Per Newtonian mechanics, a coin toss exhibits deterministic chaos theory, but could relativity cause a probabilistic outcome of a coin toss?

Question

It took me a long time to accept that a coin toss boils down to deterministic chaos theory. For example, the typical near 50/50 odds for outcomes of heads or tails results from complex initial conditions that are impossible (or nearly impossible) to control.

I also doubt that quantum mechanics would make a measurable difference on an object as massive as a US quarter, which is commonly used for coin tosses.

But I wonder if the complex calculations of general relativity could cause some indeterminism for a coin toss?

And while you answer me, please understand that I am not trained in mathematics of general relativity.

Answer 1

General relativity is a 100% deterministic theory. If you can predict the coin toss in a Newtonian environment, you can predict it in a relativistic environment.*

The key thing that makes such a discussion is the idea of Lyapunov time. A coin toss happens quickly, so we can ignore anything which creates chaos on a timescale larger than that (okay, its a bit more nuanced than that... but it's a decent first pass). Relativity affects lots of things, like the paths of the planets, but the chaotic elements of those systems are on much larger time scales.

Which leaves just the relativistic effects on the coin itself. Well, we can calculate a Lorentz factor for the coin. $\gamma=\sqrt{\frac{1}{1-\frac{v^2}{c^2}}}$. Papers have approximated the velocity of a coin somewhere on the order of 3m/s to 5m/s, so we'll use the bigger one for our calculations. Plugging in $v=5m/s$, we get $\gamma=1.000000000000000138889$. That's a relativistic effect on the order of $10^{-16}$. That's really small. You can't even use normal programs to compute those effects, because that's on the order of a IEEE-754's Unit of Least Precision. Rounding errors happen on that scale.

How small of an effect is it? Well, Jupiter's gravitational pull on the coin is on the order of $10^{-7}$ times that of Earth. So when you get to that level of determinism, you need to include Jupiter in your calculations, and you're still 9 orders of magnitude. In fact, the gravitational effect of Pluto will matter, at $10^{-9}$ or so. A micro-asteroid on the scale of a 100 nanograms hanging out in the asteroid belt will have an effect on par with the relativistic effects you are considering.

So yes, it adds some complexity. No indeterminism, because general relativity is deterministic, and the scale of its effects are quite literally "astronomically small."

*. Okay, I lied a bit. In theory the extra non-linearity could lead to better mixing of the orbits. One might be able to construct a system which is stable under Newtonian physics but chaotic under relativistic physics, but in order to physically construct the experiment, you would need to be able to measure the initial conditions to the absurd levels described here or have a coin at a non-trivial fraction of the speed of light (or flip the coin near a black hole, as Anders Sandberg suggests tongue-in-cheek)

Answer 2

Coin tosses are actually not deterministic chaos unless they start bouncing off curved surfaces, but in practice they amplify our uncertainty about initial conditions well enough to be a decent randomizer. But add a few extra forces and objects and they become chaotic.

In a chaotic dynamical system small uncertainties in initial position grow exponentially with time into big eventual uncertainty about end-states. This is described by the Lyapunov exponent $\lambda$, making them grow as $\delta(t) \approx \delta_0 e^{\lambda t}$. If you had an exponent of 1 per second, the time to make an atomic size position uncertainty ($\approx 10^{-10}$ m) grow into a centimetre uncertainty is 18.4 seconds. So even macroscopic objects can pick up quantum noise if they get to bounce around enough: astronomically small influences do matter in the not-so-long run.

General relativity doesn't change things much. It is as others pointed out a deterministic theory. It is also nonlinear: gravitational fields interact with each other, and there are chaotic solutions to the equations. But for a coin toss in normal gravity these aspects don't do much extra. Throwing coins into orbits around spinning black holes can generate chaotic trajectories, but that is not really proper coin tossing.

When we say something is indeterministic it means that that initial conditions do not contain all the information needed to determine what the outcome is. General relativity can cause this if there are singularities: you could toss your coin near a white hole, and as far as we know anything could come out and interact with the toss. But that is also why most people regard such solutions to the equations as suspect, or invoke the various censorship hypotheses. It also turns out that there are classical physics examples (Norton's dome) where determinism fails. These may be overly idealized, though. In practice we get practical indeterminism from all the unmodelled "noise" from the rest of the universe, whether gravity from remote sources, thermal noise or quantum effects, and this gets amplified by a nonlinear system.