Flying Geometrically
The geometry of long-haul air travel is intriguing, giving rise to relativistic thoughts. The other day I flew from Chicago to Hong Kong, taking off a little after mid-day in early December, and arriving just a bit later the next day (local time) but 15 hours later human being time. You think of Chicago to Hong Kong as going all the way across the Pacific. But of course we never went even remotely close to the Pacific. Our route went up over Hudson Bay (that has to be the ugliest-shaped body of water anywhere. What is it about it that looks so disgusting? What is that appendage thing?) Then over north central Canada, a chunk of Arctic Ocean, and back down over Siberia and China.
Nothing very odd about that, but think of it from the sun's perspective. If we were on the sun watching, (unlikely, but if) we would see the plane take off in the middle of the earth and hasten up to the top, going over the horizon into the polar darkness for a little while. Meanwhile the earth would turn around, the visible part moving from left to right. As the earth achieved about a half rotation, now showing its other side to the sun, our little plane would pop down again out of the darkness, finally landing in almost exactly the same place as it had started from, from the sun’s point of view, except that the earth had turned round underneath it. The plane would fly a curve of course, gradually turning 180° counter-clockwise, but in relation to the earth, as seen from the sun, it would appear to just fly up to the top, and then fly down again. In the northern hemisphere summer, it wouldn’t go into the dark, but would just turn around higher up in the light and then come back down.
Makes you think there must be an easier way of doing it; like shooting up in a rocket, hanging around for a while as the earth re-adjusts itself, and then drop down again onto Hong Kong. It would be tough, but has something to recommend itself. First, we weren’t really traveling in relation to the earth’s surface in any case, except as a very desirable side-effect. We were swimming through the air that floats above it. And that air goes around with the earth, so even though we had escaped contact with the ground, we had to swim through all the rotating air to get back to where we started from. But going up in a rocket and hanging around waiting wouldn’t work in a simple way because, in space, there is no such thing as just hanging around. If we were in space, at the point we’d like to hang around, we would immediately be in orbit around the earth anyway, since being in space means being in orbit round things. Einstein showed us that with the warping of space time. There is no fixed space to hang around in. So if we went straight up, we would be in an orbit that would take us straight down again. I suppose if we went up at the right speed and angle, we could indeed get into an orbit that would drop us back onto Hong Kong as it happened to pass by. It would be awfully expensive, and there is the problem of how not to crash too. Still, it is interesting to think that all that flying and all that burning of jet fuel just puts us back where we started.
One last puzzle about that journey. We ended up still in the middle of the day, but one day later. Where were we when the date changed? It all depends on how close to the north pole we went. If we went just to the left of it, (yes, I know, everything is left of the north pole, or west of it, or east of it too, come to that. I mean left of it from the point of view of the Chicago that we started out from, looking north.) If we went just to the left of the north pole, then the date changed when we crossed the international date line, wherever it is up there, because by going west, we were swooshing backwards through local ground time. On the other hand, if we passed just to the right of the north pole, we would not cross the date line at all. So in that case, the date would have shifted when we were flying over a bit of ground where the local time just happened to be midnight as we passed. At that point we overtook the time on the ground, and jumped on into the next day, as it was sweeping westward, irrespective of how long it took us to get there. The paradox is if we go directly over the pole itself. Then I suppose we do a sort of time travel trick, leaping 12 hours into the future at that instant of crossing. So for a tiny moment, the front of the plane would be twelve hours ahead of the tail. Just a thought.
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Then today I flew from Hong Kong to Munich. That was even stranger. The route took us up through China and over Khazahkstan, close by Moscow and on down into Germany. So we were flying across the face of the earth. Nowhere near the poles this time. But it had a similar curious geometry. We took off just after midnight, so in the darkness. We flew for a full 12 hours, and landed at 5 in the morning in Munich on the same day. It was still pitch dark, and we never saw a glimmer of daylight at any point. So only 5 hours on the ground under us had passed during our twelve hours. The flight had taken us backwards in time by 7 hours. Let’s think how this one appears from the point of view of the sun, or rather, the anti-sun. Imagine looking at the earth from the moon, for instance, if it happened to be on the other side of the earth from the sun, looking at the dark side. (Like if it was a full moon, which it wasn’t, but never mind, let’s take it as being a putative full moon.) From this fictional moon we would see Hong Kong come into view as the sun set there, and it swung round into the dark side, coming in from the left side of the earth’s disk. (North is up, as is proper for any ex-British colony.) Our plane would be seen to take off and start flying back to the left - towards the sunlight that Hong Kong had just moved out of. So now the disc of the earth would be moving from left to right, but our plane would be flying from right to left. Of course, it wouldn’t be able to fly as fast as the earth’s surface was spinning around, so relative to the moon from which we were watching, the plane would clearly be flying backwards. It would be up in the air, with its snout pointing west, but the ground under it, (and the air all around it through which it was swimming) would be moving to the east quite a bit faster. The plane couldn’t keep up, and it wasn’t using a dodge like getting up to the pole to get out of the way. So the plane, relative to the moon, and therefore the sun, was flying backwards. It was pointing left and going quite fast, so it managed to stay in the dark bit for the full 12 hours, while the land in the dark bit, including Hong Kong, whisked off to the right, and was in fact in the middle of the light side by the time we landed in Munich, which was also now on the dark side, but had been in the previous day’s sunny bit when we took off. So our plane was moving to the right, just like the earth, but facing to the left, and working damned hard not to be swept east. It thus managed to move to the east relative to the moon, but about 500 mph more slowly that the surface of the earth did. So from that point of view, it wasn’t using its engines to move, so much as using them to brake, and resist the turning of the earth. And thus it successfully managed to escape the coming of the light, and remained in night time for the whole journey.
It just all depends upon your point of view, but maybe one day the “shoot up and plop down again when the destination has arrived” method might work. If you could get enough power for lift off, and then glide until braking for touch-down, it might even end up using less energy. My math isn’t good enough to figure that out, especially not after just getting off a 12-hour flight through the darkness of the Russian steppes.
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© ajm 2007
Nothing very odd about that, but think of it from the sun's perspective. If we were on the sun watching, (unlikely, but if) we would see the plane take off in the middle of the earth and hasten up to the top, going over the horizon into the polar darkness for a little while. Meanwhile the earth would turn around, the visible part moving from left to right. As the earth achieved about a half rotation, now showing its other side to the sun, our little plane would pop down again out of the darkness, finally landing in almost exactly the same place as it had started from, from the sun’s point of view, except that the earth had turned round underneath it. The plane would fly a curve of course, gradually turning 180° counter-clockwise, but in relation to the earth, as seen from the sun, it would appear to just fly up to the top, and then fly down again. In the northern hemisphere summer, it wouldn’t go into the dark, but would just turn around higher up in the light and then come back down.
Makes you think there must be an easier way of doing it; like shooting up in a rocket, hanging around for a while as the earth re-adjusts itself, and then drop down again onto Hong Kong. It would be tough, but has something to recommend itself. First, we weren’t really traveling in relation to the earth’s surface in any case, except as a very desirable side-effect. We were swimming through the air that floats above it. And that air goes around with the earth, so even though we had escaped contact with the ground, we had to swim through all the rotating air to get back to where we started from. But going up in a rocket and hanging around waiting wouldn’t work in a simple way because, in space, there is no such thing as just hanging around. If we were in space, at the point we’d like to hang around, we would immediately be in orbit around the earth anyway, since being in space means being in orbit round things. Einstein showed us that with the warping of space time. There is no fixed space to hang around in. So if we went straight up, we would be in an orbit that would take us straight down again. I suppose if we went up at the right speed and angle, we could indeed get into an orbit that would drop us back onto Hong Kong as it happened to pass by. It would be awfully expensive, and there is the problem of how not to crash too. Still, it is interesting to think that all that flying and all that burning of jet fuel just puts us back where we started.
One last puzzle about that journey. We ended up still in the middle of the day, but one day later. Where were we when the date changed? It all depends on how close to the north pole we went. If we went just to the left of it, (yes, I know, everything is left of the north pole, or west of it, or east of it too, come to that. I mean left of it from the point of view of the Chicago that we started out from, looking north.) If we went just to the left of the north pole, then the date changed when we crossed the international date line, wherever it is up there, because by going west, we were swooshing backwards through local ground time. On the other hand, if we passed just to the right of the north pole, we would not cross the date line at all. So in that case, the date would have shifted when we were flying over a bit of ground where the local time just happened to be midnight as we passed. At that point we overtook the time on the ground, and jumped on into the next day, as it was sweeping westward, irrespective of how long it took us to get there. The paradox is if we go directly over the pole itself. Then I suppose we do a sort of time travel trick, leaping 12 hours into the future at that instant of crossing. So for a tiny moment, the front of the plane would be twelve hours ahead of the tail. Just a thought.
---------------------------------
Then today I flew from Hong Kong to Munich. That was even stranger. The route took us up through China and over Khazahkstan, close by Moscow and on down into Germany. So we were flying across the face of the earth. Nowhere near the poles this time. But it had a similar curious geometry. We took off just after midnight, so in the darkness. We flew for a full 12 hours, and landed at 5 in the morning in Munich on the same day. It was still pitch dark, and we never saw a glimmer of daylight at any point. So only 5 hours on the ground under us had passed during our twelve hours. The flight had taken us backwards in time by 7 hours. Let’s think how this one appears from the point of view of the sun, or rather, the anti-sun. Imagine looking at the earth from the moon, for instance, if it happened to be on the other side of the earth from the sun, looking at the dark side. (Like if it was a full moon, which it wasn’t, but never mind, let’s take it as being a putative full moon.) From this fictional moon we would see Hong Kong come into view as the sun set there, and it swung round into the dark side, coming in from the left side of the earth’s disk. (North is up, as is proper for any ex-British colony.) Our plane would be seen to take off and start flying back to the left - towards the sunlight that Hong Kong had just moved out of. So now the disc of the earth would be moving from left to right, but our plane would be flying from right to left. Of course, it wouldn’t be able to fly as fast as the earth’s surface was spinning around, so relative to the moon from which we were watching, the plane would clearly be flying backwards. It would be up in the air, with its snout pointing west, but the ground under it, (and the air all around it through which it was swimming) would be moving to the east quite a bit faster. The plane couldn’t keep up, and it wasn’t using a dodge like getting up to the pole to get out of the way. So the plane, relative to the moon, and therefore the sun, was flying backwards. It was pointing left and going quite fast, so it managed to stay in the dark bit for the full 12 hours, while the land in the dark bit, including Hong Kong, whisked off to the right, and was in fact in the middle of the light side by the time we landed in Munich, which was also now on the dark side, but had been in the previous day’s sunny bit when we took off. So our plane was moving to the right, just like the earth, but facing to the left, and working damned hard not to be swept east. It thus managed to move to the east relative to the moon, but about 500 mph more slowly that the surface of the earth did. So from that point of view, it wasn’t using its engines to move, so much as using them to brake, and resist the turning of the earth. And thus it successfully managed to escape the coming of the light, and remained in night time for the whole journey.
It just all depends upon your point of view, but maybe one day the “shoot up and plop down again when the destination has arrived” method might work. If you could get enough power for lift off, and then glide until braking for touch-down, it might even end up using less energy. My math isn’t good enough to figure that out, especially not after just getting off a 12-hour flight through the darkness of the Russian steppes.
My Web Home Page
My Agent
© ajm 2007
posted by Andrew Massey | Tuesday, December 11, 2007
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