# Question:algsubs in equation with multiple derivatives

## Question:algsubs in equation with multiple derivatives

Maple 18

I am trying to simplify equation 18 using equations 8 and 9. It should look a little like equation 21, but instead I get the results in equations 19 and 20.  I tried using different substituions, but algsubs gets the closest answer. A few terms are going to zero after the substitution.

When I substitute Z(X) then Zbar(X) terms vanish, and visa versa.

Initialize the metric and tetrad

 (1.1)

 (1.2)

 (1.3)

 (1.4)

 (1.5)

 (1.6)

I will try to verify the tetrad from (Kerr and Schild (1965)). However, the tetrad given in the paper seems to have the third tetrad with the wrong sign. I changed the sign and get the correct verification,

 (2.1)

 (2.2)

 (2.3)

 (2.4)

 (2.5)

 (2.6)

 (2.7)

 (2.8)

For equation 2.8 we get the following:

 (1)

Now we replicate eqn 2.16. These are the conditions for e[4,mu] to be geodesic and shear-free. The outputs are eqn 3.5.

 (2)

 (3)

 (4)

 (5)

 (6)

 (7)

Now we can define the rotation coefficients associated with rotation and expansion z = theta - i omega

 (8)

 (9)

 (10)

 (11)

We now show that the tetrad vectors are propogated parallel along each curve of the congruence of null geodesics which have e[4,~mu] as tangents.

We now use the tetrad form of the Ricci tensor. In order to use this in Maple we need to create a Ricci Tensor Tetrad function.

 (12)

 (13)

 (14)

 (15)

 (16)

 (17)

The geodesic and shear free condition given by Lemma 1 in (Goldberg and Sachs (1962)). Kerr uses the fourth tetrad instead of the third so we need to modify the Ricci tensor conditions. The equations (2) - (5) enforce the first Lemma.

Notice that none of the previous Ricci conditions can be used to solve for H.  We can use the remaining field equations to find the partial differential equations necessary to derive the metric.

 (18)

 (19)

 (20)

 (21)