Light Deflection and Gravitational Redshift

Derivation

Light Deflection and Gravitational Redshift

D-395 Home PU-403 Threads light · energy · force Depends on Schwarzschild Solution from Spherical Vacuum, Geodesic Equation from Extremal Proper Time

Statement

Starting from the Schwarzschild metric and the conserved quantities of geodesic motion, we derive two observable predictions of general relativity: the gravitational shift of a spectral line's frequency between two static radii, \( \dfrac{\nu_r}{\nu_e} = \sqrt{\dfrac{1-r_s/r_e}{1-r_s/r_r}} \), and the total deflection of a light ray grazing a spherical mass of impact parameter \(b\), \( \delta = \dfrac{4GM}{c^2 b} \), where \( r_s = 2GM/c^2 \) is the Schwarzschild radius.

Why it matters

These are the two "solar-system" tests that first distinguished general relativity from Newtonian gravity plus special relativity. The deflection is exactly twice the value a naive Newtonian photon-corpuscle calculation gives, and Eddington's 1919 eclipse measurement of the doubled value was the observation that made relativity famous. The redshift is the cleanest demonstration that gravity dilates proper time itself, not merely the trajectory of light.

Both results follow from a single idea — light travels on null geodesics of a curved metric — with no new postulates beyond the Schwarzschild solution. They are the entry point to modern applications: GPS clock corrections, gravitational lensing as a cosmological ruler, and precision spectroscopy of white dwarfs and the Sun.

Assumptions

The gravitating body is static, spherically symmetric, and uncharged (Schwarzschild).Rotation (Kerr) adds frame-dragging terms that split the deflection by polarization side and shift the redshift; a non-vacuum interior changes the exterior only through its total mass.

The field is weak along the ray: \( r_s/b \ll 1 \) and \( r_s/r \ll 1 \).Dropping this forbids the linear perturbation expansion; near the photon sphere \( r = 3GM/c^2 \) the deflection diverges and the closed-form \(4GM/c^2b\) fails badly.

Emitter and receiver of the spectral line are at rest in the Schwarzschild coordinates (no Doppler motion).Any radial or transverse velocity superimposes a special-relativistic Doppler factor that must be removed before the gravitational shift can be read off.

The photon is a test field: its own stress-energy does not perturb the metric, and geometric optics holds (wavelength \( \ll \) curvature scale).Dropping the test assumption couples the field back into Einstein's equations; dropping geometric optics means the ray picture and a single impact parameter no longer describe the light.

Derivation

\[ ds^2 = -\left(1-\frac{r_s}{r}\right)c^2\,dt^2 + \left(1-\frac{r_s}{r}\right)^{-1}dr^2 + r^2\left(d\theta^2 + \sin^2\theta\,d\phi^2\right),\qquad r_s \equiv \frac{2GM}{c^2}. \]

The Schwarzschild solution of the vacuum field equations (prior result). We restrict to the equatorial plane \(\theta=\pi/2\), legal because spherical symmetry lets any geodesic be rotated into that plane and \(\dot\theta=0\) is then preserved. A

\[ E \equiv \left(1-\frac{r_s}{r}\right)c^2\,\dot t = \text{const},\qquad L \equiv r^2\,\dot\phi = \text{const}. \]

Because the metric is independent of \(t\) and \(\phi\), the vectors \(\partial_t\) and \(\partial_\phi\) are Killing; contracting a Killing vector with the tangent \(u^\mu = dx^\mu/d\lambda\) gives a constant along any geodesic (prior result: geodesic equation). Overdot is \(d/d\lambda\) for an affine parameter \(\lambda\). B

\[ d\tau = \sqrt{1-\frac{r_s}{r}}\;dt \qquad(\text{static observer, } dr=d\phi=0). \]

For an observer fixed at radius \(r\), only \(dt\) is nonzero in \(ds^2 = -c^2 d\tau^2\); solving for \(d\tau\) converts the coordinate time \(t\) (shared by all static observers) into the physical clock time each one measures. This is the whole content of gravitational time dilation. A

\[ \nu_{\text{obs}} \propto \frac{d(\text{phase})}{d\tau} = \frac{d(\text{phase})}{dt}\,\frac{dt}{d\tau} = \frac{\nu_\infty}{\sqrt{1-r_s/r}} \;\Longrightarrow\; \frac{\nu_r}{\nu_e} = \sqrt{\frac{1-r_s/r_e}{1-r_s/r_r}}. \]

The photon's Killing energy \(E\) — its frequency measured in coordinate time — is conserved along the null geodesic, so any change in the locally measured frequency \(\nu_{\text{obs}} = E/(h\sqrt{-g_{tt}})\) comes entirely from the differing clock rates of the static emitter and receiver. Taking the ratio at \(r_e\) and \(r_r\) eliminates \(E\). B

\[ \frac{\Delta\nu}{\nu} = \frac{\nu_r-\nu_e}{\nu_e} \;\approx\; -\frac{r_s}{2}\left(\frac{1}{r_e}-\frac{1}{r_r}\right) = -\frac{GM}{c^2}\left(\frac{1}{r_e}-\frac{1}{r_r}\right). \]

Weak-field expansion \(\sqrt{1-x}\approx 1-x/2\) to first order in \(r_s/r\). For light climbing outward (\(r_r>r_e\)) the bracket is positive, so \(\Delta\nu<0\): a redshift. This closes the redshift result; the remaining steps treat the deflection. A

\[ 0 = ds^2 \;\Rightarrow\; \dot r^2 = \frac{E^2}{c^2} - \frac{L^2}{r^2}\left(1-\frac{r_s}{r}\right). \]

Light follows a null geodesic, so \(ds^2=0\). Substitute \(\dot t\) and \(\dot\phi\) from Step 2 into the equatorial line element, then multiply through by \((1-r_s/r)\). This is the radial energy equation for a photon. A

\[ u \equiv \frac1r,\quad \dot r = -\frac{L}{r^2}\frac{dr}{d\phi}\cdot(-1)= L\,\frac{du}{d\phi}\;\Rightarrow\; \left(\frac{du}{d\phi}\right)^2 = \frac{E^2}{c^2 L^2} - u^2 + r_s\,u^3. \]

Trade the affine parameter for the observable angle \(\phi\) using \(\dot\phi = L/r^2\), and substitute \(u=1/r\) so that \(\dot r = -L\,du/d\phi\). Dividing the Step 6 equation by \(L^2\) gives the photon orbit as a first-order ODE in \(u(\phi)\). B

\[ \frac{d^2u}{d\phi^2} + u = \frac{3}{2}\,r_s\,u^2 = \frac{3GM}{c^2}\,u^2. \]

Differentiate the Step 7 equation with respect to \(\phi\) and divide by the common factor \(2\,du/d\phi\). The constant \(E^2/c^2L^2\) drops out, leaving a driven harmonic oscillator whose forcing term \(3GM u^2/c^2\) is the entire relativistic effect. B

\[ u_0(\phi) = \frac{\sin\phi}{b},\qquad b \equiv \frac{cL}{E}. \]

With the right-hand side set to zero, the equation is \(u_0''+u_0=0\); the solution passing to \(r\to\infty\) (\(u\to0\)) at \(\phi=0,\pi\) is a straight line at perpendicular distance \(b\) from the centre. Matching \((du/d\phi)^2=E^2/c^2L^2-u^2\) at large \(r\) identifies \(b=cL/E\) as the impact parameter. A

\[ u = u_0 + u_1,\qquad u_1'' + u_1 = \frac{3GM}{c^2 b^2}\sin^2\phi \;\Rightarrow\; u_1 = \frac{GM}{c^2 b^2}\left(2-\sin^2\phi\right). \]

Treat the forcing term as a small perturbation (legal because \(r_s/b\ll1\)) and linearize by inserting \(u_0\) on the right. Writing \(\sin^2\phi=\tfrac12(1-\cos2\phi)\), the particular solution combines a constant with a \(\cos2\phi\) term (whose amplitude is fixed by \(u_1''+u_1=-3C\cos2\phi\)). C

\[ u=0:\quad \frac{\phi_\infty}{b} + \frac{2GM}{c^2b^2} \approx 0 \;\Rightarrow\; \phi_\infty = -\frac{2GM}{c^2 b},\qquad \delta = 2|\phi_\infty| = \frac{4GM}{c^2 b}. \]

The asymptotes are where \(u=0\) (the ray at infinity). For small \(\phi\), \(u_0\approx\phi/b\) and \(\sin^2\phi\to0\), giving one asymptote a shift \(-2GM/c^2b\) below \(\phi=0\); by the symmetry \(\phi\to\pi-\phi\) the far asymptote shifts equally, so the total bend is twice that. B

Result

\[ \boxed{\;\delta = \frac{4GM}{c^2 b}\;}\qquad\qquad \boxed{\;\frac{\nu_r}{\nu_e} = \sqrt{\frac{1-r_s/r_e}{1-r_s/r_r}} \;\approx\; 1-\frac{GM}{c^2}\!\left(\frac{1}{r_e}-\frac{1}{r_r}\right)\;} \]

Reading. A light ray grazing a mass \(M\) at closest approach \(b\) is bent toward the mass by an angle \(4GM/c^2b\) — exactly twice the Newtonian "falling corpuscle" value \(2GM/c^2b\), the extra factor coming from the curvature of space (the \(g_{rr}\) term), not just of time. Independently, a spectral line emitted deep in the potential well and received higher up is shifted to lower frequency (redshifted) by a fractional amount equal to the difference in Newtonian potential per \(c^2\). Both are first-order in \(GM/c^2\).

Units check. \(GM/c^2\) has units \(\frac{(\text{m}^3\text{kg}^{-1}\text{s}^{-2})(\text{kg})}{\text{m}^2\text{s}^{-2}} = \text{m}\), a length (the Schwarzschild radius scale). Dividing by \(b\) in metres gives a pure number — an angle in radians, as an angle must be. In the redshift, \(r_s/r\) and \(GM/c^2r\) are length/length, dimensionless, so \(\Delta\nu/\nu\) is correctly dimensionless.

Limiting cases

Newtonian / large \(b\): as \(b\to\infty\), \(\delta\to0\) like \(1/b\) — far rays are undeflected, recovering flat space.
Half the GR value: keeping only the time-curvature piece (the \(g_{tt}\) term) reproduces the 1801 Soldner/Newtonian result \(2GM/c^2b\); the space-curvature piece doubles it.
Equal radii: if \(r_e=r_r\) the redshift vanishes — a purely transverse light path between equal potentials shows no gravitational shift.
Emission to infinity: with \(r_r\to\infty\), \(\nu_\infty/\nu_e=\sqrt{1-r_s/r_e}\); the exact factor, valid even for strong fields, not just the linear approximation.
Photon sphere: as \(b\to \tfrac{3\sqrt3}{2}\,GM/c^2\) the linear formula breaks and \(\delta\to\infty\) (light can orbit); see below.

Breaks when

Strong field, small impact parameter. When \(b\) approaches the photon-sphere critical value \(b_c = \tfrac{3\sqrt3}{2}\,r_s\), the perturbation \(u_1\) is no longer small; the deflection grows logarithmically and diverges. The full elliptic-integral orbit is required, and rays can loop the mass any number of times (relativistic images).
Rotating or charged source. The Schwarzschild metric is replaced by Kerr or Kerr–Newman; frame-dragging makes the deflection depend on whether the ray passes prograde or retrograde, and the redshift acquires an angular-momentum-dependent term. The clean \(4GM/c^2b\) no longer holds.
Emitter or receiver in motion. If either endpoint has a peculiar velocity, a special-relativistic Doppler factor multiplies the gravitational ratio; the measured shift is then a product of two effects and cannot be read as pure gravitational redshift.
Cosmological / time-dependent metric. Over cosmological distances the background expands; the static Schwarzschild derivation and its conserved \(E\) no longer apply, and the observed shift mixes gravitational and expansion redshifts.

Failure modes

Halving the deflection. Using only gravitational time dilation (the Newtonian photon) gives \(2GM/c^2b\); forgetting the spatial-curvature contribution loses the factor of two that the experiment actually confirmed.
Wrong sign of redshift. Writing \(\Delta\nu/\nu = +GM/c^2(1/r_e-1/r_r)\) and calling emission from the surface a blueshift. Light climbing out of a well loses energy — it is redshifted.
Confusing coordinate and proper frequency. The Killing energy \(E\) is conserved, but the locally measured frequency is not; the shift lives entirely in the \(\sqrt{-g_{tt}}\) conversion, not in any change of \(E\) along the ray.
Using diameter instead of radius for grazing rays. The impact parameter of a limb-grazing solar ray is \(R_\odot\), not \(2R_\odot\); a factor-of-two error in \(b\) halves the predicted \(1.75''\).
Radians vs arcseconds. The formula returns radians; forgetting the \(206265''\!/\text{rad}\) conversion reports the answer \(2\times10^5\) times too small.
Applying the linear redshift near a compact object. For white dwarfs or neutron stars \(r_s/r\) is not tiny; the linear formula understates the shift and the exact square root must be used.

Discussion

The deep lesson is that both effects are one phenomenon seen twice. Gravitational redshift is what happens to a light ray's phase as it climbs a gradient of proper time; light deflection is what happens to its spatial path as it crosses a gradient of proper distance. In the weak field the metric potential \(\Phi = -GM/r\) enters \(g_{tt}\) and \(g_{rr}\) with equal magnitude, and it is precisely the equality of the "time part" and "space part" that produces the famous factor of two in the bending. Measure the deflection and you have measured that space itself is curved, not merely that clocks run slow.

The conserved quantities carry the argument. Time-translation symmetry gives \(E\), and the redshift is nothing more than the statement that \(E\) is constant while the local clock rate \(\sqrt{-g_{tt}}\) varies. Rotational symmetry gives \(L\), and the impact parameter \(b=cL/E\) is the ratio of the two conserved charges. This is why a single Schwarzschild geometry, with no extra input, predicts both numbers — and why any metric theory of gravity must reproduce them or be ruled out. Modern lensing surveys invert the deflection formula to weigh galaxy clusters and dark matter without ever seeing the mass directly.

Experimentally the two tests probe complementary regimes. Redshift is a weak-field, high-precision test: Pound and Rebka measured \(\Delta\nu/\nu\sim10^{-15}\) over 22.5 m of a Harvard tower using the Mössbauer effect, and GPS satellites must correct for it continuously or accumulate kilometres of positioning error per day. Deflection is a geometric test, refined from Eddington's \(20\%\) eclipse result to the \(10^{-4}\) precision of very-long-baseline radio interferometry tracking quasars as the Sun passes near them.

At the rigorous level the equality of the two potential contributions is not a coincidence but a consequence of the linearized Einstein equations in harmonic gauge, where a static source produces \(h_{00}=h_{ij}\delta^{ij}\) with the same coefficient \(2\Phi/c^2\). A theory with a different ratio — parametrized by the PPN parameter \(\gamma\), so that \(\delta = \tfrac{1+\gamma}{2}\cdot\tfrac{4GM}{c^2b}\) — would bend light by a different amount while leaving the Newtonian orbit unchanged. General relativity fixes \(\gamma=1\), and the Cassini-tracking bound \(|\gamma-1|<2.3\times10^{-5}\) is currently the tightest confirmation that spatial curvature contributes exactly its GR share.

Common misconceptions. The bending is not "light being pulled down like a ball" — that picture gives only half the answer and gets the mechanism wrong. Nor is the redshift the photon "losing energy to gravity" in any local sense: locally the photon's energy is whatever the local observer measures, and the shift is a comparison between two different observers' clocks. And the light never speeds up or slows down locally — its coordinate speed changes, but every local measurement returns \(c\).

Worked examples

\[ \textbf{Deflection of starlight grazing the Sun's limb.} \]

Take a ray whose closest approach is the solar radius, \(b=R_\odot\), and evaluate \(\delta=4GM_\odot/c^2b\). A

\[ \frac{GM_\odot}{c^2} = \frac{(6.674\times10^{-11})(1.989\times10^{30})}{(2.998\times10^8)^2} = 1.477\times10^{3}\ \text{m} = 1.477\ \text{km}. \]

\[ \delta = \frac{4\,(1.477\times10^{3}\,\text{m})}{6.957\times10^{8}\,\text{m}} = 8.49\times10^{-6}\ \text{rad}. \]

\[ \delta = 8.49\times10^{-6}\times(206265\ ''/\text{rad}) = 1.75''. \]

\[ \delta_\odot \approx 8.5\times10^{-6}\ \text{rad} \approx 1.75'' \]

Reading. This is Einstein's 1915 prediction and the value Eddington confirmed at the 1919 eclipse — twice the Newtonian \(0.87''\). Rays passing farther out bend proportionally less as \(1/b\).

\[ \textbf{Gravitational redshift of a solar spectral line.} \]

A line emitted at the photosphere \(r_e=R_\odot\) is received on Earth (\(r_r\approx\infty\) compared with solar terms). Use the weak-field formula. A

\[ \frac{\Delta\nu}{\nu} \approx -\frac{GM_\odot}{c^2 R_\odot} = -\frac{1.477\times10^{3}\,\text{m}}{6.957\times10^{8}\,\text{m}} = -2.12\times10^{-6}. \]

\[ \Delta\lambda = -\lambda\,\frac{\Delta\nu}{\nu};\quad \text{for the Na D line } \lambda=589.0\ \text{nm}: \ \Delta\lambda = (589.0\,\text{nm})(2.12\times10^{-6}) = 1.25\times10^{-3}\ \text{nm}. \]

\[ \frac{\Delta\nu}{\nu} = -2.12\times10^{-6},\qquad \Delta\lambda_{\text{Na D}} \approx +1.25\ \text{pm (toward red)} \]

Reading. A shift of about \(0.6\ \text{km s}^{-1}\) in equivalent velocity — small, but measured and consistent with GR once convective blueshifts are removed. On a white dwarf like Sirius B (\(\sim10^{-4}\)) the same physics gives a shift 50 times larger and easily seen.

Problems

A radio source is occulted by Jupiter, the ray grazing the cloud tops at \(b=R_J=7.15\times10^7\ \text{m}\), \(M_J=1.898\times10^{27}\ \text{kg}\). Find the deflection in microarcseconds.
Solution
\(GM_J/c^2 = (6.674\times10^{-11})(1.898\times10^{27})/(8.988\times10^{16}) = 1.409\ \text{m}\). Then \(\delta = 4(1.409)/7.15\times10^7 = 7.88\times10^{-8}\ \text{rad}\). Converting: \(7.88\times10^{-8}\times206265 = 0.0163'' = 16.3\ \mu\text{as}\). This is at the edge of VLBI sensitivity and has been detected.
Photons from a star's surface reach a distant telescope with fractional redshift \(\Delta\nu/\nu = -5.0\times10^{-4}\). If the star has one solar mass, find its radius (weak-field approximation) and comment on the object.
Solution
Weak field: \(|\Delta\nu/\nu| = GM/c^2R\), so \(R = GM/(c^2\,|\Delta\nu/\nu|) = 1.477\times10^3/5.0\times10^{-4} = 2.95\times10^{6}\ \text{m}\approx 2950\ \text{km}\). A solar mass packed into a few thousand km is a white dwarf (roughly Earth-sized). Because \(r_s/R\approx10^{-3}\) is still small, the linear formula is adequate here; for a neutron star one would need the exact square root.
Show that keeping only the \(g_{tt}\) ("time-curvature") term in the metric — equivalently, treating light as a Newtonian particle at speed \(c\) — gives half the GR deflection, \(2GM/c^2b\).
Solution
A Newtonian particle of speed \(v=c\) and impact parameter \(b\) in field \(-GM/r\) receives a transverse impulse \(\Delta v_\perp = \int F_\perp\,dt/m\). With \(F_\perp/m = (GM/r^2)(b/r)\) and \(t = x/c\), \(r^2=b^2+x^2\): \(\Delta v_\perp = \int_{-\infty}^{\infty} \frac{GM b}{(b^2+x^2)^{3/2}}\frac{dx}{c} = \frac{GM b}{c}\cdot\frac{2}{b^2} = \frac{2GM}{cb}\). The bend angle is \(\Delta v_\perp/c = 2GM/c^2b\), exactly half the relativistic value. The missing half is the spatial-curvature (\(g_{rr}\)) contribution.
A ray passes the Sun at \(b=3R_\odot\). By what angle is it deflected, and how does this compare with the limb-grazing value?
Solution
Since \(\delta\propto1/b\), tripling \(b\) divides the deflection by 3: \(\delta = 1.75''/3 = 0.583''\). Explicitly \(\delta = 4(1.477\times10^3)/(3\times6.957\times10^8) = 2.83\times10^{-6}\ \text{rad} = 0.583''\). The strong \(1/b\) falloff is why eclipse measurements target stars as close to the limb as possible.
The GPS satellite constellation orbits at \(r_r = 2.66\times10^7\ \text{m}\) from Earth's centre (\(M_\oplus=5.972\times10^{24}\ \text{kg}\)); ground clocks sit at \(r_e=R_\oplus=6.371\times10^6\ \text{m}\). Find the fractional gravitational rate difference between satellite and ground clocks, and the daily time offset (ignore the special-relativistic velocity term).
Solution
\(GM_\oplus/c^2 = (6.674\times10^{-11})(5.972\times10^{24})/8.988\times10^{16} = 4.435\times10^{-3}\ \text{m}\). The satellite clock (higher up, shallower well) runs fast relative to ground by \(\frac{\Delta\tau}{\tau} = \frac{GM}{c^2}\left(\frac{1}{r_e}-\frac{1}{r_r}\right) = 4.435\times10^{-3}\left(\frac{1}{6.371\times10^6}-\frac{1}{2.66\times10^7}\right)\). Evaluate: \((1.570\times10^{-7}-3.759\times10^{-8}) = 1.194\times10^{-7}\ \text{m}^{-1}\), so \(\Delta\tau/\tau = 4.435\times10^{-3}\times1.194\times10^{-7} = 5.29\times10^{-10}\). Over one day (\(86400\ \text{s}\)): \(\Delta t = 5.29\times10^{-10}\times86400 = 45.7\ \mu\text{s}\). Uncorrected, at \(c\) this maps to \(\sim14\ \text{km}\) of positioning error per day — which is why GPS clocks are pre-offset for general relativity.