# The Kerr Metric

Views:

Category: Entertainment

## Presentation Description

No description available.

## Presentation Transcript

### 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:

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

### Slide5:

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

Fin.