# Scale invariance in curved spacetime?

## Question

What does it mean for the metric to be scale invariant in curved spacetime (in the sense when I say a property is scale invariant in thermodynamics)? I'm confused as to how to define this. It seems to be either by means of a Weyl scaling or conformal transformation where the scaling factor is a constant? I suspect the correct way would be via means of coordinate transformations? Is there some nice mathematical condition such a metric would satisfy?

## Motivation

Consider the stress energy tensor for a perfect fluid:

$$T^{\mu \nu} = \left(\rho + \frac{p}{c^2} \right) U^{\mu} U^\nu + p g^{\mu \nu},$$

Now keeping our notation ambiguous:

$$g^{\mu \nu} \to \lambda^2 g^{\mu \nu}$$

But $$g^{\mu \nu} g_{\mu \nu} = 4$$

Thus

$$g_{\mu \nu} \to \frac{1}{\lambda^2}g_{\mu \nu}$$

We also know:

$$g_{\mu \nu} U^{\mu} U^\nu = c^2$$

Thus,

$$U^{\mu} U^\nu \to \lambda^2 U^{\mu} U^\nu$$

Thus we have effectively done the following:

$$T^{\mu \nu} \to \lambda^2 T^{\mu \nu}$$

• You want your metric to be scale invariant...but which quantity do you want to rescale in the first place so that your metric remains scale invariant? Can you elaborate on that?
– KP99
Jan 31, 2022 at 15:52
• I 'd like the stress energy tensor to be scale invariant. If that is scale invariant I suppose all my properties in thermodynamics become extensive? Jan 31, 2022 at 16:05
• I am not familiar with relativistic thermodynamics, but I guess that the thermodynamics of the matter part will become extensive, but there could be a change in entropy due to the coupled gravitational field. Conformal rescaling of metric will rescale the Weyl curvature, which encodes the gravitational entropy
– KP99
Jan 31, 2022 at 18:21
• "If that is scale invariant I suppose all my properties in thermodynamics become extensive?" I don't understand where this is coming from. $\rho$ is the density of an extensive quantity no matter what the metric is. Given an equation of state you can extract densities of other extensive quantities like entropy and conserved charges from the energy momentum tensor too. None of this has to do with scale invariance (do you want to talk about a conformal fluid?) Feb 2, 2022 at 12:10
• @MoreAnonymous, The usual way to treat thermodynamics in relativistic fluids is you effectively trade in all your extensive quantities for densities of extensive quantities (like $\rho$). A good review is Andersson, Comer (arxiv.org/abs/2008.12069) Feb 2, 2022 at 12:43

A Weyl transformation is $$g_{ab}(x) \to \Omega^2(x) g_{ab}(x)$$ Here, $$\Omega(x)$$ is an arbitrary positive function. When $$\Omega(x)$$ is a constant, we call this a scale transformation.
A conformal transformation is a diffeomorphism $$x^a \to x'^a(x)$$ such that the metric transforms as $$g_{ab}(x) \to g'_{ab}(x) = \Omega(x)^2 g_{ab}(x)$$ Here, the Weyl factor $$\Omega(x)$$ is NOT an arbitrary function. Rather it is related to the diffeomorphism $$x'^a(x)$$ [which is also not arbitrary].