The Kerr Metric for Rotating, Electrically Neutral Black Holes:

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The Kerr Metric for Rotating, Electrically
Neutral Black Holes

• The Most Common Case of Black Hole Geometry

Ben Criger and Chad Daley
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Assumptions

• Non-zero angular momentum
• Insignificant charge
• Axial symmetry
• No-Hair Theorem

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Derivation (Abridged)

• Null Tetrad Any set of four vectors (one
  timelike and three spacelike such as m) for which
  the null condition (defined below) is met.

Frame Metric
Where is our metric of choice, and is
one of the null vectors in our tetrad.
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But Whats the Point?

• To represent any metric in null tetrad / frame
  metric form.
• Meaningful Example Schwarzschild Metric

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• We perform an ingenious substitution of
  co-ordinates and obtain the following null
  vectors as valid for a new metric

We use the process detailed in the previous
slides to find the metric for these null vectors.
Drumroll please . . . .
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The Kerr Metric in Boyer-Lindquist Co-ordinates

• We present the Kerr Metric in Boyer-Lindquist
  co-ordinates (first, we present BL co-ordinates
  here in comparison with spherical co-ordinates)

Where and
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However. . .

• We dont have any physical intuition at this
  point about the metric!
• We will have to prove (or convince ourselves)
  that a is an angular momentum parameter, etc.
• We start by setting this parameter a 0 and
  seeing what happens.

Now, we can say with confidence that a represents
angular momentum (and has dimensions of radius).
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Nothing Succeeds Like Success

• Now, we try removing m from the equation, and
  leaving a fixed a.

This metric may look deceptively complex, but
this is simply the expression of flat space in
Boyer-Lindquist Co-ordinates. Here, we have
confirmed that m and a are what they appear to
be, and that our metric (chosen through a
convenient, if unintuitive method) is a valid
solution for rotating black holes.
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Just one more thing. . .

• We need to prove that the metric is flat at
  infinity.

We recover flat space in spherical co-ordinates.
We have effectively argued that this metric is
valid, and we can apply this method to the
Reissner-Nordstrom metric to obtain. . .
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The Kerr-Newman Metric

• Represents a rotating, charged, black hole
• Can devolve to any of the Schwarzschild, Kerr, or
  Reissner-Nordstrom metrics.

We use the following definitions
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Singularities and Horizons

• 2 categories, essential and coordinate
• Schwarzschild Solution
• Essential singularity at r 0
• Event horizon at Schwarzschild radius, r 2m
• Reissner-Nordström Solution
• Retain essential singularity at r 0
• 0 - 2 coordinate singularities at

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Horizons and Singularities of the Kerr Metric

• Looking at our metric we find an essential
  singularity for
• Remembering the definitions of our co-ordinates
  we find
• This corresponds to a ring of radius a

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Horizons and Singularities for a2 lt M2

• A surface of infinite gravitational red shift can
  be determined by
• Setting a 0, or ? p/2 these reduce to

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Horizons and Singularities Cont

• We can also recover two event horizons setting
  the radial coefficient to zero
• In the case of a 0, these surfaces reduce to

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Summary of Kerr Geometry (a2 lt m2)

• Essential ring singularity at
• Two surfaces of infinite
• red shift at
• Two event horizons
• at

Particle
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Possible Energy Source?

• Equip yourself with a large mass
• Within the ergophere throw the mass against the
  rotation
• Upon exiting the ergosphere you will have gained
  energy

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A word about cases of a2 m2

• For a2 gt m2 we find only the essential
  singularity at r 0
• This naked singularity violates Penroses cosmic
  censorship hypothesis
• The solution for a2 m2 is unstable

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Observed Black Holes

• Cygnus X-1 widely accepted as the first observed
  black hole candidate
• Jets observed companioning black holes termed
  quasars

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• Fin.