Beware of expectation values based on vanishing probabilities
A week ago I discussed the coin toss game ‘triple or bust‘. The game is between Alice and Bob. Alice start the game by writing a $ 1.00 IOU to Bob. Alice then makes at least six subsequent tosses with a fair coin. On each ‘heads’ Alice triples the IOU amount. On ‘tails’ she sets the IOU to zero.
The question is: how much should Bob be prepared to pay Alice to participate in this game?
As Bob can repeat this game as often as he likes, he focuses on the gains to be obtained in the long run. These are given by the expectation value for this game, which are easy to calculate. The game starts with an IOU dollar value of 1.00. On each coin toss the average IOU increases to 3/2 times the amount before the toss. That means that after the nth coin toss, the expectation value for the IOU is (3/2)^n. Alice can prevent this value to grow out of control by stopping at n=6 (completing six tosses). This sets the expectation value for the game to $ 11.39.
However, one might also reason as follows: in each game Alice can continue the tossing until tails shows. This voids the IOU and Bob will walk away empty handed. The game is worthless to Bob.
How to reconcile both reasonings?
The key issue is: what do we mean by the phrase “in the long run”? How many repeat games are required to achieve a gain per game that is close to the expectation value? The chances for Bob to win a game of n tosses is 1 out of 2^n. To reach the expectation value, N the number of repeats of the game, needs to be large enough to yield a number of wins much larger than 1. This means N >> 2^n.
Let’s tell Alice to make exactly n tosses per game, and see how Bob fares for a given number of repeats of the game. Suppose Bob has enough time to spare to reach N = 10000. For this N value, at around n = 13.3 we have 2^n equal in magnitude to N. So for n, the number of tosses per game, significantly smaller than 13 (N >> 2^n) Bob can expect to walk away with close to 1.5^n dollars per game. For n reaching values close to 13 or 14, it is completely uncertain if Bob will win a game. Effectively, after 10000 repeat games, Bob’s total return takes the shape of a lottery ticket. For n much large than 14 (N << 2^n) Bob has vanishing odds to reach a win.
The above plot shows the earnings for Bob obtained in three independent runs of 10000 games each. The horizontal axis shows n the number of coin tosses per game. The vertical axis shows Bob’s earnings per game divided by the expectation value 1.5^n. In line with the reasoning above, Bob’s earnings follow the expectation value. Around n = 14 the fluctuations in his earnings become large, and a transition happens. Above n = 14 Bob becomes progressively unlikely to make any earnings. If Alice has full freedom in increasing n, Bob is guaranteed to walk away empty handed.
In essence, the cause for the expectation value failing to represent the earnings of a participant to this game is the skeweness of the payout distribution increasing without bound. Mathematically the expected earnings for this game become ill-defined because taking the limit of N going to infinity (calculating the expectation value) followed by taking the limit of n going to infinity (allowing Alice an unlimited number of tosses) gives a different evaluation from the one based on reversing these two limits.
The same phenomenon of unbounded skeweness in the payout distribution causes the expectation values of the well known Saint Petersburg paradox to misrepresent the likely earnings.
A black hole is not some cosmic vacuum cleaner. A black hole even hardly classifies as a tangible object. The inside of any tangible object can, at least in principle, be inspected from the outside. You can scan your vacuum cleaner with X-rays and thereby reveal its internal workings. But a black hole’s ‘inner workings’ can not be inspect from the outside. The only way to peer inside a black hole is by ‘being there’. Just like you can peer into the future only by ‘being there’.
So let’s stop thinking about black holes as tangible objects. It is more insightful to think of a black hole as localized future that has separated from, and lost contact with, the future developing outside the black hole. The defining characteristic for black holes is that anyone or anything inside a black hole, can not influence the universe outside the black hole. No signal (no light, no X-ray, no gravitational wave, nothing that carries information) can be transmitted from inside the black hole to observers outside the black hole. And obviously, this also implies that any object inside can not leave the black hole.
The scenario usually considered that leads to ‘finding yourself inside a black hole’ is the scenario of a static black hole. A black hole in eternal existence. The only way to find yourself inside such a black hole is by falling into it. Yet there is an alternative scenario leading to the situation of ‘being inside a black hole’. It’s the scenario of a black hole growing from inside you. And this alternative scenario gives a better insight into the global spacetime nature of black holes.
We are going to perform a gedanken experiment. This will tell you how to grow a black hole starting from inside your belly, and quickly expanding far beyond our solar system. And… the ‘you’ in this gedanken experiment will survive all of this. Heck, you even won’t notice you’re inside a black hole.
You need some astronauts to build a cosmic wall for you. A huge spherical shell of bricks centred around you. Regular red 4 lb bricks will do fine. But you will need loads of them. A tredecillion bricks to be precise. That’s going to render you a thin spherical wall as heavy as our whole galaxy and positioned far beyond the orbits of the planets in our solar system. In preparation you start reading up on Schwarzschild radii, and conclude that in order to prevent the shell of bricks to form a black hole, the spherical wall should be large enough for a pulse of light starting at the center to travel more than 100 days before reaching the wall.
So you instruct an army of astronauts to build a spherical brick wall with a radius somewhat larger than 100 light days. For the given total mass of bricks, this radius yields the particular advantage for the astronaut bricklayers building the wall to experience a gravitational acceleration due to the wall comparable to the gravitational acceleration at earth’s surface.
Once the sphere is complete, earth is shielded from any starlight not emerging from sun. Yet, nothing changes for us in terms of gravitational phenomena. Earth continues its orbit around the sun, and people continue to walk on earth, fly in airplanes, and launch rockets. All of this is the case, thanks to Newtons’s shell theorem (or if you prefer, thanks to it’s general relativistic cousin Birkhoff’s theorem. In simple terms, this theorem states that a uniform spherical shell exerts no gravitational force on objects (Newtonian view) or doesn’t bend spacetime (Einsteinian view) anywhere inside, while on objects anywhere outside it exerts a gravitational force (bends spacetime) equal to the force (bending) that would result if the full mass of the shell would have been concentrated at it’s center.
As long as your spherical cosmic wall retains a radius larger than 100 lightdays, no black hole will ever get formed and all is safe. But guess what. Under it’s own gravitational force the sphere of bricks starts shrinking. First slowly, but then at ever larger speed. As soon as the sphere shrinks to less than 100 light days radius…
… nothing happens. People continue walking on earth, and the planets continue their orbits around the sun. Nothing whatsoever changes to the gravitational phenomena inside the cosmic wall. This cosmic wall is getting smaller but stays spherical in shape, and the shell theorem still applies.
So the daily life of people continues. But something did change. In fact, something did change 100 days earlier. Something more abstract. Earth’s future did change. Since 100 days a narrow destiny awaits mankind.
A 100 days before the cosmic wall passed the critical radius of 100 light days, an event horizon got initialized. A boundary between an inside and everything else outside that is causally disconnected from the inside. This horizon started small somewhere. It could have started in your belly or in mine. Either way, none of us would have noticed. The horizon expands at the speed of light, within a fraction of a second including all human beings, and reaching the shrinking spherical wall exactly the moment it’s radius passes the threshold of 100 light days. Once the spherical wall has fallen thru the horizon, the horizon stays constant in size measuring 100 light days in radius.
This horizon seems a boundary entirely abstract in nature. Yet, it is a boundary that acts as a watershed in spacetime delineating markedly different futures, different destinies, on either side. Your destiny becomes apparent much later and well after the spherical wall passes thru the critical size of 100 light days. In fact, the earliest you would be able to know for sure you are heading towards a grim future disconnected from the future outside, is 100 days after the spherical wall passes thru the critical radius of 100 light days. Even then you would still experience the same gravitational phenomena, and earth would still orbit around the sun as if nothing had changed.
But what then is the destiny awaiting you? Would you be teared apart? Would spaghettification happen at some point?
No, none of this is going to happen. No matter how closely the wall of bricks approaches you, the shell theorem will keep you safe from any deadly gravitation effects. So then what disaster will happen? By now that should be clear. The shell theorem doesn’t protect you from being hit by a brick. And at some moment a true giga-tsunami of bricks will crush you out of existence.
Does the above puzzle you? Does it give you an uneasy feeling of retro-causality creeping in? Isn’t it strange that a horizon would start to grow 100 days before the wall of bricks passes thru the critical radius? What if, seconds before the critical radius is reached small rockets connected to the bricks would fire thereby reversing the collapse of the wall and preventing the critical radius to be reached? Would the horizon that started growing 100 days earlier somehow be eradicated?
Such reasoning is based on the wrong intuition of a horizon being some tangible boundary. Black hole horizons are not tangible objects. They represent boundaries between distinct futures. If, at the last minute, the brick wall collapse gets reversed, we have to conclude that no horizon started to grow 100 days earlier. And no, this does not introduce any retro-causality as all that changes is in the future.