I remember a technology TV show in the mid 90’s showing a roller coaster simulator ride. The audience is shown a simulation or video of the view out the front of a roller coaster, and the seats jostle and tilt in concert with the footage. I was only 11, but I concluded that it was the coolest thing ever.

### Why they are almost convincing

There is a good physics reason why these rides are almost convincing. Galilean relativity says that inertial reference frames are indistinguishable using local experiments. In layman’s terms, if you are in an enclosed plane traveling in a straight line at a constant speed, then there is nothing you can do inside the cabin to work out how fast you are travelling^{1}. The plane could be stationary or it could be doing a thousand miles per hour, and you won’t notice any difference between walking up the aisle and down the aisle.

In a car, we gauge speed by looking out the window and watching the scenery fly past. Ride simulators can simulate a fast moving roller coaster by showing a simulation of scenery going past. They also simulate the bumps and shunts by jostling your seat – the faster your car is going, the more you will feel the small deviations from uniform motion due to potholes.

I’ve been on a few of these rides, and I’m not fully sucked in. Speed is fine, bumps are fine, but the most exciting part of a real roller coaster ride is the “stomach in your throat” feeling as you go over a crest, or being thrown to one side as you take a corner at speed. Unlike speed, acceleration can be measured locally, so it can’t be simulated with a video and a shaky chair.

### How to make them fully convincing

There is a way to simulate acceleration. Einstein’s equivalence principle roughly states that freely falling is locally indistinguishable from zero gravity. We can illustrate this point with a thought experiment. Suppose you wake up in an elevator which is freely falling (i.e. ignore wind resistance etc). There is nothing you can do inside the elevator to determine whether you are freely falling, or whether someone has turned off gravity^{2}. If you want to know what it would be like if there were no gravity, then go jump off a cliff (in your mind, of course).

In fact, the equivalence principle implies more than this. Not only can we eliminate gravity by free fall, we can create it with acceleration. Suppose you wake up in our elevator again, but this time it is sitting on the ground. There is nothing you can do inside the elevator to determine whether you are sitting still on Earth, or accelerating through empty space at . In either case, if you drop a ball, it will fall to the floor of the elevator at . A bit like this (from astronomynotes.com).

Now, how do we simulate the acceleration of a roller coaster? Suppose we want to recreate that “stomach in your throat” moment. We need to cancel the effect of the Earth’s gravitational field so that the rider experiences zero net force. We can do this by placing a large mass M above the rider’s head. Let

where is the rider’s mass, and is the distance between the large mass and rider. If the mass is 10 metres above the rider, then we need it to weigh about 15 billion tonnes. If we made it out of steel, it would need to be a sphere about 1.5 km in diameter – that’s no good. Perhaps we could snare a piece of a neutron star. A sphere about about 4 cm in diameter would do nicely.

If we put the mass on a moving arm, then we can simulate a wide range of accelerations. The fastest launch acceleration of a roller coaster is about 2.7 g. We can simulate this by moving the mass behind the rider. The mass would pull the rider into his seat, and make it feel like he is accelerating forwards. We need a slightly larger acceleration so we move the mass slightly closer by a factor of to be about 6 metres away. To simulate taking a corner at 4.5 g, move the mass to the side of the rider and about 4.7 metres away.

### A Few Practical Problems

There are a few things preventing me from racing off to the patent office. Obviously, pieces of neutron star are hard to come by. We also have to be able to swing the mass around rather quickly. To raise 15 billion tonnes by 10 metres in one second requires a million gigawatts of power – the average nuclear power station will give you one gigawatt. On the bright side, gravity is a conservative force so you will get most of that energy back when the mass drops back down.

The sphere itself will be a tad dangerous. It you touched it, you would experience a force equivalent to 300,000 g. Quite probably not good for your health.

The other problem is tidal forces. Until now, I’ve treated our rider as a point mass. However, if the mass is directly above the rider, his head will experience a stronger attractive force than his feet. When we want him to feel weightless, he will feel as though he is being stretched. This force is equivalent to about 0.3 g, which would be somewhat uncomfortable. I’m still keen.

### Footnotes

1. No cheating. Don’t look out the window. No looking at the pilot’s instruments.

2. Actually, this is only true for a small lift. If the lift is large enough, then I can drop two balls. If I’m falling towards Earth, they will fall towards the centre of the Earth, and thus from my perspective will appear to get closer. Like this

It is this effect – known as a tidal force – that cannot be disguised by free-fall. The equation of general relativity describes such effects.

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