nighty23
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This topic isn't exactly trying to achieve FTL, more about talking about why it can't be, because it's an interesting learning experience for me.

The other day, I was talking to a friend of mine about the subject, and I proposed that, say you, in a vacuum, have a setup of pistons.

There's (roughly, I did the miles to meters conversion in my head) 299,300,000 pistons, that all extend 1 meter in 1 second. If you placed 1 more piston at the end of this line of pistons, and all of the pistons were extended at the end of the year, wouldn't the head of this piston be moving at 1 m/s faster than the speed of light?

He told me that since every object has an equal and opposite reaction, they would only move, at most, half the speed of light, because they would each extend both ways away from their center of mass, so I simply proposed you double the length, plus 2- one more on each end- so that both ends would be moving ftl, and he's yet to come up with dispute, so of course I turned to the more intelligent folk I know, fig's. Got anything?
Corelis
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To get a proper answer, you'd probably have to ask someone who has dealt with relativity more than me, as I've only grazed the surface of relativity about a year and a half ago, but I suspect the problem comes from relativistic mass in your scenario.

In order to start extending with a speed of 1 meter in 1 second, a piston has to exert a certain amount of force. Naturally, the amount of force necessary for it to successfully perform its task depends on the mass of the objects it is trying to push.

There are two kinds of mass. There's your usual rest mass, which is what is what people generally think about when talking about mass, and then there is relativistic mass which depends on the speed of an object from the perspective or an observer. Since relativistic mass differs from rest mass only with very high velocities, generally it is meaningful to use only rest mass. But since you're trying to reach lightspeed, it's time to bust out special relativity.

The thing with relativistic mass is that it increases with (relative) velocity. It follows this formula:
m_relative = m_rest / sqrt(1 - (v/c)^2), where v is velocity and c is lightspeed.
As a consequence, unless the rest mass of your object isn't zero, its relative mass will approach infinity if its velocity approaches lightspeed.

So, let's consider the middle piston. As the middle piston accelerates its extension speed from zero to 1 m/s, it has to exert force which depends on the mass of the objects it is pushing. However, very quickly, from the middle piston's perspective, the masses of the pistons far away from it start reaching absurd values. Soon, the middle piston should be pushing with an infinite amount of force to be able to reach the extension speed of 1 m/s.

The same applies to all pistons, though to a lesser extent the further from the middle they are, until the effect becomes negligible near the end. Still, the middle pistons would need to be outrageously (infinitely) strong to propel the pistons in the ends into a lightspeed velocity.

So, that might be why it wouldn't work. I should point out that all that is based on a very brief glance through Wikipedia and somewhat foggy memories, so it could be incorrect.
nighty23
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nighty23 27 United States SanguinePhlegmatic INTP 58C
From what I understand, relativistic mass only applies to acceleration, that is, it only matters if you're trying to increase the energy of something that is already moving.

How about if instead of accelerating the whole extension period of each arm, the entirety of acceleration happens in the first nanosecond or so that it extends, and supposing every piston activates at the same exact time, the relativistic mass would be the rest mass, because they're all at rest.
Corelis
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It is true that the problem of infinity-approaching mass is only an issue when accelerating - if there are no other forces affecting the pistons such as gravity, then after the desired extension speed is reached, there is no need to push anything anymore, and mass becomes irrelevant.

However, no matter how quickly the process of acceleration happens, it can't be instantaneous. The extension speed cannot simply jump from 0 m/s to 1 m/s. There must be an acceleration, and if there is an acceleration, even if it takes a nanosecond, at 0.5 nanoseconds (not taking into account any other problems) all pistons have reached 0.5 m/s, which means that the last pistons are already moving at half the speed of light. The remaining 0.5 nanoseconds will become more and more difficult to manage for the middle pistons, until it becomes impossible.

The amount of time used for the acceleration is irrelevant if all the pistons accelerate at the same time. However, if that ceases to be a requirement, then relativistic mass is no longer a problem... maybe. Let's say the first piston accelerates before the rest. Then, the rest of the pistons will be moving at a speed of 1 m/s relative to the middle piston. Then the pistons connected to the middle piston start extending. Once again, the pistons will be only moving 1 m/s related to the pistons doing the pushing. This chain of successive extension proceeds until the last piston. Relativistic mass ceases to be a problem since there is never a piston trying to accelerate a piston moving faster than 1 m/s relative to its location.

This seems to make sense, but it might as well hold some flaw that I am simply unaware of. It does raise the question of what exactly is happening from the perspective of the middle piston. From its perspective, the second-last piston is pushing something the mass of which should be approaching infinity, so it should not succeed. However, from the second-last piston's perspective, the last piston isn't moving at all when it starts pushing, so it should be perfectly doable. Asking "what is the real truth" is pointless since relativity means there really isn't a single "real truth". Perhaps the push will indeed have no effect from the perspective of the middle piston. So, it fails to reach lightspeed in every perspective, if the push has no effect from the perspectives of those who would view it to move fast enough to achieve lightspeed with an increase of 1 m/s to its speed. But I should stress that I don't really know what I am talking about.

In any case, if in that scenario the relativistic mass isn't the problem, then what is? I haven't studied these things enough to tell, but Kenji's explanation about the speed of push seems reasonable. Some fascinating concept of relativity such as time dilation might also have something to say about the matter.
Kenji
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No, it doesn't work. Here's why:
∞ LINK ∞
Too long; didn't watch version: The speed of "push", that is, when you apply force to an object, how long it takes for the force to propagate through the object, is determined by the speed of sound in that material. It follows that the force will never be applied to the last piston in time (since the speed of sound is slower); the speed of light stays an upper limit for speed.
nighty23
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nighty23 27 United States SanguinePhlegmatic INTP 58C
Wouldn't the speed of sound only come into play if this was extending as a wave, one then the next then the next et cetera? Because the energy isn't supposed to be transferring through the system from one end to the other like a vibration, rather, all of the pistons are being activated at the same instant, i.e. the force only propagates through, effectively, 1 piston, relative each individual piston.
Kenji
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"Wouldn't the speed of sound only come into play if this was extending as a wave, one then the next then the next et cetera?"
It doesn't matter if the pistons are activated exactly at the same time or not. Pushing forces always cause longitudinal waves. If you want all those 1m/s speeds to add up (or half of them, or whatever), this pushing force has to propagate to the last piston all at the same time. But it cannot be for the reason I mentioned. You have to realize that pushing is just a wave and that wave travels very slowly (compared to the speed of light).
nighty23
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nighty23 27 United States SanguinePhlegmatic INTP 58C
Huh. Alright. Well this certainly gave me as much reading material as I would have wanted, and then some. Thanks!
Tama Yoshi
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Tama Yoshi 24 Canada PhlegmaticCholeric INTJ 513 472C
To complement with Kenji's link: ∞ YouTube ∞

What some people often forget, is that simple acceleration in empty space is enough to reach the speed of light. After all, since there is no friction in empty space, you will not slow down (in an optimal context).
With simple acceleration, you could experience the effect of time-space dilation (which Corelis mentionned partially with inertial mass). This would show by the fact that, given an acceleration of 10m/s at normal speeds, when you reach relativistic speeds - let's pick 0.866*c, which means a Lorentz factor of 2 - then it would take 2 seconds to gain an apparent 10m/s increase. And as you can guess, when this factor becomes huge, like infinite, you won't be accelerating even though you have a force accelerating you in a direction.

I may have scrapped the maths here, so don't take my word for it, but 0.866*c is definitely a 2 for the relativistic factor, so I'm guessing *something* has to take twice as much of *something else* to achieve the same.

Quantum mechanics is the true "speed of light" breaker, and it does it in mind-blowing ways. If you're really feeling nerdy, you can search about the double slit experiment, and further to the retrospective double slit experiment (the one where you decide to observe only after you get the information).
There is also this video on "spooky action at a distance": ∞ YouTube ∞
Onisuzume
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tl;dr

Some simple reasons:
1: Time involved using any method that doesn't encounter any of the following problems.
2: Exponentially increasing energy demands to reach sufficient acceleration. A problem even with consistant 1G acceleration.
3: Gravetic effects from the required acceleration pulping the crew. Some methods could be used to lessen or negate those effects, but there's likely still an upper limit to what could be done in that department. (Like with Event Horizon, the crew has to enter liquid tanks to avoid having their bodies pulped by the 30G acceleration that the ship is going to make. And the ship still takes some 50 or 60 days to travel between Neptune and Earth.)
4: Gravetic effects from the required acceleration being far in excess of the tensile strength of the materials involved (e.g. ship tearing itself apart). If we can't do ∞ space elevators ∞, there's no way we could possibly do near-light speeds without taking ages to reach it.