Can’t solve an equation….bypass it !!!

Radouane Gannouji

Radouane Gannouji is an associate professor at the Department of Physics, PUCV.

In general relativity, to understand how the spacetimes behave in presence of a given form of matter, we have to solve the Einstein field equations, which in general, are a set of 10 very complicated coupled nonlinear second order partial differential equations that describes the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy. Once we solve these set of field equations we get the metric of the spacetime that describes all the general important physical features of the spacetime, for example the presence of horizons, spacetime singularities, asymptotic behaviour etc.

However finding exact solutions of this complicated system is a field of research in its own right and can only be obtained for spaces of high symmetry and idealized matter content. Nevertheless, they are very important as they embody the full nonlinearity, allow studies of strong field regimes which are very useful to constrain gravitational theories, they also provide backgrounds on which perturbative analysis can be built, and therefore understand the stability of a solution like the final fate of gravitational collapse, they provide solutions for astrophysical structures like neutron stars, and they enable checks of numerical accuracy.

To find exact solutions the basic problems are:

1- what is the coordinate choice that can make the calculations simpler.

2- the complicated system cannot be generally integrated, this is principally true for realistic matter content as for example neutron stars solutions or for vacuum solution in the context of modified gravity theories where the equations are often fourth order.

Given the above problems, the key question that arises here is as follows:

Apratim Ganguly

Apratim Ganguly is a graduate student at ACRU, UKZN.

Is it possible to identify the general physical properties and global nature of a spacetime without actually solving the Einstein field equations?

In our work we tried to answer this question in a transparent manner. To bypass the problem of co-ordinate choice/ co-ordinate singularities etc. we used the local semi tetrad splitting of spacetime (which is commonly known as 1+1+2 covariant formalism) to recast the field equations into an autonomous system of covariantly defined variables. This enables us to get rid of all the coordinate singularities that may appear due to bad choice of coordinates while solving the field equations. This autonomous system simplifies considerably when we incorporate the Killing symmetries of the spacetimes. And instead of trying to find particular exact solutions of differential equations, we studied the autonomous system which gives qualitative informations on important global features of the spacetime. For that, we used all tools available in dynamical system as critical points and their stability, Poincare sphere, center manifold and so on.

It is then very easy, via this formalism, to not only obtain the nature of the central singularity of the black hole, to find whether the solution possesses horizon or not and the asymptotic behaviour of the solution but also to obtain the singularity-free nature of e.g. the Nariai solution.

This analysis, published in our recent CQG article, provides an efficient way to understand the global properties of any spacetime, by bypassing the very difficult task of solving the field equations.

Read the full article in Classical and Quantum Gravity or on arXiv:
Global structure of black holes via the dynamical system
Apratim Ganguly, Radouane Gannouji, Rituparno Goswami and Subharthi Ray
Apratim Ganguly et al 2015 Class. Quantum Grav. 32 105006

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