Measuring the stretching of space

The action of gravitational waves is sometimes characterized as a stretching of space. Eq. (9.24) makes it clear what this means: as the wave passes through, the proper separations of free objects that are simply sitting at rest change with time. However, students of general relativity sometimes find this concept confusing. A frequent question is, if space is stretched, why is a ruler (which consists, after all, mostly of empty space with a few electrons and nuclei scattered through it) not also stretched, so that the stretching is not measurable by the ruler? The answer to this question lies not in Eq. (9.24) but in the geodesic deviation equation, Eq. (9.26).

Although the two formulations of the action of a gravitational wave, Eqs. (9.24) and (9.26), are essentially equivalent for free particles, the second one is far more useful and instructive when we consider the behavior of particles that have other forces acting on them as well. The first formulation is the complete solution for the relative motion of particles that are freely falling, but it gives no way of including other forces. The second formulation is not a solution but a differential equation. It shows the acceleration of one particle (let’s call it B), induced by the wave, as measured in a freely falling local inertial frame that initially coincides with the motion of the other particle (let’s call it A). It says that, in this local frame, the wave acts just like a force pushing on B. This is called the tidal

force associated with the wave. This force depends on the separation . From Eq. (9.25), it

ξ

is clear that the acceleration resulting from this effective force is

Now, if particle B has another force on it as well, say FB, then to get the complete motion of B we must simply solve Newton’s second law with all forces included. This means solving the differential equation

where mB is the mass of particle B. Indeed, if particle A also has a force FA on it, then it will not remain at rest in the local inertial frame. But we can still make measurements in that frame, and in this case the separation of the two particles will obey

This equation allows us to treat material systems acted on by gravitational waves. The continuum version of it can be used to understand how a ‘bar’ detector of gravitational waves, which we will encounter below, reacts to an incident wave. And it allows us to answer the question of what happens to a ruler when the wave hits. Since the atoms in the ruler are not free, but instead are acted upon by electric forces from nearby atoms, the ruler will stretch by an amount that depends on how strong the tidal gravitational forces are compared to the internal binding forces. Now, gravitational forces are very weak compared to electric forces, so in practice the ruler does not stretch at all. In this way the ruler can be

corresponds to some exact solution of the nonlinear equations that has similar properties. We shall briefly derive such a solution.

Suppose the wave is to travel in the z direction. By analogy with Eq. (9.2), we might hope to find a solution that depends only on

u := t − z.

This suggests using u as a coordinate, as we did in Exer. 34, § 3.10 (with x replaced by z). In flat space it is natural, then, to define a complementary null coordinate

v = t + z,

so that the line element of flat spacetime becomes

ds2 = −du dv + dx2 + dy2.

Now, we have seen that the linear wave affects only proper distances perpendicular to its motion, i.e. in the x − y coordinate plane. So let us look for a nonlinear generalization of this, i.e. for a solution with the metric

ds2 = −du dv + f 2(u) dx2 + g2(u) dy2,

where f and g are functions to be determined by Einstein’s equations. It is a straightforward calculation to discover that the only nonvanishing Christoffel symbols and Riemann tensor components are

and others obtainable by symmetries. Here, dots denote derivatives with respect to u. The only vacuum field equation then becomes

We can therefore prescribe an arbitrary function g(u) and solve this equation for f (u). This is the same freedom as we had in the linear case, where Eq. (9.2) can be multiplied by an arbitrary f (kz) and integrated over kz to give the Fourier representation of an arbitrary function of (z − t). In fact, if g is nearly 1,

g ≈ 1 + ε(u),

so we are near the linear case, then Eq. (9.32) has a solution

f ≈ 1 − ε(a).

This is just the linear wave in Eq. (9.21), with plane polarization with hxy = 0, i.e. the polarization shown in Fig. 9.1(b). Moreover, it is easy to see that the geodesic equation

implies, in the nonlinear case, that a particle initially at rest on our coordinates remains at rest. We have, therefore, a simple nonlinear solution corresponding to our approximate linear one.

This solution is one of a class called plane-fronted waves with parallel rays. See Ehlers and Kundt (1962), § 2.5, and Stephani, et al. (2003), § 24.5.

Geometrical optics : waves in a curved spacetime

In this chapter we have made the simplifying assumption that our gravitational waves are the only gravitational field, that they are perturbations of flat spacetime, starting from Eq. (8.12). We found that they behave like a wave field moving at the speed of light in special relativity. But the real universe contains other gravitational fields, and gravitational waves have to make their way to our detectors through the fields of stars, galaxies, and the universe as a whole. How do they move?

The answer comes from studying waves as perturbations of a curved metric, of the form gαβ + hαβ , where g could be the metric created by any combination of sources of gravity. The computation of the dynamical equation governing h is very similar to the one we went through at the beginning of this chapter, but the mathematics of curved spacetime must be used. We won’t go into the details here, but it is important to understand qualitatively that the results are very similar to the results we would get for electromagnetic waves traveling through complicated media.

If the waves have short wavelength, then they basically follow a null geodesic, and they parallel-transport their polarization tensor. This is exactly the same as for electromagnetic waves, so that photons and gravitational waves leaving the same source at the same time will continue to travel together through the universe, provided they move through vacuum. For this geometrical optics approximation to hold, the wavelength has to be short in two ways. It must be short compared to the typical curvature scale, so that the wave is merely a ripple on a smoothly curved background spacetime; and its period must be short compared to the timescale on which the background gravitational fields might change. If nearby null geodesics converge, then gravitational and electromagnetic waves traveling on them will be focussed and become stronger. This is called gravitational lensing, and we will see an example of it in Ch. 11.

Gravitational waves do not always keep step with their electromagnetic counterparts. Electromagnetic waves are strongly affected by ordinary matter, so that if their null geodesic passes through matter, they can suffer additional lensing, scattering, or absorption. Gravitational waves are hardly disturbed by matter at all. They follow the null geodesics even through matter. The reason is the weakness of their interaction with matter, as we saw in Eq. (9.24). If the wave amplitudes hTTij are small, then their effect on any matter they pass through is also small, and the back-effect of the matter on them will be of the same order of smallness. Gravitational waves are therefore highly prized carriers of information from distant regions of the universe: we can in principle use them to ‘see’ into the centers of supernova explosions, through obscuring dust clouds, or right back to the first fractions of a second after the Big Bang.