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The bending of light

Space and time · level 8, lesson 4 of 6

A question to start

During a total eclipse, stars near the Sun can be seen in daytime. Why would astronomers sail to Brazil and to an island off Africa to photograph them?

The idea

If gravity is the curving of space and time, then light passing a heavy body should bend. General relativity predicts that a ray just grazing a body of mass \(M\), at a distance \(b\) from its centre, is bent by an angle

\[\theta = \frac{4GM}{c^2 b}.\]

For the Sun's edge this is 1.75″. Treating light as particles falling under Newton's gravity gives exactly half, about 0.87″. So measuring the bend was a test between the two.

Stars close to the Sun can only be seen during a total eclipse. On 29 May 1919, teams photographed the eclipse from Sobral in Brazil and from the island of Príncipe off West Africa. Comparing star positions with photographs taken at night, months apart, they found the stars shifted outwards by about the amount Einstein predicted.

A ray just grazing the planet is bent by Mars Earth
\(4GM / (c^2 R)\) about 0.00012″ about 0.00057″
where it seems to bereal starthe SunEarth

Worked examples

1. The Sun. \(GM_\odot = 1.327 \times 10^{20}\) m³/s², \(b = 6.957 \times 10^8\) m. \(\theta = \frac{4 \times 1.327 \times 10^{20}}{8.988 \times 10^{16} \times 6.957 \times 10^8} \approx 8.49 \times 10^{-6}\) rad. Times 206,265: about 1.75″.

2. Mars. \(GM = 4.283 \times 10^{13}\) m³/s², \(R = 3.39 \times 10^6\) m. \(\theta \approx 5.6 \times 10^{-10}\) rad, about 0.00012″: far too small to see.

3. Farther from the Sun. The bend falls as \(1/b\). A ray passing 10 Sun radii out is bent by about 0.175″.

Try it

  1. Why could the 1919 test only be done during an eclipse?
  2. What does Newton's gravity predict, compared with Einstein's?
  3. Which way do the stars near the Sun appear to move?
  4. A ray passes 4 Sun radii from the Sun's centre. About how much is it bent?
Answers
  1. Stars near the Sun are only visible when the Moon blocks the Sun's glare.
  2. Half as much: about 0.87″ against 1.75″.
  3. Outwards, away from the Sun.
  4. \(1.75 \div 4 \approx 0.44″\).

Common mistake

"The Sun's air bends the light." The Sun's gas is far too thin out there to bend starlight by that much, and the same bend is seen in radio waves passing the Sun. It is gravity.

Where it leads