve can travel between the exteriors because it would have to decrease one null coordinate. The spatial bridge on a time-symmetric slice is therefore not a traversable passage.

This radial spacetime diagram uses Kruskal–Szekeres coordinates to display both exteriors, the black hole and the white hole of maximally extended Schwarzschild spacetime. Each point represents a symmetry two-sphere. Radial null geodesics have slopes $\pm 1$ in the displayed coordinates; the horizons are $T_K=\pm X_K$ and the singularities satisfy $T_K^2-X_K^2=1$ in the usual normalization. A further compactification gives the corresponding Penrose diagram.

Every maximal radial timelike geodesic in positive-mass Kruskal spacetime is incomplete in at least one direction. Its equations are ${\dot{r}}^2=E^2-1+2M/r$ and $\ddot{r}=-M/r^2$. A finite turning point is a maximum, and each trajectory reaches $r=0$ at one or both ends. The remaining proper time is finite because $d\tau \sim \sqrt{r/(2M)}dr$. Complete circular timelike geodesics exist, but they have nonzero angular momentum.

In units $G=\hslash =c=k_B=1$, a Schwarzschild black hole has Hawking temperature $T=1/(8\pi M)$, so its heat capacity is $dM/dT=-8\pi M^2$. It heats up as it loses mass, preventing stable canonical equilibrium with an unlimited thermal reservoir.

Boost a Schwarzschild mass with speed $v$ and hold $p=m\gamma$ fixed as $v\to 1$. Away from the null plane $x=t$, the isotropic-coordinate expression tends toThe complete distributional limit on the null plane requires the additional coordinate treatment that produces the Aichelburg-Sexl shock wave.

The Schwarzschild-de Sitter spacetime is the spherically symmetric vacuum solution with positive cosmological constant. In static coordinates its metric function is $f(r)=1-2m/r-H^2{r}^2$, with a black-hole horizon and a cosmological horizon when $mH<1/\sqrt{27}$.

For $d{s}^2=Ad{t}^2-d{r}^2/A-r^{2}d{\Omega }^2$, $A=1-2GM/r-\Lambda r^2/3$, and $c=1$, a radial timelike geodesic has ${\dot{r}}^2=k^2-A$. The radial geodesic equation gives $\ddot{r}=-A^{\prime }/2=-GM/r^2+\Lambda r/3$, including at a turning point. This is a second derivative of the areal radius with respect to proper time; the geodesic's four-acceleration still vanishes. In the slow-motion weak-field limit, restoring units gives $d^{2}r/d{t}^2=-GM/r^2+\Lambda c^{2}r/3$.

Geodetic precession is the rotation of a parallel-transported gyroscope relative to the orbital radial direction in curved spacetime. For a circular Schwarzschild orbit its exact precession per revolution is $2\pi [1-\sqrt{1-3M/R}]$.

Painlevé–Gullstrand coordinates write the Schwarzschild metric asThe constant-$t$ spatial slices are flat, and the coordinates remain regular at the Schwarzschild radius.

Choose Kruskal–Szekeres coordinates with the right exterior at $U<0,V>0$ and $UV=(1-r/(2M))e^{r/(2M)}$. The ingoing Painlevé–Gullstrand coordinates cover that exterior and the future black-hole region $U>0,V>0$, together with their common future horizon. They exclude the other exterior, the white hole, the bifurcation sphere and the singularity. Indeed $\tau =v+F(r)-r_{\ast }(r)$, with $v=4M\log V$ and $F^{\prime }=\sqrt{2M/r}/f$; the difference $F-r_{\ast }$ is finite at $r=2M$, whereas $V=0$ requires infinite coordinate time.

The inward radial geodesic congruence with unit Killing energy has $U=f^{-1}{\partial }_t-\sqrt{2M/r}{\partial }_r$, where $f=1-2M/r$. Its metric dual is $U^{♭}=-d\tau$, so $\tau$ is a synchronized proper time for the congruence. Locally $\tau =t+2\sqrt{2Mr}+2M\log |(\sqrt{r}-\sqrt{2M})/(\sqrt{r}+\sqrt{2M})|$. This yields Painlevé–Gullstrand coordinates with flat spatial slices and a regular future horizon.

A radial observer released from rest at infinity has $dr/d\tau =-\sqrt{2M/r}$. Integration from $r=2M$ to $r=0$ gives $\Delta \tau ={\int }_0^{2M}\sqrt{r/(2M)}dr=4M/3$ in geometric units. In ordinary units it is $4GM/(3c^3)$. The endpoint is the curvature singularity, not a regular spacetime centre; this value belongs to that specified energy, not every possible infalling worldline.

An orthonormal coframe is

With the curvature conventionthe vacuum Schwarzschild metric satisfiesThe opposite Riemann-sign convention reverses this sign.

The orthonormal-frame components of Schwarzschild curvature scale as $M/r^3$. The geodesic deviation equation therefore gives radial stretching and transverse compression whose magnitude diverges as $r\to 0$.

Weyl curvature produces relative acceleration between neighbouring freely falling trajectories and therefore tidal deformation. A freely falling frame removes gravity at one event, as required by the equivalence principle, but cannot remove curvature over an extended region.

For inverse radius $y=1/r$ and azimuth $\varphi$, a Schwarzschild null geodesic obeyswhere $b=L/E$ is the impact parameter.

For a captured nonradial null geodesic with $u=1/r$, the large-$u$ orbit equation is $(du/d\varphi {)}^2\sim 2\mu u^3$. Integration on the inward branch givesThe curvature singularity is reached at finite azimuth. This does not prevent a near-critical ray from winding repeatedly near the photon sphere before its final plunge. A radial ray has constant azimuth, so this asymptotic coordinate parametrization is inapplicable to it.

For nonzero angular constant $h$, the interior null radial equation gives ${\dot{r}}^2\sim 2\mu h^2/r^3$. Its inward branch integrates to $r^{5/2}\sim \frac{5}{2}\sqrt{2\mu }|h|({\lambda }_{\ast }-\lambda )$. Thus the singularity is at finite affine parameter and $r\propto ({\lambda }_{\ast }-\lambda {)}^{2/5}$. The zero-angular-momentum case has a different affine scaling.

For an equatorial null geodesic in Schwarzschild spacetime, let $k=(1-2\mu /r)dt/d\lambda$ and $h=r^{2}d\varphi /d\lambda$ be the conserved quantities for an affine parameter $\lambda$. The null metric condition gives ${\dot{r}}^2/2+V_{\mathrm{eff}}=c^2{k}^2/2$. For $h\ne 0$, the effective potential is zero at the Schwarzschild event horizon, positive outside, and tends to zero at infinity. Its unique exterior maximum is at $r=3\mu$, with value $h^2/(54{\mu }^2)$ and second derivative $-h^2/(81{\mu }^4)$. This maximum gives the unstable photon sphere.

The impact parameter of an incoming ray is $b=|h|/(ck)$ for $k>0$. Capture requires its radial energy to exceed the maximum, namely $b<3\sqrt{3}\mu$. At equality the incoming ray approaches the circular orbit asymptotically; it does not cross the horizon. For $h=0$, the potential is identically zero and the ingoing radial ray is captured.

At the critical impact parameter $b=3\sqrt{3}M$, the null orbit equation factors as $(u^{\prime }{)}^2=(u-1/3{)}^2(2u+1/3)$ for $u=M/r$. One oriented family has the displayed solution. As $\varphi \to -\infty$, positive $A$ approaches the photon sphere from larger radii and negative $A$ from smaller radii; $A=0$ is the circular orbit. The physical domain must retain $r>0$ and the positive square root in its implicit parametrization.

A circular null geodesic remains at constant areal radius. It lies at a stationary point of the null radial effective potential and may be stable or unstable according to the sign of the second derivative there.

For $0<a<M$, the two exterior equatorial circular null geodesics of the Kerr metric have radii solving the signed equation above. The plus sign gives the prograde branch and the minus sign gives the retrograde branch. To derive it put $b=L/E$ in the double-root conditions of the equatorial Kerr null geodesic radial equation: $r^2=(b^2-a^2)/3$ and $r^3=M(b-a{)}^2$. Both exterior radii exceed $r_{+}=M+\sqrt{M^2-a^2}$ and approach $3M$ when $a\to 0$. The horizon generators are a separate case with $E=L=0$; dividing by $E$ excludes them. Constant-radius null directions at other exterior radii generally have nonzero acceleration and are not geodesics.

A photon sphere is a spherical set of circular null geodesics. In Schwarzschild spacetime it occurs at $r=3M$ and is unstable.

At $2\mu <r\le 3\mu$, measure the emission angle $\alpha$ from the outward radial direction in a static observer's orthonormal tetrad. Local photon energy is $E_{\mathrm{loc}}=E_{\infty }/\sqrt{1-2\mu /r}$, and its tangential momentum is $E_{\mathrm{loc}}\sin \alpha /c$. Thus its impact parameter is $b=r\sin \alpha /\sqrt{1-2\mu /r}$. Combining this with the null effective potential in Schwarzschild spacetime gives escape precisely for outward directions $0\le \alpha <{\alpha }_{\mathrm{crit}}$; the limiting ray approaches the photon sphere and does not escape.

As $r\downarrow 2\mu$ the cone shrinks to zero opening. A static observer does not exist on the horizon itself, and no future-directed ray starting on the Schwarzschild event horizon reaches exterior infinity. At $r=3\mu$, the critical half-angle is $\pi /2$; the exactly tangential critical ray stays on the unstable circular orbit. Escape cones for a moving observer are changed by relativistic aberration.

For a Reissner-Nordstrom black hole with $|Q|\le M$, the outer unstable circular null geodesic has radius $r_{\mathrm{ph}}=(3M+\sqrt{9M^2-8Q^2})/2$. Maximizing $f/r^2$ for $f=1-2M/r+Q^2/r^2$ gives the critical impact parameter $b_c^2=r_{\mathrm{ph}}^2/f(r_{\mathrm{ph}})=2r_{\mathrm{ph}}^3/(r_{\mathrm{ph}}-M)$. The geometrical optics capture cross-section is $\pi b_c^2$, reducing to $27\pi M^2$ in Schwarzschild spacetime and $16\pi M^2$ in the extremal charged case. Finite-wavelength absorption need not equal this geometric limit.

// Target: general-relativity.bigb

The Schwarzschild radius is $r_s=2GM/c^2$, or $2M$ in geometrized units.

The tortoise coordinate satisfies

The retarded coordinate $u=t-r_{\ast }$ puts the radial Schwarzschild metric into $d{s}^2=-fd{u}^2-2dudr+r^{2}d{\Omega }^2$, with $f=1-2M/r$. Outgoing radial null geodesics have constant $u$. These coordinates are regular at the past horizon; an infaller crossing the future horizon has $u\to +\infty$.

The advanced coordinate $v=t+r_{\ast }$ puts the radial Schwarzschild metric into the formwhich is regular at the future horizon.

These are the advanced, ingoing member of the Eddington-Finkelstein coordinates.

A Finkelstein diagram plots areal radius $r$ against $T=v-r$ in ingoing Eddington-Finkelstein coordinates. Its radial null directions show explicitly that both future light-ray families decrease $r$ inside a Schwarzschild black hole. It is not a compactified Penrose diagram.

Radial null curves comprise the ingoing family $v=\text{constant}$ and the outgoing familyOutside the horizon the outgoing family moves to larger $r$; inside it both future-directed families move to smaller $r$.

The surface $r=2M$ is the future event horizon of Schwarzschild spacetime. Its singularity in Schwarzschild coordinates is removable, whereas $r=0$ is a curvature singularity.

For two equal-mass inward timelike geodesics of unit specific Killing energy, their invariant center-of-mass energy has this finite horizon limit. If both are captured from infinity, Schwarzschild marginally bound capture gives $E_{\mathrm{com}}<m\sqrt{20}$, with $m\sqrt{20}$ a supremum. No such universal bound follows for particles prepared inside the angular-momentum barrier.

The Schwarzschild singularity is the spacelike curvature singularity at $r=0$. Curvature invariants and orthonormal tidal components diverge there.

Kruskal–Szekeres coordinates extend the Schwarzschild solution smoothly through its future and past horizons. Constant Schwarzschild radius curves are hyperbolae, while the horizons are null lines through the bifurcation sphere.

For null coordinates normalized by $UV=(2m-r)e^{r/(2m)}$, the displayed field agrees with exterior Schwarzschild time translation. Its flow rescales $U$ and $V$ inversely, preserving their product, the radius and $dUdV$. It therefore preserves the extended metric across both horizons. Its squared norm is $1-2m/r$: it is timelike in both exterior regions, spacelike in the black-hole and white-hole interiors, and nonzero null on the horizons away from the bifurcation two-sphere. At that sphere it vanishes. The zero vector should not be mistaken for a nonzero null generator. This gives an explicit symmetry that survives where the original time coordinate ceases to be regular.

For $d{s}^2=-fd{t}^2+d{r}^2/f+R^{2}d{\Omega }^2$, suppose $f$ and $R$ are analytic at $r_H$, $f(r_H)=0$, $f^{\prime }(r_H)=2\kappa >0$ and $R(r_H)>0$. The tortoise coordinate obeys the displayed expansion with analytic $h$. For $U=-e^{-\kappa (t-r_{\ast })}$ and $V=e^{\kappa (t+r_{\ast })}$ in the right exterior, $-UV=(r-r_H)e^{2\kappa h(r)}$ after fixing an additive constant. The inverse function theorem makes $r$ analytic in $UV$ through zero, while $d{s}^2=fdUdV/({\kappa }^{2}UV)+R^{2}d{\Omega }^2$ has a finite nonzero radial coefficient there. This supplies the local analytic extension and its bifurcate Killing horizon.

A radial geodesic orthogonal to the time-symmetric Kruskal slice starts at $r=2M$ with zero Schwarzschild energy and reaches $r=0$ after proper time $\pi M$.

For static observers at radii $r_A$ and $r_B$, equal Schwarzschild-coordinate time intervals correspond to proper times $d\tau =\sqrt{1-2M/r}dt$. Hence

This static-observer relation is a special case of gravitational redshift, not a definition of every gravitational frequency shift.

Two identical stationary devices counting operations at the same rate per proper time act as clocks. If the upper one sends start and finish light signals over a fixed path to the lower one in a stationary metric, both propagation delays are identical in coordinate time and cancel from the interval comparison. The lower elapsed proper time divided by the upper one is $\sqrt{g_{tt}(r_2)/g_{tt}(r_1)}$. In the weak field the relative lag is $g\Delta h/c^2$, giving the displayed operation deficit. A rate of $10^{12}$ operations per second, a week and a height difference of ten metres give a deficit of roughly $660$ operations. This observable tests the radial clock-rate factor; it does not independently determine the radial spatial metric coefficient because the common signal delay cancels.

A static observer at Schwarzschild radius $r>2M$ must maintain the outward proper accelerationIt approaches the Newtonian value $M/r^2$ at large radius and diverges at the horizon.

Every spherically symmetric vacuum solution of the Einstein field equation is locally a portion of Schwarzschild spacetime. In particular, a spherically symmetric body's exterior field depends only on its total mass.

A thin spherical shell of proper surface mass density $\rho$ held at Schwarzschild radius $r_{\mathrm{st}}$ around mass $M$ contributes, to first order in its mass,to the exterior Schwarzschild mass parameter.

A null Schwarzschild geodesic with $b\gg r_s$ is deflected through $2r_s/b=4GM/(b{c}^2)$ to first order.

This is the leading spherical-mass deflection underlying a gravitational lens.

A Schwarzschild static observer at areal radius $R$ emits an outward null geodesic at a small angle $\psi$ to the radial direction. Its impact parameter is $b=R\sin \psi /\sqrt{f}$, and the accumulated azimuth to infinity is ${\int }_R^{\infty }bdr/[r^2\sqrt{1-f(r)b^2/r^2}]=b/R+O(b^3/R^3)$. Thus weak curvature magnifies a narrow outward beam angle by $1/\sqrt{f(R)}$, rather than producing the full infinity-to-infinity deflection $4GM/(b{c}^2)$.

The 1919 solar eclipse expeditions organized by Frank Watson Dyson and Arthur Eddington compared background-star positions near the obscured Sun with reference photographs. The optical train used a slowly rotating flat mirror to send the light into a horizontal photographic telescope. A smaller telescope at Sobral provided a separate set of plates. Focus and plate-scale changes had to be distinguished from the radial pattern predicted by Schwarzschild light deflection. The experiment therefore tested a spatial pattern and its amplitude, not merely whether a star image had moved.^{[ref] [ref]}

Writing $y=b^{-1}\sin \varphi +\Delta y$ in $y^{\prime \prime }+y=3M{y}^2$ and retaining first order in $M$ givesThe displacement of the outgoing zero is $4M/b$.