# Can a conformal flat metric be curved?

• A
• Kurvature
In summary, the conversation is about determining if a given conformal metric produces zero or nonzero curvature. The metric is given as a spacetime metric with a well-behaved scale factor, and the question is whether the Ricci scalar is zero or not. The conversation also discusses the use of conformal transformations and their applications, as well as the potential use of MAXIMA to calculate the curvature.
Kurvature
TL;DR Summary
If the flat (Cartesian) metric is multiplied by a time dependent scale factor could it yield a curvature?
5/18/22
I am an MS in physics.

I need to find out if the following CONFORMAL
METRIC produces zero or nonzero curvature?

I suspect the curvature is zero, but others
have said it's probably not? MAXIMA
sometimes says it is, and other times produces
a Ricci scalar that looks like the FRW scalar
curvature ?

Does someone know the answer to this question
off the top of their head?

If not, could someone possibly plug this metric
into Mathematica and tell me if Ricci scalar
is actually zero or if not, what it actually is ?

|a 0 0 0 |
|0 a 0 0 | = the given spacetime metric
|0 0 a 0 |
|0 0 0 -a|

where: a = a(t) = "the scale factor" is a
simple well behaved function of time.

Kurvature

Penrose diagrams use a conformal transformation to map a curved metric into a flat Minkowski one (often with non-Minkowskian global topology). Hence the inverse conformal transformation maps a flat metric into a curved one. So the answer is - yes.

The curvature of ##g_{ab} = \Omega(x)^2 \eta_{ab}## is
\begin{align*}
R_{ab} = \frac{1}{\Omega^2}(4\partial_a \Omega \partial_b \Omega - 2\partial_a \partial_b \Omega -(|\nabla \Omega|^2 + \Omega \Delta \Omega)\eta_{ab})
\end{align*}where ##\Delta = \partial^a \partial_a## and ##|\nabla \Omega|^2 = \partial^a \Omega \partial_a \Omega##. Such transformations of flat spacetimes have many interesting applications. For example, a particle of charge ##q## coupling to a scalar field ##\Phi(x)## in flat spacetime, as per ##q \partial_a \Phi = u^b \nabla_b \left[ (m + q\Phi) u_a \right]##, is equivalently described by a curved spacetime with metric ##g_{ab} = (m + q\Phi)^2 \eta_{ab}## with the curvature generated by some hypothetical matter field whose properties can be explored.

dextercioby, Demystifier and vanhees71
Why do you suspect the curvature to be zero?

If you are looking at ##ds^2 = a^2(t)(-dt^2+dx^2+dy^2+dz^2)=-a^2(t)dt^2+a^2(t)dx^2+a^2(t)dy^2+a^2(t)dz^2##
you can change the variables ##t'=\int a(t)dt##, the others remain the same, then you get ##-dt'^2+a^2(t')dx^2+a^2(t')dy^2+a^2(t')dz^2## the line element of a typical cosmological solution, which will not be flat unless ##a=const##.

Also it is not that hard to caclulate curvature in terms of ##a(t)##. Even if you don't use anything that might simplify the calculations, it will not be so bad. All you need is one component of the Riemann tensor that is not zero, you don't have to compute all.

Kurvature said:
MAXIMA
sometimes says it is, and other times produces
a Ricci scalar that looks like the FRW scalar
curvature ?
What specific input are you giving MAXIMA to get these different answers? The same input will always give the same output, so it doesn't make sense to say MAXIMA "sometimes" says one thing and sometimes says another for the same input.

vanhees71

## 1. Can a conformal flat metric be curved?

Yes, a conformal flat metric can be curved. This is because conformal flatness only refers to the local geometry of a space, while curvature refers to the global geometry of a space. In other words, a conformal flat metric can have regions that are locally flat, but overall the space can still be curved.

## 2. What does it mean for a metric to be conformal flat?

A conformal flat metric means that the angles between curves in the space are preserved under conformal transformations. In simpler terms, this means that the local geometry of the space remains the same even when the space is stretched or compressed.

## 3. How is conformal flatness related to conformal symmetry?

Conformal flatness is a property of a metric, while conformal symmetry is a property of the transformations on the metric. Conformal symmetry means that the metric is invariant under conformal transformations, which preserve angles and therefore also preserve conformal flatness.

## 4. Can a conformal flat metric exist in a curved space?

Yes, a conformal flat metric can exist in a curved space. This is because conformal flatness only refers to the local geometry of a space, while curvature refers to the global geometry of a space. Therefore, even in a curved space, there can be regions that are locally conformally flat.

## 5. How is conformal flatness useful in physics?

Conformal flatness is useful in physics because it simplifies the mathematical calculations involved in studying curved spaces. By preserving angles and local geometry, conformal flatness allows for easier calculations and can reveal important symmetries in physical systems.

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