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Schwarzschild and charged endpoint algebra

Example

Take the nonrotating slice $a = 0$ of the parameter family in Smooth null screen and subextremal Kerr–Newman parameter family.

If $q = 0$ and $m > 0$ , then $r_{+} = 2 m$ , $D = 4 m^{2}$ , $A = 16 π m^{2}$ , $κ_{K N} = 1 / (4 m)$ , and $j = Ω_{K N} = Φ_{K N} = 0$ . The local horizon data are exactly those proved in Exact Schwarzschild horizon data at the zero-rotation KN point. The first-law identity becomes $d m = κ_{K N} d A / (8 π)$ , and the Smarr identity becomes $m = κ_{K N} A / (4 π)$ .
If $q^{2} < m^{2}$ , put $s = \sqrt{m^{2} - q^{2}}$ and $r_{+} = m + s$ . Then $D = r_{+}^{2}$ , $A = 4 π r_{+}^{2}$ , $κ_{K N} = s / r_{+}^{2}$ , $Φ_{K N} = q / r_{+}$ , and $j = Ω_{K N} = 0$ . The family identities reduce to $d m = κ_{K N} d A / (8 π) + Φ_{K N} d q$ and $m = κ_{K N} A / (4 π) + Φ_{K N} q$ .
At the boundary $q^{2} = m^{2}$ with $m > 0$ , the exact double-root supplier gives $r_{+} = m$ , $D = m^{2}$ , $A = 4 π m^{2}$ , $κ_{K N} = 0$ , and $Φ_{K N} = q / m$ . This is a boundary value, not an assertion about attaining the extremal parameter in a physical process.

Facts & Assumptions

Given: The real parameter family and its strict subextremal formulas, the separately checked zero-rotation zero-charge point, and the exact double-root geometric supplier.

[F1]

Smooth null screen and subextremal Kerr–Newman parameter family: $r_{+} = m + \sqrt{m^{2} - a^{2} - q^{2}}$ , $D = r_{+}^{2} + a^{2}$ , $A = 4 π D$ , $j = m a$ , $κ_{K N} = (r_{+} - m) / D$ , $Ω_{K N} = a / D$ , and $Φ_{K N} = q r_{+} / D$ on $m^{2} > a^{2} + q^{2}$ .

[F2]

Verified Kerr-Newman family first law and Smarr identity: on the strict parameter domain, $d m = \frac{κ_{K N}}{8 π} d A + Ω_{K N} d j + Φ_{K N} d q, m = \frac{κ_{K N} A}{4 π} + 2 Ω_{K N} j + Φ_{K N} q .$

[F3]

KN Killing generators, surface gravity, area and potential: for $M > 0$ and $M^{2} \geq a^{2} + q^{2} > 0$ , the selected outer horizon has $r_{H} = M + \sqrt{M^{2} - a^{2} - q^{2}}$ , area $4 π (r_{H}^{2} + a^{2})$ , $κ_{H} = \sqrt{M^{2} - a^{2} - q^{2}} / (r_{H}^{2} + a^{2})$ , and constant potential $q r_{H} / (r_{H}^{2} + a^{2})$ ; the equality case is the double root with $κ_{H} = 0$ .

[F4]

Exact Schwarzschild horizon data at the zero-rotation KN point: at $a = q = 0$ , the smooth ingoing KN specialization has the null Killing horizon $r = 2 m$ , area $16 π m^{2}$ and signed surface gravity $1 / (4 m)$ .

Verification

1.1F1F4givenalgebra

On $a = q = 0$ , $s = \sqrt{m^{2}} = m$ because $m > 0$ , so $r_{+} = 2 m$ and $D = 4 m^{2}$ . Substitution in [F1] gives $A = 16 π m^{2}$ , $κ_{K N} = 1 / (4 m)$ , and $j = Ω_{K N} = Φ_{K N} = 0$ ; [F4] independently verifies the geometric horizon data at this point.

2.1step 1.1F2algebra

Differentiating $A = 16 π m^{2}$ gives $d A = 32 π m d m$ , whence $κ_{K N} d A / (8 π) = d m$ and $κ_{K N} A / (4 π) = m$ . Thus both family identities in [F2] reduce to the stated Schwarzschild formulas.

2.2F1F2step 1.1algebra

On $a = 0$ with $q^{2} < m^{2}$ , $s = \sqrt{m^{2} - q^{2}} > 0$ , so [F1] gives $r_{+} = m + s$ , $D = r_{+}^{2}$ , $A = 4 π r_{+}^{2}$ , $κ_{K N} = s / r_{+}^{2}$ , $Φ_{K N} = q / r_{+}$ and $j = Ω_{K N} = 0$ . For Smarr, $κ_{K N} A / (4 π) + Φ_{K N} q = s + q^{2} / r_{+} = s + (m^{2} - s^{2}) / (m + s) = s + (m - s) = m$ . For the first law, differentiate $s^{2} = m^{2} - q^{2}$ to obtain $s d s = m d m - q d q$ , and hence $d r_{+} = d m + d s = (r_{+} / s) d m - (q / s) d q$ . Therefore $κ_{K N} d A / (8 π) + Φ_{K N} d q = (s / r_{+}^{2}) (8 π r_{+} d r_{+}) / (8 π) + (q / r_{+}) d q = (s / r_{+}) d r_{+} + (q / r_{+}) d q = d m$ .

3.1F1F3step 2.2algebra∎

At $q^{2} = m^{2}$ and $m > 0$ , the strict parameter formula has the continuous boundary values $s \to 0$ , $r_{+} \to m$ , $D \to m^{2}$ , $A \to 4 π m^{2}$ , $κ_{K N} \to 0$ , and $Φ_{K N} \to q / m$ . Independently, [F3] supplies this selected-model double-root data exactly, since $M = m > 0$ and $M^{2} = a^{2} + q^{2} > 0$ . This verifies the boundary formula without extending the open-family differential identity or making a process claim.

Remarks

The open-family first law and Smarr formula concern exact parameter identities. The extremal line is a boundary of that open family; the example makes no statement about how, or whether, a physical process can reach it.

Schwarzschild and charged endpoint algebra

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources