How does electric charge affect spacetime?

Question

Electric charge does not affect spacetime. Mass curves spacetime. This means that if there are two particles A and B, which are spherically symmetrical and have the same mass, even if A is charged and B is uncharged, they both must curve the spacetime in the same way.

There is RN metric for non-rotating charged black hole and KN metric for rotating charged black hole. The Schwarzschild metric and the Kerr metric should be enough.

However, this is not the case. This means that electric charge affects spacetime.

How does electric charge affect spacetime?

Answer 1

The equations of general relativity for spacetime curvature are not directly sensitive to charge. But the stress-energy tensor (that acts as a source term for curvature) is sensitive to energy, and a static electric field has some intrinsic energy. The result is that the Reissner-Nordström metric has curved spacetime even if there is no mass.

For any kind of conserved charge in a field theory there is a component of the stress-energy tensor, and that causes spacetime curvature.

Answer 2

Charge does effect spacetime. Einstein's Field Equations relate gravity not to strictly mass, but the stress-energy tensor. Mass is energy via $E=mc^2$. There's also other energy contributions, eg. motion of the source particles. Charged particles also imply stresses.