The Reissner-Nordstrom metric

[Orion1]

Two citations referenced by Wikipedia with respect to General Relativity models for spinless charged neutron stars, reference 2 - eq. 24, states that the Reissner-Nordstrom metric relativistic Einstein-Maxwell Gauss law for the electric system charge is:
Q(r) = \int_0^r 4 \pi j^0 e^{\frac{(\nu + \lambda)}{2}} dr

However, according to reference 3 - eq. 5, the electric system charge is:
Q(r) = \frac{1}{r^2} \int_0^r 4 \pi r^2 \rho_{ch} e^{\frac{\lambda}{2}} dr

What Einstein-Maxwell charge effects with respect to the metric components \nu and \lambda are these equations describing?

Why does the reference 2 equation have two metric components and the reference 3 equation has only one metric component?

Why is the reference 2 equation missing the r^2 dimensions?

And what are the International System of Units (S.I.) for j^0 and \rho_{ch}?

Please post links to this thread for subject equations cross-reference.
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Reference:
http://en.wikipedia.org/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric"
http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.2708v2.pdf"
http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332004000200038"

 

[Orion1]

total mass of charged neutron star...


How was this equation generated?

Total mass of spinless charged neutron star at a radial distance r:
\frac{dm(r)}{dr} = \frac{4 \pi r^2 \epsilon(r)}{c^2} + \frac{Q(r)}{c^2 r} \frac{dQ(r)}{dr}

How exactly does charge contribute to mass this way?
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Reference:
http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.2708v2.pdf"

 

[Orion1]

Please post links to this thread for subject equations cross-reference.
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[Orion1]

nu metric component...



Given that F^{\mu \kappa} satifies the covariant Maxwell field equation:
\left[\sqrt{-g} F^{\kappa \mu} \right]_{, \mu} = 4 \pi J^{\kappa} \sqrt{-g}

The quantity J^{\kappa} denotes the four-current which represents the electromagnetic sources in the star. For a static spherically symmetric system, the only non-zero component of the four-current is J^1, which implies that the only non-vanishing component of F^{\kappa \mu} is F^{01}:

In this case I derived this formula for the relativistic electric field:
F^{01}(r) = E(r) = \frac{Q(r)}{r^2} e^{- \frac{ \left( \nu + \lambda \right)}{2}}

The formula suggested by the remaining reference papers:
F^{01}(r) = E(r) = \frac{Q(r)}{r^2}

Electric system charge:
Q(r) = \int_0^r 4 \pi j^0 e^{\frac{ \left( \nu + \lambda \right)}{2}} dr

The formula suggested by the remaining reference papers:
Q(r) = \int_0^r 4 \pi r^2 \rho_{ch} e^{\frac{\lambda}{2}} dr

Therefore:
\boxed{j^0 = \rho_{ch} r^2}

Integration by substitution:
F^{01}(r) = E(r) = \frac{e^{- \frac{ \left( \nu + \lambda \right)}{2}}}{r^2} \int_0^r 4 \pi j^0 e^{\frac{ \left( \nu + \lambda \right)}{2}} dr

Why does this solution have a \nu metric component and two metric components relativistic electric_field modifier e^{- \frac{ \left( \nu + \lambda \right)}{2}}?
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Reference:
http://en.wikipedia.org/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric"
http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.2708v2.pdf"
http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332004000200038"
http://www.google.com/url?sa=t&sour...o.pdf&usg=AFQjCNG7gtMPDuwpTrqYwoZ2NuS1lk1nRw"