One would think that an experiment that confirms a prediction made by a scientific theory is extremely interesting and useful.
I just dropped some toast on the floor. That confirms a prediction of gravity. When do I get my PhD?
I was, rather obviously, talking about confirming a prediction that hasn't been confirmed experimentally before.
That's a nebulous concept. What counts as a prediction which hasn't been confirmed experimentally before?
I just dropped a piece of toast weighing exactly 82.569321 g on the floor. It hit the ground. That confirms a prediction of gravity which has never been confirmed before.
Of course, nobody would expect that a slight variation in toast mass would change the results significantly. The point is that although the triple-slit experiment had not been confirmed before, enough other, related experiments had not shown any anomalies to suggest very strongly that we already knew how it was going to end up.
For the triple-slit experiment to produce anything other than the result it did, there would have to be a new kind of radio wave interference operating on a three-way basis independent from a superposition of two-way interferences. In other words, interference would have to result from something other than a linear superposition of component waves. But radio transmitters already operate using exactly this assumption, with transmitters in numbers of far greater than three, and we haven't run into any problems yet of three-way or higher-way nonlinear interference effects. I would therefore say that the results of this experiment have been indirectly confirmed already. But as I said, it's a nebulous concept.
The kind of experimental observation of a theoretical prediction which is interesting is one which hasn't previously been observed in any form - such as the finding of antimatter after Dirac predicted it, or the confirmation of special relativity by meson atmospheric penetration. These experiments demonstrated that a whole new section of a theory was viable. The triple-slit experiment has far more in common with my toast experiment than with these great experiments.
I was, rather obviously, talking about confirming a prediction that hasn't been confirmed experimentally before.
That's a nebulous concept. What counts as a prediction which hasn't been confirmed experimentally before?
I just dropped a piece of toast weighing exactly 82.569321 g on the floor. It hit the ground. That confirms a prediction of gravity which has never been confirmed before.
Even you ought to acknowledge that there's a very substantial difference between the two-slit and the three-slit experiments, compared to dropping different objects whose masses differ in some micrograms. The two-slit experiment has been performed to death in numerous different variations (eg. trying to measure which slit the particle goes through, the quantum eraser variation, the delayed choice quantum eraser variation, and so on, all of which give interesting results), but doing it with three slits to confirm a theoretical prediction is a significant step.
I know next to nothing about quantum mechanics, but I find it curious and interesting why the particle behaves like it passed through two of the slits even if there are more available. As I mentioned earlier, there's this interpretation of quantum mechanics where the particle takes "all possible" paths from the emitter to the receiver, and that's the reason for the interference pattern. However, to my layman mind, that seems contradictory with the concept that the particle behaves like going only through two slits regardless of how many there are, which would indicate that the particle does not take all possible paths. It's more like the particle takes two paths through two random slits. I'm wondering if I could understand why particles behave like that.
Even you ought to acknowledge that there's a very substantial difference between the two-slit and the three-slit experiments, compared to dropping different objects whose masses differ in some micrograms.
Actually, I think this is the crux of our disagreement. I don't think there's a substantial difference, given everything else we know about electromagnetic radiation. In particular, none of the observations from our use of multiple radio transmitter broadcast has resulted in anomalous results.
I know next to nothing about quantum mechanics, but I find it curious and interesting why the particle behaves like it passed through two of the slits even if there are more available.
I don't think this is an accurate description of the situation, from what I can see. The particle acts like it went through all three slits. The resultant interference pattern is a linear combination of the single-slit and double-slit experiments for each slit and pair of slits in the triple-slit experiment. This isn't a surprising result.
EDIT: the abstract of the paper on arxiv says "Born's rule predicts that quantum interference, as shown by a double slit diffraction experiment, occurs from pairs of paths." - which I think is basically saying that interference patterns occur as linear combinations of individual waves. In effect, it's saying that A+B+C == (A+B)+C == A+(B+C) - ie a three-way interference is equivalent to a two-way interference plus an extra wave, and doesn't behave in any "special" way. It's *not* saying that the particle only goes through two slits.
EDIT2: wait, my physics teacher in secondary school did an analogous experiment - in the late 1990s, I might add. She shot a laser into a diffraction grating which is basically a multiple slit experiment. I expect physics teachers everywhere are showing their kids this supposedly amazing result.
I still have hard time understanding or accepting that notion. Imagine General Relativity predicting light bending when passing close to massive objects or the perihelion of the orbit of mercury precessing (in a non-newtonian way) or time changes caused by gravity wells, and nobody bothering to actually check if those are true by actual measurements in 80+ years because it would just confirm the prediction and hence it would be extremely boring and uninteresting. I just don't buy that.
But this is not an experiment where a null result was expected -- there were arguments that we should observe light deflection even in Newtonian theory. Sure a null result was a possible result, but there were other results which had to be ruled out even if GR was not being tested.
Warp wrote:
I know next to nothing about quantum mechanics, but I find it curious and interesting why the particle behaves like it passed through two of the slits even if there are more available.
Ah, but it is nothing of the sort; this is just Huygen's principle in the context of quantum mechanics -- each slit acts as if it were a particle source, and the waves (probability amplitudes) are linearly superposed. The interference effects are seen only when you calculate probabilities from the (linearly superposed) amplitudes, and the squared-norm relation between amplitudes and probabilities naturally splits out the slits pairwise. This is perfectly consistent with Feynman's path integral interpretation.
Moreover, and like the quantum eraser, this it is not even a quantum thing -- a macroscopic beam of light going through a 3-slit experiment will give the exact same thing because the electromagnetic fields are linearly superposed after the slits and the field intensity is proportional to the squared-norm of the electric field.
There's something that I don't quite understand about General Relativity, so I was hoping if someone could clarify.
If I understand correctly, GR does not forbid two points in space (and hence eg. two particles at those points) receding from each other faster than c (caused, for example, by the metric expansion of space). Effectively what this causes is a horizon between these two points: One point cannot see or in any other way transmit information to the other point.
Point A will thus have its own observable universe, which would be the part of the universe which is receding from A slower than c, and point B will be outside this. Likewise point B will have its own observable universe and A will be outside of it.
So my question could be summarized as: Can these two observable universes overlap?
Either of the two possible answers is baffling.
If the answer is "yes", then it would introduce a contradiction: A could send a probe at sublight speed to the overlapping part, which is perfectly possible because it's inside A's observable universe, but since it is also inside B's observable universe (after all, the observable universes are overlapping here), the probe could now go to B at sublight speed. Hence you just transmitted a probe at sublight speed from A to B, which is a contradiction to the original premise.
If the answer is "no" then it introduces a different problematic situation: Assume that B is just barely outside A's observable universe (in other words, A and B are receding from each other just slightly faster than c). If B's observable universe cannot overlap with A's, it would mean that B's observable universe would have to be very small to not to "touch" A's. However, from B's point of view it would have to be the other way around (B has a "normal" observable universe and A has a tiny one).
Obviously there must a third option I'm not seeing here. (I bet it's a similar thing to the classic "if you are in a train traveling at 0.9c and walk at 0.2c from the rear to the front of the train, you just exceeded c" conundrum. However, I don't know what the proper explanation is in this case.)
So my question could be summarized as: Can these two observable universes overlap?
Yes. For any event horizon (apparent or not), you can separate the universe into the following (possibly empty) sets:
Observable universe: that portion of the universe that can influence the observer in some way (e.g., light). This is the causal past of the observer.
Interaction universe: that portion of the universe that can be influenced by the observer in some way (e.g., light). This is the causal future of the observer.
Post-horizon universe: that portion of the universe that can neither influence nor be influenced by the observer. This is the (set theoretical) complement of the (set theoretical) union of the causal past and the causal future of the observer.
(except for the observable universe, these names are not "standard")
Both observers can see the light coming from (the same) distant stars located in directions orthogonal to their relative motion, so their observable universes overlap. Their interaction universes will also overlap in a similar manner, but in the future instead of in the past; but the two observers' will be outside each other's observable and interaction universes. The observable and interaction universes may overlap in some regions and they may be disjoint in others.
That explanation is enough to answer the conundrum you presented; I can explain it later if you can't work it out.
Joined: 11/30/2008
Posts: 650
Location: a little city in the middle of nowhere
I think put simply, if A and B are travelling faster than the speed of light away from each other, then it would be impossible to send a probe or a signal from A to B, because by the time the signal reached a point where the two observable universes intersected, then it wouldn't be part of both A nd B's observable universe anymore. This of course you would need to observe from a third reference frame, where both A and B are observable.
Both observers can see the light coming from (the same) distant stars located in directions orthogonal to their relative motion, so their observable universes overlap. Their interaction universes will also overlap in a similar manner, but in the future instead of in the past; but the two observers' will be outside each other's observable and interaction universes. The observable and interaction universes may overlap in some regions and they may be disjoint in others.
I didn't quite understand the explanation. If a star is observable from both A and B, couldn't an observer at the star transmit information from A to B (which ought to be impossible)? The star can observe light coming from A and rely the information it gets to B.
andymac wrote:
I think put simply, if A and B are travelling faster than the speed of light away from each other
I think it's technically not correct to say that "A and B are traveling faster than the speed of light" because that's a violation of special relativity (and hence also general relativity). Rather "the distance between A and B grows faster than the speed of light". At a quick glance it might sound like the same thing, but it's not, which is why I like to make the distinction as it is, AFAIK, relevant. The latter is caused by a change in the geometry of space (which is allowed by GR).
then it would be impossible to send a probe or a signal from A to B, because by the time the signal reached a point where the two observable universes intersected, then it wouldn't be part of both A nd B's observable universe anymore.
Both A and B can observe the same star. This would mean that the star could eg. tell B about the existence of A (because the star can observe both) eg. by sending a photograph of it, which seems to violate the principle that it's impossible for A and B to know about each other. This is what confuses me. Obviously something else is happening here.
Joined: 11/30/2008
Posts: 650
Location: a little city in the middle of nowhere
You seem to have forgotten that there are certain conditions required in order for two objects to recede faster than the speed of light. You just say that two objects ARE receding faster than the speed of light, and then forget to mention how, and exactly what occurs during this time. I think you are making the false assumption that you are in this scenario: you are at a referance point D, which isn't accelerating. You see point B travelling at 0.75c to the left and you see point A travelling at 0.75c to the right. Of course they will be able to communicate with each other, B sends a laser to point A at D, it will eventually reach A because A is only traveling 0.75c. From B's perspective, A is travelling slower than c, because of time dilation. However, in reality, D is accelerating, so different things occur. For example, the displacement between two objects can't increase faster than the speed of light forever for example (even if you are accelerating at 10^-50ms-2 and the object is a googolplex km away and travelling at 0.99999999999c directly away). Take these sorts of things into account, and you will find that there are no contradictions. I haven't really explained anything, because I can't visualise such a situation accurately, however, I know the these apparent contradictions come from the fact that you are no longer in an inertial reference frame.
Also, I admit, my previous post was wrong in so many differnt ways. Probably so is this one, but I'll give it a try.
Both observers can see the light coming from (the same) distant stars located in directions orthogonal to their relative motion, so their observable universes overlap. Their interaction universes will also overlap in a similar manner, but in the future instead of in the past; but the two observers' will be outside each other's observable and interaction universes. The observable and interaction universes may overlap in some regions and they may be disjoint in others.
I didn't quite understand the explanation. If a star is observable from both A and B, couldn't an observer at the star transmit information from A to B (which ought to be impossible)? The star can observe light coming from A and rely the information it gets to B.
A point in the observable universe is a point in space-time, not simply a point in space. Since as marzojr said, the observable universe of A is defined as being every point which could have affected A's current state, then a point in A's universe is a point in A's past. As a result, A cannot transmit something to the star in A's observable universe, because this would involve sending the signal backwards in time!
A' B'
\uAB /
\ /
uA \/ uB
/\
/ \
/ u \
A B
In the diagram above, A is A at some point in the past. B is B at some point in the past. A' is A some time later; B' is B some time later. uA is the universe that A can affect. uB is the universe that B can affect. uAB is the overlap. u is the universe that neither can affect.
A can affect B' but not B.
The same diagram can demonstrate observable universes: uA and u are the observable universe of A'; uB and u are the observable universe of B'.
rhebus is absolutely right: your observable universe is in your past, and you cannot send anything into it without some for of time travel. Your observable universe is constantly expanding as time passes because there will be a bigger region of space-time which could have affected you, just as your interaction universe is constantly shrinking (even though it remains the same size -- infinite -- because of the universe's accelerated expansion).
As we all know, in most RPGs and some other game types with a world map, the world depicted by the map is usually torus-shaped. (In other words, if you go over the left edge, you will appear at the same altitude on the right edge and, more relevantly, if you go over the upper edge, you will appear on the lowe edge. This is only possible if the map represents the surface of a torus.)
This got me thinking: If by some magic a planet was indeed torus-shaped, how would gravity work on its surface? More specifically, where would gravity point to at different points on the surface of this planet?
Consider this cross-section of a torus:
What would be the direction of gravity at points A and B, for example? (If this direction depends on the major and minor radii of the torus, take them into account in the answer.)
Joined: 11/22/2004
Posts: 1468
Location: Rotterdam, The Netherlands
Warp wrote:
This got me thinking: If by some magic a planet was indeed torus-shaped, how would gravity work on its surface? More specifically, where would gravity point to at different points on the surface of this planet?
Consider this cross-section of a torus:
What would be the direction of gravity at points A and B, for example? (If this direction depends on the major and minor radii of the torus, take them into account in the answer.)
I am not capable by far to give a proper answer to this question but it would appear to me that gravity should be less strong in the "inside" of the torus (point A). On the "outside" of the torus (let's imagine a point C existing on 270 degrees of the circle where point B is at 180 degrees; to the left, that is), gravity points exclusively inwards, whereas at point A gravity would be affected by the portion of the torus that is "above" A.
It also depends on how large the planet is. The smaller the torus, the stronger the relative difference between the inside and the outside of the torus would be. An extremely large and thin torus would behave more like a rectangle, whereas you would be nearly weightless in the inside of a small and thick torus.
At least, that's how I envision it, and I don't know if any of it is correct. I'm pretty sure there was something interesting going on outside when the teacher was explaining about gravity.
It also depends on how large the planet is. The smaller the torus, the stronger the relative difference between the inside and the outside of the torus would be. An extremely large and thin torus would behave more like a rectangle, whereas you would be nearly weightless in the inside of a small and thick torus.
Clearly at point A gravity either points directly inwards of outwards, but which and how much, I don't dare to even guess. In fact, without doing the actual math (I believe it probably requires a volume integral) I wouldn't even dare to guess if the direction depends on the major/minor radii of the torus.
These things sometimes work in surprising ways. For example, one could easily think that the gravity outside of a sphere (ie. at some distance x from the center of the sphere) might depend on the radius of the sphere. After all, if the radius is larger (and hence the surface of the sphere closer to the point where we are measuring) but with the same mass, the mass of the sphere will be more spread to the sides, spreading the pull of gravity to a larger area. Likewise if the sphere is smaller (but with the same mass), the mass is concentrated on a smaller space and hence the gravity (when measured from distance x) is likewise more concentrated on one direction.
However, when one does the math, it results that the radius of the sphere has no effect on the strength of gravity as measured from distance x (as long as the radius is smaller or equal to x). It doesn't matter if the radius is x, x/1000, or even 0, if the mass of the sphere is the same, the force of gravity as measured from x will be identical.
Of course with the torus, and the measurement point being at A, things may be different, but nothing would surprise me.
The direction of gravity at point B is even harder to imagine.
Joined: 11/22/2004
Posts: 1468
Location: Rotterdam, The Netherlands
Warp wrote:
Dada wrote:
It also depends on how large the planet is. The smaller the torus, the stronger the relative difference between the inside and the outside of the torus would be. An extremely large and thin torus would behave more like a rectangle, whereas you would be nearly weightless in the inside of a small and thick torus.
These things sometimes work in surprising ways. For example, one could easily think that the gravity outside of a sphere (ie. at some distance x from the center of the sphere) might depend on the radius of the sphere. After all, if the radius is larger (and hence the surface of the sphere closer to the point where we are measuring) but with the same mass, the mass of the sphere will be more spread to the sides, spreading the pull of gravity to a larger area. Likewise if the sphere is smaller (but with the same mass), the mass is concentrated on a smaller space and hence the gravity (when measured from distance x) is likewise more concentrated on one direction.
However, when one does the math, it results that the radius of the sphere has no effect on the strength of gravity as measured from distance x (as long as the radius is smaller or equal to x). It doesn't matter if the radius is x, x/1000, or even 0, if the mass of the sphere is the same, the force of gravity as measured from x will be identical.
That's true, but when we stand atop the surface of a sphere, our distance to the center depends on the radius. That's (partly) why there are minuscule gravity differences at various altitudes on Earth. If the Earth were a lot smaller, but had the same mass as it has now, gravity wouldn't be stronger but it would affect us much more because of how much closer to the center we would be. But you are right that the net force itself would be the same.
This is why I mentioned the difference between a small and thick torus and a large and thin torus. A person at point A should be able to jump a lot higher than a person at the point C I mentioned earlier if the torus is small and thick because he is relatively quite close to the other side of the torus, which has its own gravity pointing towards its surface. In the case of a large and thin torus, this effect wouldn't be as strong because one is further away from the other side.
That's true, but when we stand atop the surface of a sphere, our distance to the center depends on the radius. That's (partly) why there are minuscule gravity differences at various altitudes on Earth. If the Earth were a lot smaller, but had the same mass as it has now, gravity wouldn't be stronger but it would affect us much more because of how much closer to the center we would be. But you are right that the net force itself would be the same.
Yes, but I was talking about the direction of the gravity at A (rather than its strength). I don't know if it depends on the dimensions (and/or mass) of the torus, and I don't dare even guess.
This is why I mentioned the difference between a small and thick torus and a large and thin torus. A person at point A should be able to jump a lot higher than a person at the point C I mentioned earlier if the torus is small and thick because he is relatively quite close to the other side of the torus, which has its own gravity pointing towards its surface. In the case of a large and thin torus, this effect wouldn't be as strong because one is further away from the other side.
But the major question is whether gravity is pointing up or down at point A (from A's perspective). It's not at all clear to me which way it would (and whether changing the dimensions or the mass of the torus would invert the direction).
Assuming an equal distribution of mass, the answer is quite simple. Take all points with distance r (r being the large radius of the torus) and call this circle c. Now, gravity on any point on the inside or outside of the circle will point to the closest point on c.
Points like b, however, will have gravity in a direction that's a bit more towards the center of the torus.
Joined: 11/22/2004
Posts: 1468
Location: Rotterdam, The Netherlands
Warp wrote:
But the major question is whether gravity is pointing up or down at point A (from A's perspective). It's not at all clear to me which way it would (and whether changing the dimensions or the mass of the torus would invert the direction).
Like Scepheo mentioned, A's gravity would keep him grounded. The hollow center of the torus does not have any mass, so it is not part of the gravity well. Thus, gravity points towards the ground at A's position.
The other side of the torus (the part which is "above" A), however, does have mass. Since it's further away, it does not affect a person standing at point A all that much, but it does mean he would be able to jump higher (this phenomenon is exacerbated the thicker the torus becomes).
It would be pretty hard to calculate the exact direction of gravity for point B. That depends on a number of variables (such as the mass of the torus and the exact size of the minor radius). Someone who actually knows physics should answer that one.
Like Scepheo mentioned, A's gravity would keep him grounded. The hollow center of the torus does not have any mass, so it is not part of the gravity well. Thus, gravity points towards the ground at A's position.
That's not so clear to me. After all, the majority of the torus' mass is above A. It thus becomes a question of how the distance of this mass (from A) affects the end result.
As said, this would probably require solving a volume integral or something similarly complicated. (I have gone through the solution for a sphere at the university, but it was so many years ago that I don't remember any details.)
Wouldn't gravity make the torus collapse into a sphere unless you make up some forces that prevent it from doing so?
Forces like rigid structural forces? Donuts don't fall in under their own gravity because the strength of the baked batter is stronger than the gravitational force; just upsize that to larger sizes and you're good to go...assuming, of course, that the rigid strength needed is actually possible. Niven's Ringworld had to be composed of unobtanium for similar reasons; the centifugal stresses involved in spinning a torus with radius 1AU fast enough to simulate 1g were tremendous. In this case, though, the gravitational force of the torus itself was ignored, I think; in any event it paled in comparison to the rotational forces.
Pyrel - an open-source rewrite of the Angband roguelike game in Python.
Probably, there's no way to escape the triple integral...
If I missed no calculations, for point A, the only function which has a computable primitive is when we use spheric coordinates. Let \mu be the density of the torus (assumed to be constant), d the distance of the center of the revolved circle to the center of the torus and R the radius of the torus, the value gravitational field at A in the radial direction (the only one which is not zero) is the beautiful integral:
No way I'm taking it further, it's very time-consuming, I'll try point B later...
The thought process behind my ignorant assumption was that atoms/objects with low momenta should effectively get trapped in the middle of the donut, so a lot of mass should accumulate in its center pretty quickly.
Kuwaga: that depends entirely on if the zero point in the center of the torus is stable or not. If it is stable, then yes, over time the torus would fill with random debris, assuming you didn't clean it out from time to time.
Niven's Ringworld was famously unstable, so he had to retcon attitude jets onto the perimiter of the world. How valid a comparison between the Ringworld-Sun system and a torus-debris system is, I don't know.
Pyrel - an open-source rewrite of the Angband roguelike game in Python.
Wouldn't gravity make the torus collapse into a sphere unless you make up some forces that prevent it from doing so?
Anything would collapse into a sphere if the density is increased enough (and if it's increased even more, it would collapse into a point). However, if the torus is small enough (and not too dense) and has enough structural strength, it will obviously retain its shape. (Note that the torus having gravity is not dependent on its mass, as long as it's nonzero. Any mass has gravity, even an electron. It's just that you need a lot of mass before the gravity becomes significant and not overwhelmed by other forces.)
No way I'm taking it further, it's very time-consuming, I'll try point B later...
