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?


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.

  • $\begingroup$ @bbgodfrey I included my problem in the question. $\endgroup$
    – user108687
    Apr 15, 2017 at 1:15
  • $\begingroup$ @bbgodfrey I am sorry for my mistake: I decided to rename it before sending here. Kerr = Metric. I will edit the question. $\endgroup$
    – user108687
    Apr 15, 2017 at 1:47
  • $\begingroup$ What is your definition of "simplified"? If you mean eliminating two of {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 minimize LeafCount. By the way, I do not get what you have in the question for ` InvKerr`. $\endgroup$
    – bbgodfrey
    Apr 15, 2017 at 1:50
  • $\begingroup$ @bbgodfrey Yes, my definition is the one you mention. I was not aware about the mechanism of simplifying. So you suggest doing such things by hand? $\endgroup$
    – user108687
    Apr 15, 2017 at 1:53
  • $\begingroup$ If by your last comment you mean eliminating two of {a, e, m, r, θ} which two? $\endgroup$
    – bbgodfrey
    Apr 15, 2017 at 1:56

1 Answer 1


The following works for your example:

    Cos[2 θ]==2Cos[θ]^2-1 && ρ^2==r^2+a^2 Cos[θ]^2 && Δ==r^2-2 m r+a^2+e^2
] //TeXForm

$\left( \begin{array}{cccc} \frac{1}{\rho ^2} & 0 & 0 & 0 \\ 0 & \frac{\Delta }{\rho ^2} & \frac{a}{\rho ^2} & \frac{a^2+r^2}{\rho ^2} \\ 0 & \frac{a}{\rho ^2} & \frac{\csc ^2(\theta )}{\rho ^2} & \frac{a}{\rho ^2} \\ 0 & \frac{a^2+r^2}{\rho ^2} & \frac{a}{\rho ^2} & \frac{a^2 \sin ^2(\theta )}{\rho ^2} \\ \end{array} \right)$


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