It so happens that we are forced to put the negative of the above into gravitational equations if it is to be possible to obtain static spherically symmetric solutions of the equations, free from singularities, which could represent electrical particles. Making the change of sing one finds as a required solution

φ₁ = φ₂ = φ₃ = 0, φ₄ = ε / 4,

ds² = - 1 / ( 1 - 2m/r - ε²/(2r²)) dr² - r² * (dθ² + sin²θ * dΦ²) + (1 - 2m/r - ε²/(2r²)) dt² (8)

Here m has obviously the significance of a gravitationg mass, e of that of an electrical charge.
It turns out that also in this case there is no difficulty in forming a solution without singularity corresonding to the solution just given. ^{note 4}
Curiously enough, one finds that the mass m is not determined by the electrical charge e, but that e and m are independent constants of integration. It also turns out that the removal of the singularity it is not necessary to take the ponderable mass m positive. In fact, as we shall show immediately, there exists a solution free from singularities for which the mass constant m vanishes.
Because we believe that these massless solutions are the physically important ones we will consider here the case m = 0.
The field equations without denominators can be written

φ_mn = φ_m,n - φ_n,m, g²φ_mn;sg^ns = 0,

g²(R_ik + φ_iaφ_k^a - 1/4 * g_ik*φ_abφ^ab) = 0, (9)

where in the last equation the term in R has been omitted because it vanishes in consequence of (7), by which T_a^a is zero.
If in Eq.(8) (with m = 0) one replaces r by the variable u according to the equation

u² = r² - ε²/2

one obtains

φ₁ = φ₂ = φ₃ = 0, φ₄ = ε / (u² + ε²/2)^1/2,

ds² = - du² - (u² + ε²/2)(dθ² + sin²θ*dΦ²) + (2u²/(2u²+ε²))*dt², (8a)


This solution is free from singularities for all finite points in the space of two sheets
-A.Einstein, N.Rosen - 1935

With all constants in place (8a) becomes :

ds² = - du² - (u² + e²G/(2πεc⁴))(dθ² + sin²θ*dΦ²) + (2u²/(2u²+e²G/(πεc⁴)))*dt²

e = electric charge
ε = electric permitivity constant
c = speed of light
G = gravitation constant
ds = proper time
θ = angle 1
Φ = angle 2
r = radius
t = time
Ε = (e²G/(πεc⁴))