Scale invariance in curved spacetime?

Question

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:

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

Answer 1

You can't say that your metric is scale "invariant". What you perhaps meant is to say that the theory is scale invariant and the metric is scale covariant.

Assuming you mean the latter, scale invariance of a theory simply implies that if I scale the metric, velocities, stress tensor, etc. as you have shown, then the physics of the system remains unchanged.

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].