Light Bending and Classical Tests
Experimental confirmations and coordinate extensions of Schwarzschild spacetime
Gravitational Light Deflection
One of the three classical tests of general relativity is the bending of light by a massive body. A photon passing a mass M with impact parameter b is deflected by an angle that can be computed from the null geodesic equation.
Starting from the orbit equation for photons (using u = 1/r):
$$\frac{d^2u}{d\phi^2} + u = 3Mu^2$$
The zeroth-order solution (straight line) is uβ = sin(Ο)/b. Substituting this into the right-hand side and solving perturbatively gives the first-order correction. The total deflection angle, computed as the change in the asymptotic angle of the trajectory, is:
Einstein's Light Deflection Formula
$$\Delta\phi = \frac{4GM}{c^2 b}$$
Exactly twice the Newtonian prediction
For light grazing the Sun (b β Rβ):
$$\Delta\phi = \frac{4GM_\odot}{c^2 R_\odot} = \frac{4 \times 1.474 \text{ km}}{6.96 \times 10^5 \text{ km}} = 1.75''$$
This was confirmed by Eddington's 1919 solar eclipse expedition (1.98" Β± 0.16"), making Einstein world-famous.
The derivation in detail: expand the integral for the total angular change:
$$\Delta\phi = 2\int_0^{u_{\max}} \frac{du}{\sqrt{\frac{1}{b^2} - u^2 + 2Mu^3}} - \pi$$
Expanding to first order in M/b (weak-field approximation):
$$\Delta\phi \approx 2\int_0^{1/b} \frac{du}{\sqrt{1/b^2 - u^2}}\left(1 + \frac{Mu^3}{1/b^2 - u^2}\right) - \pi = \frac{4M}{b}$$
Shapiro Time Delay
The fourth classical test of GR, proposed by Irwin Shapiro in 1964, is the time delay of radar signals passing near a massive body. A radar pulse sent from Earth, reflected off another planet, and returned will arrive later than expected from flat-space geometry.
For a photon traveling in the equatorial plane, the coordinate velocity satisfies:
$$\frac{dr}{dt} = \pm\left(1 - \frac{2M}{r}\right)\sqrt{1 - \frac{b^2}{r^2}\left(1 - \frac{2M}{r}\right)}$$
The coordinate time for a round trip from radial coordinate rβ to rβ (passing through closest approach rβ) contains an excess over the flat-space travel time:
Shapiro Time Delay
$$\Delta t = \frac{4GM}{c^3}\left[\ln\left(\frac{4r_1 r_2}{r_0^2}\right) + 1\right]$$
For a radar signal passing near the Sun to a planet at distance rp:
Viking Mars mission (1977): Measured delay of ~250 ΞΌs, agreeing with GR prediction to 0.1% accuracy. This remains one of the most precise tests of GR in the solar system.
Gravitational Redshift
A photon emitted at radius re and observed at radius ro (with ro > re) is redshifted. The relationship between emitted and observed frequencies follows from the constancy of E = (1 - 2M/r)dt/dΟ along the photon's path:
Gravitational Redshift Formula
$$\frac{\nu_o}{\nu_e} = \frac{\sqrt{1 - 2M/r_e}}{\sqrt{1 - 2M/r_o}}$$
$$z = \frac{\nu_e - \nu_o}{\nu_o} = \sqrt{\frac{1 - 2M/r_o}{1 - 2M/r_e}} - 1$$
In the weak-field limit (2M/r βͺ 1), this reduces to:
$$z \approx \frac{GM}{c^2}\left(\frac{1}{r_e} - \frac{1}{r_o}\right) \approx \frac{GM}{c^2 r_e} \quad (\text{for } r_o \to \infty)$$
Experimental Verifications:
- Pound-Rebka experiment (1960): Measured redshift in a 22.6 m tower at Harvard, confirming GR to 10%
- Pound-Snider (1964): Improved to 1% accuracy
- Gravity Probe A (1976): Hydrogen maser on rocket, confirmed to 0.007%
- GPS satellites: Clocks run 45 ΞΌs/day fast due to gravitational blueshift; corrected in real time
Eddington-Finkelstein Coordinates
The coordinate singularity at r = 2M in Schwarzschild coordinates can be eliminated by introducing the tortoise coordinate r* and then defining new time coordinates. The tortoise coordinate is:
$$r^* = r + 2M \ln\left|\frac{r}{2M} - 1\right|$$
As r β 2M, r* β -β (the horizon is pushed to minus infinity)
Defining the advanced null coordinate v = t + r*:
Ingoing Eddington-Finkelstein Metric
$$ds^2 = -\left(1 - \frac{2M}{r}\right)dv^2 + 2\,dv\,dr + r^2\,d\Omega^2$$
This metric is perfectly regular at r = 2M. The determinant g = -rβ΄sinΒ²ΞΈ is non-zero everywhere except the true singularity at r = 0. Ingoing light rays are simply v = const (straight lines in (v,r) coordinates), while outgoing light rays satisfy dv/dr = 2/(1 - 2M/r). At r = 2M, outgoing rays have dv/dr β β (they are "stuck"), confirming the one-way membrane nature of the event horizon.
Kruskal-Szekeres Maximal Extension
The Eddington-Finkelstein coordinates remove the coordinate singularity but only cover part of the maximal extension. The Kruskal-Szekeres coordinates (T, X) cover the entire maximally extended Schwarzschild spacetime. They are defined by:
For r > 2M:
$$T = \left(\frac{r}{2M} - 1\right)^{1/2} e^{r/(4M)} \sinh\left(\frac{t}{4M}\right)$$
$$X = \left(\frac{r}{2M} - 1\right)^{1/2} e^{r/(4M)} \cosh\left(\frac{t}{4M}\right)$$
For r < 2M:
$$T = \left(1 - \frac{r}{2M}\right)^{1/2} e^{r/(4M)} \cosh\left(\frac{t}{4M}\right)$$
$$X = \left(1 - \frac{r}{2M}\right)^{1/2} e^{r/(4M)} \sinh\left(\frac{t}{4M}\right)$$
In these coordinates, the metric becomes:
Kruskal-Szekeres Metric
$$ds^2 = \frac{32M^3}{r}\,e^{-r/(2M)}\left(-dT^2 + dX^2\right) + r^2\,d\Omega^2$$
where r is implicitly defined by TΒ² - XΒ² = (1 - r/(2M))er/(2M). This metric is manifestly regular at r = 2M. Light cones are at 45Β° everywhere (as in Minkowski space), making causal structure immediately visible.
Penrose Diagram for Schwarzschild
The Penrose (conformal) diagram compactifies the entire spacetime into a finite region while preserving causal structure (light rays travel at 45Β°). The maximally extended Schwarzschild spacetime has four regions:
Region I β Exterior
Our universe: r > 2M. Bounded by future and past horizons and by future and past null infinity. Asymptotically flat. This is the region described by the original Schwarzschild coordinates.
Region II β Black Hole Interior
Future of the event horizon: r < 2M. Contains the future singularity at r = 0 (a spacelike hypersurface). All timelike curves end at the singularity.
Region III β White Hole Interior
Time-reverse of Region II. Contains the past singularity. All timelike curves originate from the singularity. Considered unphysical (not formed in gravitational collapse).
Region IV β Parallel Universe
A second asymptotically flat exterior region, causally disconnected from Region I for external observers. Connected via a non-traversable Einstein-Rosen bridge (wormhole).
Key features of the Penrose diagram:
The singularities (r = 0) are horizontal lines at top (future) and bottom (past)
$$\bullet \text{ Each point represents a 2-sphere } S^2$$
$$\bullet \text{ The horizons } (r = 2M) \text{ are diagonal lines at } 45Β°$$
$$\bullet \text{ Null infinity } \mathscr{I}^\pm \text{ and spatial infinity } i^0 \text{ form the boundary}$$