What does G mean in general relativity?

[vanhees71]

That simply is not true for dimensions. Those can and do change with different unit systems. In some unit systems $v$ is dimensionless, in some it has dimensions of $L^{1}\ T^{-1}$, and in some it has its own base dimension of $V^{1}$.

In most systems of units you would only have two of those as base dimensions, and the other would be derived. But it is possible to have a system of units where $V$, $L$, and $T$ are all independent base dimensions. That would be weird, but in principle it is no different than how SI and Gaussian units treat charge.

In all unit systems $[V]=[L]/[T]$. In some unit systems (HEP natural units as well as Planck units) [L]=[T] and thus [V]=1.

 

[Dale]

In all unit systems $[V]=[L]/[T]$.

See post 49 for a counterexample

 

[Vanadium 50]

See post 49 for a counterexample

That counter-example works by takling an external constant and running it into (or, I suppose, out of) the unit definition. This is actually fine and not a counter-example at all, because everything measurable still has the relationship between measurable quantities.

By the way, I see a definite confusion in this thread about units and dimensions. They are not the same thing: torque and energy have the same dimensions (in MKSA) but are different things with different units.

 

[vanhees71]

In the SI torque and energy have the same dimensions. Of course this doesn't imply that they are the same as a physical quantity. In natural units everything is measured in dimensionless numbers but that doesn't imply at all that all quantities have the same physical meaning.

 

[Dale]

This is actually fine and not a counter-example at all, because everything measurable still has the relationship between measurable quantities.

It is a counter-example for the specific claim that "in all unit systems [V]=[L]/[T]". In the Dale2 units $$v=k\frac{l}{t}$$$$[V]=[V^{1}\ L^{-1}\ T^{1}]\frac{[L]}{[T]}$$$$[V]=[V]$$ so $V$ is an independent base dimension.

Yes, of course it is fine and everything measurable still has the relationship between measurable quantities. The point is exactly that the dimensionality of a quantity is NOT something measurable because the dimensionality can be revised by convention without affecting anything measurable.

When you measure a charge, $q$, in SI vs in Gaussian units you can use the same experiments and the same measurements even though the formulas you use are different and the dimensionality of the resulting $q$ is different.

 

[ohwilleke]

For example in SI units Newton’s 2nd law is $\Sigma \vec f =m\vec a$ but you could make a system of units (Dale units) where force is its own base dimension. In those units Newton’s 2nd law is $\Sigma \vec f = k m \vec a$ where $k$ is a dimensionful universal constant with dimensions of $F^{1}\ M^{-1}\ L^{-1}\ T^{2}$.

Dimensionless doesn't distinguish between a "base dimension" and a "derived dimension."