# What are physics constant in Kerr metric?

• zerop
In summary, the Kerr metric is a mathematical representation of the spacetime around a rotating black hole. The values of physics constants in the Kerr metric include G (the gravitational constant), M (mass), c (the speed of light), a (angular momentum), r (radius), and others. Some books may assume G = c = 1, which means they are using geometric units where 1 unit of mass is equal to G/c^2 and 1 unit of angular momentum is equal to G/c. The element [1,4] in the g_compts array represents the radial parameter and can also be written as [4,1]. Theta refers to the latitude approach in the Kerr metric. There is a metric table for element

#### zerop

1. What are the value of physics constant in Kerr metric, including G, M, c, a, r, or others?
I expect to simplify Gamma

2. why g_compts[1,4] has element and not [4,1]?

3. Some book assume G = c = 1, what is the meaning of this setting?

4. Different material have different metric, are there a metric table for element table?

5. What is theta in Kerr metric?

************** Kerr metric *****************
t r theta phi
t
r
theta
phi

with(tensor):
coord := [t, r, theta, Phi]:

g_compts:=array(sparse,1..4,1..4):

G := 6.67*10^(-11)

triangle := r^2 - 2*G*M*r/c^2 + a^2:
p2 := r^2 + ((cos(theta))^2)*a^2:
A := (r^2+a^2)^2 - (a^2)*triangle*(sin(theta))^2:

g_compts[1,1]:= (triangle - (a^2)*(sin(theta))^2)*(c^2)/p2:
g_compts[1,4]:= 4*G*M*a*r*(sin(theta))^2/(c*p2):
g_compts[2,2]:= -p2/triangle:
g_compts[3,3]:= -p2:
g_compts[4,4]:= -A*(sin(theta)^2)/(p2):

g1 := create([-1,-1], eval(g_compts)):
g1_inv := invert( g1, 'detg' ):

D1g := d1metric( g1, coord ):

Cf1_1 := Christoffel1(D1g):
Cf2_1 := Christoffel2(g1_inv, Cf1_1):
displayGR(Christoffel2,Cf2_1):

G is the gravitational constant in units m^3/(kg s^2) (1 in geometric units)

m is mass in kg where M is the geometric unit for mass (M=Gm/c^2) in metres

c is the speed of light in m/s (or 1 in geometric units)

a is the geometric units for angular momentum in metres (a=J/mc where J is angular momentum in SI units)

Delta (or triangle as you call it) is the radial parameter in m^2.

when writing delta, you have written delta=r^2-2*G*m*r/c^2+a^2. If geometric units are used, you can simply write delta=r^2-2M+a^2 where M=*G*m*r/c^2, the answers are the same.

g_compts[1,4] does include for [4,1], they've just substituted the 2*(2*.. with a 4*.., it can be rewritten-

g_compts[1,4]=2*(2*M*a*r*(sin(theta))^2/(p2)), [1,4] & [4,1] being the same, another way of writing it is 2*g_compts[1,4].

theta is the latitude approach, 90 degrees (or pi/2) at the equator and 0 at the poles.

You may also find this web page useful-
http://www.astro.ku.dk/~milvang/RelViz/000_node12.html

The text above relating to delta should read-

'when writing delta, you have written delta=r^2-2*G*m*r/c^2+a^2. If geometric units are used, you can simply write delta=r^2-2Mr+a^2 where M=G*m/c^2, the answers are the same.'

## 1. What is the Kerr metric?

The Kerr metric is a mathematical model that describes the geometry of space and time around a rotating, uncharged black hole. It is named after New Zealand mathematician Roy Kerr, who first proposed it in 1963.

## 2. What are physics constants in Kerr metric?

The physics constants in Kerr metric are the parameters that determine the properties of the black hole, such as its mass and angular momentum. They include the mass of the black hole (M), its angular momentum (J), and the speed of light (c).

## 3. How are these constants related to the Kerr metric?

The Kerr metric is a solution to Einstein's field equations in general relativity, which describe the curvature of spacetime due to the presence of matter and energy. The values of the physics constants in Kerr metric determine the specific geometry of the black hole spacetime.

## 4. Are these constants the same for all black holes?

No, the values of the physics constants in Kerr metric can vary depending on the properties of the black hole. For example, a black hole with a larger mass will have a different value for its mass constant than a smaller black hole.

## 5. How do these constants affect the behavior of matter near a black hole?

The physics constants in Kerr metric determine the strength of the gravitational pull and the curvature of spacetime near the black hole. This affects the motion of matter and light, as well as the formation and evolution of structures such as accretion disks around the black hole.