# Einstein field equations and scale invariance

• johne1618
In summary, the Einstein's field equations are scale invariant if the stress-energy density in the Universe is zero. However, the addition of the cosmological constant breaks the scale invariance.

#### johne1618

Hi,

Are Einstein's field equations without the cosmological constant scale invariant?

If so does the addition of the cosmological constant break the scale invariance?

John

After looking at the web it seems that the Einstein's field equations would only be scale invariant if the stress-energy density in the Universe was zero. Is that right?

if the actual density equals the critical density the universe would be static

$\rho_{crit} = \frac{3c^2H^2}{8\pi G}$

Mordred said:
if the actual density equals the critical density the universe would be static

$\rho_{crit} = \frac{3c^2H^2}{8\pi G}$

Hi Mordred, I rather think the universe would be spatially flat in that case.
$\rho_{crit} = \frac{3c^2H^2}{8\pi G}$ combined with the first Friedmann equation yields k = 0.

Last edited:
Einstein's field equations are invariant under coordinate transformation. To my knowledge this is generally true regarding the laws of physics.

timmdeeg said:
Hi Mordred, I rather think the universe would be spatially flat in that case.
$\rho_{crit} = \frac{3c^2H^2}{8\pi G}$ combined with the first Friedmann equation yields k = 0.

as far as I know the definition of critical density is the density at which stops expansion granted its also related to the curvature. Mind you dark energy complicates this definition.

http://www.astro.virginia.edu/~jh8h/glossary/criticaldensity.htm
http://www.collinsdictionary.com/dictionary/english/critical-density

Its also the definition in my textbooks which I can't post one of them being Barbera Ryden's introductory to cosmology

more accurately its the density at which expansion stops without the cosmological constant. Ignoring the cosmological constant it would describe the fate of the universe a flat is static, positive curvature is open and expanding forever, negative would be collapsing. However the cosmological constant complicates this older reasoning.

http://map.gsfc.nasa.gov/universe/uni_fate.html

see the link above for further details on the effect of geometry has on expansion

Last edited:
Mordred said:
as far as I know the definition of critical density is the density at which stops expansion granted its also related to the curvature. Mind you dark energy complicates this definition.

http://www.astro.virginia.edu/~jh8h/glossary/criticaldensity.htm

Mordred

I'm afraid that timdeeg is right in post #4, and you are wrong. The definition of the critical density is the density at which the universe is neither spatially open nor closed, but flat (zero curvature). In the link you posted above, you seem to have missed a very important part of the description:

The mass density of the universe which just stops the expansion of space, after infinite cosmic time has elapsed.

Indeed, (for the zero lambda case), if the density is critical, the universe expands forever, so it is incorrect to say that the expansion ever stops. The rate of expansion slows, of course, but for the critical case it never reaches zero. It approaches zero asymptotically as t → ∞. That's what the statement in boldface above is saying.

EDIT: also this whole discussion is kind of off topic, it seems unrelated to what the OP was asking.

fair enough, good to know and yes I did miss that aspect thanks for the clarification

I think you should be more focused on the Friedmann equations, not Einstein's field equations. Einstein inserted lambda mainly to avoid the obvious problems posed by a static universe.

Let me address the original question since the friedmann issue has been cleared up:

The EFE do not single out a length scale, hence are 'scale invariant' in that sense. Simply put, gravity on its own knows only about c and G, out of which a length scale cannot be formed. Given matter fields in the form of a stress energy tensor, one can again construct a length scale: $\ell \sim \left(G c^{-4} < T > \right)^{-1/2}$

The addition of Lambda does also introduce a length scale (one can after all think of it as a stress energy). The dimensions of Lambda are L^-2, so obviously 1/sqrt(Lambda) is a preferred length.

Just to add some context after the previous good reply to the OP, precisely this lack of scale invariance of GR in the presence of mass-energy was what prevented from succeding Weyl's first attemp of unification of gravity with EM in 1918. But his flawed idea was the seed of a more fruitful "gauge incariance" in the years to come.

Also to answer #2, it is kind of true that in the absence of matter-energy scale invariance should be recovered, however in practice this is not so at least in the Schwarzschild case as it is usually interpreted physically (weak field), as mass-energy manages to sneak in thru the boundary condition(central mass) that makes it lose the scale invariance.
But the equation by itself is trivially scale invariant.