## 125 Reputation

15 years, 290 days

## expand causing lhs to be zero...

Maple 18

I am trying to expand out the terms  of equation 13.  The expand command causes the lhs to be zero?

 (1.1)

 (1.2)

 (1.3)

 (1.4)

 (1.5)

 (1.6)

 (1)

 (2)

 (3)

 (4)

 (5)

 (6)

 (7)

 (8)

 (9)

 (10)

 (11)

 (12)

 (13)

 (14)

Why does the expand command cause the lhs to be zero?

## How do I get dsolve to find the solution...

Maple 18

 (1)

 (2)

 (3)

When I try to solve this system of ODE, Maple does not factor out f4(r) in eqn4.  How do I get Maple to solve eqn4? So that I can get the correct result for f1(r) below:

 (4)

Thanks.

## algsubs in equation with multiple deriva...

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.

 (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)

Maple 18

Does anyone know how to incorporate the tetrad with the directional derivative? I tried using the SumOverIndices, but get crazy results. I know Maple can find the answer easily because I have done the same thing by hand. What am I missing?

The directional derivative should take the form f,1 = eaμ df/dxμ . The answer is Y,1 = dY/dζ – Ybar dY/du.  I obviously do not get this result.

 (1)

 (2)

 (3)

 (4)

 (5)

 (6)

 (7)

## Length of Output Exceeded Error when usi...

Maple 18

I am using the SumOverRepeatedIndices and get a Length of Output Exceeded error. Sometimes if I close the file and restart the program then I get a result and no error.  However, if I recalculate then I get the error.

 (1)

 (2)

 (3)

 (4)

 (5)

 (6)

 (7)

 (8)

 (9)

 (10)

 (11)