General Relativity and Gravitational Waves

Index

CHAPTER 1

The Equivalence Principle

This assumption of exact physical equivalence makes it impossible for us to speak of the absolute acceleration of the system of reference, just as the usual theory of relativity forbids us to talk of the absolute velocity of a system.

A. Einstein

1.1 The Eötvos Experiment

From the time of Newton it had been assumed that the ratio of the inertial mass to the weight (gravitational mass) of a body is the same for all substances. If we assume this and write the equations of motion for a body in the earth’s gravitational field, the mass cancels out and all freely falling bodies have the same acceleration.

In 1890 Eötvos (¹) performed an ingenious experiment designed to test the ratio of inertial mass to weight. Consider a mass on the earth’s surface (Fig. 1.1). There is a gravitational force G acting toward the earth’s center, and an inertial force I which is the centrifugal force associated with the earth’s rotation. The ratio of the two magnitudes, and of the corresponding components of these forces, depends on the ratio of the gravitational to the inertial mass. Eötvos suspended two masses from a torsion balance, as shown in Fig. 1.2, at a latitude about midway between the equator and the pole. Suppose matters are arranged so that the balance is in equilibrium with the rod connecting the masses in the observer’s horizontal plane and pointing in the east-west direction. We can first conclude that the net torque component resulting from the vertical components of the resultant forces G + I on the two bodies is zero. If the ratio of inertial to gravitational mass is not the same for both, then the horizontal components of G + I will give rise to a torque which is canceled by an opposite torque of the suspension wire. If now the entire apparatus is rotated through the angle π, the bodies are interchanged and the sign of the torque associated with the horizontal components of G + I will reverse. The torque of the suspension wire, however, remains the same. The result is that an angular deflection of the rod and masses relative to the frame of the apparatus will be observed if the ratio of inertial to gravitational mass is not the same for both bodies.

Fig. 1.1

Fig. 1.2

Let the gravitational mass of one of the bodies be M1 and let its inertial mass be m1. Let ir be a unit vector from the body to the center of the earth, and let im be a unit vector in the plane of the meridian, normal to the earth’s axis of rotation. Let ge be the magnitude of the earth’s gravitational field. Then the gravitational force G1 is given by

(1.1)

Let a be the earth’s radius, let ω be its angular velocity, and let ϕ be the latitude. Then the inertial (centrifugal) force I1 is given by

(1.2)

Suppose the second body has gravitational mass M2 and inertial mass m2. We compare the forces on it with those on the first body, by means of the torsion balance. Assume that M1 and M2 are so chosen that the rod can be suspended in the center. Let the rod be represented by the vector b, and let the torque be denoted by T. We can write

(1.3)

The resultant of the four forces must be in the direction of the thin wire which supports the rod, and is given by

(1.4)

The component of the torque parallel to the supporting wire will tend to cause an observable rotation. Employing the preceding expressions enables us to write for the effective torque

(1.5)

In (1.5) we have omitted the centrifugal force in the denominator since it is very small in comparison with the gravitational force. Evaluating (1.5) and making the substitutions,

(1.6)

gives, for the effective torque,

(1.7)

Expression (1.7) vanishes if α1 = α2, and for α1 ≠ α2 it will have a value which depends on the orientation of the rod b, with respect to the vector ir × im, which is normal to the meridian plane. It has its maximum when b points in the east-west direction. As we remarked earlier, the torsion balance is brought to equilibrium by turning it until the rod points in the east-west direction, in a plane tangent to the earth’s surface. Then the apparatus is rotated through the angle π, reversing the sense of b; if α1 ≠ α2 there will be a torque which may then give a rotation of the rod relative to the frame which supports the balance. Eötvos observed no rotation and concluded that within one part in 10⁸, α1 = α2 for all the materials which were tested. This experiment has been repeated. (², ³) The work of Southerns was done with pendulums and demonstrated the equality of α for radioactive materials. Professor R. H. Dicke (⁴) is now repeating the Eötvos experiment with greatly refined apparatus employing three bodies, and a threefold axis of symmetry, to minimize local disturbances. At this time his results agree with those of Eötvos, and the equality of α for certain substances is established to a few parts in 10¹⁰.

The Eötvos experiment enables certain conclusions to be drawn concerning the elementary particles. The ratio of mass to weight for an electron plus a proton may be shown to be the same as for the neutron to one part in 10⁷, and the reduction in mass of a nucleus resulting from nuclear binding forces can be shown to one part in 10⁵ to be accompanied by a similar reduction in weight. With an accuracy of five parts in a thousand it can be concluded that the binding energy of the orbital electrons is accompanied also by a corresponding change in weight.

Bondi (⁵) notes a possible distinction between mass which is acted upon and mass which is the source of a gravitational field. The mass which is acted upon he calls passive gravitational mass, and a mass which is a source is called active gravitational mass. The Eötvos experiment, in this view, determines the equality of the ratio of inertial and passive gravitational mass.

1.2 Negative Mass

Nothing in either Newtonian or relativistic gravitation theory precludes the existence of negative mass, but it is an empirical fact that it has never been observed. Both Newtonian gravitation theory and general relativity indicate a quite different behavior for negative mass than for the corresponding situation in electrodynamics. If a small negative mass interacts with a large positive mass, again the (negative) mass cancels out on both sides of the equation of motion and the acceleration is still toward the positive mass. Thus a positive mass attracts all other masses, both positive and negative. A small negative mass would be expected to fall in the earth’s gravitational field. Similarly a negative mass repels all other masses, regardless of their sign. For a pair of bodies, one with positive mass and one with negative mass, with magnitudes about equal, we should expect the positive mass to attract the negative mass and the negative mass to repel the positive mass so that one chases the other! If the motion is confined to the line of centers the pair is expected to move with uniform acceleration. This problem has been discussed by Bondi (⁵).

Schiff (⁶) has recently considered the possibility that the gravitational mass of an anti particle, the positron, might be negative. His arguments are based on the renormalized quantum electrodynamics. The Coulomb field of an atomic nucleus produces a polarization of the vacuum. This effect, first calculated by Uehling (⁷) for hydrogen, produces a 27 Mc contribution to the Lamb shift of the 2S state in hydrogen. The virtual electron-positron pairs associated with vacuum polarization would be expected to contribute to the renormalized mass of atoms. We know from experiment that the inertial mass of the positron is positive. If the gravitational mass were negative, different atoms would be expected to have slightly different ratios of inertial to (passive) gravitational mass. This follows because the relative contribution of virtual pairs to the mass would depend on the nuclear charge and its distribution. This would vary for different atoms. For the case where the gravitational rest mass of a positron is assumed equal in magnitude and opposite in sign to that of a negative electron, but its kinetic energy is acted upon normally by a gravitational field, the difference between gravitational mass and renormalized inertial mass is finite and approximately equal (⁶) to

Here m is the electron mass, μ = mc/ , Z is the atomic number, and F(q) is the Fourier transform for the nuclear charge distribution, normalized to unit total charge. This expression has a ratio to the atomic mass of 10–7 , 2 × 10–7 , and 4.3 × 10–7 for aluminum, copper, and platinum, respectively. Since these numbers are larger than the uncertainties in the mass ratios determined by Eötvos, Schiff concludes that the possibility that the gravitational mass of the positron is negative is ruled out.

It is likely that some experiment to see if anti neutrons fall in the earth’s gravitational field may be attempted. As we remarked earlier, if existing theories of gravitation are accepted they will be expected to fall, in any case.

1.3 Equivalence of Different Frames of Reference

The empirical fact that the two kinds of mass are equivalent did not fit anywhere in theoretical physics until Einstein (′, whose z ′ ′ be at rest at t = 0 when a light pulse of energy Eα is emitted from point α. This light is absorbed at point β′ is gl/c.

Fig. 1.3

The light had energy Eα and momentum Eα/c. The energy at point β can be obtained by use of the Lorentz transformation and is given by

(1.8)

where

Evaluating (1.8) gives

(1.9)

frame. Imagine a mass M then moved to β. Light of energy Eα is emitted at α and absorbed by M at β. The total gravitational mass of M plus the absorbed light is M′. Now lift M′ back to α and re-emit light so that the mass at α is again M. There is no net change of energy during the process, so the change in energy going from α to β can be set equal to the energy change on the return.

(1.10)

Making use of (1.9) then leads to

(1.11)

Expression (1.11) states that the increment in gravitational mass is the change in the inertial mass, and it therefore follows that the equivalence of mass and weight can be considered as a consequence of the equivalence of an accelerated frame to a gravitational field.

1.4 Gravitational Red Shift of Spectral Lines

It follows also from the equivalence principle that we should expect a gravitational red shift of spectral lines. For consider again the emission of light at α ′, which is momentarily at rest. The light of frequency ν is received by an observer at β who measures the frequency in units of his own proper time. The Doppler shift in the frequency at β gives rise to

(1.12)

and

(1.13)

In the equivalent gravitational field, (1.13) again holds, and

(1.14)

The quantity gl in (1.14) is the change in the gravitational potential, and we write the frequency shift νβ–να as

(1.15)

In (1.15) ϕβ is the gravitational potential (a negative quantity) at the point where the light is received and ϕα is the gravitational potential at the point where the light is emitted. For light received on earth from a star, (ϕβ > ϕα• If M is the star’s mass,