Figure 5.1. The Penrose diagram corresponding to the eternal Schwarzschild black hole. Five astronauts start out at rest at R = 1.1. Four of them head towards the black hole. Blue freefalls, Green freefalls with Blue but switches her rockets on after crossing the horizon and attempts to accelerate away from the singularity. Magenta also travels with Blue and Green until she reaches the horizon, after which she accelerates towards the singularity. Red accelerates away from the black hole from the moment her journey begins and succeeds in escaping to infinity. Orange also accelerates away from the black hole, but not enough to escape.
We followed Blue in the previous chapter. He sets out from rest at R = 1.1, freefalls into the black hole and ends up at the singularity. His worldline is marked in units of one hour by his watch and nothing unusual happens from his perspective at the event horizon – he sails through it oblivious. There are 20 dots on his worldline inside the horizon, which corresponds to almost a day inside the black hole before the end of time.†
Green begins her journey alongside Blue and freefalls towards the horizon with him, but on crossing the horizon she panics, shouts ‘Burma’‡ and turns on her rocket engine in a vain attempt to escape. There are only 16 dots along her worldline once she crosses the horizon, which means that accelerating away has resulted in the end of time arriving sooner.§
Magenta takes what might seem, at first sight, to be a more fatalistic view. She decides to fall together with Green and Blue until the horizon, and then she gently accelerates towards the end of time at 480g (which is five times less acceleration than Green). She presses play on Joy Division’s ‘Unknown Pleasures’ and flicks the switch. Perhaps irritatingly for her, this lengthens her stay inside the horizon and she lives longer than Green – her worldline has 17 dots. The spacetime geometry inside the black hole is certainly counter-intuitive.
The maximum time anyone can spend inside the horizon of a black hole corresponds to someone who starts out from rest on the horizon and does absolutely nothing but fall freely to the singularity. Apathy pays. This corresponds to just over a day (28 dots) inside the horizon of the supermassive black hole in M87.
On passing through the horizon, Green and Magenta accelerate away from the apathetic Blue, who’s probably relaxing to Miles Davis. Green accelerates away from the singularity and Magenta accelerates towards it. Blue sees Green recede into the distance above him in the direction of the horizon, and Magenta heads away in the direction of the singularity. They both get smaller and smaller from Blue’s point of view as they disappear into the distance in opposite directions. So far so normal. It certainly looks like Magenta is heading towards her doom at the singularity and Green is doing her best to stay close to the horizon. None of this seems different to the way things would be in any other region of spacetime.
In Figure 5.1, we’ve drawn two light beams that Magenta shines out. The first is emitted just under nine hours after she passed through the horizon and the other at just under 14 hours. Drawing light beams like this is what we should do if we want to investigate what each of our astronauts actually sees. Remember that the beauty of Penrose diagrams is that the light cones all point vertically upwards and open out at 45 degrees. Notice that the earlier light beam intersects Blue’s trajectory. That means Blue sees Magenta emit the beam of light (he receives the light at the point on his trajectory where the beam intersects it). Now for the fun bit. Look at the second beam Magenta shines out. It never intersects Blue’s trajectory, and therefore Blue never sees Magenta turn on her torch to emit that second beam. In other words, Blue never sees Magenta’s final moments, even though she is accelerating away from him. The light from Magenta’s final four dots simply doesn’t have time to enter Blue’s eyes before he reaches the end of time. He would see her way down below him as the last light reached him before he winked out of existence.
If Blue turns around, he’ll see Green way above him attempting to accelerate away from the singularity. Again, he’ll reach the end of time before he sees Green end her days. What’s interesting is that every astronaut has the same experience. Nobody ever sees anyone else hit the singularity. The reason is the horizontal nature of the singularity on the Penrose diagram. It is a moment in time, and we can never see events that are simultaneous with a moment in time. We always see things slightly in the past, because it takes light time to travel to our eyes. This means that nobody falling into the black hole sees anyone else reach the singularity before they themselves reach it – they quite literally never see it coming. If you are struggling to see this, imagine drawing 45-degree light beams all over the diagram. They’ll tell you what each astronaut can and can’t see.
The singularity doesn’t arrive entirely unannounced to the unsuspecting astronauts though because as they approach the singularity they get stretched out by tidal forces: they get spaghettified. Standing on the Earth, the pull of gravity is slightly greater at your feet than at your head, but not so much that you notice that you are being stretched. The gravitational pull of the Moon on the Earth has a similar stretching effect, causing the more noticeable twice daily tides. We can see how tidal forces arise using the Penrose diagram in Figure 5.2.
Figure 5.2. Spaghettification.
The dotted lines correspond to the worldlines of two balls falling into the hole. One ball starts out at R = 2 and the other a little closer in at R = 1.8. Once again, the dots correspond to regular ticks of a clock (imagine a clock glued to each ball). The dots are close together to make it easier for us to see the tidal effects (though harder to count the dots). The 45-degree lines correspond to a light beam bouncing back and forth between the balls. We can use the bouncing light as a ruler to measure the distance between the balls, just as you might use a laser tape measure at home if you enjoy putting up shelves. The numbers are the number of ticks between successive bounces as measured on the watch attached to the lower ball. These numbers correspond to the roundtrip travel time of the light pulses. The key thing to notice is that the time between bounces increases as the two balls fall towards the horizon. This means that the balls are moving apart as they fall towards the black hole.
Imagine now that you are falling into the black hole feet first. Your head and feet will try to move apart but, since they are connected by your body, you’ll instead feel like you are being stretched. For the black hole in M87, the tidal effects at the horizon would be unnoticeable, but you’d begin to feel uncomfortable inside at around R = 5 million kilometres. At some point around 3 million kilometres your head would come off. You’d have been spaghettified. Closer to the singularity, your constituent atoms would be ripped apart. Even more dramatically, for a typical stellar mass black hole you’d be spaghettified before you even reached the horizon.
Let’s now return to Figure 5.1 and ask what things look like to an observer who remains outside the black hole. Red sets out alongside Blue, Green and Magenta but wisely decides to switch on her engines in good time and accelerate away from the black hole. In good time is a ‘relative’ term here – she pulls 864g until Schwarzschild t = 1.5, at which point she decides enough is enough and switches off her rocket engine. Red’s 864g acceleration should probably be accompanied by Kenny Loggins’s ‘Danger Zone’. You can see the moment when Red switches off her rockets because that’s when her worldline takes an abrupt left turn and heads for the apex of the diamond. Red has escaped the black hole and, being immortal, her worldline will continue to future timelike infinity at the top apex of the Penrose diagram. Remember that nothing peculiar happens to anyone, at any point before they start to experience tidal forces and Red manages to avoid those. She does feel the acceleration of her rocket for the first portion of her journey, but once that’s done, she floats around happily inside her spaceship as she coasts into infinity.