Relativistic orbits and Gravitational Waves from gravitomagnetic corrections Fabio Garufi , Salvatore Capozziello, Mariafelicia De Laurentis, Luca Forte, Leopoldo Milano Dipartimento di Scienze Fisiche, Università di Napoli “Federico II” and INFN Sez. di Napoli

Gravitomagnetic effects: the weak field limit The weak field approximation of the gravitational field is

Numerical approach to the solution

The stress–energy tensor for perfect-fluid matter is given by

Our aim is to study how gravitomagnetic effects modify the orbital shapes and what are the parameters determining the stability of the problem. The energy, the mass and the angular momentum, essentially, determine the stability. Beside the standard periastron precession of General Relativity, a nutation effect is induced by gravitomagnetism and stability depends on it. The solution of the above system of differential equations presents some difficulties since the equations are stiff. For our purposes, we have found solutions by using the so called Stiffness Switching Method to provide an automatic mean of switching between a non-stiff and a stiff solver coupled with a more conventional explicit Runge-Kutta method for the non-stiff part of differential equations.

which, in the weak field approximation p<<ρc2, gives From the Einstein field equations, one obtains

To achieve these equations , the harmonic conditions have been used Integrating

GW amplitude with gravitomagnetic corrections The metric is determined by the gravitational Newtonian potential

The total power of emitted GW in a given solid angle Ω is given by

and by the vector potential the metric tensor in terms of scalar and vector

the geodesic equation follows

Blue: No gravitomagnetic corrections. Red: with gravitomagnetic corrections. Projection on the axis of maximal variance (singular value decomposition).

α μ ν &x&α + Γμν x& x& = 0

1 ∂Φ 2 ⎛ ∂ Vi ∂ V j ⎞ 0 0 0 ⎟ ; Γij = − 3 ⎜⎜ j + Γ00 = 0; Γ0 j = 2 j i ⎟ c ∂x c ⎝ ∂x ∂x ⎠ dots indicate differentiation with respect to the affine parameter ⎞ 1 ⎛⎜ ∂ Φ k ∂ Φ k ∂ Φ k ⎟ δ δ δ Γij = − 2 + + i j ij i ⎟ c ⎜⎝ ∂ x j ∂x k ∂x ⎠ k Considering only the spatial components, we obtain the orbit ∂V j ⎞ ⎛ 1 2 V ∂ ∂ Φ k k ⎟⎟ − ; Γ0 j = 3 ⎜⎜ equations. Calling dleuclid=ΦΒijdxidxj, and ek=dxk/dleuclid we have: Γ00 = 2 j c ∂xk c ⎝ ∂x ∂xk ⎠

retaining terms up to the orders Φ³/c2 and Vl/c3 but the procedure can be iterated up to higher order terms

de 2 4 = − 2 [∇φ − e(e ⋅ ∇φ )] + 3 [e × (∇ × V)] dleuclid c c The gravitomagnetic term is the second one and it is usually discarded since considered not relevant. This is not true if v/c is quite large as in the cases of tight binary systems or point masses approaching to black holes

While the GW amplitude is: where Qij are the quadrupole moments and eij the polarizations. Summing over the polarizations and integrating over the solid angle :

Time series of both dr(t)/dt and d2r(t)/dt2 together with the phase portrait {r, dr/dt} are shown assuming as initial values of the angular precession and nutation velocities integer ratios with the radial velocity: dφ/dt={1/10;1/2} dr/dt dθ/dt={1/10;1/2} dφ/dt The stiffness of the equations is evident from the peaks in the t vs r’,r’’ plots

The numerical simulations have been performed in two cases: i) a 1.4 solar masses (MO) neutron star orbiting around a Super-MBH (106 MO) e.g. Sagittarius A*) ii) a 10 MO BH orbiting around a Super-MBH. Computations are performed with orbital radii measured in mass units. Initial distances are sampled to show orbits from high eccentricity up to circularity. rmax − rmin Mm e= μ= rmax + rmin M +m

From the Lagrangian

starting from the Euler-Lagrange equations we derive the orbital equations of motion Plots along the panel lines from left to right: field velocities, total gravitational waveform and waveform polarizations for a neutron star of 1.4 MO

φ 0 = 0; θ 0 = r&0 = −

1 ; 100

π

r0 = ( 20 ÷ 1000 ) μ

2 1 r&0 φ&0 = − 10

E = 0 . 95

θ0 =

μ = 1.4 M O ; r0 = (1500 ÷ 2500 )μ

corresponding to the spatial components of the geodesic equation, and from

Difference between total GW amplitudes with and without gravitomagnetic correction. Same initial conditions as above.

which is the constraint equation related to the energy.

μ = 10 M O r&0 = −

Amaldi 8 8 Edoardo Amaldi Conference on Gravitational Waves Columbia University New York, NY June 21-26, 2009 th

φ 0 = 0; θ 0 =

1 1 ; φ&0 = − r&0 100 10

π

r0 = ( 20 ÷ 1000 ) μ

2 π E = 0 .95 θ 0 = 2

Blue line: the foreseen LISA sensitivity OPTIMISTIC CURVE (one year integration + white dwarf background noise). Red diamonds ( 1.4 MO) green circles (10 MO) are the h values for the systems we have studied.

π 2