# Curvature tensor in all flat space coordinates

• A
hi, I am just curious about, and really wonder if there is a proof which demonstrates that curvature tensor is 0 in all flat space coordinates. Nevertheless, I have seen the proofs related to curvature tensor in Cartesian coordinates and polar coordinates, but have not been able to see that zero curvature tensor dominates all flat space coordinates ( besides the polar and cartesian coordinates ). Could you please explain or help me reach the proof ???

ShayanJ
Gold Member
hi, I am just curious about, and really wonder if there is a proof which demonstrates that curvature tensor is 0 in all flat space coordinates. Nevertheless, I have seen the proofs related to curvature tensor in Cartesian coordinates and polar coordinates, but have not been able to see that zero curvature tensor dominates all flat space coordinates ( besides the polar and cartesian coordinates ). Could you please explain or help me reach the proof ???

The Riemann tensor transforms under a coordinate transformation as ## R'^\sigma_{ \ \rho \mu \nu}=(S^{-1})^\sigma_\alpha S^\beta_\rho S^\gamma_\mu S^\delta_\nu R^\alpha_{ \ \beta \gamma \delta}## which means its components in the new coordinate system are linear combinations of its components in the old coordinate system. So if the components are all zero in the old coordinate system, they'll also be zero in the new coordinate system. That's why you only need to prove they're zero in only one coordinate system to prove that the space is flat.

Orodruin
Staff Emeritus
Homework Helper
Gold Member
To add to Shyan's reply, this is true for any tensor. If it is zero in one set of coordinates, it is zero in all coordinates.

As Shyan and Orodruin have already pointed out, the proof lies in the fact that Riemann is a tensor. As geometric objects, tensors are by their nature and definition independent of the choice of coordinate basis - if you perform a transformation into a different coordinate system, then the expressions for each individual component of the tensor may change, but the relationships between the various components will not, meaning the overall geometric object remains the same.

In a perhaps more tangible sense, a change in coordinates just signifies that you choose to label the same events in spacetime in a different way - and since labels are completely arbitrary, these events will still remain related in the same ways, so the overall geometry cannot change simply by your act of "re-labelling". Therefore, you need to calculate Riemann only once, in any suitable coordinate system of your choice, and the result ( flatness or not ) will be applicable for all other equivalent choices of coordinate basis as well.

That is the beauty of tensors.

pervect
Staff Emeritus