3.3 The Static Limit in Kerr-Newman Space-Time |
49 |
which is compatible with result (3.2.20) for the Ricci tensor of the Kerr-Newman metric if
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F012 |
+ F232 = |
q2 |
(3.2.28) |
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2κρ4 |
We conclude that the Kerr-Newman metric provides a consistent solution of the coupled Maxwell Einstein field equations if there exist two functions F01(r, θ ) and F23(r, θ ) of the coordinates r, θ such that:
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(3.2.28) is verified, |
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the 2-form: |
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F ≡ 2F01V 0 V 1 + 2F23V 2 V 3 |
(3.2.29) |
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is closed: |
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dF = 0 |
(3.2.30) |
3. |
and also coclosed, namely: |
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d F = 0 |
(3.2.31) |
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where |
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F ≡ 2F23V 0 V 1 + 2F01V 2 V 3 |
(3.2.32) |
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is the Hodge dual of the 2-form F . |
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Indeed (3.2.30) and (3.2.31) are the two Maxwell equations.
By direct evaluation one can verify that all the above conditions are met by the following two functions:
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√ |
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r2 − α2 cos2 θ |
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F |
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2√ |
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ρ4 |
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It follows from (3.2.33) that the Kerr-Newmann solution has both an electric and a magnetic field, while the non-rotating spherical symmetric limit (α → 0) which is the Reissner Nodström solution has only an electric field. The electric and magnetic charges of the rotating black-hole are rigidly related to each other in order to obtain a consistent solution of Maxwell Einstein equations.
3.3 The Static Limit in Kerr-Newman Space-Time
The key feature of the Kerr-Newman space-time is that it describes the gravitational field of a rotating black hole. This will become evident by studying the properties of the world-lines of test particles or observers around the hole.
50 |
3 Rotating Black Holes and Thermodynamics |
Fig. 3.2 An observer rotating in the equatorial plane around the hole has an angular velocity Ω = dφdt with respect to the fixed stars
Differently from the Schwarzschild metric, which is both static and spherically symmetric, the KN-metric is static and only axial symmetric. Indeed, instead of four, the KN-space-time admits only two Killing vector fields corresponding to translations in the time variable t and in the axial angle φ respectively. In the coordinate patch we have utilized these Killing vectors have the following simple form:
k |
= |
∂ |
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k |
∂ |
(3.3.1) |
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∂t |
∂φ |
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˜ = |
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and their norms and scalar products are directly related to the metric coefficients in the following way:
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tt = |
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− |
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ρ |
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g |
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2mr − q2 |
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sin2 θ |
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+ |
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− Δα |
2 |
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α |
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sin |
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tφ = |
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2mr − q2 |
α sin2 θ |
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Consider next an observer which rotates around the black hole along a circular orbit lying in its equatorial plane. The trajectory of such a test-particle is characterized by the following simple equation:
r = const; |
θ = |
π |
; t = s; φ = Ωs |
(3.3.5) |
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where s is an affine parameter and Ω is the angular velocity of the particle perceived by an observer that is at rest with respect to the distant fixed stars (see Fig. 3.2). The 4-velocity of such a test-particle is given by:
k |
(3.3.6) |
u = (1, Ω, 0, 0) = k + Ω ˜ |
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For a physical particle the norm of the 4-velocity is necessarily non-negative and this yields the following interesting quadratic condition on the angular velocity Ω:
(u, u) |
≥ |
0 |
(k, k) + 2(k, ˜ |
+ |
˜ ˜ |
2 |
≥ 0 |
(3.3.7) |
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k)Ω |
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3.3 The Static Limit in Kerr-Newman Space-Time |
51 |
The roots of the above quadratic form are given by:
Ω± = |
−gtφ ± gt2φ − gtt gφφ |
(3.3.8) |
gφφ |
and we have physical non-tachyonic observers as long as their angular velocity lyes in the range between the two roots:
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Naming: |
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ω |
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gtφ |
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α |
2mr − q2 |
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≡ − gφφ |
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(r2 + α2)2 − Δα sin2 θ |
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the two roots of the quadratic form (3.3.7) can also be rewritten as:
Ωmin ≡ Ω− = ω − ω2 |
− gφφ |
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gtt |
Ωmax ≡ Ω+ = ω + ω2 |
gtt |
− gφφ |
(3.3.9)
(3.3.10)
(3.3.11)
(3.3.12)
The interest of this rewriting comes from the fact that the quantity ω has a distinctive physical interpretation, namely it is the angular velocity of a locally non-rotating observer.
Locally Non-rotating Observers Which observers deserve the name of locally non-rotating? Clearly those whose angular momentum vanish! We now prove that this happens for those test-bodies whose 4-velocity is orthogonal to the constant time hypersurfaces t = const, so that they are at rest on them. For a similar observer the 4-velocity is just the gradient of time, namely:
uμ = μt = gμν ∂ν t = gμt |
(3.3.13) |
Taking into account the specific form of the Kerr-Newman metric we obtain that the 4-velocity of a locally non-rotating observer is:
u = gtt , gφt , 0, 0 |
(3.3.14) |
Consider now the general problem of computing time-like geodesics for the KNmetric. Just as in the case of the Schwarzschild metric we can address such a problem starting from the effective Lagrangian1 and writing the corresponding EulerLagrange equations:
0 = |
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(3.3.15) |
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1Compare with (3.8.5)–(3.8.9) of Volume One.
52 |
3 Rotating Black Holes and Thermodynamics |
Also here we immediately derive two-conserved quantities associated with the cyclic Lagrangian coordinates t and φ. The first integral of motion associated with the azimuthal angle φ is the angular momentum and (3.3.15) provides its definition. For a test-body moving on a world-line of type (3.3.5) we find
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+ |
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(3.3.16) |
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According to (3.3.14) the angular momentum of a locally non-rotating observer (LNRO) vanishes since we obtain:
LNRO = gφφ gφt + gφt gtt ≡ δφt = 0 |
(3.3.17) |
and this concludes the proof of our statement.
The crucial point, however, is that a locally non-rotating observer has a nonvanishing angular velocity with respect to the reference frame of the fixed stars. In other words a test-body with a null angular momentum is perceived to rotate around the hole by a distant observer who is at rest in the asymptotic flat geometry. What is the actual angular velocity of such a locally non-rotating test body? It is given by the quantity ω which we introduced in (3.3.10). Indeed the equation:
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φ |
g |
t |
g |
φφ |
Ω |
+ |
g |
φt |
(3.3.18) |
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0 = gφφ ˙ + |
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φt ˙ = |
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is solved by Ω = ω. Hence |
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ω |
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(q2 − 2mr)α sin2 θ |
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(3.3.19) |
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= q2 |
+ r2 + α2 − α2 sin2 θ − 2mr |
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is the angular velocity with which rotates with respect to the fixed stars an observer which is at rest with respect to its local geometry. The behavior of this angular velocity with respect to the radius r and to the declination angle θ is displayed in Fig. 3.3.
Static Observers We have seen that those observers who have zero angular momentum and are not rotating with respect to the local geometry have a non-vanishing angular velocity Ω = ω with respect to the fixed stars. We can now consider the case of the static observers defined as those whose angular velocity in the fixed star frame vanishes, namely Ω = 0. The angular momentum of the static observers does not vanish. It is equal to:
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= |
g |
tφ = − |
(q2 − 2mr)α sin2 θ |
(3.3.20) |
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r2 + α2 cos2 θ |
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The physical interpretation of this fact is clear. The black hole rotates and drags all reference frames along its rotation. In order to stand still, a test-body needs to have an angular momentum which counterbalance the dragging of the inertial frames. The question is: how far can the dragging be opposed? The answer is simple: as long as the 4-velocity of a static observer is time-like. For a static observer (3.3.6)