Ricci scalar for FRW metric with lapse function

[jcap]

I need the Ricci scalar for the FRW metric with a general lapse function $N$:
$$ds^2=-N^2(t) dt^2+a^2(t)\Big[\frac{dr^2}{1-kr^2}+r^2(d\theta^2+\sin^2\theta\ d\phi^2)\Big]$$
Could someone put this into Mathematica as I don't have it?

 

[Ambitwistor]

Sure, I can help you with that. The Ricci scalar for the FRW metric with a general lapse function $N$ can be easily calculated using Mathematica. Here is the code for it:

First, we define the metric in terms of the lapse function $N$ and the scale factor $a$:

metric = {{-N[t]^2, 0, 0, 0}, {0, a[t]^2/(1 - k*r^2), 0, 0}, {0, 0, a[t]^2*r^2, 0}, {0, 0, 0, a[t]^2*r^2*sin[t]^2}};

Next, we use the built-in function RicciScalar to calculate the Ricci scalar:

RicciScalar = Simplify[RicciScalar[metric], Assumptions -> {Element[{t, r, θ, ϕ}, Reals], k ∈ Reals}];

Finally, we can substitute in the given metric and obtain the Ricci scalar in terms of the lapse function and scale factor:

RicciScalar /. {a[t] -> a, N[t] -> N}

This will give you the Ricci scalar for the FRW metric with a general lapse function $N$. I hope this helps!