# Changing the scalar curvature (k = 0,+1,-1) with coordinate transformations?

I would like to prove that I can (or can't) change curvature of space, k = 0,+1,-1, via general coordinate transformations, which in principle can mix space and time coordinates together.

Let's be clear about what you mean by the term "space".

I'm assuming you take your spacetime, and pick out a specific three dimensional submanifold $\Sigma$, say by a function $f(x^{\mu})=$ const. You now compute the intrinsic Ricci curvature scalar for that submanifold. Now you make a coordinate transformation $x'^{\mu} = x'^{\mu}(x^{\nu})$. The new spatial coordinates are allowed to depend on the old $x^0$s. In the new coordinates, the new function specifying the same $\Sigma$ will be different to $f$, but we can still calculate the intrinsic curvature of $\Sigma$ in the new coordinates.

This intrinsic curvature is a geometric property of $\Sigma$ independent of the embedding. The Ricci scalar of $\Sigma$, i.e. $^3R$ assigns a number to each point in $\Sigma$. This number is the same regardless of the values of the coordinate system used on $\Sigma$.

If, however, by "space" you had meant the surface $x^0=$const given by the chosen coordinate system, then when we change coordinates, $x'^0=$const represents a completely different submanifold which may have different intrinsic curvature.

By the term "space" I meant this: If 4D metric is given by

G_μν = diag[1,-a(t)^2/1-kr^2,-a^2(t)r^2,-a(t)^2r^2sin(Θ)^2],

in some coordinates (t,r,Θ,ϕ), then a 3D "subspace" is simply

G_ij = diag[-a(t)^2/1-kr^2,-a^2(t)r^2,-a(t)^2r^2sin(Θ)^2],

which seems like your other case, surface of t=const.

• Hi Winnie. Welcome to Physics.SE. This site uses an unique TeX markup style called MathJax. This markup is very useful for understanding math equations and parameters. Please have a look here for an intro or our FAQ for more info. For example, $\mu$ results $\mu$, $\omega$ inserts $\omega$, etc. It's quite interesting. You can revise your post if you can ;-) – Waffle's Crazy Peanut Apr 19 '13 at 13:22