Monday, 30 March 2020

Gravitational redshift

The second piece of evidence for general relativity we examine is gravitational redshift in section 5.5. That's when the wavelength (or frequency) of light changes as it moves to stronger or weaker parts of a gravitational field.

Apparently Pound and Rebka were the first to measure it using gamma rays going up 72 feet (that's 22m in new money). They did it in the Jefferson laboratory (pictured)  at Harvard in 1959. That's about 40 years after Einstein predicted it. The change in the wavelength was 2 parts in a thousand trillion (##2## in ##10^{15}##). They measured it by wiggling the source of the gamma rays about in a speaker cone and seeing when the Doppler shift cancelled the gravitational shift!

The calculations are quite simple (and I got the right answer without cheating!) but I really needed to understand a few other things which sent me right back to chapters 3 and then 1. It all concerns the energy-momentum vector for a massive particle and then a massless e.g. gamma ray) particle and how the energy and therefore frequency fits into that. Light dawned.

The basic calculations: Commentary 5.5 Gravitational Redshift.pdf (2 pages)
On four-momentum and energy: Commentary 3.4 Particle energy.pdf (6 pages)

Monday, 23 March 2020

Precession of perihelion of Mercury

In 5.5, we continue our adventures from the previous section 5.4 where we calculated a few things from the Killing vectors and geodesics of the Schwarzschild metric. I learnt a good lesson about Killing vectors then that went over my head previously.

There are several amazing things in this section one group being mathematical dexterity and the other being the feats of astronomers. The latter have measured the precession of the perihelion of Mercury at ##{44}^{\prime\prime}  \text{per century}##. ##{44}^{\prime\prime}## is 44 seconds of an arc or 44 times 1/3600 degrees. That's about the angle that a soccer ball would subtend if it was one kilometer away and if you could see it. How did they do that?

Apparently we are following d'Inverno 1992, not Einstein, and we start with a differential equation for the orbit in terms of radial distance ##r## and affine parameter ##\lambda## and turn it into a simpler differential equation in terms of ##x\propto1/r## and azimuth ##\phi##. A crafty differentiation then makes the equation further collapse into the equation of an ellipse with a small perturbation, as witnessed by astronomers. Using more trigonometrical tricks we then solve the GR perturbation part and find a term in that which must correspond to the precession. A further trig trick gets us to "the equation for an ellipse with an angular period that is not quite ##2\pi##" and from that we extract the precession of the perihelion of Mercury. How did anybody think up all that?

The image shows the path of a Mercury that precesses about a million times faster than our own Mercury. I plotted it to check the  "the equation for an ellipse with an angular period that is not quite ##2\pi##". It is surprisingly accurate for such a large precession.

Read my attempts to follow Carroll at
Commentary 5.5 Precession of perihelia.pdf (7 pages)
Commentary 5.4 Geodesics of Schwarzschild.pdf (8 pages)


Monday, 9 March 2020

The Schwarzschild metric

Event horizon

In this section 5.1 Carroll rediscovers the Schwarzschild metric and I discover a little fib of Carroll's which leads us to an event horizon, which he doesn't even mention. What a cheek!

It might be interesting to some that Schwarzschild was German and in German Schwarz means black and Schild means shield so his name means 'black shield'. Like Rotschild (red shield) the great bankers. Schwarzschild published his metric in the same year that Einstein published his theory of general relativity. Quick work. He died the same year. Perhaps the effort killed him, but I bet he was pleased.

FYI  the Schwarzschild metric  is$$

{ds}^2=-\left(1-\frac{2GM}{r}\right){dt}^2+\left(1-\frac{2GM}{r}\right)^{-1}{dr}^2+r^2{d\Omega}^2
$$where ##{d\Omega}^2## is the metric on a two sphere$$
{d\Omega}^2={d\theta}^2+\sin^2{\theta}{d\phi}^2
$$and, of course, the spatial coordinates are spherical polar, in all they are ##\left\{t,r,\theta,\phi\right\}##.

The Schwarzschild radius is given by$$

R_S=2GM
$$That is also the radius of the event horizon of a black hole of mass ##M## so we are able to calculate how compressed the Sun would have to be to be a black hole: radius 3 km.

Read all about it: Commentary 5.1 The Schwarzschild metric.pdf (6 pages including the Riemann tensor)

Lagrange Formulation of General Relativity

Section 4.3 on the Lagrange formulation of General Relativity was pretty tough. I had to start by reading and understanding section 1.10 on classical field theories of which I knew nothing. That took about a month. Back on this section  I got very confused about small variations, which are vital for this branch of calculus. I was misled (even lied to) by JG on math.stackexchange and then helped by Physics Forums. I had to collect all my new knowledge about variations in a separate commentary. Carroll threw several interesting  challenges at me. We end with a new definition of the energy momentum tensor.

Read my thoughts: Commentary 4.3 Lagrange Formulation of GR.pdf (7 pages)
And nuggets on variations: Commentary Variations of objects as in calculus of variations.pdf (3 pages)

Thursday, 5 March 2020

Planck dimensions

At equations 4.91-4.95 Carroll gives the Planck's set of four dimensioned quantities: Planck's mass, length, time and energy. I wanted to compare them with actual things. Planck first noticed these way back in 1899.
\begin{align}

m_p=\sqrt{\frac{\hbar c}{G}}&=2.18\times{10}^{-8}\rm{kg}&\rm{{10}^{7}\ E. coli}\phantom {100000000000000000000}&\phantom {10000}(1)\nonumber\\

l_p=\sqrt{\frac{\hbar G}{c^3}}&=1.63\times{10}^{-35}\rm{m}&\rm{Radius\ of\ proton\ ={10}^{-15}\ m}\phantom {10000}&\phantom {10000}(2)\nonumber\\

t_p=\sqrt{\frac{\hbar G}{c^5}}&=5.39\times{10}^{-44}\rm{s}&\rm{Cosmic\ inflation\ ends\ at\ {10}^{-32}s}\phantom {10000}&\phantom {10000}(3)\nonumber\\

E_p=\sqrt{\frac{\hbar c^5}{G}}&=1.95\times{10}^9\rm{J}&\rm{Sun\ emits{\ 10}^{26}\ Js^{-1}.  \text{ A-bomb}\rm={10}^{12}}\ J&\phantom {10000}(4)\nonumber\\

&=1.22\times{10}^{19}\rm{GeV}&

&\phantom {10000}\nonumber

\end{align}
Max Planck 1858-1947
He then says "Most likely, quantum gravity does not become important until we consider particle masses greater than ##m_p##, or times shorter than ##t_p##, or lengths smaller than ##l_p##, or energies greater than ##E_p##; at lower scales classical GR should suffice. Since these are all far removed from observable phenomena, constructing a consistent theory of quantum gravity is more an issue of principle than of practice."

Whilst it is unimaginable that we will see things shorter than ##t_p## or smaller than ##l_p##, particle masses ('point masses') greater than ##m_p## are commonplace in GR and energies greater than ##E_p## are happening all the time. Can anybody help me make sense of the ##m_p,E_p## parts? And why are those 'greater than' and the others 'less than'?

I found out when I asked on physics forums. Read it at Commentary 4.5 Planck dimensions.pdf (2 pages).

Tuesday, 25 February 2020

Einstein's equation

Einstein age 18. Credit.
We are now on section 4.2 which has some very shady approximations. However we do get to Einstein's equation for general relativity if we tolerate that. The equation is$$
R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi GT_{\mu\nu}
$$where ##R_{\mu\nu},R## are the Ricci tensor and scalar which tell us about the curvature of spacetime, ##g_{\mu\nu}## is the metric, ##G## is Newton's constant and ##T_{\mu\nu}## is the energy-momentum tensor. So the equation tells us how the curvature of spacetime reacts to the presence of energy-momentum (which includes mass). Newton is not forgotten altogether😊.

The equation can also be written as $$
R_{\mu\nu}=8\pi G\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right)
$$where ##T=g_{\mu\nu}T_{\mu\nu}## and in empty space where ##T_{\mu\nu}=0## that gives us$$
R_{\mu\nu}=0
$$The equation is a field equation for the metric and the Newtonian gravity field equation is Poisson's equation$$
\nabla^2\Phi=4\pi G\rho
$$where ##\Phi## is the gravitational potential and ##\rho## the mass density.

The section starts by plausibly guessing that GR field equation must be of the form $$
R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\kappa T_{\mu\nu}
$$where ##\kappa## is a constant we must find. The GR field equation must be the same as Poisson's equation in almost-flat spacetime. So we use a small perturbation ##h_{\mu\nu}## on the flat metric: $$
g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}
$$and discarding second order terms in ##h_{\mu\nu}## eventually work out that ##\kappa=8\pi G## to bring the two equations into line.

However! Carroll's 4.38 is wrong. It says ##T_{00}=\rho## and in fact ##T_{00}=\rho\left(1-h_{00}\right)## and  if 4.38 were right then 4.39 would be wrong, but in fact it is right. Carroll is sort of having it both ways and we only get ##\ \kappa\approx8\pi G## at best. It contains first order terms in ##h_00##. Hopefully the next section using the Lagrangian formulation will do better!

See Commentary 4.2 Einsteins equation.pdf (5 pages) for the details.

Sunday, 23 February 2020