The Metric

Kruskal from From Stanford

I started to write this in May 2019 when I read section 2.5 and it has remained a work in progress until now. I have quite a collection of metrics: 3D Euclidean, Plane polar, Spherical polar, Surface of sphere (S2), Minkowski, Spherical Minkowski, Schwarzschild, Eddington-Finkelstein, Kruskal. I will add more as I find them.

We start with a bit of general information about a metric, then list each of the metrics. There's also something on null, spacelike and timelike coordinates; converting metrics to coordinate systems with an example on spherical metrics; four velocities and mysterious interchange of  ##\mathrm{d}x## and ##dx##.

On this page the coordinate infinitesimals (which are really one forms) should be written with an unitalicized ##\rm{d}## for example ##\mathrm{d}x## not ##dx##. That is very fiddly in Latex so I have not bothered except in the very first occurrence! (Most texts don't bother at all).

The glory of the metric

The metric ##g_{\mu\nu}## is a symmetric (0,2) tensor and Carroll lists the following to appreciate its "glory". He is following Sachs and Wu (1977).

1. The metric supplies a notion of "past" and "future".
2. The metric allows the computation of path length and proper time.
3. The metric determines the "shortest distance" between two points, and therefore the motion of test particles.
4. The metric replaces the Newtonian gravitational field ##\phi##.
5. The metric provides a notion of locally inertial frames and therefore a sense of "no rotation".
6. The metric determines causality, by defining the speed of light faster than which no signal can travel.
7. The metric replaces the traditional Euclidean three-dimensional dot product of Newtonian mechanics.
8. The (inverse) metric lowers (raises) indices: ##U_\mu=g_{\mu\nu}U^\nu\ ,\ U^\mu=g^{\mu\nu}U_\nu##

I added the last one. It is related to the second last.

Two and three dimensional geometry metrics

These are called Euclidean or Riemannian metrics.

3D Euclidean$${ds}^2=\mathrm{d}x^2+\mathrm{d}y^2+\mathrm{d}z^2$$$$g_{ij}=\left(\begin{matrix}1&0&0\\0&1&0\\0&0&1\\\end{matrix}\right)$$we explain why the equation and matrix formulations are equivalent. 

Plane polar coordinates

Coordinates ##(r,\theta)## radial, angle to ##x## axis$$g_{ij}=\left(\begin{matrix}1&0\\0&r^2\\\end{matrix}\right)$$Spherical polar 

Coordinates ##(r,\theta,\phi)## radial, polar, azimuthal (= longitude)$$g_{ij}=\left(\begin{matrix}1&0&0\\0&r^2&0\\0&0&r^2\sin^2{\theta}\\\end{matrix}\right)$$Surface of sphere (S2)

Coordinates ##(\theta,\phi)## polar, azimuthal$$g_{ij}=\left(\begin{matrix}1&0\\0&\sin^2{\theta}\\\end{matrix}\right)$$This metric is often written $${d\Omega}^2={d\theta}^2+\sin^2{\theta}{d\phi}^2$$

Relativistic metrics

These are called Lorentzian or pseudo-Riemannian metrics. 

We use a ##-+++## signature and the speed of light ##c## is conveniently set to 1.

Minkowski

Coordinates ##(t,x,y,z)## $$g_{\mu\nu}=\left(\begin{matrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\\\end{matrix}\right)$$Spherical Minkowski

Coordinates ##(t,r,\theta,\phi)## ##r,\theta,\phi## as in spherical$$g_{\mu\nu}=\left(\begin{matrix}-1&0&0&0\\0&1&0&0\\0&0&r^2&0\\0&0&0&r^2\sin^2{\theta}\\\end{matrix}\right)$$We could also write$${ds}^2=-d\tau^2=-{dt}^2+{dr}^2+r^2{d\Omega}^2$$Schwarzschild metric (spherical polar)

Coordinates are ##(t,r,\theta,\phi)## time, radial, polar, azimuthal$${ds}^2=-d\tau^2=-\left(1-\frac{2GM}{r}\right){dt}^2+\left(1-\frac{2GM}{r}\right)^{-1}{dr}^2+r^2\left({d\theta}^2+\sin^2{\theta}{d\phi}^2\right)$$We often replace ##2GM## (twice Newton's gravitational constant times the central mass) by ##R_s## the Schwarzschild radius which is the radius of event horizon and the last part by ##r^2{d\Omega}^2##. 

Eddington-Finkelstein metric

The Eddington-Finkelstein metric is for the same spacetime as Schwarzschild but with coordinates ##\left(v,r,\theta,\phi\right)## $${ds}^2=-\left(1-\frac{R_s}{r}\right)dv^2+dvdr+drdv+r^2{d\Omega}^2$$$$v=t+r+R_s\ln{\left|\frac{r}{R_s}-1\right|}$$This is the first metric we have listed which is not diagonal.

Kruskal predecessor

On the way to the Kruskal metric we get coordinates ##\left(v^\prime,u^\prime,\theta,\phi\right)## with metric equation$${ds}^2=-\frac{2{R_s}^3}{r}e^{-\frac{r}{R_s}}\left(dv^\prime du^\prime+du^\prime dv^\prime\right)+r^2{d\Omega}^2$$where ##r## is implicitly defined in terms of ##u^\prime,v^\prime## as$$u^\prime v^\prime=-\left(\frac{r}{R_s}-1\right)e^{r/R_s}$$Kruskal

Kruskal coordinates are ##\left(T,R,\theta,\phi\right)## where in terms of Schwarzschild ##t,r## $$T=\left(\frac{r}{R_s}-1\right)^{1/2}e^\frac{r}{2R_s}\sinh{\left(\frac{t}{2R_s}\right)}$$$$R=\left(\frac{r}{R_s}-1\right)^{1/2}e^\frac{r}{2R_s}\cosh{\left(\frac{t}{2R_s}\right)}$$and they give a metric equation$${ds}^2=\frac{4{R_s}^3}{r}e^{-\frac{r}{R_s}}\left(-dT^2+dR^2\right)+r^2{d\Omega}^2$$with ##r## implicitly defined from$$T^2-R^2=\left(1-\frac{r}{R_s}\right)e^\frac{r}{R_s}$$

Read all that and more on  at Commentary 2.5 The Metric.pdf (11 pages)

Kruskal coordinates and the maximally extended Schwarzschild solution

Kruskal coordinates ##\left(T,R,\theta,\phi\right)## are called 'maximally extended' because they cover the whole spacetime except the true singularity at ##r=0##. Indeed, they find some remarkable new regions of spacetime! The history of the discoveries spans 45 years from Einstein and Schwarzschild (1915) to Kruskal (1960).

The Kruskal coordinates are related to Schwarzschild coordinates ##\left(t,r,\theta,\phi\right)## by $$T=\left(\frac{r}{R_s}-1\right)^{1/2}e^\frac{r}{2R_s}\sinh{\left(\frac{t}{2R_s}\right)}$$

$$R=\left(\frac{r}{R_s}-1\right)^{1/2}e^\frac{r}{2R_s}\cosh{\left(\frac{t}{2R_s}\right)}$$and give a metric equation$${ds}^2=\frac{4{R_s}^3}{r}e^{-\frac{r}{R_s}}\left(-dT^2+dR^2\right)+r^2{d\Omega}^2$$with ##r## implicitly defined from$$T^2-R^2=\left(1-\frac{r}{R_s}\right)e^{r/R_s}$$From these we can draw a Kruskal diagram  showing lines of constant ##t## and ##r## and light cones which miraculously are always at 45° just like in flat spacetime.

The regions above the upper ##r=0## line and below the lower ##r=0## line are not part of spacetime. (##r<0## and ##t>+\infty## or ##t<-\infty## in them). The rest of the diagram is divided into four regions.

I    Right quarter. Normal space time outside the event horizon.👍

II   Below the upper ##r=0## line and above the upper ##r=R_s## lines. Inside the event horizon.👎

III  Above the lower ##r=0## line and below the lower ##r=R_s## lines. The white hole.💣

IV  Left quarter. The unreachable mirror image of normal space time.👻

The red light cones, which are always at 45°, are informative. In region I you can always maintain a fixed ##r## and you can always move up towards and into region II. You can never get into regions III or IV. Once in region II it is impossible to maintain constant ##r## because lines of constant ##r## are always flatter than 45° so you inevitably arrive at ##r=0##. Region IV is like region I and you can only get to region II from it. Carroll says region III is a the time reverse of II and can be thought of as a white hole. "There is a singularity in the past, out of which the universe appears to spring". Things can only come out of it. They can just get directly to II through the origin, more likely they will go into I or IV.

Regions II and III are allowed even though ##t>+\infty## in II and ##t<-\infty## in III.

Although region I and region IV are mutually unreachable, if an intrepid explorer went from region I into region II, they would be able to see some things that had happened in region IV. Likewise an explorer from region IV could have a brief look at region I before perishing.

Find out how to get to Kruskal and a few other things in Commentary 5.7 Kruskal coordinates.pdf (11 pages).

Time travel inside a black hole

We can now use the geodesic that we have found and tested to plot a spacetime diagram all the way to the centre of a black hole. The graph shows the result for our famous beacon, or even for a foolish astronaut.

As we saw before they dawdle at the event horizon for ever (##t\rightarrow\infty## as ##r\rightarrow2GM##). But we know you can cross an event horizon and when that happens in the distant future they hurry backwards in time and soon get to a reasonable ##r,t##! 

Meanwhile the astronaut looking at a wristwatch sees proper time, ##\tau##, ticking by steadily and reaches the centre in finite time.

Luckily the astronauts time travel cannot be observed from the outside. Our red faces are saved.

Given everything we have done before the calculations are simple. It's a one pager in Commentary 5.6 Time Travel.pdf.

Schwarzschild Black Holes - The Geodesic

(1) and (2) are the ##t## and ##r## geodesic equations in the Schwarzschild metric which we found in section  5.4. (2) becomes a bit simpler on a radial path. That's (3). Geodesic equations are meant to give you trajectories of freely falling particles parametrised by ##\lambda##. (4) and (5) are the equation of a particle (or beacon) falling along a radius into a black hole from a distance ##r_*##. We used them to plot a beacon's path here. They come from (6) which we calculated in exercise 5.5 where we also calculated (7). 

##t## is coordinate time, ##r## is the distance from the centre. They are the coordinates.

##\lambda## is an affine parameter (it is proportionate to the length along the line).

##R_s## is the Schwarzschild radius (radius of event horizon).

##\theta,\phi## are the other spherical polar coordinates (polar and azimuth), which we can ignore.

##r_*## is the radial distance from which the test particle, or beacon, is dropped.

##\tau## is the proper time, which can be an affine parameter for a massive particle.

The question is: Is the path given by (4) and (5) a real geodesic? That is, does it satisfy (1) and (3)?

And the answer is YES.

I studied these geodesics three months ago and they have never yet been useful. This is the first time that they have even clicked with anything!

Here's how they click: Commentary 5.4#1 Geodesics of Schwarzschild.pdf (only 2 pages really)

Proper acceleration, Spaghettification and G2

After that Exercise 5.5 I thought I was an expert on falling into a black hole and and that I could calculate the 'proper acceleration' and spaghettification which is the term used for what happens when acceleration differs so much in different parts of your body that you get stretched out like the doomed lady on the right.

Then there is there is the matter of the gas cloud known as G2, which was discovered heading towards Sagittarius A*, the black hole at the centre of our galaxy, in 2011 by some folk at the Max Planck Institute. G2 was destined to come closest to Sagittarius A* in Spring 2014 "with a predicted closest approach of only 3000 times the radius of the event horizon". There was great excitement because spaghettification and great fireworks were expected. However nothing much happened and G2 continues on its way, orbiting Sagittarius A*.

I attempted to do some calculations and tested them on Physics Forums and got adverse comments from PeterDonis. Ibix was more positive "I think your maths is correct ...". PeterDonis showed me the 'correct' way of calculating proper acceleration and then dragged me back to the geodesic deviation equation which is the right way to calculate spaghettification. But I still don't fully understand said equation and how to use it😭. I also learnt a bit more about units: 'natural' and 'geometric'. PeterDonis is a hard task master.

The correct way to calculate proper acceleration gives infinite acceleration at the event horizon. One benefit of that is that it tells you that you cannot escape falling through it, once close enough. That is true. The drawback is that it makes the radial change in acceleration also infinite. So you will get ripped up at the event horizon. That is not true given a big enough black hole.

Here are my calculations (about four pages) and what I learnt (another three).
Commentary 5.6#4 Proper acceleration.pdf 

Beacon falling into a black hole, revisited

Having done Exercise 5.5 and discovered lots of things about a beacon being dropped into a black hole we can now revisit Commentary 5.6 where we struggled with Carroll's figure 5.8 and Carroll's claims about increasing intervals observed at a safe distance ##r_\ast## from the centre of a black hole. Back then I ran into several problems: 1) I could not calculate the equation for the radial geodesic, 2) When I used some invented curve, that was more or less the right shape, I had to estimate ##\Delta\tau_1##'s along the beacon path, 3) Having done that the intervals measured by the observer did often not increase.

Now we have an equation of motion for the beacon, that is the radial geodesic. I thought it would never be useful. We now find that it is useful it, and it is the intimidating$$
t=\frac{\sqrt{r_\ast-R_S}}{\sqrt{R_S}}\left[\sqrt{r_\ast-r}\sqrt r-\left(r_\ast+2R_S\right)\sin^{-1}{\left(\frac{\sqrt r}{\sqrt{r_\ast}}\right)}\right]
$$$$
+R_S\ln{\left|\frac{\left(r_\ast-R_S\right)\sqrt r+\sqrt{R_S}\sqrt{r_\ast-R_S}\sqrt{r_\ast-r}}{\left(r_\ast-R_S\right)\sqrt r-\sqrt{R_S}\sqrt{r_\ast-R_S}\sqrt{r_\ast-r}}\right|}
$$$$
+\frac{\pi\left(r_\ast+2R_S\right)\sqrt{r_\ast-R_S}}{2\sqrt{R_S}}
$$(##R_s=2GM## is the Schwarzschild radius) so we can plot that on a graph and the first problem is resolved. Moreover we found the proper speed of the beacon$$
\frac{dr}{d\tau}=-\sqrt{\frac{R_s\left(r_\ast-r\right)}{rr_\ast}}
$$By inverting and integrating that we have an expression for ##\tau## along the path and we can calculate ##\Delta\tau_1##'s which fixes the second problem. We already knew how to plot the return flight of the photon and when we do so, we find that the intervals measured by the observer do

increase for successive signals. So problem 3 was fixed! However, we are still not out of the woods. My second attempt is shown on the right with amounts in natural units. The observer is at ##r_\ast=15## and the event horizon at ##2GM=10##. We consider three photons emitted at ##a,b,c## separated by ##\Delta\tau_1=6##. We can measure the intervals seen by the observer and they do increase but the photon world line is disappointingly flat, unlike Carroll's. Spacetime is almost flat up to event ##b##. So we have to start very close to the event horizon with ##r_\ast=12## and zoom in to the area marked by the red rectangle. Eventually we achieve something like Carroll's (with a bonus photon from ##d##) as shown below. The diagram on the left is fairly bare like Carroll's the same one on the right has numbers added in natural units. ##\Delta\tau_1## was a very small ##0.5##  and the intervals observed at ##r_\ast## were an order of magnitude larger and increasing as Carroll predicted. The observer might have wanted to subtract out the, easily calculated, flight times of the returning photons. The intervals between emissions still increase as can be seen by the increasing vertical distances of ##a,b,c,d## on the diagram.

As an added bonus, from the equation for the proper speed of the beacon, we can calculate it's finite proper time to the event horizon and to the centre of the black hole. The increasing length along the geodesic for fixed proper time helps us intuit a resolution to the apparent paradox that the beacon 'never seems to get into the black hole'

Moreover if we get inside the event horizon, the proper time from there to ##r=0## is ##\pi GM## which is the maximum possible value that we calculated in exercise 5.3.

Get the details here: Commentary 5.6#3 Schwarzschild Black Holes.pdf (5 pages)

More intriguing puzzles remain:

• To relate the monster formula for ##t## to the geodesic equation from section 5.4; 
• To think about the beacon's (free falling) inertial coordinate system, its forward and backward light cones and its relationship to the Schwarzschild coordinate system 
• Spaghettification and the mysterious G2 gas cloud

Exercise 5.5 Observer and beacon outside black hole

Question

Consider a comoving observer sitting at constant spatial coordinates ##\left(r_\ast\ ,\theta_\ast\ ,\ \phi_\ast\right)## around a Schwarzschild black hole of mass ##M##. The observer drops a beacon onto the black hole (straight down along a radial trajectory). The beacon emits radiation at a constant wavelength ##\lambda_{em}## (in the beacon rest frame).

a) Calculate the coordinate speed of the beacon as a function of ##r##.
b) Calculate the proper speed of the beacon. That is, imagine there is a comoving observer at fixed ##r##, with a locally inertial coordinate system set up as the beacon passes by, and calculate the speed as measured by the comoving observer. What is it at ##r=2GM##?
c) Calculate the wavelength ##\lambda_{obs}##, measured by the observer at ##r_\ast##, as a function of the radius ##r_{em}## at which the radiation was emitted.
d) Calculate the time ##t_{obs}## at which a beam emitted by the beacon at radius ##r_{em}## will be observed at ##r_\ast##.
e) Show that at late times, the redshift grows exponentially: ##\lambda_{obs}/\lambda_{em}\propto e^{t_{obs}/T}##. Give an expression for the time constant ##T## in terms of the black hole mass ##M##.

Answers

This question is a fascinating can of worms. I needed the help of an unwitting mentor Jeriek Van den Abeele from the University of Oslo. The first crucial help was to use the timelike killing vector constant $$
E=\left(1-\frac{2GM}{r}\right)\frac{dt}{d\tau}
$$This came in useful not only for the beacon velocities but also for the  travel time of the photon back to the observer at ##\ r_\ast## in question d.

I also needed help from Physics Forums on comoving coordinates and I think I understand them now, although there was some dispute about how the (second)  comoving observer in question (b) would actually achieve that state. When people talk about comoving coordinates for the Universe, they are talking about something quite different from here: Coordinates comoving with the Universe are growing with the Universe.

Part of the worminess arises from confusion about what coordinates are referring to what. There was a profusion of subscripts: ##r_{em},r_\ast,\ R_s,t_\gamma,t_{em},t_{obs},t_b##. The last of those was introduced gratuitously by my mentor who I refer to as Oslo. In my opinion ##t_b\equiv t_{em},\ \ r_b\equiv r_{em}##. I try to avoid these subscripts as much as possible.

I got two answers to question (b). One was the proper speed calculated in the strange comoving inertial coordinate system, as Carroll asked, and the other was the proper speed in Schwarzschild coordinates. I called them ##dr^\prime/d\tau^\prime## and ##dr/d\tau##. The first was what Carroll asked for and can be used (but is not essential) in question (c). Apart from that I am not sure how useful it is. ##dr/d\tau## is what would be experienced by an astronaut falling with the beacon and it does not reach the speed of light at the event horizon.

Interestingly I calculated the answer to (c) using Doppler redshift + gravitational redshift and Oslo did it in one leap - which was more complex and contained a small error which had no effect. The formulas in the two answers looked quite different but when plotted gave the same lines. Eventually I proved that the formulas were in fact the same.

As usual the actual geodesic equations are not used to find out about all these geodesics. Nevertheless I will have a try to see if I can do better with that beacon in section 5.6.

Read it all at Ex 5.5 Observer and beacon.pdf (11 pages not including other documents)