## Friday, 17 January 2020

### Physics in curved spacetime

I've now started chapter 4 on Gravitation just in time for 2020. Very exciting!
 Fools straight line
Carroll first states two formulas of Newtonian Gravity his 4.1 and 4.2$$\mathbf{a}=-\nabla\Phi$$where $\mathbf{a}$ is the acceleration of a body in a gravitational potential $\Phi$. And Poisson's differential equation for the potential in terms of the matter density $\rho$ and Newton's gravitational constant $G$:$$\nabla^2\Phi=4\pi G\rho$$I had a long pause thinking about the various formulas for the Laplacian $\nabla^2$ here.
How to these tie up with the old-fashioned laws? Newton's law of gravity is normally stated as$$F=G\frac{m_1m_2}{r^2}$$which combined with Newton's second law $F=m\mathbf{a}$ gives us the acceleration of a mass in the presence of another as$$\mathbf{a}=G\frac{M}{r^2}$$In exercise 3.6 we were given 'the familiar Newtonian gravitational potential'$$\Phi=-\frac{GM}{r}$$A bit of rough reasoning shows these are equivalent.

At his 4.4 Carroll states that the next equation gives the path of a particle subject to no forces$$\frac{d^2x^i}{d\lambda^2}=0$$If we solve it in polar coordinates for $r,\theta$ instead of $x,y$ Carroll says we get a circle and he cheekily suggests that we might think free moving particles follow that path. But the solution is $$r=m\theta+k$$where $m,k$ are constants. We can plot that and, obviously if $m=0$ we get a circle of radius $k$ but if $m\neq0$ we get other more interesting lines which are equally wrong. See above. Another error by Carroll, but only minor 😏. The next one is a corker.

Then we examine the equations in a near Newtonian environment and equation 4.13 $g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu}$ is wrong. The actual equation is obviously$$g^{\mu\nu}=\eta^{\mu\nu}+h^{\mu\nu}$$Properly 4.13 might be
$$g^{\mu\nu}=\eta^{\mu\nu}-h_{\rho\sigma}\eta^{\mu\sigma}\eta^{\nu\rho}$$which is true to first order and gives $h^{00}=-h_{00}$ which is used in the next section. If  one accepts the approximation that $\eta$ can be used to raise and lower indices on any object of order $h$ then that also gives us$$g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu}$$which says $h^{\mu\nu}=0$. Oops! But it turns out it turns out that the sign on $h^{\mu\nu}$ is immaterial in this section. There is a full analysis in the pdf.

We soon arrive at the conclusion that in the near Newtonian environment we have the time, time component of the metric is $$g_{00}=-1-2\Phi$$which is also what we were given in Exercise 3.6.