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Laguna and Wolszczan have produced a numerical model which describes the timing resid-uals caused by a rotating black-hole companion in a binary system with a pulsar. In partic-ular, the aim of their model is to identify the additional timing delay caused by black-hole rotation, which would be observed in addition to the Shapiro delay. The parameters of their system include black-hole mass, black-hole angular momentum, orbital inclination, binary orbital period and eccentricity. The approach taken differs from the numerical model in this thesis, as they specify a detector position and size, and they subsequently integrate photon trajectories along Kerr null geodesics backwards towards the binary system. They use Monte Carlo methods in conjunction with a fourth order Runge-Kutta integrator, with a simple form of adaptive stepsize scaling. They use a sample of 20000 photons to generate the timing residual data, describing the magnitude of the effect that rotation has on the timing delay measured for pulses. In this section I discuss their calculations and results.

They begin with the 1984 result presented by Dymnikova, [18], which gives the time taken for a photon travelling in the equatorial plane of the Kerr metric, to reach its maximum proximity to the black-hole: where MH is the mass of the black-hole, a is the rotational parameter, r is the point of origin for the incident photon and d its maximum proximity to the black-hole. Note that r >> d for this approximate integral of the equatorial Kerr null geodesic to hold true.

The t+ solution corresponds to the case of a photon corotating with the black-hole spin, so that the propagation time is reduced by the negative sign of the final term. The t− case corresponds to a contra-rotating photon, which must move against the flow of space-time rotating about the black-hole. Accordingly the positive sign for the last term increases the propagation time to the point of maximum proximity. I have tested the numerical model in part II of this thesis against this analytical result. The comparison is discussed in chapter 9.

Returning to the Laguna et al. paper, they describe the difference in propagation times on either side of the rotating black-hole. To do this, they isolate the final term in (2.2.1), which is the only term dependent on the rotational parameter of the black-hole, representing the contribution that the dragging of inertial reference frames makes to the delay or advance

of the signal. They calculate propagation times from emitter to maximum proximity, and subsequently maximum proximity to observer by using the results for t+ and t− cases, with the result trivially just adding a multiple of the final term in (2.2.1) for each portion of photon trajectory.

∆t =16aMH

d (2.2.2)

Laguna et. al. provide an order of magnitude calculation describing system parameters which enhance the detectability of this rotational effect without reducing the probability of finding such a system too significantly. They describe an elliptical orbit which is edge-on with respect to the observer, a black-hole of mass agreeing with the literature for proposed pulsar/black-hole binary systems, quoting Narayan et al. [19]. They scale the system by choosing a value for the semi-major axis which does not reduce the lifetime of the system below 107 years before merger due to emission of gravitational radiation. The superior conjunction occurs with the pulsar at the periastron of the ellipse, minimizing distance between pulsar and black-hole to maximize relativistic effects. The size of the semi-major axis was selected to be ap = 5R^J and ap = 10R^J. 20000 Photons were then integrated backwards, via 4th order Runga-Kutte (RK), from a distant detector to the plane of the pulsar’s orbit. Only photons which moved within 10⁻⁴apof the pulsar’s orbit were analyzed, with the timing residuals formed by comparison of propagation times through flat spacetime (i.e. the first term in (2.2.1)). They reduce step size in the vicinity of the black-hole, but no details were given as to the mechanism of scaling.

Their numerical analysis uses a substantial post-processing routine. They form a uniform orbital phase grid by iterpolation of timing data, which is recorded by folding data about the azimuthal angle reached at closest approach for pair points (on either side of the blackhole) which would have identical travel times without the presence of the black-hole (flat space-time). The propagation times are then subtracted. Photons are allowed to travel slightly out of the equatiorial plane. Finally, they show that their results agree with the estimate of equation (2.2.2), after having re-formulated the impact parameter d for the case of an elliptical orbit. Note that equation (2.2.2) was derived from Dymnikova’s result, and is thus specific to equatorial photons. The photon trajectories integrated numerically do however allow for slight deviation out of the equatorial plane, hence a slight discrepancy may be expected. Ultimately, their data shows micro-second level differences in the propagation delays of pulses, caused by the rotation of the black-hole. The model presented in part II

could be used, with sufficient adjustment, to replicate the measurements of Laguna et al.

with even greater precision.