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) r is radius in metres 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.'