Metrics Wormhole
for combined field, gravity , electricity, einstein , rosen derived following schwarzschild static spherically symmetric solution
ϕ
1
=
ϕ
2
=
ϕ
3
=
0
,
ϕ
4
=
ϵ
4
,
{\displaystyle \phi _{1}=\phi _{2}=\phi _{3}=0,\phi _{4}={\frac {\epsilon }{4}},}
d
s
2
=
−
1
(
1
−
2
m
r
−
ϵ
2
2
r
2
)
d
r
2
−
r
2
(
d
θ
2
+
sin
2
θ
d
ϕ
2
)
+
(
1
−
2
m
r
−
ϵ
2
2
r
2
)
d
t
2
{\displaystyle ds^{2}=-{\frac {1}{(1-{\frac {2m}{r}}-{\frac {\epsilon ^{2}}{2r^{2}}})}}dr^{2}-r^{2}(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2})+(1-{\frac {2m}{r}}-{\frac {\epsilon ^{2}}{2r^{2}}})dt^{2}}
(
ϵ
{\displaystyle \epsilon }
= electrical charge)
the field equations without denominators in case when m = 0 can written
ϕ
μ
ν
=
ϕ
μ
,
ν
−
ϕ
ν
,
μ
{\displaystyle \phi _{\mu \nu }=\phi _{\mu ,\nu }-\phi _{\nu ,\mu }}
g
2
ϕ
μ
ν
;
σ
g
ν
σ
=
0
{\displaystyle g^{2}\phi _{\mu \nu ;\sigma }g^{\nu \sigma }=0}
g
2
(
r
i
k
+
ϕ
i
α
ϕ
k
α
−
1
4
g
i
k
ϕ
α
β
ϕ
a
b
)
=
0
{\displaystyle g^{2}(r_{ik}+\phi _{i\alpha }\phi _{k}^{\alpha }-{\frac {1}{4}}g_{ik}\phi _{\alpha \beta }\phi ^{ab})=0}
in order eliminate singularities, if 1 replaces r u according equation:
u
2
=
r
2
−
ϵ
2
2
{\displaystyle u^{2}=r^{2}-{\frac {\epsilon ^{2}}{2}}}
and m = 0 1 obtains
ϕ
1
=
ϕ
2
=
ϕ
3
=
0
,
ϕ
4
=
ϵ
/
(
u
2
+
ϵ
2
2
)
1
2
{\displaystyle \phi _{1}=\phi _{2}=\phi _{3}=0,\phi _{4}=\epsilon /(u^{2}+{\frac {\epsilon ^{2}}{2}})^{\frac {1}{2}}}
d
s
2
=
−
d
u
2
−
(
u
2
+
ϵ
2
2
)
(
d
θ
2
+
sin
2
θ
d
ϕ
2
)
+
(
2
u
2
2
u
2
+
ϵ
2
)
d
t
2
{\displaystyle ds^{2}=-du^{2}-(u^{2}+{\frac {\epsilon ^{2}}{2}})(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2})+({\frac {2u^{2}}{2u^{2}+\epsilon ^{2}}})dt^{2}}
the solution free singularities finite points in space of 2 sheets
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