An example of a discretization in spherical coordinates for a 2-dimensional simulation is that used by Müller & Janka [208] (see also [306, 201]). Here they define the gravitational quadrupole radiation field ():

where is the distance between the observer and the source, is the angular coordinate in our coordinate system, and is given by: where , are the velocities along the ordinate axes in spherical coordinates and is the Newtonian potential. In spherical coordinates, the partial derivatives in and are easily definedIn three dimensions, we can simplify this equation:

where and , where where i and j are the coordinate axes.For smooth particle hydrodynamics (SPH), this formalism becomes [45]:

where is the distance to the source. The angle brackets in these equations denote averaging over all source orientations, so that for example The quantities are the second time derivatives of the trace-free quadrupole moment of the source. The diagonal elements are given by, for example, where is the mass of an SPH particle, and is its coordinate location. The other diagonal elements can be obtained using Equation (18) plus cyclic permutation of the coordinate labels: via , , ; via , , . The off-diagonal elements are given by, for example, the remaining off-diagonal elements can be found via symmetry () plus cyclic permutation. This approach provides only averaged square amplitudes, instead of the specific amplitude of the GW signal.For GWs from neutrinos, many groups use [208] (based on the original analysis of Epstein [76] and Turner [317]):

where is the gravitational constant, is the speed of light and is the distance of the object. denotes the emergent strain for an observer situated along the source coordinate frame’s z-axis (or pole) and is the comparable strain for an observer situated perpendicular to this z-axis (or equator).In SPH calculations, these expressions reduce to:

where is the timestep, the summation is over all particles emitting neutrinos that escape this star (primarily, the boundary particles). There are some assumptions about the emission of the neutrinos in this model, but Fryer et al. [107] argued that these assumptions lead to errors smaller than a factor of 2. The correspond to observers along the positive/negative directions of each axis: for instance, in the polar equation, this corresponds to the positive/negative z-axis. In the equatorial region, the x,y-axis is determined by the choice of x or y position for the (x,y) coordinate in the equation.The signal arising from neutrino emission is an example of a GW burst “with memory”, where the amplitude rises from a zero point and ultimately settles down to a value offset from this initial value. The noise sources and optimal data analysis techniques for these signals will differ from, for example, signals from bounce that do not have memory [25, 80].

Other methods have been developed to calculate the GW emission (e.g., Regge variables [328, 121], Zerilli equations [346], Moncrief metric [202], see [214] for a review). Shibata and collaborators (e.g., [277]) have used the gauge-invariant Moncrief variables in flat spacetime:

where where are functions of and and are two-sphere integrations with dependencies on the coordinates and the Lorentz factor . Most of these are defined in Shibata & Nakamura [280]. With these gauge-invariant variables, we can derive the energy emitted in GWs: Shibata & Sekiguchi [282] define a number of other useful quantities for determining GW emission using this formalism.

Living Rev. Relativity 14, (2011), 1
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