## I2 The Deflection of Light in the Gravitational Field of the Sun

We now want to do the calculations for the experiment, whose outcome made Einstein so famous in 1919: The deflection of light at the solar limb. The effect on the public and the clamor of the press concerning this result can only be understood against the backdrop of the just ended disaster of the First World War (see [32-232ff] or [38-191ff]). In **H6**, we already stressed that the outcome of this experiment speaks in favor of the GTR and against Newton’s theory which expected only half the value predicted by the GTR.

We take the following approach: a light beam of width Δx passes the sun at a distance D along the y-axis. Thereby, according to our formulas in **G5**, the inner side of the beam which is closer to the sun, moves by a little bit slower than the outer side, so the wave front is tilted by a small angle Δß:

To calculate the sum of all these small changes of direction Δß, we will integrate from y = +∞ to y = -∞ using the constant value D for x. In so doing, we require that the entire change in direction is small. In principle, this is ‘gravitation by refraction’!

**Step 1**

We determine a workable expression for the infinitesimal change in direction Δß:

We must integrate over the y-axis. We are not in error when we write dy = c

_{0}• dt , even if the light beam (as seen from OFF) does not quite advance with velocity c

_{0}. This really only means that our time slices dt are not all of the same size. Still we may overall write

So if we knew the partial derivative of the speed of light with respect to x, then we could write our integral:

**Step 2**

We determine the partial derivative of c(r, ∞, φ) with respect to x. The calculation is a bit tedious:

**Step 3**

We evaluate the integral:

On the light’s path D = x = 2.33 light seconds is constant. In place of +∞ to -∞ one could also integrate from 2000 to -2000 light seconds (the earth is about 500 light seconds from the sun). If so then note that α must also be in these units: α = G • M / c

^{2}= G’ • M ≈ 4.9261 • 10

^{-6}light seconds! Using the TI-89 calculator to do the integration delivers a result of 8.4571 • 10

^{-6}. This is the whole ß deflection in radians. Converting to arc seconds, we get 8.4571 • 10

^{-6}• 180 • 3600 / π ≈ 1.74 arc seconds. Using Mathematica ® with twice the precision gives the same value, namely 1.7518 arc seconds.

For our integral, [27-143] gives the anti-derivative (y / r + (y / r)

^{3}) / D. This can easily be checked by differentiating! The limit of y / r as y approaches infinity is simply 1. Thus for the total deflection in radians one obtains the simple formula

What experimental data are available to test this formula?

[27-145] presents the analysis of photographs of star fields during solar eclipses up till 1952. The values lie between 1.61" and 2.01" with uncertainties of 0.10” to 0.45” (I have omitted so-called ‘outliers’). This is enough to give GTR preference over Newton's theory, but the uncertainty is greater than 10%. The ‘phone book’ [29-1105] gives data from measurements with radio telescopes. Every year on October 8, the sun – as seen from the earth - moves over the quasar 3C279. Another quasar 3C273 is in the vicinity and allows a precise measurement of the angle between these two objects. In 1970 the GTR could be confirmed to within 5% accuracy. Measurements using VLBI (very long baseline interferometry) could confirm GTR in 1995 to an accuracy of 0.9996 ± 0.0017, i.e., to 1.7 parts per thousand. In 1999 an analysis of 2 million VLBI measurements was published, which delivered an accuracy of 0.99992 ± 0.00014. This data are taken from the following website on December 26, 2006: http://relativity.livingreviews.org/open?pubNo=lrr-2001-4&page=node10.html

The page http://relativity.livingreviews.org due mainly to Clifford M. Will has rendered outstanding service and provides the most comprehensive and current information about ongoing research and experimental trials in the field of GTR.

Another confirmation of GTR comes from the data of ESO's Hipparcos satellite. This had the task of precisely measuring the position of 118,000 stars (up to size class 12.5), so that they could later be used as reference. Hipparcos (after a very inauspicious start, see Wikipedia) performed this task brilliantly. The angular resolution of the position measuring instruments was 0.001 arc seconds. Thus, it could confirm the predictions of GTR across the entire sky with an accuracy of about 0.3%. Light deflection can already be measured, if the light moves past the sun at a distance of one astronomical unit, i.e., a distance of 150 million kilometers! When we direct our sight from the earth, along the y-axis, i.e., in a direction perpendicular to the segment connecting the earth to sun, then the light beam has already suffered exactly half of the total ß deflection. ß = 2 • R_{S} / D, where D = 1 AU has the value 0.0081 arc seconds and half is still 4 thousandths of an arc second, i.e., four times the accuracy of the Hipparcos satellite!

"He denies the Big Bang Theory !"

Oswald Huber, Neue Zürcher Zeitung, Sunday Edition, November 12, 2006

© Oswald Huber & NZZ