Diagonalizing a metric by a coordinate transformation

[Lilian Sa]

Homework Statement

gravity

Relevant Equations

metric transformation

hey there :)

So I had a homework, and I was asked to diagonalize the metric ${ds}^2=-{dt}^2+{dx}^2+2a^2(t)dxdy+{dz}^2$ and to find the coordinate transformation for the coordinates of the new metric.
so I found the coordinate transformation but the lecturer said that what I found is a specific solution and not the general solution.
And he said that the function F (I attached my solution) is a function that should be dependent on the other parameter of the problem.
and I don't know how to translate this.
I started to solve it from the start but I got entangled >_<

thank you :)

 

[Nate810]

It sounds like your lecturer wants you to find a more general solution. To do this, you need to find a function F that is dependent on all of the parameters in the problem. For example, if you have a parameter a, then your function F should depend on a as well. In addition, your solution should work for any value of a (not just a specific one). If you are struggling to find the general solution, it might be helpful to look up some examples of how to solve similar problems. This could give you an idea of what the general solution should look like.