# Amateur's question on Black Holes [duplicate]

Black holes are caused by massive curvature of the fabric of space-time. Is it right in believing theoretically that forces of electromagnetic origin could also lead to distortion of the fabric of space-time, (though it may not be as tremendous as the extent to which distortion is brought about by gravitational forces)? If it is right, then could we venture on the existence of tiny holes in space-time due to electromagnetic effect on space-time fabric?

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## marked as duplicate by John Rennie, Brandon Enright, jinawee, DavePhD, Emilio PisantyMay 14 '14 at 11:28

See the question I've linked: charge does curve spacetime. However because charge comes in +ve and -ve signs it tends to average out to zero and it's not significant on the large scale. – John Rennie May 14 '14 at 6:16
as an amateur, I fail to grasp the statement because "charge comes in +ve and -ve signs". Can charges, in their unpaired configuration, lead to comparatively larger curvature, on the small scale? – abstract May 14 '14 at 6:21
Electrons have charge -1 and protons have charge +1, and as far as we know there are equal numbers of both so the net charge in the universe is zero. A very large charge can significantly curve spacetime, but it's hard to build up such a large charge. On the small scale charge has no effect on spacetime curvature as the charge of a single electron or proton far too small to have any significant effect. – John Rennie May 14 '14 at 6:28
I get it. Thanks. – abstract May 14 '14 at 6:31

Electromagnetic effects do lead to a curvature of spacetime, as gravitation couples to any quantity in the stress-energy tensor, as dictated by the Einstein field equations. Specifically, the tensor is given by,

$$T^{ab}=-\frac{1}{\mu_0}\left( F^{ac} F_{c}^b +\frac{1}{4}g^{ab}F_{cd}F^{cd}\right)$$

where $F$ is the field-strength of the electromagnetic $4$-potential $A$, which the electric and magnetic fields depend on. The corresponding field equations are,

$$R^{ab}-\frac{1}{2}g^{ab}R + g^{ab}\Lambda = \frac{8\pi G}{\mu_0}\left(F^{ac} F_{c}^b +\frac{1}{4}g^{ab}F_{cd}F^{cd} \right)$$

The theory is often referred to as 'Einstein-Maxwell theory.' Analytic and approximate black hole solutions to the theory are known, c.f. Spherically symmetric black hole solutions to Einstein-Maxwell theory with a Gauss-Bonnet term by D.L. Wiltshire. From their abstract:

The only spherically symmetric solutions of the theory are shown to be generalisations of the Reissner-Nordstrom and Robinson-Bertotti solutions. The “Reissner-Nordstrom” solutions have asymptotically flat and asymptotically anti-de Sitter branches, however, the latter are unstable.

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