GR Lagrangian

  • Thread starter Malamala
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Homework Statement


The Lagrangian density for the ##h=h^{00}## term of the Einstein gravity tensor can be simplified to: $$L=-\frac{1}{2}h\Box h + (M_p)^ah^2\Box h - (M_p)^b h T$$ The equations of motion following from this Lagrangian looks roughly like (I didn't calculate this, they are given in the problem): $$\Box h = (M_p)^{a}\Box(h^2)-(M_p)^bT$$ For a point source ##T=m\delta^3(x)##, solve the equation for h to first order in the source T, with ##M_p=\frac{1}{\sqrt{G_N}}##. This result should reproduce the Newtonian potential.

Homework Equations




The Attempt at a Solution


So to first order, we can drop the ##h^2## term and we are left with $$\Box h = -(M_p)^bT $$ $$h = \frac{1}{\Box} (-(M_p)^bT)=-(M_p)^b\frac{1}{\Box} (T)$$ where ##\frac{1}{\Box}## is the propagator (Green function) associated with the field. Based on some calculations and properties of the Green function I got $$h=-\frac{M_p^b m }{4 \pi r}$$ I am pretty confident of my calculations so far. Now, to actually get Newton potential I need ##M_p^b=4\pi G##. It is not mentioned, but I assume ##G_N=4\pi G## so the only thing I have to show is that ##b=-2## to reproduce the classical result. I just don't get that value... I tried to do a dimensional analysis of the Lagrangian, and I have ##[L]=4##, ##[\Box] = 2## so ##[h]=1##. As T is the stress energy tensor ##[T]=4## and ##[M_p]=1## so we are left with ##b=-1## I just don't know where I am missing a factor of 2. Also, assuming my calculations above were wrong, the whole time ##(M_p)^b## was just a constant so that should be the same, regardless of the rest of the solution. So I guess I am doing something wrong with the dimensional analysis. Can someone help me please? Thank you!
 

Answers and Replies

  • #2
Orodruin
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Assuming that ##h_{\mu\nu}## is the weak field correction to the Minkowski metric, it is dimensionless. The Einstein-Hilbert action is not always written on dimensionless form, but the appropriate factors of ##M_P## are then included as an overall factor when icluding matter fields.
 

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