DoctorDick
02-December-2001, 08:44 PM
A Puzzle,
I apologize for being so slow at posting this but it was the weekend and I do have other things I had to do and I wanted to be as clear as possible. It’s a problem I dreamed up some 40 years ago when I was first learning about General Relativity. It is made very clear in most presentations of General Relativity that it is impossible to define a rigid object. Seeing how most physics laboratories make a lot of use of so called “rigid” equipment, the inherent difficulty of the suggested conflict aroused my interest.
Let us suppose that you are completely familiar with special and general relativity. You are in charge of programming the navigation computer on a ship which is a member of a large intergalactic fleet of fighting ships. Now I am sure most of you are well aware of the existence of idiots in command positions who have no idea of what constraints control your work (if you don’t know what I am talking about, see most any Dilbert cartoon). It turns out that your commander (in charge of the entire fleet) does not believe in relativity at all. He is aware of the finite speed of light so he is aware of the fact that there are distortions in what he actually sees during a maneuver, but beyond that, he is ignorant of the entire phenomena and has no interest in learning.
Your problem is that the commander doesn’t care about the fact that you cannot define a rigid object; as far as he is concerned, his entire fleet is to maneuver as if it is a rigid object. Now he may not be able to check things during an actual maneuver, as his fleet is far too large, but he can certainly review the computer logs of what your ship did during the maneuver. What he can do is to make sure everyone is in formation after their rockets are shut down, and, he can tell from the computer logs if anyone made any changes in position after the maneuver was completed.
The rules you have to obey are as follows: first, everyone in the fleet must start their rockets simultaneously (this is well defined as, prior to the maneuver, they are all in the same inertial frame), and second, everyone in the fleet must shut down their rockets simultaneously (this is also well defined as, after the maneuver, they are all in the same inertial frame).
When any maneuver is finished, the commander may tour the fleet and examine the computer logs as to when the rockets on a given ship were fired, in what direction they were fired and exactly how long they were fired. Since he has been the commander for several thousand years, he is quite familiar with the firing logs for most every position in the fleet and for every common maneuver performed. He will notice something unusual immediately. If he thinks you allowed your ship to get out of position during the maneuver, the punishment is the death penalty.
Now we know the other navigators have managed to fake the guy out for a thousand years or so we ought to be able to figure it out too. Restating the problem, given our initial position in the fleet (relative to the flag ship - set up a coordinate system with the flagship at the origin) and the intended maneuver to be followed (expressed in the commanders instantaneous local coordinate system) find the best relativisticly correct path which your ship should follow.
What you need to come up with is an algorithm which will carry you from your start position to your finish position for each and every maneuver the commander may specify. Hint: you will clearly have to include adjustment of the times on your computer clock during the maneuver as the clocks will not read the same as the commanders clock during that phase. (All clocks on board your ship are synchronized with the computer so you don’t have to worry about any other clocks.) On the other hand, no adjustment on the time can be made after the maneuver is completed (let’s say that is just thrown in to keep you honest).
Start with some simple maneuvers and see if you can figure out what your ship should be doing. Try a straight line acceleration first. You all should be able to work that one out. Once you get that worked out, try a simple turn with no acceleration for the flagship; that gets a tad more complex. If you get that worked out, you can try a moving turn: i.e., a turn with acceleration in the direction of the turn (centripetal force on the flagship) actually a rather trivial complication.
Now an accelerating turn gets difficult (at least for me). I did it 40 years ago but it wasn’t easy and I am not entirely sure it was without error. I had some major timing problems. At any rate, I think this ought to keep you all busy thinking for a while and I think it will show some significant insights into the dynamics of general relativistic motion. I am curious as to what kind of results you guys can come up with.
Have fun -- Dick
I apologize for being so slow at posting this but it was the weekend and I do have other things I had to do and I wanted to be as clear as possible. It’s a problem I dreamed up some 40 years ago when I was first learning about General Relativity. It is made very clear in most presentations of General Relativity that it is impossible to define a rigid object. Seeing how most physics laboratories make a lot of use of so called “rigid” equipment, the inherent difficulty of the suggested conflict aroused my interest.
Let us suppose that you are completely familiar with special and general relativity. You are in charge of programming the navigation computer on a ship which is a member of a large intergalactic fleet of fighting ships. Now I am sure most of you are well aware of the existence of idiots in command positions who have no idea of what constraints control your work (if you don’t know what I am talking about, see most any Dilbert cartoon). It turns out that your commander (in charge of the entire fleet) does not believe in relativity at all. He is aware of the finite speed of light so he is aware of the fact that there are distortions in what he actually sees during a maneuver, but beyond that, he is ignorant of the entire phenomena and has no interest in learning.
Your problem is that the commander doesn’t care about the fact that you cannot define a rigid object; as far as he is concerned, his entire fleet is to maneuver as if it is a rigid object. Now he may not be able to check things during an actual maneuver, as his fleet is far too large, but he can certainly review the computer logs of what your ship did during the maneuver. What he can do is to make sure everyone is in formation after their rockets are shut down, and, he can tell from the computer logs if anyone made any changes in position after the maneuver was completed.
The rules you have to obey are as follows: first, everyone in the fleet must start their rockets simultaneously (this is well defined as, prior to the maneuver, they are all in the same inertial frame), and second, everyone in the fleet must shut down their rockets simultaneously (this is also well defined as, after the maneuver, they are all in the same inertial frame).
When any maneuver is finished, the commander may tour the fleet and examine the computer logs as to when the rockets on a given ship were fired, in what direction they were fired and exactly how long they were fired. Since he has been the commander for several thousand years, he is quite familiar with the firing logs for most every position in the fleet and for every common maneuver performed. He will notice something unusual immediately. If he thinks you allowed your ship to get out of position during the maneuver, the punishment is the death penalty.
Now we know the other navigators have managed to fake the guy out for a thousand years or so we ought to be able to figure it out too. Restating the problem, given our initial position in the fleet (relative to the flag ship - set up a coordinate system with the flagship at the origin) and the intended maneuver to be followed (expressed in the commanders instantaneous local coordinate system) find the best relativisticly correct path which your ship should follow.
What you need to come up with is an algorithm which will carry you from your start position to your finish position for each and every maneuver the commander may specify. Hint: you will clearly have to include adjustment of the times on your computer clock during the maneuver as the clocks will not read the same as the commanders clock during that phase. (All clocks on board your ship are synchronized with the computer so you don’t have to worry about any other clocks.) On the other hand, no adjustment on the time can be made after the maneuver is completed (let’s say that is just thrown in to keep you honest).
Start with some simple maneuvers and see if you can figure out what your ship should be doing. Try a straight line acceleration first. You all should be able to work that one out. Once you get that worked out, try a simple turn with no acceleration for the flagship; that gets a tad more complex. If you get that worked out, you can try a moving turn: i.e., a turn with acceleration in the direction of the turn (centripetal force on the flagship) actually a rather trivial complication.
Now an accelerating turn gets difficult (at least for me). I did it 40 years ago but it wasn’t easy and I am not entirely sure it was without error. I had some major timing problems. At any rate, I think this ought to keep you all busy thinking for a while and I think it will show some significant insights into the dynamics of general relativistic motion. I am curious as to what kind of results you guys can come up with.
Have fun -- Dick