# Ring -singularity (Determinant of the Kerr metric)

• RestlessRiver
In these cases, we have:det(g) = 0This tells us that the metric becomes degenerate at the poles, meaning that there are no well-defined directions in these regions.In summary, the determinant of the Kerr metric helps us understand the properties of the spacetime around a rotating black hole. It becomes infinite at the event horizon, indicating a singularity, and goes to zero at large distances, showing the metric approaches flat spacetime. It also becomes degenerate at the poles, indicating a loss of well-defined directions. I hope this helps you better understand this problem.
RestlessRiver
"Ring"-singularity (Determinant of the Kerr metric)

My problem is as follows:
"Calculate the determinant of the Kerr metric. Locate the plac where it is infinite. (In fact, this gives the "ring"-singularity och the Kerr black hole, which is the only one)

I got the determinant to :

7a2r4sin4θ+7a4r2sin4θ-8a2r2sin4θ-16Ma2r3sin4θ+16M2a2r2sin4θ-2Ma4rsin4θ+2a2r4sin2θ+a4r2sin2θ+r6sin2θ+2Ma4rsin2θ-4M2a2r2sin2θ-2Mr2sin2θ

all devided by r2 + a2 - 2Mr

where
a=angular momentum/M
M=Mass

and I talked to my prefessor and he told me that the answer should be the equation of a ring (x2+y2 = constant) in spherical coordinates, I have all this in Boyer-Lindquist coordinates I believe, and according to wikipedia

{x} = \sqrt {r^2 + a^2} \sin\theta\cos\phi
{y} = \sqrt {r^2 + a^2} \sin\theta\sin\phi
{z} = r \cos\theta

(http://en.wikipedia.org/wiki/Boyer-L...st_coordinates )

I don't get it to be an eq of a ring (or circle) .. please help =)

Last edited by a moderator:

I can understand your confusion and frustration with this problem. The Kerr metric is a complex mathematical equation that describes the spacetime around a rotating black hole. Its determinant is an important quantity that can tell us about the properties of the black hole, including the location of its singularity.

To start, let's define the Kerr metric in Boyer-Lindquist coordinates as:

ds^2 = -\left(1-\frac{2Mr}{\Sigma}\right)dt^2 - \frac{4Mar\sin^2\theta}{\Sigma}dtd\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma d\theta^2 + \left(r^2+a^2+\frac{2Ma^2r\sin^2\theta}{\Sigma}\right)\sin^2\theta d\phi^2

where M is the mass of the black hole, a is its angular momentum, and \Sigma = r^2 + a^2\cos^2\theta and \Delta = r^2 - 2Mr + a^2. Now, the determinant of this metric is given by:

det(g) = -\Sigma^2\sin^2\theta

This is a negative quantity, which means that the metric is not positive definite and therefore cannot be interpreted as a distance or a measure of length. However, we can still analyze the behavior of this determinant to understand the properties of the Kerr metric.

First, let's look at the behavior of the determinant as r approaches the event horizon, which is defined by \Delta = 0. This gives us:

det(g) = -\frac{4Ma^2r\sin^4\theta}{\Sigma^2}

As r gets closer to the event horizon, this determinant approaches infinity, indicating a singularity at the horizon. This is the "ring"-singularity that you mentioned in your problem.

Next, let's consider the behavior of the determinant as r approaches infinity. In this case, we have:

det(g) = -r^2\sin^2\theta

This determinant goes to zero as r goes to infinity, indicating that the metric becomes flat in this limit. This makes sense, as the spacetime around a black hole should approach flat spacetime at large distances.

Now, let's examine the behavior of the determinant as \theta approaches 0 or \pi (

## 1. What is a ring singularity?

A ring singularity is a type of singularity that is predicted by the Kerr metric in general relativity. It is a region in spacetime where the curvature becomes infinite, causing the laws of physics to break down. In the Kerr metric, the ring singularity is a circular ring rather than a point, which is a unique feature of this type of singularity.

## 2. How is the ring singularity related to black holes?

The ring singularity is a feature of the Kerr metric, which is a mathematical solution that describes the spacetime around a rotating black hole. It is located at the center of the black hole and is responsible for the extreme gravitational forces that make black holes so fascinating and mysterious.

## 3. Can the ring singularity be observed or measured?

No, the ring singularity cannot be observed or measured directly. It is a mathematical concept that arises from the Kerr metric and is not accessible to experimental observation. However, its effects on the surrounding spacetime can be indirectly measured through the study of gravitational waves and other phenomena.

## 4. What happens if an object falls into the ring singularity?

Since the ring singularity is a region where the laws of physics break down, it is impossible to predict what would happen to an object that falls into it. The object would likely be torn apart by the extreme gravitational forces, but beyond that, we cannot make any definitive statements about its fate.

## 5. Is the existence of the ring singularity confirmed by observations?

While the Kerr metric and the concept of the ring singularity have been extensively studied and have proven to be very accurate in describing the behavior of rotating black holes, their existence has not been confirmed by direct observations. However, the evidence from gravitational waves and other phenomena strongly supports the existence of ring singularities and the Kerr metric.

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