There is a matrix $M$, and as far as it has quite a complicated form, there are several algebraic equations defining some of its parameters. I try to find the inverse of $M$ and simplify it. Unfortunately, $Mathematica$ gives the answer in the form, differing from the one I want: precisely, it substitutes parameters in the inverse matrix instead of giving the answer in terms of parameters (where it is possible).
Concretely, I have $$\rho = \sqrt{r^2 + a^2 Cos[\Theta]^2}$$ $$\Delta = r^2 - 2 m r + a^2 + e^2$$ and want the answer to be given in terms of $\Delta$ and $\rho$, where it is possible.
I tried to use $\mathsf{Eliminate[]}$ instead of $\mathsf{FullSimplify[]}$, but did not succeed. Moreover, there is a problem with $\mathsf{Eliminate[]}$: there may be no possibility to fully eliminate the parameters.
Is there any way to make $Mathematica$ work as I want?
EDIT:
I have following problem:
\[Rho] = Sqrt[r^2 + a^2 Cos[\[Theta]]^2];
\[CapitalDelta] = r^2 - 2 m r + a^2 + e^2;
Metric = {{\[Rho]^2, 0, 0, 0}, {0, 0, - a Sin[\[Theta]]^2,
1}, {0, -
a Sin[\[Theta]]^2, \[Rho]^-2 ((r^2 +
a^2)^2 - \[CapitalDelta] a^2 Sin[\[Theta]]^2) \
Sin[\[Theta]]^2, - a \[Rho]^-2 (2 m r - e^2) Sin[\[Theta]]^2}, {0,
1, - a \[Rho]^-2 (2 m r -
e^2) Sin[\[Theta]]^2, -(1 - \[Rho]^-2 (2 m r - e^2))}};
InvMetric = Inverse[Metric] // Simplify // MatrixForm
And what I get is:
,
although I want to get $\rho^{-2}$ instead of (0; 0) entry, for example.
{a, e, m, r, θ}
in terms of{Δ, ρ}
, doing so is fairly straightforward but you may not be happy with what you get. Simplification is one of the hardest things to accomplish with symbolic algebra programs, in part because there is no clear definition of what it means, and everyone means something different. Mathematic attempts to minimizeLeafCount
. By the way, I do not get what you have in the question for ` InvKerr`. $\endgroup${a, e, m, r, θ}
which two? $\endgroup$