Preface to curvature

5.1 O n t h e re l a t i o n o f g ra v i t a t i o n t o c u r va t u re

Until now we have discussed only SR. In SR, forces have played a background role, and we have never introduced gravitation explicitly as a possible force. One ingredient of SR is the existence of inertial frames that fill all of spacetime: all of spacetime can be described by a single frame, all of whose coordinate points are always at rest relative to the origin, and all of whose clocks run at the same rate relative to the origin’s clock. From the fundamental postulates we were led to the idea of the interval s2, which gives an invariant geometrical meaning to certain physical statements. For example, a timelike interval between two events is the time elapsed on a clock which passes through the two events; a spacelike interval is the length of a rod that joins two events in a frame in which they are simultaneous. The mathematical function that calculates the interval is the metric, and so the metric of SR is defined physically by lengths of rods and readings of clocks. This is the power of SR and one reason for the elegance and compactness of

tensor notation in it (for instance the replacement of ‘number density’ and ‘flux’ by N). On a piece of paper on which had been plotted all the events and world lines of interest in some coordinate system, it would always be possible to define any metric by just giving its components gαβ as some arbitrarily chosen set of functions of the coordinates. But this arbitrary metric would be useless in doing physical calculations. The usefulness of ηαβ is its close relation to experiment, and our derivation of it drew heavily on the experiments.

This closeness to experiment is, of course, a test. Since ηαβ makes certain predictions about rods and clocks, we can ask for their verification. In particular, is it possible to construct a frame in which the clocks all run at the same rate? This is a crucial question, and we shall show that in a nonuniform gravitational field the answer, experimentally, is no. In this sense, gravitational fields are incompatible with global SR: the ability to construct a global inertial frame. We shall see that in small regions of spacetime – regions small enough that nonuniformities of the gravitational forces are too small to measure – we can always construct a ‘local’ SR frame. In this sense, we shall have to build local SR into a more general theory. The first step is the proof that clocks don’t all run at the same rate in a gravitational field.

The gravitational redshift experiment

Let us first imagine performing an idealized experiment, first suggested by Einstein. (i) Let a tower of height h be constructed on the surface of Earth, as in Fig. 5.1. Begin with a particle of rest mass m at the top of the tower. (ii) The particle is dropped and falls freely with acceleration g. It reaches the ground with velocity v = (2gh)1/2, so its total energy E, as measured by an experimenter on the ground, is m + 12 mv2 + 0(v4) = m + mgh + 0(v4).

(iii)The experimenter on the ground has some magical method of changing all this energy into a single photon of the same energy, which he directs upwards. (Such a process does not violate conservation laws, since Earth absorbs the photon’s momentum but not its energy, just as it does for a bouncing rubber ball. The student skeptical of ‘magic’ should show how the argument proceeds if only a fraction ε of the energy is converted into a photon.)

(iv)Upon its arrival at the top of the tower with energy E , the photon is again magically changed into a particle of rest mass m = E . It must be that m = m; otherwise, perpetual

motion could result by the gain in energy obtained by operating such an experiment. So we are led by our abhorrence of the injustice of perpetual motion to predict that E = m or, for

the photon,

We predict that a photon climbing in Earth’s gravitational field will lose energy (not surprisingly) and will consequently be redshifted.

Although our thought experiment is too idealized to be practical, it is possible to measure the redshift predicted by Eq. (5.1) directly. This was first done by Pound and Rebka (1960) and improved by Pound and Snider (1965). The experiment used the Mössbauer effect to

m

γ

h

obtain great precision in the measurement of the difference ν − ν produced in a photon climbing a distance h = 22.5 m. Eq. (5.1) was verified to approximately 1% precision.

With improvements in technology between 1960 and 1990, the gravitational redshift moved from being a small exotic correction to becoming an effect that is central to society: the GPS navigation system incorporates vital corrections for the redshift, in the absence of which it would not remain accurate for more than a few minutes. The system uses a network of high-precision atomic clocks in orbiting satellites, and navigation by an apparatus on Earth is accomplished by reading the time-stamps on signals received from five or more satellites. But, as we shall see below, the gravitational redshift implies that time itself runs slightly faster at the higher altitude than it does on the Earth. If this were not compensated for, the ground receiver would soon get wrong time-stamps. The successful operation of GPS can be taken to be a very accurate verification of the redshift. See Ashby (2003) for a full discussion of relativity and the GPS system.

This experimental verification of the redshift is comforting from the point of view of energy conservation. But it is the death-blow to our chances of finding a simple, specialrelativistic theory of gravity, as we shall now show.

Nonexistence of a Lorentz frame at rest on Earth