Nonlinear gravity as a classical field theory

[spaghetti3451]

Homework Statement

In this problem, you will calculate the perihelion shift of Mercury simply by dimensional analysis.

(a) The interactions in gravity have

$\mathcal{L}=M^{2}_{Pl}\Big(-\frac{1}{2}h_{\mu\nu}\Box h_{\mu\nu}+(\partial_{\alpha}h_{\mu\nu})(\partial_{\beta}h_{\mu\alpha})h_{\nu\beta}+\cdots\Big)-h_{\mu\nu}T_{\mu\nu},\qquad\qquad (1)$

where $M_{Pl}=\frac{1}{\sqrt{G_{N}}}$ is the Planck scale. Rescaling $h$, and dropping indices and numbers of order $1$, this simplifies to

$\mathcal{L}=-\frac{1}{2}h\Box h+(M_{Pl})^{a}h^{2}\Box h-(M_{Pl})^{b}hT.\qquad\qquad (2)$

What are $a$ and $b$ (i.e. what are the dimensions of these terms)?

(b) The equations of motion following from this Lagrangian are (roughly)

$\Box h = (M_{Pl})^{a}\Box (h^{2})-(M_{Pl})^{b}T.\qquad\qquad (3)$

For a point source $T=m\delta^{(3)}(x)$, solve Eq. (3) for $h$ to second order in the source $T$ (or equivalently to third order in $M_{Pl}^{-1}$). You may use the Coulomb solution we already derived.

(c) To first order, $h$ is just the Newtonian potential. This causes Mercury to orbit. What is Mercury's orbital frequency, $\omega=\frac{2\pi}{T}$? How does it depend on $m_{\text{Mercury}}$, $m_{\text{Sun}}$, $M_{Pl}$ and the distance $R$ between Mercury and the Sun?

(d) To second order, there is a correction that causes a small shift Mercury's orbit. Estimate the order of magnitude of the correction to $w$ in arcseconds/century using your second-order solution.

(e) Estimate how big the effect is of other planets on Mercury's orbital frequency. (Dimensional analysis will do - just get the right powers of masses and distances.)

(f) Do you think the shifts from either the second-order correction or from the other planets should be observable for Mercury? What about for Venus?

(g) If you derive Eq. (3) from Eq. (2), what additional terms do you get? Why is it OK to use Eq. (3) without these terms?

Homework Equations

The Attempt at a Solution

(a) I understand what it means to rescale $h$ and drop indices.

In particular, for the first term, $h\Box h = M^{2}_{Pi}h_{\mu\nu}\Box h_{\mu\nu}$ so that $h=M^{2}_{Pi}h_{\mu\nu}.$

Therefore, for the third term, $h_{\mu\nu}T_{\mu\nu}=(M_{Pl})^{b}hT$ which implies $h_{\mu\nu}T_{\mu\nu}=(M_{Pl})^{b}(M_{Pl}h_{\mu\nu})T$ so that $b=-1.$What does it mean to drop numbers of order $1$? What are the numbers of order $1$ here anyway?

 

[nrqed]

Homework Statement

In this problem, you will calculate the perihelion shift of Mercury simply by dimensional analysis.

(a) The interactions in gravity have

$\mathcal{L}=M^{2}_{Pl}\Big(-\frac{1}{2}h_{\mu\nu}\Box h_{\mu\nu}+(\partial_{\alpha}h_{\mu\nu})(\partial_{\beta}h_{\mu\alpha})h_{\nu\beta}+\cdots\Big)-h_{\mu\nu}T_{\mu\nu},\qquad\qquad (1)$

where $M_{Pl}=\frac{1}{\sqrt{G_{N}}}$ is the Planck scale. Rescaling $h$, and dropping indices and numbers of order $1$, this simplifies to

$\mathcal{L}=-\frac{1}{2}h\Box h+(M_{Pl})^{a}h^{2}\Box h-(M_{Pl})^{b}hT.\qquad\qquad (2)$

What are $a$ and $b$ (i.e. what are the dimensions of these terms)?

(b) The equations of motion following from this Lagrangian are (roughly)

$\Box h = (M_{Pl})^{a}\Box (h^{2})-(M_{Pl})^{b}T.\qquad\qquad (3)$

For a point source $T=m\delta^{(3)}(x)$, solve Eq. (3) for $h$ to second order in the source $T$ (or equivalently to third order in $M_{Pl}^{-1}$). You may use the Coulomb solution we already derived.

(c) To first order, $h$ is just the Newtonian potential. This causes Mercury to orbit. What is Mercury's orbital frequency, $\omega=\frac{2\pi}{T}$? How does it depend on $m_{\text{Mercury}}$, $m_{\text{Sun}}$, $M_{Pl}$ and the distance $R$ between Mercury and the Sun?

(d) To second order, there is a correction that causes a small shift Mercury's orbit. Estimate the order of magnitude of the correction to $w$ in arcseconds/century using your second-order solution.

(e) Estimate how big the effect is of other planets on Mercury's orbital frequency. (Dimensional analysis will do - just get the right powers of masses and distances.)

(f) Do you think the shifts from either the second-order correction or from the other planets should be observable for Mercury? What about for Venus?

(g) If you derive Eq. (3) from Eq. (2), what additional terms do you get? Why is it OK to use Eq. (3) without these terms?

Homework Equations

The Attempt at a Solution

(a) I understand what it means to rescale $h$ and drop indices.

In particular, for the first term, $h\Box h = M^{2}_{Pi}h_{\mu\nu}\Box h_{\mu\nu}$ so that $h=M^{2}_{Pi}h_{\mu\nu}.$

Therefore, for the third term, $h_{\mu\nu}T_{\mu\nu}=(M_{Pl})^{b}hT$ which implies $h_{\mu\nu}T_{\mu\nu}=(M_{Pl})^{b}(M_{Pl}h_{\mu\nu})T$ so that $b=-1.$What does it mean to drop numbers of order $1$? What are the numbers of order $1$ here anyway?

They mean the terms of order $M_{Pl}^0$, i.e. with no factors of the Planck mass. These would appear if one would include more of the terms that are not shown (where the three dots are). These three dots contain terms that contain factors $\frac{1}{M_P^2}, \frac{1}{M_P^4}$ and so on. Since they get multiplied by $M_P^0$, they start at order $M_P^0$. They therefore basically mean that one should neglect all the terms that are implicitly contained in the three dots.