Shapiro's Delay

In 1964 Irwin Shapiro published a paper with the elegantly descriptive title "Fourth Test of General Relativity". While attending a lecture in 1961 he had discerned the fact that I pointed out in my discussion of gravitational refraction of light - that a line drawn between two established points spans more kilometers if it passes close to the sun than it spans if it passes far from the sun. Shapiro gained from that fact the idea that radio signals passing the sun as they travel between planets would be delayed a short interval beyond the time that astronomers would normally calculate from the positions of the two planets and an assumption that space is not warped. To avoid confusion, I'll point out here that Einstein did not discover this effect: he worked out the description of the bending of light by using a highly abstract mathematical treatment that did not involve acknowledging, even tacitly, that the shrinkage of space by gravity increases the optical path length of the light passing the sun.

As we did with the gravitational deflection of starlight, we must account two separate phenomena in calculating the amount of time by which a radio signal passing close to the sun lags behind the transmission time calculated from purely Newtonian considerations. First we must account the shrinkage of space, as I indicated above, the more obvious of the two phenomena we must take into our account. And second we must take into our account the effect due to the gravitational dilation of time elapsed on clocks deep within the sun's gravitational field.

Over the past four centuries astronomers have so refined their knowledge of the planets' orbits that they can calculate the location of any given planet in space accurately enough to send spacecraft to the planet and to land the craft on that planet at a previously chosen location. Of course those calculations use Newtonian physics, so they provide us with an expectation against which we can test General Relativity. We might, for example, calculate the distance between Earth and Mars at some instant when the sun lies more or less between both planets and then use that calculation in an experimental test of the theory. Newtonian physics tells us that so many kilometers lie between the two planets and General Relativity tells us that somewhat more kilometers lie between the two planets. Because gravity shrinks space, in essence it pulls more kilometers into the gap between the given points.

In working out the shrinkage of space for calculating the deflection of starlight by gravitational refraction, I showed that the difference between the Newtonian length of a line drawn from some perihelion point to a point a distance X from it in a direction perpendicular to the radial line from the center of the sun to the perihelion point has the mathematical expression

(Eq'n 1)

Because the original line has shrunk by that much, we must insert that much extra distance between the two planets. But light crosses a kilometer in the same time, 3.33564 microsecond, regardless of whether the kilometer has shrunk or not, so that extra distance adds a delay to the time a radio signal takes in going from one planet to the other.

If, for example, we want to send a radio signal from Earth to Mars when the sun's limb lies just on the line between the two planets, then the signal will require an extra amount of transit time equal to

(Eq'n 2)

in which equation L_{1} represents
the extra distance that the sun's gravity inserts between Earth and the radio
signal's perihelion and L_{2} represents the extra distance that the
sun's gravity inserts between the signal's perihelion and Mars. For the
Earth-perihelion distance we have the mean value X_{1} = 149,570,000
kilometers, for the Mars-perihelion distance we have the mean value X_{2}
= 227,840,000 kilometers, and we have for the perihelion distance R = 696,000
kilometers. For the sun MG/c^{2} = 1.478 kilometer, so when we make the
appropriate substitutions into Equation 1, we obtain L_{1} = 8.96
kilometers and L_{2} = 9.58 kilometers for a total insertion of 18.54
kilometers. Multiplying that distance by 3.33564 microseconds per kilometer
gives us a calculated delay of 61.84 microseconds or 123.68 microseconds for a
round trip.

In addition to that delay we must include a delay due to the distortion of time within the sun's gravitational field. Pick some one kilometer along our radio signal's path, one close to the sun, and imagine a brief pulse of light traversing it, illuminating synchronized clocks at its ends. We know that the times that those clocks display must differ by 3.33564 microseconds. Now move far away from the sun along a line perpendicular to that measured kilometer so that you reach a point equidistant from both clocks and in a region of space effectively outside the sun's gravitational field. Because of gravitational time dilation flashes of light reflected from the clocks when the pulse passes them will reach that point a little over 3.33564 microseconds apart. That discrepancy accumulates over the length of the radio signal's path and adds a further delay to the one due to shrinkage of space.

Again bearing in mind that the radio signal's path lies almost entirely on our chosen x-axis, we can use the Schwarzschild Transformation of time to calculate how the time elapsed across the interval dX will appear to observers far from the sun. We have

(E'qn 3)

But that integrand yields an equation essentially the same as Equation 1; in particular, one that will yield the same amount of delay that we calculated via Equation 2. Thus, in our example, gravitational time dilation adds 61.84 microseconds to the one-way trip and 123.68 microseconds to the round-trip time elapsed between transmission of the signal from Earth and its reflection's return to Earth.

General Relativity thus tells us that a radio signal going from Earth to Mars and back should be delayed 247.36 microseconds over the time calculated from purely Newtonian considerations. On 1976 Nov 26 Mars, as seen from Earth, just grazed the southern limb of the sun. At that time two groups of researchers, on at the Massachusetts Institute of Technology and the other at the Jet Propulsion Laboratory in Pasadena, California, used the Project Viking landers on Mars as transponders in efforts to measure the Shapiro delay. By making measurements from July 1976 through November 1977, they were able to gather enough data to confirm the equality between the delay that Shapiro calculated and the actual delay to one part in one thousand, making this observation the best confirmation of General Relativity available so far.

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Sources

McVittie, G.C., "General Relativity and Cosmology", University of Illinois Press, 1965.

Will, Clifford M., "Was Einstein Right?", Basic Books, Inc., New York, 1986, ISBN 0-465-09088-5.

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