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

Understand

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 $r_{NCS}>r_{S} >r_{RN}$.