# lib_Sch_GSF

### A C library for fast interpolation of the Gravitational Self Force (GSF)

This library interpolates the Lorenz-gauge gravitational self-force (GSF) for bound geodesic orbits about a Schwarzschild black hole using a model fitted to numerical GSF data. The data used to form the fit consists of 1100+ data points spaced between orbital eccentricities of $e=0$ and $e=0.2$ and semi-latus rectums of $p=6+2e$ and $p=12$ (the line $p=6+2e$ marks the edge of the space of stable orbits). Details of the fitting formula used can be found in Ref. [1]. The library will be updated as more accurate self-force data is computed across a wider range of the orbital parameter space.

#### Orbital parametrization

#### Download

The library can be downloaded here: lib_Sch_GSF.tar.bz2 (last updated 24 Nov 2011)#### Usage

The library makes use of the Gnu Scientific Library (GSL) so this will need to be installed. The library is documented with comments and is very straightforward to use. When using the library the most important thing is to remember to load the model coefficients first using:

lib_Sch_GSF_load_model();

After that the dimensionless contravariant components of the conservative and dissipative GSF for a given orbital eccentricity ($e$), semi-latus rectum ($p$) and orbital phase ($\chi$) are computed with:

double lib_Sch_GSF_Fr_cons(double e, double p, double chi); double lib_Sch_GSF_Fr_diss(double e, double p, double chi); double lib_Sch_GSF_Fphi_cons(double e, double p, double chi); double lib_Sch_GSF_Fphi_diss(double e, double p, double chi);

where we have taken $\chi=0$ to be at periastron passage. To restore the correct dimensions the $F^r$ and $F^\phi$ components should be multiplied by $\eta^2$ and $M\eta^2$ respectively where $\eta=\mu/M$. If the t-component of the GSF, $F^t$, is required it can be computed from the orthogonality condition $F^\alpha u_\alpha = 0$.

The alpha correction required to convert between Lorenz gauge time ($t$) and 'asymptotically flat time' ($\hat{t}$), where the conversion is given by $\hat{t} = (1+\alpha)t$ [3], is calculated with

double lib_Sch_GSF_alpha(double e, double p);

A simple example program called GSF_from_model is included that demonstrates the use of the library. It can be compiled with:

make clean all

#### Accuracy

Defining the relative accuracy as

$dF = max_\chi\left[\frac{F_{model}(\chi)-F_{data}(\chi)}{F_{data}(\chi)}\right]$

for a given $(p,e)$, the model reproduces the GSF for all data points used in forming the model to within the following accuracies

F^r_cons: dF < 5e-5 F^r_diss: dF < 6e-4 F^phi_cons: dF < 1e-3 F^phi_diss: dF < 8e-4Note this is the maximum deviation found between the model and the current data. In general the average deviation is a lot better. Details of the model used to fit to the numerical GSF data can be found in Ref. [1]. Note that these accuracies are only guaranteed for $\epsilon=p-6-2e>0.2$, below this you are on your own. Improvements in this region will be forthcoming.

#### Credits

This library was written by Niels Warburton. The codes used to produce the GSF data were developed by Sarp Akcay, Niels Warburton and Leor Barack [2] and Leor Barack and Norichika Sago [5]. If you make use of this library in your work please cite Ref. [1].#### References

[1] N. Warburton, S. Akcay, L. Barack, J. Gair, N. Sago, PhysRevD.85:061501, (2012), arXiv:1111.6908[2] S. Akcay, N. Warburton, L. Barack, arXiv:1308.5223

[3] N. Sago, L. Barack, S. Detweiler, Phys.Rev.D78:124024, (2008), arXiv:0810.2530

[4] C. Cutler, D. Kennefick, E. Poisson, Phys.Rev.D50:3816 (1994)

[5] L. Barack and N. Sago, Phys.Rev.D81:084021, (2010), arXiv:1002.2386