Analogy between the Schwarzschild solution in a noncommutative gauge theory and the Reissner-Nordstr\"om metric

Abstract

We study modifications of the Schwarzschild solution within the noncommutative gauge theory of gravity. In the present analysis, the deformed solutions are obtained by solving the field equations perturbatively, up to the second order in the noncommutativity parameter $\Theta$, for both exterior and interior solutions of the equations of motion for $e_\mu ^a \left(x\right)$. Remarkably, we find that this new noncommutive solution is analogous to the Reissner-Nordstr\"om solution in the ordinary spacetime, in which the square of electric charge is replaced by the square of the noncommutativity parameter, but with opposite sign. This amounts to the noncommutative Schwarzschild radius $r_{NCS}$ becoming larger than the usual radius $r_S =2M$, instead of smaller as it happens to the Reissner-Nordstr\"om radius $r_{RN}$, implying that $r_{NCS}>r_{S} >r_{RN}$. An intuitive interpretation of this result is mentioned.