Take a classic game that makes heavy use of a simple physics engine (such as
Asteroids), and add in general relativity to the physics engine (and set the speed of light so that the game has very noticeable relativistic effects). For the Asteroids example, asteroids traveling at high speeds relative to you would exhibit Lorenz contraction, asteroids
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(Your twin defects to the other side, and later turncoats and comes back. If they traveled to the enemy's base at 4/5c, will you be aged differently when you get back.)
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On further thought, however, Asteroids would not be a good candidate for this at all: it's played in toroidal space, so "going towards" or "going away" from an object would be ill-defined (the way that when you go into the woods, halfway through you instantaneously switch from going into them to coming back out, and your velocity relative to the edge of the woods has a discontinuity), so Lorenz contraction and Doppler shift as I understand them wouldn't work. I was then thinking about something like StarFox, but it doesn't allow enough movement control to be interesting. Joust is played in cylindrical space and would have the same problems as Asteroids. Something like Elastomania has only stationary objects (besides the bike itself) and would be boring.
I can't think of a good candidate game for this, but I still think the idea itself is a good one.
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I may be remembering this wrong, but if you're doing it from the ship's point of view, can't it not see the asteroids until after they've passed it? (Amount of time spent pondering that: three seconds. Feel free to demonstrate that I'm wrong)
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Seems like the way to build it would be to just do it all exactly the way relativity demands, and see if the gameplay is any good, then tweak accordingly.
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