paradox I thought of (relating to relativity)

[`hýsøŕ]

Now there are still things I don't know about in relativity but I thought of this the other day (by that i mean many, many months ago) and I thought it might be post worthy xD

 

this will sound a little unusual but bear with me

 

So say you have a spacecraft with a group of incompetent, suicidal aliens inside, trying to develop a nuclear weapon to kill themselves with.

They're doing this by trying to squeeze a piece of nuclear 'fuel' so that it reaches a particular density where, if you fire a neutron into the fuel, you could get a successful nuclear fission reaction going, triggering the nuclear fuel and causing it to destroy the spacecraft and them inside it.

 

Now I'm hanging around on in space still relative to this spacecraft, which is moving relative to me at a constant velocity. Now because it's moving, the principle of length contraction comes into play, saying that from my perspective, the alien spacecraft is compressed in its direction of motion. So if I were to measure the density of the nuclear fuel that the aliens are using, I'd get a larger answer, because the length of the block of the fuel is reduced, smaller volume, so a higher fuel density.

 

So say that the aliens make several attempts to blow themselves up with the nuclear fuel, but because these alien scientists are somewhat incompetent, they fail consistently, because they haven't squeezed the fuel enough for a successful reaction to get going. However I'm watching from my safe vantage point in space and I see them moving past make making an attempt to 'ignite' the nuclear fuel. Because I see the fuel as being at a higher density, the reaction IS able to get going (the atoms in the fuel are just close enough for the neutrons to, on average, hit another uranium nucleus say, and continue the reaction), and the experiment works and the ship gets destroyed and they all die.

 

How is this possible if from their frame of reference, the fuel wasn't able to ignite because it wasn't compressed enough, and so they are still alive from their frame of reference?

[swansont]

If an event happens in one frame, it happens in all frames. (or the negation of that). So your motion will not cause the nuclear material to explode. Why? Because of relativity. The density/shape requirement for criticality that holds in the rest frame holds in that frame only. It will be different in your frame. As will any parameter that depends on time, length or energy or any other value that is not invariant.

[`hýsøŕ]

You mean that whatever the critical density is, it will depend on other things that change between reference frames, and not just the length contraction effect? If so I guess that'd make sense, since it'd be a crazy violation of .. loads of things .. if the ship both got destroyed and didn't, ofc. thanks

[Endercreeper01]

In relativity, anything that happens in one point of view applies in all points of view (reference frames).

The density requirements would have to increase in your reference frame in order for the same thing to happening both frames of reference.

[SamBridge]

Well he does have a point, density can be relativistic. I think the paradox can be resolved by the notion that the length contraction applies to all components of the nuclear device making everything scale down by the same proportion, even the neutron fired at the nuclear fuel. When this happens, the neutron will appear to travel less distance between the neutron gun and the fuel over your measured interval of time and thus have lesser relative kinetic energy that adds to the energy from the motion you observe as the ship heads towards you near the speed of light, so if the neutron doesn't have enough energy to do it on their end, it won't on your end either. The total amount of energy in the system is still conserved, so if its not enough energy, its not enough energy, and if it is, it is, and the only difference is how you measure it. Instead of measuring the neutron being fired from a standing igniter device, it already has a boost in its velocity from the entire ship traveling near light, so when you add addition velocity like when its fired from a gun, the units of distance it measures must be shorter in order not go past the speed of light, and so it will appear to strike the fuel storage will less energy even though its still traveling near the speed of light.

In contracted space, the proportion of energy it takes to ignite a specific density of fuel in proportion to the velocity of the neutron stays the same. Or in other words, If you put a ruler near a black hole or accelerate it to near the speed of light, that contracted ruler will still show that twelve of its own contracted inches make a total of one of its contracted feet, so something traveling 2 meters per second in its frame will still appear to you as traveling two of its own contracted meters per second, just not two of your meters per second.

 

Or at least that's a start, it probably isn't perfect, but something to work with.

Edited April 20, 2014 by SamBridge

[swansont]

Well he does have a point, density can be relativistic. I think the paradox can be resolved by the notion that the length contraction applies to all components of the nuclear device making everything scale down by the same proportion, even the neutron fired at the nuclear fuel. When this happens, the neutron will appear to travel less distance between the neutron gun and the fuel over your measured interval of time and thus have lesser relative kinetic energy that adds to the energy from the motion you observe as the ship heads towards you near the speed of light, so if the neutron doesn't have enough energy to do it on their end, it won't on your end either. The total amount of energy in the system is still conserved, so if its not enough energy, its not enough energy, and if it is, it is, and the only difference is how you measure it. Instead of measuring the neutron being fired from a standing igniter device, it already has a boost in its velocity from the entire ship traveling near light, so when you add addition velocity like when its fired from a gun, the units of distance it measures must be shorter in order not go past the speed of light, and so it will appear to strike the fuel storage will less energy even though its still traveling near the speed of light.

In contracted space, the proportion of energy it takes to ignite a specific density of fuel in proportion to the velocity of the neutron stays the same. Or in other words, If you put a ruler near a black hole or accelerate it to near the speed of light, that contracted ruler will still show that twelve of its own contracted inches make a total of one of its contracted feet, so something traveling 2 meters per second in its frame will still appear to you as traveling two of its own contracted meters per second, just not two of your meters per second.

 

Or at least that's a start, it probably isn't perfect, but something to work with.

 

Please show the equations that support the highlighted section.

[SamBridge]

 

Please show the equations that support the highlighted section.

Before I even attempt to do that, read what I said at the bottom. If it isn't perfect which means I was expecting an error and if you only have something wrong with that component, I'm assuming the rest of it is correct and that's what was wrong. So, considering the fact that I already admitted part of it isn't completely right, instead of merely complaining, why not just do you part to answer the question in the visual way they he actually presented the situation by building off of what I already said and easily, more efficient knock away all the things you're suppose to do by pointing out what's wrong then pointing out how its wrong in relation to the OP that explains how the scenario visually works?

 

But anyway, here's the math.

Let's assume it takes x joules of energy in a neutron from an inertial frame to ignite the fuel at critical density, or from the frame of the aliens and the frame of the outside observer if they observed the ship standing still. From the frame of the aliens, they will observe the neutron striking the nuclear fuel at whatever the speed they observe if leaving the neutron igniter gun which is probably 20,000 k/s, and the neutron's mass is 1.6749286*10^-27 kg, so E=(m)(v^2) gives us a total of 6.68*10^(-19) joules. Now, let's say x = 7*10^(-19) joules. From the aliens frame, the neutron barely doesn't have enough energy to ignite the fuel. So with what everyone's saying which is "it just happens that events are measured the same order if they are causally connected", it must not have enough energy from the other frame, or from hysors outside frame that observes the ship traveling at near the speed of light or we'd have that crazy duality violation.

 

So, in order to keep hysor's frame from observing the near-luminal ship from exploding at critical density, here's what happens: Let's say the ships is moving at 80% the speed of light (239833.96 km/s) and the ship is 10 meters long, so the ship will contract to 6 meters giving us the ratio of 6/10, so we know the distance between the neutron gun and the fuel will shrink by the same proportion as well, a ratio of 6/10 making the fuel that much denser if we treat it as a cubical object and it takes less energy, say y joules, enough to create what would be critical density from the aliens frame. From the frame of reference of the nuclear fuel storage, the neutron will strike it more slowly, at a velocity of the difference between the neutron's added and dilated/contracted velocity and the velocity an outside frame observes the ship to be moving because of the fact the ship and the neutron gun is already attempting to move away from the neutron because the neutron is no longer attached to the ship when it leaves the gun, relativistic-ally of course, not just by adding two regular vectors, but using the formula

[math]w = (u+v)/(1+(uv)/(c^2))[/math] W is the velocity hysor will measure that we'll use to get the difference, u is the speed of the ship that already adds to the neutron's initial velocity and v is the extra speed the neutron gained from being fired from the gun, which will turn out to be [ 239833966m/s + 20000000m/s ] divided by something that looks awfully close to length contraction but without the square root which is (1+ (239833966*20000000)/(299792458^2)) = 1.053 so we divide (23983396 + 20000000) by (1.053) to get 24666169.5 which makes sense considering 20000000 isn't that fast compared to the speed of light. So that's the velocity hysor observes the neutron. But now, we use the difference in those speeds to figure out the speed that hysor and the fuel storage agree on the relative velocity that the neutron strikes the fuel storage which is the same as the relative velocity that the neutron gun measures the neutron heading away from it which from the gun's frame is 20,000km/s (200000000m/s)...which turns out to be 239833966m/s and I spent a long time typing this so I'm taking a break now and ill come back to it later since this was the last thing I was working on.

 

 

The neutron from hysor's frame will still move 80%c + contracted 20,000km/s, but but hysor and the fuel storage will agree that the neutron still only strikes the fuel storage with the relative contracted difference and not the total 80%c + contracted 20,000km/s as per the fact that the fuel storage is confirmed to be moving away from the neutron in some amount in hysor's frame.

 

 

It's a shorter distance to travel, so the neutron will have more relative energy from the frame of reference of the fuel storage right? I know you know that's wrong, the ship is moving, and if the neutron gun is at the back of the ship pointing forward (let's assume it is) and the nuclear fuel is in the front, from the neutron's frame and from hysors frame, hysor and the fuel storage will observe the neutron striking the fuel at what must be a slow enough speed with a low enough relative energy that the 10/6 times greater density will still not be ignited by the neutron traveling the 6/10 shrunk distance in the same metric time between the gun and the fuel with the increased velocity from the gun because the neutron will appear to travel slower than it should have from hysor's frame but still travel the same proportion of contracted units per second in a moving ship as non-contracted units per second in a stationary ship. It's just what I said with a ruler. If I take a ruler in a stationary frame and role a marble by it that says the marble travels 2 inches per second my second, then it moves at 80% the speed of light (ignoring acceleration that would cause the marble to fall to the back of the ship) and someone says they role the marble at 2 inches per second, I'm still going to see that contracted ruler measure the marble as traveling two of its own contracted inches per second even if its not traveling two of my inches per second, so I can still say from my own frame that the marble has the same relative energy from the frame of the person who rolled it before and after traveling near the speed of light if I take into account the length contraction.

 

However it works out, it has to be something like that, because otherwise if we don't consider the length contraction from a visual standpoint, there's no reason to say the same event must happen in both frames which allow the neutron to travel slower than it should even with the seemingly increased energy from the neutron gun and thus still not have enough energy from velocity to ignite the fuel from hysor's frame and the fuel storage's frame outside of "that's just how its causally connected" which doesn't explain anything.

Edited April 20, 2014 by SamBridge

[`hýsøŕ]

I am reading these but will wait for the final conclusion you guys reach before I make any concrete replies or questions, if you don't mind hehe

[SamBridge]

I am reading these but will wait for the final conclusion you guys reach before I make any concrete replies or questions, if you don't mind hehe

Well I only explained it that way because Swan asked for it. But basically, the same event happens because the relativistic effect affects all components of the situation, not just the nuclear fuel.

Edited April 21, 2014 by SamBridge

[swansont]

 

But anyway, here's the math.

 

 

I asked for the equations, i.e. a general case.

[SamBridge]

 

I asked for the equations, i.e. a general case.

ok?

w=(u±v)/(1±(u*v/c^2), that's one relevant equation,

L = Lo * sqrt( 1 - v^2/c^2), that's another

t = to / sqrt( 1 - v^2/c^2), yet another

I don't know what you want me to do with them if you didn't like my first attempt.

 

The volume of the fuel storage is scaled down by a factor of 10/6, but so is everything else, at 80% c anyway. So if in the alien's frame the neutron doesn't have enough energy to ignite the fuel at it's current density, from hysor's frame it won't either because the neutron will appear to be traveling slower towards the fuel storage by the amount necessary to keep the relative difference in energy between the neutron's energy and the energy required to ignite the fuel lower in a similar relative difference due to length contraction and time dilation and the fact that the storage is trying to move away from the neutron, otherwise it should ignite the fuel. So the relative energy required to ignite the fuel decreases due to the higher density, but the neutron still travels slow enough towards the fuel storage from hysor's frame to not ignite it.

If I put rulers next to everything, hysor would see the neutron still traveling the same amount of contracted units per contracted second as the aliens said it traveled in normal units from their frame, but hysor would say they are smaller units compared to his own and thus the neutron would appear to travel slower than it "should" from hysor's frame, and it happens more so when you add time dilation and say the neutron counts units of time slower compared to hysor. The "proportionality" I'm talking about is the length contraction/dilation. As I said, if I put a ruler near a black hole, it will still say 12 of its own inches equals one of its own feet even though the entire ruler itself will be smaller than mine, and you could do a similar thing with time itself if you treat time as just another plain dimension in which its metric contracts, which Einstein did like to do and is also probably why the time dilation equation is so similar to the length contraction equation. In fact, I'm pretty sure there's a way to measure time in meters because of that phenomena.

 

If someone else has a better explanation I'm all for it, but I don't think the answer "that's just how it is, you can't see it any other way" is good enough so that's why I bothered posting.

Edited April 21, 2014 by SamBridge

[swansont]

ok?

w=(u±v)/(1±(u*v/c^2), that's one relevant equation,

L = Lo * sqrt( 1 - v^2/c^2), that's another

t = to / sqrt( 1 - v^2/c^2), yet another

I don't know what you want me to do with them if you didn't like my first attempt.

 

 

 

"the proportion of energy it takes to ignite a specific density of fuel in proportion to the velocity of the neutron stays the same"

 

To me that sounds like speed and (kinetic energy)/density transform the same way in all frames. This doesn't work classically, so it's not apparent to me that relativistically it should magically work. Classically, if I look in a frame where speed has doubled, the KE will be up by a factor of 4, and density will be unchanged.

 

If your statement means something else, show what parameters you think stay the same.

[SamBridge]

 

 

 

To me that sounds like speed and (kinetic energy)/density transform the same way in all frames. This doesn't work classically, so it's not apparent to me that relativistically it should magically work. Classically, if I look in a frame where speed has doubled, the KE will be up by a factor of 4, and density will be unchanged.

Well classical non-relativistic mechanics doesn't work like that. Increasing one's speed to an outside frame while near the speed of light takes a lot more energy than if you're going at 2 miles per hour. And, the density may not be changed from the alien's frame, but it can still "appear" dense from someone like hysor's frame. You're semi-right about the density being unchanged, as I keep saying, the same proportion of length contraction applies to everything on the ship, and that means that even though the distance between atoms of the fuel storage from hysor's frame appears closer, if a rulers put up to the fuel storage then hysor would still see that the distance between atoms that creates the density counts the same number of its own contracted units as the aliens do with their normal units that they view.

[swansont]

Well classical non-relativistic mechanics doesn't work like that. Increasing one's speed to an outside frame while near the speed of light takes a lot more energy than if you're going at 2 miles per hour.

 

So why would the proportions stay the same?

[SamBridge]

 

So why would the proportions stay the same?

I don't necessarily think they'd stay exactly the same in terms of (a-b)/a = (c-d)/c, I just think the new difference would be proportional to the seemingly reduced energy it took from the seemingly increased density minus the slower speed of the neutron relative to the fuel, they both decrease. I don't know by how much exactly because I don't have an equation to relate density to the energy required to ignite the fuel, but I'd still assume both the aliens and hysor would say "it barely wasn't enough" due to the previous factors I mentioned, and there should be a function that can relate the ratio if we ever find that missing equation. Everything in the ship contracts by the same length factor, but the neutron also appears to travel slower in relation to the fuel, so that's the best answer I can come up with without more information. Maybe it isn't (a-b)/a -> (c-d)/c, it could be (a-b)/a -> (c-d)^2/c^2 or something, but it will be directly related to the factors I mentioned and it will have to be true in some way in order for the fuel to not show two different outcomes. If it wasn't that way and (c-d)/c > (a-b)/a, it could easily imply c was greater than d and thus the neutron did have enough energy from outside frames. If it takes less energy to ignite the fuel, the neutron better not be traveling with a fast velocity that would transfer that much energy or more.

Edited April 21, 2014 by SamBridge

[SamBridge]

Alright, we have a better answer thanks to bickering in some other thread.

 

If we assume the aliens never saw the nuclear fuel ignite, then from hysor's frame even though the nuclear storage does gain density as it contracts and atoms appear to move closer together, the physical dimensions of the atoms themselves also shrink by the same proportion from the length contraction, therefore they will never come into a proportionally closer contact that allows pieces of nuclei to travel with enough relative energy and with enough probability of hitting another nucleus to allow the reaction to go critical.

To boil it down, throwing a beach ball in a room can have the same probability of ending up somewhere in its domain as throwing a marble in a small box, so in this case the probability of a smaller neutron/proton hitting a smaller nucleus in a smaller volume yields the same or at least in some way similar probability as a big neutron/proton hitting a big nucleus in a big volume, thus not allowing the reaction to go critical no matter how great the contraction is.

Edited April 23, 2014 by SamBridge

[`hýsøŕ]

I guess that makes sense, thanks a lot hehe

 

So say if it wasn't nuclear fuel at all, if it was simply some experiment which gives an outcome depending on the density (which the nuclear fuel thing as an example of) would there always be some compensating effect that would prevent a sort of, 'fracturing of reality' where one observer sees an outcome because of his reference frame and another observer sees some other outcome because they're in a different reference frame? Or is this a special case, due to the geometry of the nuclear fuel and the atoms changing enough to account for the probabilities of collisions and things staying the same?

[swansont]

The nuclear fuel example isn't just a matter of density, though, it's also a matter of geometry. You can have a cube that goes critical whereas the same mass and density reshaped into a rectangular slab will not.

 

The underlying point is that these parameters are frame-dependent. Absent a rigorous demonstration that the various effects cancel/compensate, one should not simply assume that they do.

[SamBridge]

The nuclear fuel example isn't just a matter of density, though, it's also a matter of geometry. You can have a cube that goes critical whereas the same mass and density reshaped into a rectangular slab will not.

 

The underlying point is that these parameters are frame-dependent. Absent a rigorous demonstration that the various effects cancel/compensate, one should not simply assume that they do.

Well relativity still has classical components. If I put 8 1*1*1 cubes inside a 10*10*10 cube, then I shrink it such that I have 8 1*1*.5 cubes in a 10*10*5 cube, the first ratio of volumes is 8u^3;1000u^3, the second ratio of volumes is 4u^3:500u^3, the same exact proportion. Both 8/1000 and 4/500 yield the same number, thus the probability of finding a certain smaller cube within the respective larger cube stays the same. And, a reaction going critical doesn't necessarily require an increased density, just a high enough probability of nucleons running into various nuclei which coincidentally can be achieved with physical force compression but not length contraction.

Edited April 23, 2014 by SamBridge

[swansont]

Well relativity still has classical components. If I put 8 1*1*1 cubes inside a 10*10*10 cube, then I shrink it such that I have 8 1*1*.5 cubes in a 10*10*5 cube, the first ratio of volumes is 8u^3;1000u^3, the second ratio of volumes is 4u^3:500u^3, the same exact proportion. Both 8/1000 and 4/500 yield the same number, thus the probability of finding a certain smaller cube within the respective larger cube stays the same. And, a reaction going critical doesn't necessarily require an increased density, just a high enough probability of nucleons running into various nuclei which coincidentally can be achieved with physical force compression but not length contraction.

 

The volume fraction is not the issue, so its constance is irrelevant. In a given frame, where the 1x1x1 cubes would represent a critical geometry, the same configuration of 1x1x.05 cubes would not be. You can't apply the same criteria from different frames.

[SamBridge]

 

The volume fraction is not the issue, so its constance is irrelevant. In a given frame, where the 1x1x1 cubes would represent a critical geometry, the same configuration of 1x1x.05 cubes would not be. You can't apply the same criteria from different frames.

Did you actually read the explanation? It's not critical geometry because the probability stays the same due to constant geometric proportion, that's the reason why the configuration of 1x1x.5 isn't critical, I'm agreeing with you that the geometry is why it's not critical.

[swansont]

Did you actually read the explanation? It's not critical geometry because the probability stays the same due to constant geometric proportion, that's the reason why the configuration of 1x1x.5 isn't critical, I'm agreeing with you that the geometry is why it's not critical.

 

But if it's critical in one frame it must be critical in the other. The point is the criteria are relative.

[SamBridge]

 

But if it's critical in one frame it must be critical in the other. The point is the criteria are relative.

Yeah, that's what 8 other people already said, and I merely provided an explanation for how it physically happens to be the case beyond "that's just how it is."

Edited April 23, 2014 by SamBridge

[swansont]

Yeah, that's what 8 other people already said, and I merely provided an explanation for how it physically happens to be the case beyond "that's just how it is."

Your explanation contradicts the conclusion, which is why this keeps getting pointed out.

[SamBridge]

Your explanation contradicts the conclusion, which is why this keeps getting pointed out.

No it doesn't because my conclusion is that the fuel didn't ignite in the aliens frame and it didn't in hysor's frame either, which many people agree is would happen equally as much as if it DID ignite in one frame THEN it would ignite in the other. The difference is I assumed in one frame the fuel didn't ignite in one frame and drew a conclusion to show how it might not physically happen in the other, no contradiction.

Edited April 24, 2014 by SamBridge