## Tests of general relativity

Over the years since Einstein's publication of general relativity, a number of exacting tests have been carried out to test observational predictions of the theory. Some of the tests are really only tests of the principle of equivalence, while others are true tests of the full theory.

A direct test of the principle of equivalence involves the measurement of the attraction of two different objects by some third body. A class of such experiments are called Eotvos experiments, after the person who devised the original experiment around the turn of the 20th century. The most accurate recent versions of the experiment were carried out by a group at Princeton University in the 1960s and a group at Moscow University in the 1970s. Their findings indicate that the principle of equivalence is accurate to one part in 1011.

The equivalence principle we have discussed applies strictly to objects that are so small that we can ignore the differences from one side to the other in the gravitational field they feel. We can treat them as point objects. However, there is a stronger form of the principle of equivalence that says that it also applies to objects with substantial gravitational binding energy, such as planets or stars. This has been tested by closely measuring the motion of the Moon (Fig. 8.6). A series of mirrors have been left on the Moon by the Apollo astronauts. Laser signals can be sent from Earth, bounced off these small mirrors, and then detected as very weak return signals. By timing the round trip we can measure the distance to the Moon very accurately, to within a few centimeters. These studies have indicated that the Earth and Moon fall towards the Sun with the same acceleration to within seven parts in 1012.

8.3.1 Orbiting bodies

One series of tests of general relativity involves the behavior of orbiting bodies. The paths are slightly different than predicted by Newtonian

The 2.7 m telescope of the McDonald Observatory,Texas, has been used to fire a laser beam at a reflector on the Moon, then they detect the weak return. By timing the round trip, the distance to the Moon is very accurately determined. [McDonald Observatory]

gravitation. An important feature involves elliptical orbits. In an elliptical orbit, the distance of the orbiting body from the body exerting the force is changing. The orbiting body is therefore passing through regions of different space-time curvature. (See Fig. 8.7, which may help in visualizing this.) The effect of the changing curvature is to cause the orbit not to close. After each orbit, the position of perihelion (closest approach) has moved around slightly.

The effect will be greatest for orbits of highest eccentricity, since the widest range of curvatures will be covered. Also, the smaller the semi-major axis, the greater the effect. This is because the gravitational field changes faster with distance when you are closer to the object exerting the force. In the Solar System, both of these points make the effect most pronounced for Mercury. (a) Curved space-time for Mercury's orbit around the Sun.The closer to the Sun you get, the greater the curvature of space-time. Since Mercury's orbit is elliptical, its distance from the Sun changes. It therefore passes through regions of different curvature. (b) This causes the orbit to precess.We can keep track of the precession by noting the movement in the perihelion, designated P|, P2 and P3 for three successive orbits. (The amount of the shift is greatly exaggerated.)

(a) Curved space-time for Mercury's orbit around the Sun.The closer to the Sun you get, the greater the curvature of space-time. Since Mercury's orbit is elliptical, its distance from the Sun changes. It therefore passes through regions of different curvature. (b) This causes the orbit to precess.We can keep track of the precession by noting the movement in the perihelion, designated P|, P2 and P3 for three successive orbits. (The amount of the shift is greatly exaggerated.)

at this point it appears that there is not enough solar flattening to challenge Einstein's results.

### 8.3.2 Bending electromagnetic radiation

Einstein's chance to predict an effect that had not been seen came in the bending of light passing by the edge of the Sun. He said that the warping of space-time alters the path of light as it passes near the source of a strong gravitational field. According to general relativity, photons follow geodesics. The light will then appear to be coming from a slightly different direction. If the light is coming from a star, the position of the star will appear to be slightly different than if the bending had not taken place, as indicated in Fig. 8.8.

According to Einstein, the angle 0 (in radians) through which the light passing a distance b from an object of mass M is given by

It is closest to the Sun, and, except for Pluto, has the most eccentric orbit.

The perihelion of Mercury's orbit advances by some 5600 arc seconds per century. However, of this, all but 43 arc seconds per century can be accounted for by Newtonian effects and the perturbations due to motions of other planets. The Newtonian effects could be calculated accurately and subtracted off. Einstein was able to explain the 43 arc seconds per century exactly in his general relativity calculations. This was considered to be an interesting result for general relativity, but not a crucial test, since Einstein explained something that had been observed. A crucial test involves predicting things that haven't been observed yet.

In recent years a controversy has grown out of this test of general relativity. A group at Princeton in the 1960s measured the shape of the Sun and found a slight flattening. A flattened Sun would also have an effect on the orbit of Mercury, reducing the general relativistic effects by enough to say that Einstein's calculation is wrong. Further measurements have indicated that the original experiment on the Sun's shape was in error, but some experiments suggest that there is some flattening. While some of this research is continuing,

If we set b equal to the radius of the Sun (6.96 X 1010 cm) we get an angle of 8.47 X 10~6 rad, which is equal to 1.74 arc seconds. This is a very small angle and is hard to measure.

The measurement is made even more difficult by the fact that we cannot see stars close to the

Actual Apparent position of star

Sun position of star

Observer

Bending of starlight passing by the Sun.The observer thinks that the star is straight back along the received ray.

Sun on the sky. Therefore, the test must be made during a total eclipse of the Sun, when the sky is photographed, and then the same part of sky is photographed approximately six months later. The positions of the stars on the two photographs are then compared. The first attempt to carry this out was by a German team trying to get to a Russian viewing site for a 1914 eclipse. They were thwarted by the state of war between the two countries. The next try was in 1919, in an effort headed by Sir Arthur Eddington. In the intervening years, Einstein had found an error in his calculations, so it is probably just as well that the observations weren't done until the theoretical prediction was finalized. The result was a confirmation of Einstein's prediction. The recognition of the magnitude of Einstein's contribution was immediate, both among physicists and the general public.

The solar eclipse experiment is a hard one, and the original one had a 10% uncertainty associated with it. More recent tries have reduced the uncertainty to about 5%. Different types of experiments are needed for greater accuracy. A major improvement can be made by using radio waves. The bending applies equally to electromagnetic radiation of all wavelengths. The advantage of radio waves is that the Earth's atmosphere does not scatter them. We can observe any radio source as the Sun passes in front of it and watch the position of the source change. These tests have confirmed Einstein's predictions to greater accuracy than the eclipse experiments.

There is another effect related to the bending of light. The longer path that results from the curvature of space-time around the Sun causes a delay in the time for a signal to pass by the Sun. Two types of observations have been done to test this. One involves the reflection of radio waves from Mercury and Venus as they pass behind the Sun. We know the positions of the planets very accurately, so we know how long it should take for the signal to make a round trip. The other type of experiment involves spacecraft that have been sent to various parts of the Solar System, especially Mariners 6, 7 and 9, and Viking orbiters and landers on Mars. We simply follow the signals from the spacecraft. Since we know where the spacecraft should be, we can determine the time delay as the spacecraft pass behind the Sun. Using this technique, Einstein's predictions have been confirmed to an accuracy of 0.1%.

There is another interesting result related to the bending of the paths of electromagnetic waves. A massive object can bend rays so well that it can act as a gravitational lens. Physicists have speculated on this possibility for some time. Recent observations of quasars, to be discussed in Chapter 19, have revealed a number of sources in which double images are seen as a result of this gravitational lens effect (e.g. Fig. 8.1).

### 8.3.3 Gravitational redshift

The wavelengths of photons change as they pass through a gravitational field. This effect is called the gravitational redshift (Fig. 8.9). It is really a consequence of the principle of equivalence.

We can make a plausibility argument to estimate the magnitude of the effect. We have already seen in the previous section that the gravitational effect of some mass is to alter the trajectories of photons (i.e. they follow geodesics that are not straight lines). This makes it plausible that the gravitational field can do work on the photon, changing its energy. In order to estimate the gravitational potential energy of a photon (—GMm/r) we assign an "effective mass", E/c2, and since E = hc/X, this effective mass is h/cX. So if a photon moves from r1 out to r2, conservation of energy would give us hc _ GMh _ hc _ GMh Ai rtcAt A2 r2cA2

Solving for the ratio of the wavelengths gives