K. Rajesh Nayak

Algebraic approach to time-delay data analysis for LISA

Cancellation of laser frequency noise in interferometers is crucial for attaining the requisite sensitivity of the triangular three-spacecraft LISA configuration. Raw laser noise is several orders of magnitude above the other noises and thus it is essential to bring it down to the level of other noises such as shot, acceleration, etc. Since it is impossible to maintain equal distances between spacecrafts, laser noise cancellation must be achieved by appropriately combining the six beams with appropriate time delays. It has been shown in several recent papers that such combinations are possible. In this paper, we present a rigorous and systematic formalism based on algebraic geometrical methods involving computational commutative algebra, which generates in principle all the data combinations canceling the laser frequency noise. The relevant data combinations form the first module of syzygies, as it is called in the literature of algebraic geometry. The module is over a polynomial ring in three variables, the three variables corresponding to the three time delays around the LISA triangle. Specifically, we list several sets of generators for the module whose linear combinations with polynomial coefficients generate the entire module. We find that this formalism can also be extended in a straightforward way to cancel Doppler shifts due to optical bench motions. The two modules are in fact isomorphic. We use our formalism to obtain the transfer functions for the six beams and for the generators. We specifically investigate monochromatic gravitational wave sources in the LISA band and carry out the maximization over linear combinations of the generators of the signal-to-noise ratios with the frequency and source direction angles as parameters.

Fundamentals of the LISA stable flight formation

The joint NASA–ESA mission, LISA, relies crucially on the stability of the three-spacecraft constellation. Each of the spacecraft is in heliocentric orbit forming a stable triangle. In this paper we explicitly show with the help of the Clohessy–Wiltshire equations that any configuration of spacecraft lying in the planes making angles of ±60° with the ecliptic and given suitable initial velocities within the plane, can be made stable in the sense that the inter-spacecraft distances remain constant to first order in the dimensions of the configuration compared with the distance to the Sun. Such analysis would be useful in order to carry out theoretical studies on the optical links, simulators, etc.

Black holes in nonflat backgrounds: the Schwarzschild black hole in the Einstein Universe

As an example of a black hole in a non-flat background a composite static spacetime is constructed. It comprises a vacuum Schwarzschild spacetime for the interior of the black hole across whose horizon it is matched onto the spacetime of Vaidya representing a black hole in the background of the Einstein universe. The scale length of the exterior sets a maximum to the black hole mass. To obtain a non-singular exterior, the Vaidya metric is matched to an Einstein universe. The behavior of scalar waves is studied in this composite model.

Gyroscopic Precession and Inertial Forces in Axially Symmetric Stationary Spacetimes

We study the phenomenon of gyroscopic precession and the analogues of inertial forces within the framework of general relativity. Covariant connections between the two are established for circular orbits in stationary spacetimes with axial symmetry. Specializing to static spacetimes, we prove that gyroscopic precession and centrifugal force both reverse at the photon orbits. Simultaneous non-reversal of these in the case of stationary spacetimes is discussed. Further insight is gained in the case of static spacetime by considering the phenomena in a spacetime conformal to the original one. Gravi-electric and gravi-magnetic fields are studied and their relation to inertial forces is established.

Gyroscopic precession and inertial forces in the Kerr - Newman spacetime

The phenomenon of gyroscopic precession in the Kerr - Newman spacetime is studied using the Frenet - Serret formalism. General formulae are obtained for circular quasi-Killing trajectories. The motion on the equatorial plane and along the geodesics are investigated as special cases. Expressions are obtained for the general relativistic analogues of inertial forces such as gravitational, Coriolis - Lense - Thirring and centrifugal forces in the Kerr - Newman spacetime. Reversal of gyroscopic precession and the centrifugal force is considered on the equatorial plane. These phenomena are also examined in the Reissner - Nordstrom spacetime by setting the angular parameter equal to zero. In this case the Coriolis force vanishes identically and both gyroscopic precession and the centrifugal force reverse at the circular photon orbit.

Time-delay interferometry for LISA with one arm dysfunctional

In order to attain the requisite sensitivity for LISA, laser frequency noise must be suppressed below the secondary noises such as the optical path noise, acceleration noise etc. In a previous paper (Dhurandhar et al., Class. Quantum Grav., 27, 135013, 2010), we have found a large family of second generation analytic solutions of time delay interferometry with one arm dysfunctional and also estimated the laser noise due to residual time-delay semi-analytically from orbit perturbations due to Earth. Since other planets and solar-system bodies also perturb the orbits of LISA spacecraft and affect the time delay interferometry (TDI), we simulate the time delay numerically in this paper for all solutions with n \leq 3. To conform to the actual LISA planning, we have worked out a set of 3-year optimized mission orbits of LISA spacecraft starting at June 21, 2021 using CGC2.7 ephemeris framework. We then use this numerical solution to calculate the residual optical path differences in the second generation solutions of our previous paper, and compare with the semi-analytic error estimate. The accuracy of this calculation is better than 1 cm (or 30 ps). The maximum path length difference, for all configuration calculated, is below 1 m (3 ns). This is well below the limit under which the laser frequency noise is required to be suppressed.

Gyroscopic Precession and Centrifugal Force in the Ernst Spacetime

The phenomenon of gyroscopic precession in the Ernst spacetime is studied within the framework of the Frenet-Serret formalism. General formulae are obtained for circular orbits. At the same time general relativistic analogues of inertial forces such as gravitational and centrifugal forces are also investigated in the Ernst spacetime. Reversal of gyroscopic precession as well as centrifugal force is considered at the circular photon orbits. These phenomena are examined in the Melvin universe as a special case of the Ernst spacetime by setting the mass parameter equal to zero.

Equilibrium of a charged test particle in the Kerr - Newman spacetime: force analysis

In a previous work, we studied the equilibrium of a charged test particle placed in the combined gravitational and electromagnetic fields of a Kerr - Newman black hole. Recently, general relativistic analogues of inertial forces have been formulated. In this paper we analyse the inertial and the electromagnetic forces acting on the stationary test particle. This leads to an understanding of the behaviour of these forces as well as the mechanism of equilibrium.

General relativistic treatment of LISA optical links

LISA is a joint space mission of the NASA and the ESA for detecting low-frequency gravitational waves in the band 10−5 to 1 Hz. In order to attain the requisite sensitivity for LISA, the laser frequency noise must be suppressed below the other secondary noises such as the optical path noise, acceleration noise, etc. This is achieved by the technique called time delay interferometry (TDI) in which the data are combined with appropriate time delays. In this paper we approximately compute the spacecraft orbits in the gravitational field of the Sun and Earth. We have written a numerical code which computes the optical links (time delays) in the general relativistic framework within an accuracy of ~10 m, which is sufficient for TDI. Our computation of the optical links automatically takes into account the effects such as the Sagnac, Shapiro delay, etc. We show that by optimizing LISA orbits, and using the symmetries inherent in the configuration of LISA and in the physics, the residual laser noise in the modified first-generation TDI can be adequately suppressed. We demonstrate our results for some important TDI observables.

Algebraic approach to time-delay data analysis: orbiting case

The laser phase noise is one of the dominant noises for the gravitational wave detector LISA. Since it is impossible to maintain equal distances among spacecraft, the time-delay interferometric (TDI) techniques are used to eliminate the laser phase noise along with optical bench noise. In this work, we estimate the effects due to the Sagnac phase by taking the realistic model for LISA orbital motion. We extend the algebraic formalism to include the effects due to the Sagnac phase.