# Examining the Kerr metric

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#### jeffreyH

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##### Examining the Kerr metric
« on: 19/03/2015 01:16:16 »
The Kerr metric describes a rotating, uncharged and spherical black hole. It was a solution to the Einstein field equations discovered by Roy Kerr in 1963. Details can be found in the wikipedia article.

http://en.wikipedia.org/wiki/Kerr_metric

Of interest is the first set of equations in the mathematical form section. This is a complicated equation but can be viewed in a simpler manner. We can view the spacetime around the black hole like a set of Russian dolls made up of concentric spheres moving away from the event horizon. This description does not account for the flattened sphere that describes the ergosphere. These distortions of the spacetime can be derived from this initial view.

The Russian doll model can however be used to show how the spacetime rotates at the surface of each successive sphere and why this would result in frame dragging and co-rotation of objects within the ergosphere. This can be thought of as a curl in the gravitational field.
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#### jeffreyH

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##### Re: Examining the Kerr metric
« Reply #1 on: 19/03/2015 01:29:41 »
Another situation can be considered involving a rotating body. If this body is spherical and the circumference of the equator matches exactly the distance traveled by light in one second the rotation can be relativistic at the equator and non-relativistic at the poles. If one revolution of the body takes 1.0001 seconds then objects at the equator will experience extreme relativistic effects whilst those at the poles will experience non-relativistic effects. This means that time and distance vary greatly between the equator and the poles.

In the case of the Kerr black hole the poles should be completely outside the influence of the ergosphere. While the gravitational influence is still extreme at the poles, objects are not locked into the ergosphere. Hence the freedom of relativistic jets to escape at the poles.
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#### jeffreyH

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##### Re: Examining the Kerr metric
« Reply #2 on: 19/03/2015 01:44:31 »
The gravitational field around a Kerr black hole will have directionality normal to the surface at the poles and will deviate away from normal the nearer to the equator. At the equator the angle of deviation will be at its maximum. This effect increases with the speed of angular momentum.

Details of normal vectors can be found here:

http://mathworld.wolfram.com/NormalVector.html

This deviation towards the equator describes the type and extent of the ergosphere and why co-rotation occurs.

EDIT: Because of this field profile objects moving past a black hole with relativistic rotation can never be said to be moving perpendicular to the field. This also appears to suggest that the direction of approach determines whether or not an object will be consumed by the black hole.
« Last Edit: 19/03/2015 01:56:32 by jeffreyH »
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#### jeffreyH

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##### Re: Examining the Kerr metric
« Reply #3 on: 20/03/2015 20:32:50 »
The Kerr metric looks very daunting upon first view. However two sets of coordinate systems can start to help explain exactly what may be going on. These are:

Polar coordinates:

http://nrich.maths.org/2755

And spherical coordinates:

http://mathinsight.org/spherical_coordinates

Reviewing the Kerr metric in light of this information then can give clues as to exactly what is going on.

http://en.wikipedia.org/wiki/Kerr_metric#Mathematical_form
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#### jeffreyH

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##### Re: Examining the Kerr metric
« Reply #4 on: 21/03/2015 15:17:51 »
In the Kerr metric J represents the angular momentum. What about alpha equation.

$$\alpha = \frac {J}{Mc}$$

Since the momentum p = Mv and since c is also a velocity Mc is an upper limit on momentum. Therefore J/(Mc) is a proportionality of the angular momentum of the mass to its upper limit.
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#### jeffreyH

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##### Re: Examining the Kerr metric
« Reply #5 on: 21/03/2015 15:51:36 »
The rho equation $$\rho^2 = r^2 + \alpha^2 COS^2 \theta$$ is the spin coefficient.
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#### jeffreyH

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##### Re: Examining the Kerr metric
« Reply #6 on: 21/03/2015 16:38:21 »
It is useful to go back to first principles before continuing and reead through an introduction to angular momentum.

http://en.wikipedia.org/wiki/Introduction_to_angular_momentum
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#### jeffreyH

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##### Re: Examining the Kerr metric
« Reply #7 on: 02/05/2015 14:39:16 »
In the Kerr metric the left hand side is $$c^2d\tau^2$$ and has the speed of light and the proper time in the product. Proper time needs to be understood to see what the metric expresses.

http://en.wikipedia.org/wiki/Proper_time
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#### jeffreyH

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##### Re: Examining the Kerr metric
« Reply #8 on: 02/05/2015 17:34:50 »
This then needs to be compared to the coordinate time and the distinction made between the none variable nature of proper time along a path in spacetime and the variance of coordinate time.

http://en.wikipedia.org/wiki/Coordinate_time
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#### Colin2B

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##### Re: Examining the Kerr metric
« Reply #9 on: 10/05/2015 22:12:22 »
Is this only applicable to black hole theory or are there other applications?
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#### jeffreyH

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##### Re: Examining the Kerr metric
« Reply #10 on: 11/05/2015 18:27:17 »
Is this only applicable to black hole theory or are there other applications?

This deals with extreme conditions. In what we may term 'normal' spacetime the differences are often very small. You could look at neutron stars in a similar way although without the event horizon. I am in the process of reading Galaxy Formation Second Edition by Professor Malcolm Longair. It isn't only gravity that needs to be considered. His book discusses the COBE and WMAP data and the early history of astrophysics and cosmology. I would need to read the section on relativistic gravity before I can answer your question.
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#### Colin2B

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##### Re: Examining the Kerr metric
« Reply #11 on: 11/05/2015 18:58:20 »
Thanks, I ask because noted  PmbPhy in another thread talking about eigenvalues which I have used in vibration analysis. As the great man said, it's all connected.
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#### jeffreyH

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##### Re: Examining the Kerr metric
« Reply #12 on: 11/05/2015 19:34:38 »
Thanks, I ask because noted  PmbPhy in another thread talking about eigenvalues which I have used in vibration analysis. As the great man said, it's all connected.

I'll let you know when I've waded through Galaxy Formation.
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#### jeffreyH

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##### Re: Examining the Kerr metric
« Reply #13 on: 12/05/2015 13:48:17 »
Thanks, I ask because noted  PmbPhy in another thread talking about eigenvalues which I have used in vibration analysis. As the great man said, it's all connected.

I have something to add here but will need to copy the relevant section from Galaxy Formation.
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#### jeffreyH

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##### Re: Examining the Kerr metric
« Reply #14 on: 12/05/2015 19:06:40 »
From Malcolm Longair's Galaxy Formation Second Edition pages 249-250 when discussing the cosmological principle we have the following.

"A key step in the development of these models was introduced by Hermann Weyl in 1923 of what is known as Weyl's postulate (Weyl, 1923). To eliminate the arbitrariness in the choice of coordinate frames, Weyl introduced the idea that, in the words of Hermann Bondi (Bondi, 1960):

'The particles of the substratum (representing the nebulae) lie in space-time on a bundle of geodesics diverging from a point in the (finite or infinite) past.'

The most important aspect of this statement is the postulate that the geodesics, which represent the world lines of galaxies, do not intersect, except at a singular point in the finite, or infinite past. Again, it is remarkable that Weyl introduced this postulate before Hubble's discovery of the recession of the nebulae. By the term 'substratum' Bondi meant an imaginary medium which can be thought of as a fluid which defines the overall kinematics of the system of galaxies. A consequence of Weyl's postulate is that there is only one geodesic passing through each point in space-time, except at the origin. Once this postulate is adopted, it becomes possible to assign a notional observer to each world line and these are known as fundamental observers. Each fundamental observer carries a standard clock and time measured on that clock from the singular point is called cosmic time.

One further assumption is needed before we can derive the framework for the standard models. This is the assumption known as the cosmological principle and it can be stated:

'We are not located at any special location in the universe.'"

Colin, this is also a situation in which proper against coordinate time can be applied. However the starting point is the big bang and there are still huge gaps in knowledge about this. What does strike me is that the voids between galaxies should have vastly different coordinate time.
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#### jeffreyH

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##### Re: Examining the Kerr metric
« Reply #15 on: 08/11/2015 23:24:46 »
As an addition to this thread the Schwarzschild equation from the metric tensor can be expressed as $$\left( \frac{E}{c} \right)^2\left( 1 - \frac{r_s}{r} \right) - \rho_x^2 \left( 1 - \frac{r_s}{r} \right)^{-1} - \rho_y^2 r^2 - \rho_z^2 \left( r^2 sin^2 \theta \right)$$. Comparing this with the Kerr metric shows just how simplified the Schwarzschild solution is.
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