Why Does the Right Hand Side of 5.18 Only Have Two Derivatives?

[binbagsss]

Homework Statement

I am trying to follow my lecture notes, attached below, an argument to the number of degrees of freedom from the number of constraints of he Einstein field equations

The bit I a man stuck on is the second thumbnail, and the deduction that he righ hand side of 5.18 only has two derivatives at most. I see that the terms the connection multiplied by G must be a first derivative of the metric multiplied by a second derivative, since $R \approx \partial\Gamma \implies G \approx \partial^2 g$

However for the first term I would deduce a third derivative of the metric, since there's another derivative acting on $G$

So I am unsure how to deduce the right hand side must at most have two derivatives of any type. Do I need to carry out some sort of dimensional analysis ? Obviously these terms must be consistent with each other, in turn consistent with the left hand side? From the second and third term I could agree with the statEment but not the first ? If so, I am unsure how to go about this...

Many thanks

Homework Equations

Above

The Attempt at a Solution

Above[/B]

 

[binbagsss]

bump?

 

[TSny]

The notes make the statement:
"But we know that the right hand side has at most two derivatives, of any kind."

I agree with you that this is not correct. The first term of the right hand side would have third-order derivatives of metric components with respect to spatial coordinates $x^i$. I believe the statement should have said something like:
"But we know that the right hand side has at most two time derivatives of any of the $g_{\mu \nu}$."

With this correction, the next statement of the notes is still correct. That is, "there cannot be any term in $G^{t \nu}$ with two time derivatives in it".