The Kerr Metric

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The Kerr Metric for Rotating, Electrically Neutral Black Holes:: 

The Kerr Metric for Rotating, Electrically Neutral Black Holes: The Most Common Case of Black Hole Geometry Ben Criger and Chad Daley


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

Derivation (Abridged): 

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.

But What’s the Point?: 

But What’s the Point? To represent any metric in null tetrad / frame metric form. Meaningful Example: Schwarzschild Metric


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 . . . .

The Kerr Metric in Boyer-Lindquist Co-ordinates: 

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

However. . . : 

However. . . We don’t 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).

Nothing Succeeds Like Success: 

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.

Just one more thing. . .: 

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. . .

The Kerr-Newman Metric: 

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:

Singularities and Horizons: 

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

Horizons and Singularities of the Kerr Metric: 

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

Horizons and Singularities for a2 < M2: 

Horizons and Singularities for a2 < M2 A surface of infinite gravitational red shift can be determined by; Setting a = 0, or θ = π/2 these reduce to;

Horizons and Singularities Con’t…: 

Horizons and Singularities Con’t… We can also recover two event horizons setting the radial coefficient to zero In the case of a = 0, these surfaces reduce to:

Summary of Kerr Geometry (a2 < m2): 

Summary of Kerr Geometry (a2 < m2) Essential ring singularity at: Two surfaces of infinite red shift at : Two event horizons at: Particle

Possible Energy Source?: 

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

A word about cases of a2 ≥ m2: 

A word about cases of a2 ≥ m2 For a2 > m2 we find only the essential singularity at r = 0 This naked singularity violates Penrose’s cosmic censorship hypothesis The solution for a2 = m2 is unstable

Observed Black Holes: 

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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