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Forum » SpaceEngine » Science and Astronomy Discussions » Science and Astronomy Questions
Science and Astronomy Questions
PlutonianEmpireDate: Friday, 14.10.2016, 19:55 | Message # 856
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Watsisname, so to get the partial pressure, I just multiply the percentages and ppm by the new atm pressure?




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WatsisnameDate: Friday, 14.10.2016, 20:53 | Message # 857
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Precisely. smile Partial pressure is the part of the total pressure caused by that particular gas. Usually it's the partial pressure that matters for things like toxicity or breathability.




 
PlutonianEmpireDate: Friday, 14.10.2016, 22:45 | Message # 858
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Thank you! smile




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HuesudoDate: Saturday, 15.10.2016, 11:50 | Message # 859
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Quote Watsisname ()
At several ppm concentration, continuously? I guarantee you don't wink SO2 gas becomes sulfuric acid on contact with water. That includes in your eyes and mucous membranes.

Hm... that doesn't sound so good surprised
 
WatsisnameDate: Saturday, 15.10.2016, 12:35 | Message # 860
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PlutoniamEmpire: Sure thing. smile

I was thinking back to the visual appearance of a faster-than-light particle, and decided to work out the appearance of its flyby a bit more rigorously. I wanted to gain some further insight to the problem, and there are a few interesting cases to explore. There's some math involved, so feel free to skip to the end if you just want to know the result.

To start, the time at which the observer receives photons from the particle, emitted from where the particle was at a position x, is



where 'b' is the 'impact parameter', or how much the particle missed the observer by, and β is the particle's speed as a fraction of the speed of light (β=v/c).

If we let b=1 (like one light second) and β=2 (twice the speed of light), then we get the following curve:



To interpret this, think of time (for the observer, in seconds) running vertically up the graph, and the apparent position of the particle is the horizontal axis (in light seconds). Notice the particle is never seen for times less than ~0.866 seconds. The particle first appears there at that time, at a position of -0.577 light seconds. In other words the particle appears just before its closest approach point. As time progresses forwards, the particle splits and is seen both in front of and behind that point. The steeper the curve, the more time passes for a given change in its apparent position. So a shallower slope represents a faster apparent speed. In this case, to the left (backwards) is faster.

Now it might be easier to visualize what's happening if I plotted its apparent position over time, but that turns out being messier. It must result in two functions (we see two particles), and isolating x requires solving a quadratic. Just to show it:



We could use that, but it's more of a hassle. Better to stick with time as a function of apparent position.

Let's think about the behavior of that function (the simpler one). For speeds much slower than light (β<<1), x/β will be huge and dominate over the radical. But x/β is a line, so the particle would have virtually constant apparent speed... as we expect for non-relativistic particles. The slope of this line is 1/β, and the inverse of its slope is the apparent speed, which is β. Makes perfect sense. For particles moving much slower than light, their apparent speeds are essentially their true speeds.

But what about as β increases -- the particle getting faster and faster, up to and beyond the speed of light? Can we find apparent speeds for any true speed? Yes we can. Let's use calculus. Differentiating time with respect to position (very backwards from what we normally do in calculus tongue ), we get



Its apparent speed, then, is the inverse of that:


That's the other reason why I began with t(x) instead of x(t)... differentiating x(t) is a lot less fun than differentiating t(x) and then inverting.

In the case where the particle moves right at or away from you (impact parameter b=0), this becomes super simple:



Minus signs for when the particle is approaching. And here's the graph to visualize it. This is apparent speed vs. true speed, now. Green for approach, yellow for receding.



Now we can figure out some interesting things very quickly. For instance, at what true approach speed is its apparent speed equal to the speed of light? Set v=1, and β ends up being 0.5. Half the speed of light. Its apparent speed after passing you is 1/3 of c.

As the speed goes up toward the speed of light, then its apparent approach speed goes to infinity, and its apparent recession speed rises to 1/2 of c.

Then for increasing speeds which are greater than the speed of light, the apparent approach speed decreases (but is backwards), and the apparent recession speed continues increasing. In the limit that the true speed goes to infinity, its apparent speed goes to the speed of light... in both directions simultaneously.

Neat-o!

[Of course, no particle or information can ever move faster than the speed of light -- that would violate causality for reasons of relativity. But we might instead imagine a long line of flash bulbs, preset to flash with a propagation speed arbitrarily faster than c. No information is traveling faster than c in this case, yet to an observer it looks like a light-emitting particle moving faster than light. In fact, thinking of it that way helps explain why the infinite-speed particle would seem to appear right in front of you and splits off in both directions at the speed of light. It's exactly like looking at an infinitely long chain of flash bulbs which all went off simultaneously. You see the flashes from successively more distant bulbs, with the flashes reaching you at the speed of light.]





 
FastFourierTransformDate: Sunday, 16.10.2016, 11:43 | Message # 861
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WOW, Watsisname, take your fake internet points for making such an effort in calculating all of this and making the plots. Very elegant aesthetics by the way smile

Quote Watsisname ()
[Of course, no particle or information can ever move faster than the speed of light -- that would violate causality for reasons of relativity. But we might instead imagine a long line of flash bulbs, preset to flash with a propagation speed arbitrarily faster than c. No information is traveling faster than c in this case, yet to an observer it looks like a light-emitting particle moving faster than light. In fact, thinking of it that way helps explain why the infinite-speed particle would seem to appear right in front of you and splits off in both directions at the speed of light. It's exactly like looking at an infinitely long chain of flash bulbs which all went off simultaneously. You see the flashes from successively more distant bulbs, with the flashes reaching you at the speed of light.]


My question is goig to sound a bit dumb but wouldn't some Cherenkov Radiation appear in this situation?
 
WatsisnameDate: Sunday, 16.10.2016, 15:46 | Message # 862
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Quote FastFourierTransform ()
Very elegant aesthetics by the way


Thanks. I've taken a liking to optimizing things for the black forum background. smile

Quote FastFourierTransform ()
My question is goig to sound a bit dumb but wouldn't some Cherenkov Radiation appear in this situation?


Not dumb at all!

For those unfamiliar, Cherenkov radiation is emitted by charged particles moving through a medium faster than the speed of light in that medium. A classic example is the blue glow in the water around nuclear reactors.

Supposedly, this would also apply to tachyons in the vacuum (the vacuum is a medium and it has the required properties, like an electric constant). However, this is a bit less than theorizing, since have never seen a tachyon, and according to relativity they can't exist. But we can pretend and assume our understanding of sub-light physics works the same for faster-than-light.

Right away we come up with some weird implications. For example, instead of slowing tachyons down like it does to normal particles, the Cherenkov radiation would speed the tachyons up! Tachyons of lower energy move faster, and a tachyon with zero energy moves infinitely fast.

Charge-less tachyons would not be free from this accelerating effect, either. While they wouldn't interact with light, they would instead produce a gravitational Cherenkov radiation, since they go through the vacuum faster than the speed of gravitational radiation. Even if they have no rest mass, they still produce a gravitational field due to their momentum (photons do this, too). So all tachyons would emit this radiation, which would accelerate them to infinite speeds.

Pretty crazy stuff!





 
DoctorOfSpaceDate: Sunday, 16.10.2016, 16:01 | Message # 863
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Watsisname, great post and very informative, about what I was looking for. Not that Destructor1701 wasn't helpful but that much I had down smile

Quote Watsisname ()
and according to relativity they can't exist.


Do you mean simply because nothing can go faster than light or have I missed something that specifically says tachyons can't exist?





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WatsisnameDate: Monday, 17.10.2016, 06:52 | Message # 864
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Short answer is that they violate causality. But I'll take some time to explain, as best as I can anyway, what that means and why that is. Much of this I'm sure you and others already know, but let's run through it anyway. This will be a 'Why is the speed of light the cosmic speed limit?' post. smile

Let's start from the statement "nothing can go faster than light" specifically. It is an assertion, and an observation, but where does it come from? What is it that makes it impossible to go faster than c?

The first thing we might try is a 'brute force' method of accelerating a particle to faster and faster speeds. Surely, if we just accelerate it enough, it would break the speed of light. But it doesn't. What we find is that it takes infinite energy to accelerate any massive particle to c. Massless particles move at (and only at) c.

So it is hopeless to get anything which was sub-light up to the speed of light, let alone past it (unless we suppose warping the space-time like with an Alcubierre drive). But what if something was already faster than light, and was always that way? There is no problem with passing the infinite energy requirements if it never had to be accelerated past c. That's where the idea of tachyons comes in.

But there is another problem. This one is causality, and it's a bit more subtle and more fundamental to the laws of the universe. To understand it, we'll need to look at some space-time diagrams. Here's one (shamelessly taken from lecture notes from University of Toronto).



Time runs vertically up the graph and has units ct (speed of light times time to give a distance). Position in space is the horizontal. An 'event' is a point in space-time. It says 'something happens here and now'. A world line shows the history of an object through space and time. If an object is not moving through space, then it makes a vertical line. If it does move through space, then its line becomes tilted -- more tilted the faster it moves. The speed of light is a line tilted 45° (we usually make it be that angle out of convenience, by choosing units where the speed of light is 1 unit of distance for 1 unit of time, like one light year per year). Things moving faster than light are then closer to horizontal than that 45°.

Finally, something moving backward in time traces a path downward rather than upward.

It is often said that tachyons move backward in time. But that's not quite right. For instance, a 60° angle to the vertical is faster than light, but still going forwards in time. So why would that violate causality?

The answer lies in the way which intervals of space and intervals of time transform from one frame of reference to another. Space and time are relative, and they are relative in a specific way which is described by the Lorentz transformation. This transformation ultimately arises because the speed of light must be the same for everyone. For the speed of light to be a constant, space and time must be relative in a way which preserves that constant. This is Einstein's great discovery.

Here is a bit goofy-looking animation showing how the Lorentz transformation works:



It shows the location of events (the small white dots) in space and time, according to an observer who changes their speed (they follow the curved world line). The big white dots along the world line represent equal intervals of proper time, which is the time the observer measures. There's some math behind this which we won't worry about here, but the basic idea is that the transformation is hyperbolic. The more important point is to visually understand what is happening:

Notice that the distance between any two points changes when the observer's speed changes. And even more than that, their sequence can change. Two events which were on the same horizontal line don't necessarily stay on that line. This is one of the startling implications of relativity: two events which are simultaneous in one frame are not simultaneous in other frames! And the relative order of two events can even be different, if they are separated more in space than in time. This means they are not causally related, since information can't propagate from one event to the other. In the space-time diagram, the future light cone starting from one event does not include the other event. The other event is "elsewhere" from it, which we also describe as being a "space-like" separation.

For an event which is contained in the other event's light cone (a "time-like" separation), there is no question about which event happens first. The two are in causal contact -- information from one event reaches the other event. All observers will agree on which event happened first, even if they disagree on how much time elapsed between them, or how far apart they are in space. And we see this in the transformation -- no event ever crosses that 45° line from another event.

This potential for flipping the sequence of events separated by space-like intervals is the heart of the problem of tachyons. The reason is because tachyons allow the transfer of information from one event to another event in elsewhere. Since the relative sequence of those two events depends on frame of reference, there will exist frames of reference where a faster-than-light particle would arrive at the destination at an earlier time. This leads to situations with logical paradox, like a signal returning to its origin before it was sent.

Suppose it's the future. Humans have colonized Mars. But a twisted Bond villain wants to undo it all. He has placed planet-busting bombs on both Earth and Mars, and will set them off simultaneously unless his demands are met. To make sure they go off at the same time, the bombs are detonated when they receive a radio signal, which he will send from a point equidistant from the two planets.

Such is his plan, anyway. The people of Earth and Mars know about the bombs, but not that he intends to detonate them simultaneously. So as soon as one explodes, a tachyon signal will be sent to the other planet to warn them to disable the bomb there.

Of course, this plan would fail. Both planets blow up simultaneously. There's no hope for saving either of them.

Or is there?

You are Bond. You are currently flying from Earth to the midway point to confront the villain. Too late -- the signal has been sent. It sweeps past you. What do you see?

Remember you see the same speed of light as everyone else. You can say you are staying still but it is the Earth flying away behind you and Mars coming towards you. And since you see the same speed of the detonation signal coming towards you as away from you, Mars is rushing into the signal, while Earth is moving away from it. Mars hits the signal and blows up first.

Simultaneity is relative.

That seems preposterous. But it's true and it's not a problem (for physics -- morality of destroying two planets and everyone on them notwithstanding). The two events (Earth and Mars blowing up) are not causally related to one another. They are more separated in space than in time. Instead, they are causally related to the event which is the signal sent from half-way in between. No observer will disagree that Mars and Earth blew up after the signal was sent.

Now let's throw in the tachyons.

You see Mars blow up first. The moment the bomb-detonating signal arrived there, the Martians frantically sent their faster-than-light signal to Earth to warn them to disable the bomb. In your frame of reference, this signal reaches Earth before the detonating signal does. Earth is saved.

Paradox. Are both planets destroyed, or is Earth saved? Or is Mars saved? You could have been flying from Mars to Earth, and you'd see Earth blow up first.

Relativity tells us that separation of events in space and in time is relative, but events themselves are physical. So we can't have a planet either destroyed or not destroyed depending on your reference. That's an event. It can't both happen and not happen. Furthermore, relativity says that there is no 'special' frame of reference. We must treat them all as being equally valid. So this paradox tells us that this situation cannot occur. Information cannot be sent faster than light, therefore there cannot be tachyons. If they exist, then they can't interact with anything, which is equivalent to not existing.





 
steeljaw354Date: Monday, 17.10.2016, 10:01 | Message # 865
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Watsisname, So speed of light isn't the limit?
 
WatsisnameDate: Monday, 17.10.2016, 13:22 | Message # 866
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No, I'm showing that it is. Unless wormholes or Alcubierre drives are possible (and they probably aren't), then this is what relativity tells us. You cannot accelerate something to faster than the speed of light, nor can there be any particle moving faster than the speed of light to begin with. If we supposed that such a thing could exist, it would lead to paradoxes.

This limit also has less to do with light, and more to do with causality. There is a limit to how fast information can travel between events, and it happens to be the speed of light. It's a property of space-time. smile





 
spacerDate: Monday, 17.10.2016, 16:01 | Message # 867
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Watsisname, how will you define teleportation? will teleportation work like worm hole?
smile





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DoctorOfSpaceDate: Monday, 17.10.2016, 16:33 | Message # 868
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Watsisname, even when it is a subject I am fairly well versed in your posts are always a treat to read.

I still do not see how this invalidates tachyons completely. What about the reinterpretation principle where by trying to detect a tachyon generates that tachyon, it is perceived as traveling forward in time, and observers can't distinguish between the emission and absorption?

Secondly isn't there still the possibility that tachyons are either minimally interacting or non interacting and completely causally disconnected? (This is getting into Russell's teapot scenario)

Regarding light cones, what about scenarios where no FTL is required for paths that are on tilted trajectories?



I would be interested to see your view on this page here
http://www.bibliotecapleyades.net/ciencia....l01.htm





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AerospacefagDate: Monday, 17.10.2016, 21:32 | Message # 869
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Quote DoctorOfSpace ()
I would be interested to see your view on this page here


Quote
In a curved space-time it is possible to "tilt" or "tip over" light cones. With a sufficient amount of space-time curvature a particle or person could continue to move into their own local future at sub-light speed but actually travel along a world line that loops back on itself as shown in the illustration.

Impossible, actually. The "cone" is not a fixed figure and is in fact a result of series of equations, interpreted in a certain way. From my knowledge of physics, I suppose the cone will change it's shape according to the difference of equations, but without having actual equations on hand, I would need to work on that to prove it. And it is pretty hard analytical problem at that.

Thus, the cone will never "tilt over", like an actual goddamn coffee cup and there's no violation of causality at all, as well as speed of light violation and reversed flow in time.

Quote DoctorOfSpace ()
Secondly isn't there still the possibility that tachyons are either minimally interacting or non interacting and completely causally disconnected? (This is getting into Russell's teapot scenario)

I have the faint idea that further study in the direction of "dark matter/energy" theory can prove that, but the problem with them is pretty much the same as with tachyons - they don't like to interact with normal matter too.
 
steeljaw354Date: Monday, 17.10.2016, 21:38 | Message # 870
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Watsisname, Maybe other civilizations found ways to circumvent the time dilations, let's say we have a ship that leaves it's home planet. It takes 1 year to go it's destination and 1 year passes at it's destination for example. But we haven't found those circumventions yet.
 
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