I3    The Shapiro Experiment

In I2 we saw that a beam of light that passes close to the sun barely changes its direction. On the other hand, the time needed for the traversal of a given path diameter grows significantly, because the light, in accordance with the formulas of G5, moves more slowly than it would when outside a gravitational field. Problem 5 of H7 illustrates the effect of spatial curvature, but a virtually equal proportion is also due to the curvature of space-time. It is again an effect that can be well understood viewed as ‘gravitation by refraction’.

In 1962 Irwin Shapiro suggested that this delay be measured by sending some strong radio signals to Venus, when it is in opposition to the Earth, and then measuring the time it takes for the (extremely weak) reflected signals to arrive.

When in 1964 the 120 foot Haystack antenna in Westford, US was left by the military to MIT, Shapiro and his team began plans to carry out the experiment. The experiment first took place from November 1966 until August 1967. "It would have been nice to prove Einstein wrong," said Shapiro later. That has not been granted him since all experiments up till 2006 have confirmed the GTR within the specified accuracy.

Shapiro lowered the imprecision of his initial measurements from over 3% to less than 1% in subsequent years. Newer versions of this experiment work with transponders on space probes. These receive the signal from the earth and after a precisely known delay send it with increased intensity back to earth. Thus with the Viking Mars probe of 1979 the predictions of the GTR for this delay in the gravitational field of the sun could be confirmed to an accuracy of 0.1%. In 2003, with the space probe Cassini an accuracy of 0.0012% was achieved!

An observer in OFF would in the situation presented on the left measure values of a = -498.67 and b = 370.70 (as in I2, we calculate everything in units of light seconds, so that c0 = 1 and α ≈ 4.9261 • 10-6). Without gravity one would expect a duration of 2 • (b - a) / c0 ≈ 2 • (370.70 + 498.67) / 1 ≈ 1738.74 seconds. In the following we calculate the difference in time that arises because the light near the sun travels a little bit slower.

With gravity the duration in both directions (with c0 = 1!) is

This integral is numerically very unstable. A substitution using 1 / (1-x) = (1 + x) / (1-x2) helps since we then eliminate in the denominator the very small x2 term:

This is the entire light travel time there and back with gravity. The difference to the expected value without gravity is

This integral can be evaluated both numerically and analytically. For r the root of D2 + y2 is used. For a path that passes directly through the solar limb,
D ≈ 2.33, and using a = -499 and b = 371 the calculator delivers for the total ‘delay’ the value of the signal ∆T ≈ 213.3 µs ≈ 0.000,213,3 seconds - an easily measurable value.

With help of an integral table, or a computer algebra system one can find an anti-derivative:

Setting the limits of integration and using additional symbols 

aV ~ Sun-Venus distance, yV ~ y-coordinate of Venus, yV > 0 
aE ~ Sun-Earth distance, yE ~ y-coordinate of Earth, yE < 0 
φV ~ angle   yAxis-Sun-Venus
φE ~ angle   negative yAxis-Sun-Earth

we get the following expression for the total amount of delay which also provides good values for a path at a great distance from the sun:

The 120 foot radio antenna at MIT in Westford / USA with which
Shapiro in 1966/67 carried out his first experiment.

In opposition the two angles φE and φV are very small and we may set the cosine value to 1. For this special situation this gives us the simpler formula

Even this simple formula provides for D = 2.33 (solar limb), aE = -499 and aV = 371 a delay of 213.3 µs.

These values ∆T are actually calculated for an observer in OFF. For earthlings we must multiply by a factor (1 - α / r) ≈ (1 - 4.9 • 10-6 / 491) because on the earth the clocks run slightly slower than in OFF. This correction affects only the eighth decimal place and can therefore be omitted.

For the Venus opposition of January 25, 1970 my astronomy program gives the values D ≈ 8.47, aE ≈ 491 and aV ≈ 363. Plugging these into our formula we get a delay of 160.4 µs. Shapiro measured, however, a maximum of 180 µs. This difference indicates that my value of D was slightly too big.

The Shapiro effect is also interesting because it decreases only slowly with increasing distance D from the central mass. The light deflection is in accordance with the formula in I2 proportional to 1 / D. The Shapiro delay, however, is essentially proportional to 1 / ln(D), as is seen from the formula above. At a distance of 100 solar radii the value of the deflection of light decreases to 1%, but the delay, there is still 21% of the maximum effect at the solar limb. One speaks, therefore, of a ‘long-range effect’.