A Thought Experiment and Newtonian Paradoxes.

[espritch]

The Newtonian model of the universe is often described as a clock work universe; everything works in a way that matches our everyday experience. Gravity is a force pulling everything together, time marches forward in a consistent manner for everyone, and a few elegant principles can explain things as divergent as a falling apple or the orbit of the moon. It's a comfortable place, easy to get your head around. Unfortunately, it isn't the universe we inhabit. So along came Einstein and the relativistic universe. Suddenly, gravity is a curvature of space-time, time itself is relative, and the only constant is the speed of light. This is a universe not nearly so consistent with everyday experience. It's a universe where thing like black holes and the infamous twins paradox can occur. And it's a lot harder to wrap your head around.

My purpose here is to propose a thought experiment that I hope will show that the Newtonian universe may not actually be all that neat and tidy and can lead to some rather paradoxical results itself; results that conveniently go away if we accept some fundamental premises of relativity. Of course I must preface this by saying that I am not a physicist and may be skating on very thin ice. But it was either post it and get some feedback or else keep bouncing it around in my brain in an annoyingly distracting fashion. So feel free to comment, correct me, or just tell me I'm all wet.

I'll start with two spaceships (Spaceship One and Spaceship Two) which are coasting along in nice flat Newtonian space at an equal velocity. Spaceship One is moving from point A to point B, Spaceship Two from point X to point Y. The distance between A and B is equal to the distance between X and Y. So they will both make their respective journeys in an equal period of time. The only way one could cover the distance faster than the other is for some force to be applied to it so that it accelerates and obtains a greater velocity. So far, so obvious.

Now, I'll introduce a planetoid. This particular planetoid is plopped down right in the middle of the path of Spaceship Two dead center between point X and point Y. Luckily for the crew of Spaceship Two, the planetoid has a hole drilled all the way through it big enough to let a spaceship fly right through. I will also note that Spaceship One is far enough away from the planetoid that the gravitational effect on it is negligible.

The behavior of Spaceship One is pretty simple. It just costs along at a constant velocity from point A to point B. The behavior of Spaceship Two is more complex. The planetoid exerts a gravitation tug on the ship that causes it to accelerate. This acceleration increases until the ship reaches the mouth of the tunnel at which point it begins to diminish until the ship reaches the center where the acceleration reaches zero (and the velocity of the ship reaches it's maximum). The acceleration then reverses and becomes negative acceleration reducing the velocity of the ship so that by the time it reaches point Y on the other side, the velocity of Spaceship Two is exactly the same as it was at point X.

If we graph the velocity of our two ships with respect to time over the course of the journey, the graph for Spaceship One is a flat line while the graph for Spaceship Two is a bell curve with it's highest point corresponding to the moment the ship passed through the center of the planetoid and it's lowest values corresponding to the end points at X and Y. Since these end point velocities are equal to the velocity of Spaceship One, it is evident that over the whole course of the trip, Spaceship Two was moving at a greater velocity than Spaceship One. So naturally, you would conclude that Spaceship Two will cover the distance from X to Y in less time than it takes Spaceship One to get from A to B. But this "obvious" conclusion has a problem: what was the force that allowed Spaceship Two to outrace it's sister ship? The planetoid certainly exerted a gravitational force on the ship. But the net acceleration for that force over the entire course of the trip sums up to zero. If it didn't, Ship Two would not have arrived at point Y with exactly the same velocity it had at point X. So did Ship Two actually get to point Y sooner than Ship One reached B? And if not, how do we account for all that extra velocity?

In a Newtonian universe this would seem to be a paradox. In a relativistic universe, however, it really isn't a problem. Gravity curves space. The path from X to Y for Spaceship Two is actually longer than the path from A to B for Spaceship One because Spaceship Two is plunging through a gravity well. It has to move faster just to get to point Y at the same time as Spaceship One reaches point B.

[chance]

I’ll attempt an answer. First why, would ship two not arrive earlier it has a gravity boost, and for some of the journey is travelling faster than ship one.

If this is not the case, and IRL both ship arrive simultaneously, then perhaps as ship 2 is accelerating to the planetoid, the planetoid is also moving towards ship 2, they meet at some point before halfway, the remainder of the journey is now longer under deceleration, and ship 2 arrives at a slower speed than ship 1

[TeaBag420]

Saying "there's a paradox" doesn't mean there is one.

Even in what you are calling a "Newtonian" universe, there is no paradox.

To paraphrase David Brinkley, "get ready for 14 pages of more goddamn nonsense."

When Newton and Einstein take the stage, can Heaviside be far behind?

And of course, "it"'s "its" is "its" not "it's".

[scribble]

Originally posted by espritch
But this "obvious" conclusion has a problem: what was the force that allowed Spaceship Two to outrace it's sister ship? The planetoid certainly exerted a gravitational force on the ship.


There you go!


But the net acceleration for that force over the entire course of the trip sums up to zero.


Calculus will help you understand the proper way to compute the area under your curve.

The only other thing I can add is that videogames using plain old newtonian physics duplicate this effect flawlessly. I'm not a physicist, so I can't comment on it more directly.

[Soapy Sam]

I question your premise, sir!. (Removes goldfish).
I think it does get there first. And it uses less fuel to accelerate to cruise speed in the process.

[DaveW]

Net acceleration might be zero, but the average velocity for Spaceship 2 will definitely be higher than Spaceship 1 in this example, so it would arrive at its destination first. I don't see the paradox, either.

[wittgenst3in]

In a Newtonian universe this would seem to be a paradox. In a relativistic universe, however, it really isn't a problem. Gravity curves space. The path from X to Y for Spaceship Two is actually longer than the path from A to B for Spaceship One because Spaceship Two is plunging through a gravity well. It has to move faster just to get to point Y at the same time as Spaceship One reaches point B.

Wait a sec, the amount that this matter curves space is really really tiny compared to other stuff. The path length isn't changed significantly compared to the extra average speed obtained. Hence the paradox still exists in a relativistic universe.

I see what you are getting at with the idea of a paradox, but I don't think it's paradoxical. Take this example:

Spaceship A is completely motionless 100km above planet donut. It is being held in the jaws of some imaginary fixed thingy. The jaws open and spaceship A accellerates from 0 to say 100km/s at the centre of the planet.

It whips through the centre and starts decelerating again. At the same distance on the opposite side of the planet is another set of fixed jaws. At the exact moment when velocity slows back to zero the jaws close. Spaceship A has now made the trip with no initial velocity and only gravity assist.

So I think the only problem is with our conception that energy need be expended to get a faster trip.

P.s. Neat thing to do if you are ever flying through planet donut is to do a burn in the centre. This means you are accellerating mass of (you+propellant) on the way in, and only decelerating your mass on the way out.

[scribble]

Originally posted by wittgenst3in
So I think the only problem is with our conception that energy need be expended to get a faster trip.


When I was in (grade) school, we were taught a concept - I'll probably get the name wrong. Stored kinetic energy? The idea being that there's some energy in being suspended over a gravity well, inherently (bad explanation, I know). The energy required to get you up the well, I guess, whether you had to make that trip yourself or not.

I'm not a physicist by a longshot, so I've no idea about the depths of that concept -- but it seems to apply here.

[TillEulenspiegel]

scribble , I was taught the phrase "potential" energy.

The example that is given...

The spaceship that fell through the planet is subject to gravitational attraction . That would accelerate the object even more until the inertia of initial velocity and imparted acceleration due to G attraction is effected by the ever increasing drag ( backward G attraction ) upon divergence from the center of the mass and in a perfect non-loss system in a vacuum would result in an acceleration and a deceleration with the final product = 0 in speed. That cannot happen because of the laws of thermodynamics, so I'm seeing a small net loss in an ideal scenario and a greater loss in an actual environment.

[espritch]

chance:
I’ll attempt an answer. First why, would ship two not arrive earlier it has a gravity boost, and for some of the journey is traveling faster than ship one.

If this is not the case, and IRL both ship arrive simultaneously, then perhaps as ship 2 is accelerating to the planetoid, the planetoid is also moving toward ship 2, they meet at some point before halfway, the remainder of the journey is now longer under deceleration, and ship 2 arrives at a slower speed than ship 1


There is such a thing as gravity assist. The Cassini probe that recently reached Saturn used such gravity assists from Earth, Venus, and Jupiter to gain speed for it's voyage to Saturn. However, this is not equivalent to the situation I describe in my thought experiment. In the case of Cassini, the probe gained momentum while each planet lost some (but so little as to be negligible). In my scenario, there is no net gain in momentum; both the ship and the planetoid have the same momentum when the ship reaches point Y that they had when it was at point X.

TeaBag420
Saying "there's a paradox" doesn't mean there is one.

I would agree, which is why I not only said there was a paradox, but tried to show why I thought that there was one. Just saying "there is not a paradox", doesn't mean there is not one.

scribble
Calculus will help you understand the proper way to compute the area under your curve.

Calculus would indeed be required to properly calculate the area under the curve. But calculating the area under the curve is not necessary to the experiment. It is only necessary to recognize that Spaceship Two will have a higher velocity than Spaceship One everywhere except at the start and end points.

Soapy Sam
I question your premise, sir!. (Removes goldfish).
I think it does get there first. And it uses less fuel to accelerate to cruise speed in the process.

Actually neither ship uses any fuel. At the start point both ships have already achieved an equal velocity and coast the remainder of the journey on pure inertia. If the ship does get there first, it still did so with a zero net gain in momentum and a zero net gain in energy. This would still appear to be a paradox.

Consider the case where there is no planetoid and Spaceship Two fires it's rockets to accelerate. Near the end of the trip it fires it's retro rockets to slow to it's initial velocity. In this case Spaceship Two will certainly arrive first, but there is a difference. For the rocket to fire, energy had to be converted from chemical energy to momentum. Half was imparted to the spaceship, half to the exhaust gas. When the ship fired the retro rocket to slow down, it again converted chemical energy to momentum, negative for the ship, positive for the exhaust gas. The total momentum for the ship and it's exhaust gas increased by the amount of chemical energy released.

In my scenario, there is a zero net change in momentum and energy.

wittgenst3in
Wait a sec, the amount that this matter curves space is really really tiny compared to other stuff. The path length isn't changed significantly compared to the extra average speed obtained. Hence the paradox still exists in a relativistic universe.

Well, I may be guilty of over simplification. It isn't just distance that is changed by gravity, it is also time. Some of what Spaceship Two experiences as acceleration, Spaceship One views as just a slowing down of the clock on Spaceship Two. We measure velocity as distance covered in a given period of time. If you cover the same distance in a shorter time period, you appear to be moving faster. But maybe time is just moving slower for you. I don't really understand relativity well enough to fully analyze it's full effect on my scenario, but I'm guessing that it will be sufficient to explain the paradox (assuming there really is one).

Spaceship A is completely motionless 100km above planet donut. It is being held in the jaws of some imaginary fixed thingy. The jaws open and spaceship A accelerates from 0 to say 100km/s at the center of the planet.

It whips through the center and starts decelerating again. At the same distance on the opposite side of the planet is another set of fixed jaws. At the exact moment when velocity slows back to zero the jaws close. Spaceship A has now made the trip with no initial velocity and only gravity assist.


Hmmm...this still strikes me as paradoxical. The ship gets from point A to point B with no net expenditure of energy or any net change in the energy of the of the system. But I must admit I don't see how relativity would resolve it either. Which would kind of defeat my who purpose here. :confused: I guess I'll just have to ignore it and hope it goes away. ;)

[wittgenst3in]

Originally posted by espritch


Well, I may be guilty of over simplification. It isn't just distance that is changed by gravity, it is also time. Some of what Spaceship Two experiences as acceleration, Spaceship One views as just a slowing down of the clock on Spaceship Two. We measure velocity as distance covered in a given period of time. If you cover the same distance in a shorter time period, you appear to be moving faster. But maybe time is just moving slower for you. I don't really understand relativity well enough to fully analyze it's full effect on my scenario, but I'm guessing that it will be sufficient to explain the paradox (assuming there really is one).
[/B]
Another valid point, but the effects are again vanishingly small. Another thought experiment will help here, but this time it can be done at home!

I set up two synched writstwatches as follows: watch B is in a frictionless cart moving from the start of the track to a point 3 meters away at a constant velocity of say 0.5ms-1 . The time taken for B to arrive is therefore 6 seconds.

Watch A is set up on a long pendulum arranged so the swing passes through both the starting point and another point horizontally 3 meters away. Give A a small initial velocity, or even just release it and again it beats watch B.

Measure both watches against one another. No significant time dialation has occured.


Originally posted by espritch
Hmmm...this still strikes me as paradoxical. The ship gets from point A to point B with no net expenditure of energy or any net change in the energy of the of the system. But I must admit I don't see how relativity would resolve it either. Which would kind of defeat my who purpose here. :confused: I guess I'll just have to ignore it and hope it goes away. ;) [/B]

It seems to me that the only way this occurs is that the spaceships/watches move along paths of equal energy (i.e. potential + kinetic).

For example draw a graph kind of like a topographical map, except the lines represent equal energy, rather than equal height. To move to another parallel path on the map would require a change in the amount of energy in the system.

If memory serves these diagrams are called Lorentz attractors a google search for them should give you an idea of what I'm talking about.

(Geez, I better study this as well. I'm doing a subject next semester for the second time and I just know this is going to come up.)

Edit: After checking around, it'd be better to look for attractors (http://en.wikipedia.org/wiki/Attractor) rather than lorentz attractors. Attractors are the graphs I described above, and lorentz attractors are a type of them which don't settle down into a recognizable pattern, and are considered 'strange'.

[Pragmatist]

Originally posted by wittgenst3in
Another valid point, but the effects are again vanishingly small. Another thought experiment will help here, but this time it can be done at home!

I set up two synched writstwatches as follows: watch B is in a frictionless cart moving from the start of the track to a point 3 meters away at a constant velocity of say 0.5ms-1 . The time taken for B to arrive is therefore 6 seconds.

Watch A is set up on a long pendulum arranged so the swing passes through both the starting point and another point horizontally 3 meters away. Give A a small initial velocity, or even just release it and again it beats watch B.

Measure both watches against one another. No significant time dialation has occured.




It seems to me that the only way this occurs is that the spaceships/watches move along paths of equal energy (i.e. potential + kinetic).

For example draw a graph kind of like a topographical map, except the lines represent equal energy, rather than equal height. To move to another parallel path on the map would require a change in the amount of energy in the system.

If memory serves these diagrams are called Lorentz attractors a google search for them should give you an idea of what I'm talking about.

(Geez, I better study this as well. I'm doing a subject next semester for the second time and I just know this is going to come up.)

Edit: After checking around, it'd be better to look for attractors (http://en.wikipedia.org/wiki/Attractor) rather than lorentz attractors. Attractors are the graphs I described above, and lorentz attractors are a type of them which don't settle down into a recognizable pattern, and are considered 'strange'.

Look up Hamilton's principle. You'll find that objects move along paths of stationary action.

Also remember that special relativity is only valid for inertial reference frames. An accelerating frame is not inertial. Your pendulum is an accelerating frame. So is the ship under the influence of gravity.

[TeaBag420]

Originally posted by espritch
chance:


There is such a thing as gravity assist. The Cassini probe that recently reached Saturn used such gravity assists from Earth, Venus, and Jupiter to gain speed for it's voyage to Saturn. However, this is not equivalent to the situation I describe in my thought experiment. In the case of Cassini, the probe gained momentum while each planet lost some (but so little as to be negligible). In my scenario, there is no net gain in momentum; both the ship and the planetoid have the same momentum when the ship reaches point Y that they had when it was at point X.


Yes, and the Enterprise used to achieve time travel by swinging around the sun...


TeaBag420


I would agree, which is why I not only said there was a paradox, but tried to show why I thought that there was one. Just saying "there is not a paradox", doesn't mean there is not one.


You said it "[struck you] as paradoxical" but you didn't even prove that the premise was true. The math (aye, there's a harsh word now, laddy) shouldn't be a big deal for a fractal expert.



scribble


Calculus would indeed be required to properly calculate the area under the curve. But calculating the area under the curve is not necessary to the experiment. It is only necessary to recognize that Spaceship Two will have a higher velocity than Spaceship One everywhere except at the start and end points.




Hmmm...this still strikes me as paradoxical. The ship gets from point A to point B with no net expenditure of energy or any net change in the energy of the of the system. But I must admit I don't see how relativity would resolve it either. Which would kind of defeat my who purpose here. :confused: I guess I'll just have to ignore it and hope it goes away. ;)


Your assertion is that one ship gets there in less time than the other. Prove that first, using math, then we can worry about your (imagined) paradoxes.

I can't help you with your physics homework, but I can help you with your English homework. "It's" is not correct the way you use it.

[Robin]

From Teabag420
Your assertion is that one ship gets there in less time than the other. Prove that first, using math, then we can worry about your (imagined) paradoxes.


I agree, I would need to see the calculations. Would the velocity take the shape of a bell curve?

[scribble]

Originally posted by espritch
[B]chance:


There is such a thing as gravity assist. The Cassini probe that recently reached Saturn used such gravity assists from Earth, Venus, and Jupiter to gain speed for it's voyage to Saturn. However, this is not equivalent to the situation I describe in my thought experiment. In the case of Cassini, the probe gained momentum while each planet lost some (but so little as to be negligible). In my scenario, there is no net gain in momentum; both the ship and the planetoid have the same momentum when the ship reaches point Y that they had when it was at point X.


They have no "extra" momentum, but they've both moved, just as much as in your cassini example.

What makes you think the planet doesn't move the opposite direction of the ship in your example -- albiet a negligible amount?

From all I can figure, it should...

[TeaBag420]

Scribble, doesn't any of this sound familiar to you? To repeat my paraphrase of David Brinkley, "Fourteen pages of more goddamn nonsense".

[scribble]

Originally posted by TeaBag420
Scribble, doesn't any of this sound familiar to you? To repeat my paraphrase of David Brinkley, "Fourteen pages of more goddamn nonsense".

I don't know, I always though espritch was a pretty reaonable guy in the past. He's probably just missing something simple here. Or I am.

We'll see.

[wittgenst3in]

Originally posted by scribble
They have no extra momentum, but they've both moved, just as much as in your cassini example.

What makes you think the planet doesn't move the opposite direction of the ship in your example -- albiet a negligible amount?

From all I can figure, it should...

The planet would obviously move, but not a significant amount. Even if the planet was 'clamped' somehow the result would still be (practically) the same.


Originally posted by TeaBag420
Your assertion is that one ship gets there in less time than the other. Prove that first, using math, then we can worry about your (imagined) paradoxes.

Consider my example using the jaws before. It's obvious that a spaceship released above (i.e. not in a tangential orbit) planet donut will plummet through and out the other side, reaching the same height as it was released at (neglecting atmosphere, intersteller hydrogen, eddy currents with metal, etc.)

Now run the experiment again but with the spaceship having some velocity towards the planet before it is released. It will pick up speed and plummet under gravity, and when it passes the same height at the other end, it will have the same velocity it started with (Don't believe me, do a conservation of momentum calculation).

Spaceship A will always get there faster than spaceship B. This shouldn't be an issue, what should be is the preconception we have that it will end up slower.

Edit: Last sentence would be better read as:
This shouldn't be an issue, what should be is the preconception we have that they will be the same.

[scribble]

Originally posted by wittgenst3in
The planet would obviously move, but not a significant amount. Even if the planet was 'clamped' somehow the result would still be (practically) the same.


Yes, but according to espritch, the big mystery here is in that the planet wouldn't be affected at all. Which is untrue. Sure, unless we imagine a truly massive spaceship, you probbaly couldn't measure the difference. But it's there.

That's all I was getting at.

[wittgenst3in]

Originally posted by scribble
Yes, but according to espritch, the big mystery here is in that the planet wouldn't be affected at all.

Which bit of his are you referring to? I can't see where he says that.

He does say that there is no net change in the momentum of the planet and ship, but this is not the same thing as saying they are unaffected.

The planet accelerates towards the starting point initially, then after the ship has passed through it accelerates in the other direction. But it has the same momentum at the start as at the end.

[Earthborn]

Look at it with a more mundane model than spaceships in space with planetoids:

http://member.prolinea.org/~joan/newton.gif

Cart 2 has an extra force that accelarate it at first but slows it down later. Cart 1 has only its initial speed, whatever it is (here it is 0). Which one arrives at the end sooner? I'm no physisist, so don't take my word for it, but I think it is a no-brainer. I also see no paradox.

[SGT]

Expenditure of energy is only needed to do some work. A satelite turns around a planet without expenditure of energy (I am assuming a frictionless movement).
In the example given by wittgenst3in with the jaws, the ship had an initial potential energy. This potential energy is turned into kinetical energy, while the ship moves across planet donut and finishes with the same initial kinetic energy. There has been movement but no net work.
In the original problem, both spaceships start and finish the trip with the sam kinetic energy, so there is no net gain in traversing the planet, but spaceship two arrives first, since during part of the trajectory it has a greater velocity.
I see no paradox at all.

[espritch]

TeaBag420
Scribble, doesn't any of this sound familiar to you? To repeat my paraphrase of David Brinkley, "Fourteen pages of more goddamn nonsense".

You seem to have jumped to an unfounded conclusion either about my intent or my identity. I am not an anti-relativist, nor for that matter an anti-Newtonian. I have never participated on a relativity related topic on this forum before and have no idea who Heaviside is. I have no desire for "14 more pages of Goddamn nonsense". I started this thread simply to have a discussion of what I considered an interesting topic and to get some feedback as to whether I had, indeed, missed something. To quote Freud, "Sometimes a cigar is just a cigar."

That being said, I do understand your reaction. I've hung out at Bad Astronomy some and I do know how relativity threads can get rather strange. If you really want strange, try debating a geocentrist some time. :D

Robin
I agree, I would need to see the calculations. Would the velocity take the shape of a bell curve?

The bell curve represents the change in velocity as the ship accelerates to the center of the planetoid and then decelerates on the other side. I agree that it generally better to work out the equations, but I don't think it will be necessary this time - I think I've already figured out where the problem with my thought experiment lies (I'll get to that a little later).

Scribble
I don't know, I always though espritch was a pretty reasonable guy in the past.

Thanks for the vote of confidence. :)

They have no "extra" momentum, but they've both moved, just as much as in your cassini example.

What makes you think the planet doesn't move the opposite direction of the ship in your example -- albeit a negligible amount?

From all I can figure, it should...

You misunderstood. I never assumed that the planetoid didn't move. Just as the planetoid accelerates the ship on the first half of the trip, the ship accelerates the planetoid (though to a much lesser degree). On the second leg, the planetoid decelerates the ship and the ship decelerates the planetoid. The net change in momentum for both the ship and the planetoid at the end point is zero. In the case of Cassini, the total momentum is also unchanged but there is a permanent transfer of momentum from the planet to the probe.

I have, as previously noted, decided my thought experiment was flawed and there is no paradox. I agree that Spaceship Two does get to point Y before Spaceship One gets to point B. My original problem with this was that if this was the case, Spaceship Two appeared to be getting a free lunch. It was beating Spaceship One despite the fact there was no net expenditure of energy. The problem is that I was looking at the scenario only within a arbitrary limit (i.e. the distance from point X to point Y). If you considered what happens after the ships pass the end point, the apparent paradox quickly resolves itself. Spaceship One continues sailing along at the initial velocity. But Spaceship Two continues to decelerate. Even though Spaceship Two covered the distance from point X to Y faster, in the long run Spaceship One will catch up with and even pass Spaceship Two so there is
no free lunch and all is well with the universe.

Oh well, another thought experiment goes down in flames. Thanks to everyone who responded. Your feed back was helpful.

[TeaBag420]

Well, it appears I was overly suspicious (a sissy way of saying "I was wrong").

To refine your original scenario, you might want to stipulate that both craft start out traveling faster than the escape velocity of the planetoid.