Binary puls ars

Another system in which the pericenter shift is observable is the Hulse-Taylor binary pulsar system PSR B1913+16 that we introduced in § 9.3. While this is not a “test particle” orbiting a spherical star, but is rather two roughly equal-mass stars orbiting their common center of mass, the pericenter shift (in this case, the periastron shift) still happens in the same way. The two neutron stars of the Hulse-Taylor system have mean separation 1.2 × 109 m, so using Eq. (11.40) with M = 1.4 M+ = 2.07 km gives a crude estimate of φ = 3.3 × 10−5 radians per orbit = 2◦.1 per year. This is much easier to measure than Mercury’s shift! In fact, a more careful calculation, taking into account the high eccentricity of the orbit and the fact that the two stars are of comparable mass, predict 4◦.2 per year.

For our purposes here we have calculated the periastron shift from the known masses of the star. But in fact the observed shift of 4.2261◦ ± 0.0007 per year is one of the data which enable us to calculate the masses of the neutron stars in the PSR B1913+16 system. The other datum is another relativistic effect: a redshift of the signal which results from two effects. One is the special-relativistic ‘transverse-Doppler’ term: the 0(v)2 term in Eq. (2.39). The other is the changing gravitational redshift as the pulsar’s eccentric orbit brings it in and out of its companion’s gravitational potential. These two effects are observationally indistinguishable from one another, but their combined resultant redshift gives one more number which depends on the masses of the stars. Using it and the periastron shift and the Newtonian mass function for the orbit allows us to determine the stars’ masses and the orbit’s inclination (see Stairs 2003).

While PSR B1913+16 was the first binary pulsar to be discovered, astronomers now know of many more (Lorimer 2008, Stairs 2003). The most dramatic system is the so-called double pulsar system PSR J0737-3039, the first binary discovered in which both members are seen as radio pulsars (Kramer et al. 2006). In this system the pericenter shift is around 17◦ per year! Because both pulsars can be tracked, this system promises to become the best testing ground so far for the deviations of general relativity from simple Newtonian behavior. We discuss some of these now.

Post-Newtonian gravity

The pericenter shift is an example of corrections that general relativity makes to Newtonian orbital dynamics. In going from Eq. (11.37) to Eq. (11.38), we made an approximation that the orbit was ‘nearly Newtonian’, and we did a Taylor expansion in the small quantity

2 ˜ 2 −

M /L , which by Eq. (11.20) is (M/r)(1 3M/r). This is indeed small if the particle’s orbit is far from the black hole, r M. So orbits far from the hole are very nearly like Newtonian orbits, with small corrections like the pericenter shift. These are orbits where, in Newtonian language, the gravitational potential M/r is small, as is the particle’s orbital velocity, which for a circular orbit is related to the potential by v2 = M/r. Far from even an extremely relativistic source, gravity is close to being Newtonian. We say that the pericenter shift is a post-Newtonian effect, a correction to Newtonian motion in the limit of weak fields and slow motion.

We showed in § (8.4) that Newton’s field equations emerge from the full equations of general relativity in this same limit, where the field (we called it h in that calculation) is weak and the velocities small. It is possible to follow that approximation to higher order, to keep the first post-Newtonian corrections to the Einstein field equations in this same limit. If we did this, we would be able to show the result we have asserted, that the pericenter shift is a general feature of orbital motion. We showed in that section that the metric of spacetime that describes a Newtonian system with gravitational potential φ is, to first order in φ, given by the metric

For comparison, let us take the limit for weak fields of the Schwarzschild metric given in Eq. (11.1:

where we have expanded grr and kept only the lowest order term in M/r. These two metrics look similar – their g00 terms are identical – but they appear not to be identical, because if we transform the spatial line element of Eq. (11.43) to polar coordinates in the usual way, we would expect to see in Eq. (11.44) the spatial line element

(Readers who want to be reminded about the transformation from Cartesian to spherical coordinates in flat space should see Eq. (6.19) and Exer. 28, § 6.9.) Is this difference from the spatial part of Eq. (11.44) an indication that the Schwarzschild metric does not obey the Newtonian limit? The answer is no – we are dealing with a coordinate effect.

Coordinates in which the line element has the form of Eq. (11.45) are called isotropic coordinates because there is no distinction in the metric among the directions x, y, and z. Isotropic coordinates are related to Schwarzschild coordinates by a change in the definition of the radial coordinate. If we call the isotropic coordinates (t, r¯, θ , φ), where (t, θ , φ) are the same as in Schwarzschild, then we make the simple transformation for large r given by

Then to lowest order in M/r or M/r¯, the expression 1 + 2M/r in Eq. (11.44) equals 1 − 2M/r¯. But the factor in front of d 2 is r2, which becomes (again to first order) r¯2(1 + 2M/r). It follows that this simple transformation changes the spatial part of the line element of Eq. (11.44) to Eq. (11.45) in terms of the radial coordinate r¯. This demonstrates that the far field of the Schwarzschild solution does indeed conform to the form in Eq. (8.50).

There are of course many other post-Newtonian effects in general relativity besides the pericenter shift. In the next paragraph we will explore the deflection of light. Later in this chapter we will meet the dragging of inertial frames due to the rotation of the source of gravity (also called gravitomagnetism). It is possible to expand the Einstein equations beyond the first post-Newtonian equations and find second and higher post-Newtonian effects. Physicists have put considerable work into doing such expansions, because they

provide highly accurate predictions of the orbits of inspiralling neutron stars and black holes, which are needed for the detection of the gravitational waves they emit (recall Ch. 9).

The post-Newtonian expansion assumes not just weak gravitational fields but also slow velocities. The effect of this is that the Newtonian (Keplerian) motion of a planet depends only on the g00 part of the metric. The reason is that, in computing the total elapsed proper time along the world line of the particle, the spatial distance increments the particle makes (for example dr) are much smaller than the time increments dt, since dr/dt 1. Recall that g00 is also responsible for the gravitational redshift. It follows, therefore, that Newtonian gravity can be identified with the gravitational redshift: knowing one determines the other fully. Newtonian gravity is produced exclusively by the curvature of time in spacetime.

Spatial curvature comes in only at the level of post-Newtonian corrections.

Gravitational deflection of light

In our discussion of orbits we treated only particles, not photons, because photons do not have bound orbits in Newtonian gravity. In this section we treat the analogous effect for photons, their deflection from straight-line motion as they pass through a gravitational field. Historically, this was the first general-relativistic effect to have been predicted before it was observed, and its confirmation in the eclipse of 1919 (see McCrae 1979) made Einstein an international celebrity. The fact that it was a British team (led by Eddington) who made the observations to confirm the theories of a German physicist incidentally helped to alleviate post-war tension between the scientific communities of the two countries. In modern times, the light-deflection phenomenon has become a key tool of astronomy, as we describe in a separate paragraph on gravitational lensing below. But first we need to understand how and why gravity deflects light.

We begin by calculating the trajectory of a photon in the Schwarzschild metric under the assumption that M/r is everywhere small along the trajectory. The equation of the orbit is the ratio of Eq. (11.8) to the square root of Eq. (11.12):

where we have defined the impact parameter,

In Exer. 1, § 11.7, it is shown that b would be the minimum value of r in Newtonian theory, where there is no deflection. It therefore represents the ‘offset’ of the photon’s initial trajectory from a parallel one moving purely radially. An incoming photon with L > 0 obeys the equation

The initial trajectory has y → 0, so φ → φ0: φ0 is the incoming direction. The photon reaches its smallest r when y = 1/b, as we can see from setting dr/dλ = 0 in Eq. (11.22) and using our approximation Mu 1. This occurs at the angle φ = φ0 + 2M/b + π/2. It has thus passed through an angle π/2 + 2M/b as it travels to its point of closest approach. By symmetry, it passes through a further angle of the same size as it moves outwards from its point of closest approach (see Fig. 11.5). It thus passes through a total angle of π + 4M/b. If it were going on a straight line, this angle would be π , so the net deflection is

To the accuracy of our approximations, we may use for b the radius of closest approach rather than the impact parameter L/E. For the sum, the maximum effect is for trajectories for which b = R+, the radius of the Sun. Given M = 1 M+ = 1.47 km and R+ = 6.96 × 105 km, we find