# On the characteristic initial value problem for the spherically symmetric Einstein–Euler equations

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## Abstract

We consider the Einstein equations for a barotropic irrotational perfect fluid with equation of state \(p=\rho \), where *p* is the pressure of the fluid, and \(\rho \) its mass-energy density. We express \(\rho \) in terms of the potential introduced thanks to the irrotationality of the fluid, obtaining therefore the relativistic analogue of Bernoulli’s law well known in classical Fluids Mechanics. Under the spherical symmetry assumption we reduce the problem to a highly nonlinear evolution partial integrodifferential equation which we solve globally by a fixed point method, under smallness condition on the initial datum and regularity assumption at the center of symmetry. All the investigation is performed in Bondi coordinates so that the problem dealt with is a characteristic initial value problem with initial datum prescribed on a light cone with vertex at the center of symmetry.

## Keywords

Characteristic initial value problem Global solution Einstein equations Irrotational perfect fluid Spherical symmetry Bondi coordinates## Mathematics Subject Classification

35A01 35A02 35A09 35B06 83C10 83C20 83C55## Notes

### Acknowledgements

I acknowledge support from the Abdus Salam International Centre for Theoretical Physics (ICTP, Trieste-Italy) where this work was finalized at the beginning of one of my research visits as Senior Guest Scientist in the Mathematics Group of ICTP. I also express my thankfulness to the anonymous reviewers for their suggestions that helped improve the paper.

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