Hard Light Productions Forums
Off-Topic Discussion => General Discussion => Topic started by: jr2 on September 18, 2014, 03:01:40 pm
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Ran across this, not sure if anything new included but learned of a few new claimed aspects of good news (if true).
How about this article that says it is possible using the basics of the Alcubierre drive? Initially needed negative mass energy and enough positive energy as an entire star. But there is research suggesting this *might* not be the case.
http://blogs.discovermagazine.com/crux/2014/09/17/close-star-trek-propulsion/#.VBsQnvmwK8M
>Similarly, there are warp bubbles that would be much easier to achieve energetically than the one Alcubierre used in his calculations. The Alcubierre warp bubble has walls with a thickness on the scale of what’s called a Planck length (~1.62 x 10-35 meters), but if you increase the wall thickness up to a few hundred nanometers, meaning the size range of the wavelengths of visible light, it turns out that the energy requirement plummets.
>And not only does the technology become more feasible from a quantitative standpoint (i.e. the amount of energy needed), but from a qualitative standpoint as well. “Not only does the thicker bubble wall mean that we’d need a lot less energy to generate the warp, but it also means we might do it with electromagnetic technology,” explains Eric Davis, a breakthrough propulsion physicist at the Institute for Advanced Studies in Austin, Texas. “And that’s precisely the kind of technology that we humans have developed.” Just think about that cell phone.
>In other words, changing a few numbers in Alcubierre’s calculations makes warp at least thinkable in terms of doable technology. Based on similar tweaked calculations, White has also figured out that space can be softened to a certain extent, like changing the wooden table into foam, making it that much easier to compress. Another tweak, discovered by Davis, is that if the warp generator is pulsed, i.e. turned on and off really fast, that reduces the energy requirement further. **And, by the way, even the need for negative energy that I mentioned earlier need not be a show-stopper, since there is a kind of negative energy — negative vacuum energy — that could be created by certain capabilities that we have, including lasers and a technology known as quantum optics.**
There's also links to show that there are ways to bend space without negative energy
http://en.wikipedia.org/wiki/Warp-field_experiments#mediaviewer/File:Spacetime_expansion_boost.jpg
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Yes, those results are well-known and were discussed in the last thread we had on the Alcubierre drive. They still don't cancel out some of the more esoteric requirements of the drive (such as the need for exotic matter) and the accompanying question about feasibility.
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well, supposedly there is some mention of a way to get our alcubire drives without negative mass (or otherwise using some sort of arrangement of positive energy to get the same local effect as negative). I cannot find any detailed explanation of this anywhere so it might just be a misunderstanding of a few reporters but it that is a thing that would be big.
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You can make a perfectly good Alcubierre drive without negative energy, but it won't be able to go FTL. Sub-light Alcubierre would still be a pretty big deal, since it'd mean freedom from the Tsialkowsky's rocket equation - the drive would not expel any reaction mass, and therefore it'd have (potentially) infinite Isp, under "KSP terms" (not that you could really apply them to such a drive). You could run it as long as you can keep it powered, and you should also be able to get the bubble pretty close to the speed of light, even if you won't exceed it. Alcubierre drive is essentially the coveted "reactionless drive". Charlatans from around the world sometimes invent those, but it's the only one with a scientific basis.
The bottom line is, there'd be no time dilation inside the bubble (at least, according to the way I understand the effect), so a 4-LY interstellar flight would still take 4 years both inside and outside. With a normal drive, an interstellar journey can still be as short as you like, but beyond a certain point it only becomes shorter from your point of view (there's nothing preventing you from getting to Alpha Centauri in one day if you've got fuel for that. It's just that 4 years will pass on Earth while you're doing it. Relativity is weird, though this makes perfect sense if you get into it). In a way, lack of FTL travel is not your problem. It's the problem of all those stuck on a planet you're taking off from. :)
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Ah, not too mention the particle blast of radioactive solar system shattering doom!
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A sub-light version shouldn't have that (though any space dust encountered along the way can and will shoot out in a somewhat similar way), but yeah, an FTL Alcubierre would indeed cause a great big flash of radiation in front of it when shut off. I don't know what magnitudes we're talking about, but I think that it'd be sufficient to either avoid pointing directly at your destination, or just shut the drive off in interplantary space and do that last few light-minutes using a normal drive. When it comes to Earth, you might want to do this even with STL version. Nobody wants a space dust-based shotgun to take out his satellite. :)
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Dragon, I'm curious, where exactly did you find a variant of the Alcubierre drive that does not require exotic matter?
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yeah, I've heard mention of that a few times and that is real interesting but I haven't been able to find a source on it.
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Dragon, I'm curious, where exactly did you find a variant of the Alcubierre drive that does not require exotic matter?
Everything dr. Harold White does is based on sub-light version of the effect, and he's not looking for exotic matter or anything like that. The FTL Alcubierre drive is actually an extrapolation of an effect that should be possible to induce and measure with normal matter, in STL conditions. It's kind of analogous with how you can have FTL particles just fine, it's just that their mass would have to be imaginary. While I'm not sitting in this like White does, I presume that it has to do with a squared Lorenz factor being in the energy equations somewhere (wouldn't be surprising, it pops up all over the place in relativity). Generally, aside from singularities at v=c itself, relativistic equations generally make sense at v>c, but something is bound to end up either imaginary or negative, due to the aforementioned Lorenz factor being everywhere (it's 0 when v=c and imaginary when v>c. An imaginary number gives a negative number if squared). The above assessment might, of course, be totally wrong, but if it is, then it's likely hard to explain on a forum without LaTeX support.
Of course, it's all assuming it works like we think it does. Last time I checked, experiments were inconclusive on the issue. While finding out it doesn't work would be interesting on it's own, it'd be not nearly be as exciting as finding out that a reactionless drive is possible.
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yeah, one of those rare times when "we were right" would be the more interesting result of the experiment.
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Well, that depends on who "we" are. :) This theory is somewhat divisive in the scientific community, with various very good theoretical physicists presenting equally convincing "for" and "against" arguments. That said, dr. White is the only experimental physicist who seems to have anything to publish on this effect, and he seems optimistic. Once he gets conclusive results, some theories will need serious revising (or even outright discarding) either way. Of course, every SF fan (myself included) likely hopes he's right. :) Also, aside from interstellar drives, I can imagine applications of this effect in research on gravity (it can be though of as curvature of space, and if White is right, then we'll have a convenient way to mess with this curvature).
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yeah, there's lots of applications for this, i mean artificial gravity?
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Well, this depends on how malleable this "bubble" would be, but I don't think so. It might give us insight into nature of gravity, though. The big problem with gravity is that not only that it's weak, we've got no real way of affecting it and measuring the effects (we can move mass around, but to get a measurable effect, we'd need a huge amount of it). This could be such a way, if we can artificially curve spacetime, then we could be able to find something out about it's nature.
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The big problem with gravity is that not only that it's weak, we've got no real way of affecting it and measuring the effects (we can move mass around, but to get a measurable effect, we'd need a huge amount of it).
I wouldn't say that. The effects of gravity (http://en.wikipedia.org/wiki/Cavendish_experiment) are measurable between fairly small (albeit macroscopic) objects, and you don't even need supercomputers or lasers to get measurable readings.
However if you meant piling up enough matter to meaningfully alter the geometry of space-time (ie. cause a measurable distortion from locally euclidian curvature), that's a bit of a different matter.
In fact the view that gravity is "weak" is a sort of problematic statement since it depends so wildly on context. If you consider it a force, then yes it's locally much weaker than electroweak or strong interactions. However in large scale it's the dominant force in the universe.
And if you consider gravity from general relativity point of view, then it isn't actually weak at all. If you think about it, absolutely minuscule changes in space-time are enough to cause measurable gravitational effects, like an apparent attractive force between objects of matter - or bending of light as it passes next to a star. In that sense, I think gravity is actually quite a strong "force", or effect, from something that is really difficult to perceive.
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The big problem with gravity is that not only that it's weak, we've got no real way of affecting it and measuring the effects (we can move mass around, but to get a measurable effect, we'd need a huge amount of it).
And if you consider gravity from general relativity point of view, then it isn't actually weak at all. If you think about it, absolutely minuscule changes in space-time are enough to cause measurable gravitational effects, like an apparent attractive force between objects of matter - or bending of light as it passes next to a star. In that sense, I think gravity is actually quite a strong "force", or effect, from something that is really difficult to perceive.
That's a very interesting way to view it -- small curvatures produce huge effects. For the field at Earth's surface, the radius of curvature is something like a light-year. Totally weak. Yet the consequences are obvious. :)
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Well to elaborate - "huge effects" is all relative, isn't it?
I mean, if the curvature at Earth's surface produces apparent acceleration due to gravity that is 9.81 m/s2, it's still a very small effect compared to something extreme - like the gravitational acceleration* at an event horizon...
In the scale from "zero" to "event horizon", "small" curvatures produce "small" accelerations and extreme curvatures produce extreme accelerations (and other gravitational effects).
The only reason why the small accelerations produced by small curvatures seem so significant to us is because... they are very significant to us, and part of our every day life, and it is one of the defining factors of the cosmos on the level that we apes observe it. It's sort of the same thing as with quantum mechanics, but in reverse. The extremes of our universe are not intuitive to us.
*Which is a funny thing since it's not exactly a constant - the more massive black hole, the smaller the "surface" gravity is, if you define event horizon to be the surface.
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Indeed, and well said. 'Measurable' was the right word, not 'huge'. I remark on how, when considered in the context of curvature, the gravitational fields we are familiar with seem incredibly insubstantial, yet we are intimately familiar with their effects. Which is the whole idea, perspective matters. We are used to weak fields, weak accelerations, and slow velocities, relative to the range of possibilities. As you say, physics in this range is intuitive to us. We know it hurts to fall and hit the ground. We know its hard to climb a mountain. We know it's hard to launch something into space. That damned ever-present nuisance, gravity. But we don't usually think about how a tiny magnet can overcome the gravitational pull of the entire planet. How crazy is that? Magnets are like magic, every child is at one time fascinated by them. But it's just because it is a stronger field than what we are more used to, and it can both attract and repel.
Or how crazy is it that gravity doesn't ever actually hurt us here on Earth. It is the impact, not the fall, as so many have pointed out. The field accelerates us, but we don't feel it. We are too small to feel it. The curvature seems flat to us. We only feel the electrostatic forces when we intersect something 'solid', and they are not so forgiving. But the gravity of a black hole can kill you, rapidly, unavoidably, before you even reach the singularity. The curvature itself tears you apart in one way, crushes you in the other.
It's probably for the best that the extremes of the universe are not intuitive to us; they're usually lethal. :p
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Magnets are like magic, every child is at one time fascinated by them. But it's just because it is a stronger field than what we are more used to, and it can both attract and repel.
Magnets are crazy awesome in the sense that they always exist in dipoles. The force between two magnetic poles is analogous to force between two electric charges, so the force between magnetic dipole is also analogous to electric dipole - which means, magnets naturally provide us with a hands-on visualization about how electric dipoles behave, as well! Meaning that the force between two magnets (small bar magnets) is inversely proportional to the fourth power of the distance.
This means, basically, that when you're handling a bunch of bar magnets, their interactions are actually somewhat analogous to van der Waals forces between some simple diatomic molecules. They don't even need to be polar molecules in the chemical sense (like carbon monoxide) - every diatomic molecule behaves like a dipole to some extent because the electrons in the molecule are almost never perfectly evenly distributed... which creates continuously fluctuating dipoles in some direction. Which enables van der Waals forces even in non-polar substances, like H2 and N2, enabling them to have liquid phase at reasonably high temperatures... even though the effect is much weaker than with truly polar liquids. They fill a container, you can pour them, you can submerge objects in them... by contrast, liquid helium is some really un-intuitive stuff!
With some effort, you could build a retaining frame to hold some magnets in the shape of water molecule as well. That might be a pretty interesting visualization for physics classes. A box of small and strong enough water-molecule shaped magnets should, in theory, exhibit some similar bonding as water molecules do, with positive poles snapping onto negatives.
Actually now that I think about it, it should be possible to demonstrate other stuff as well with magnets. Like nuclear forces. With a clever configuration of magnets, you could create macroscopic objects that would behave somewhat like protons, others that would behave like neutrons, and you could build simple nuclei from them. You could even demonstrate unstable and stable nuclei, and the nuclear bond energy stored into the system...
Or how crazy is it that gravity doesn't ever actually hurt us here on Earth. It is the impact, not the fall, as so many have pointed out. The field accelerates us, but we don't feel it. We are too small to feel it. The curvature seems flat to us. We only feel the electrostatic forces when we intersect something 'solid', and they are not so forgiving.
I can feel electrostatic forces with the the hair on my arms! Or rather, I can use them to sense electric fields... but that's just another way of saying that an existing electric field exerts a force on the strands of hair on my arms, which I feel by the hair bending...
But the point about touch interaction is another interesting topic.
Turns out that repulsive Coulomb force between the electrons in two objects is just one part of the story. The aforementioned electronic fluctuation in matter causes small dipoles to form continuously, and those dipoles first produce an attractive force (van der Waals again) between the molecules as they approach each other, hence the gekko has enough adhesion to climb the wall and even stay on a roof.
There is also a balance point where the attractive dipole force is overcame by repulsive Coulomb force. However the story doesn't stop there, because it turns out that beyond that balance point, the repulsive force actually increases quite a bit faster than simple electric repulsion would suggest...
...which is fundamentally caused by Pauli exclusion principle.
Basically, when you touch something solid, the molecules on your skin first arrive at the balance point where there's neither attraction or repulsion between them and the foreign object.
If pressure is applied, your fingers don't sink into the object (much), partly because there's an electrostatic repulsive force that pushes against them, but the electrons also repulse each other because they simply cannot share the same quantum state.
And that's really sort of spooky interesting to me, because it turns out a fundamentally quantum characteristics of particles is, at least partially, responsible for something as basic as touch interaction.
About the only thing more freaky is that, apparently, we can smell the difference between molecules that have different quantum vibration characteristics...
...so if that's true, maybe quantum physics isn't actually so hopelessly inaccessible to our intuition as we've thought.
Maybe we just need to name quantum properties based on olfactory qualities instead of visual qualities like spin or colour charge? :drevil:
But the gravity of a black hole can kill you, rapidly, unavoidably, before you even reach the singularity. The curvature itself tears you apart in one way, crushes you in the other.
Not if the black hole is big enough! ;7
But hey, talking about visualizing or measuring the curvature of space...
What if you were to build a reasonably large container of known dimensions (a spherical container would be nice for its symmetry), and you would have a way to measure the volume of that container somehow, shouldn't you see a slight decrease in the volume as you send the container from Earth's surface to high orbit, like geostationary orbit?
Only problem is, I'm not sure how to measure the difference accurately enough to get reliable results. One way I thought of would just be gauging the internal diameter of the container by a laser distance meter; another would be to use the container as a microwave resonant cavity and measuring the changes in the resonant frequency.
The point is making the difference in the curvature of space measurable in a way that is sort of intuitively understandable. The question is - how accurately would you need to measure the volume, or cavity diameter, in order to be able to get meaningful results that can be separated from noise and actually correlated with the gravitational potential?
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Actually now that I think about it, it should be possible to demonstrate other stuff as well with magnets. Like nuclear forces. With a clever configuration of magnets, you could create macroscopic objects that would behave somewhat like protons, others that would behave like neutrons, and you could build simple nuclei from them. You could even demonstrate unstable and stable nuclei, and the nuclear bond energy stored into the system...
This is a really cool idea. I fear the weight of the structure might become a problem as its size/complexity increases, but then you could just build it in orbit on the ISS or something. I would love to see that!
I can feel electrostatic forces with the the hair on my arms!
You certainly can! Every particle in your body does not respond to the electrostatic field in the same way. A charge builds up over the hairs on your arm and you can feel the resultant forces. Not so with gravitational field. All test particles falling together in some small region of a gravitational field will fall with the same acceleration. Doesn't matter if they are feathers or hammers, hydrogen nuclei or gold. And that's the whole point of the equivalence principle -- a local experiment cannot distinguish between freefall in a grav field, or weightlessness far from one. The only clue is that for real gravitational fields there are tidal forces, but these vanish in sufficiently small reference frames. So you cannot feel gravitation, unless the field is sufficiently strong and you are a sufficiently extended body (yes I am calling you fat). :)
if that's true, maybe quantum physics isn't actually so hopelessly inaccessible to our intuition as we've thought.
Whoa. I knew of some articles about quantum mechanical effects that might be apparent to biological systems (I think I remember in birds or something, it's been a while), but I didn't hear about us smelling them. That reminds me of how special relativistic effects explain the particular yellowish hue of gold, which is pretty cool. But then again, how much does it really help us, if our goal is to understand it so that we can do physics? We can relabel phenomena with names that are more intuitive, but there's still that confusing, unintuitive logical framework we have to learn, and I think that's why things like QM are so hard. For instance, we are used to thinking that if we understand the entirety of something, then we know everything about all of its components. Or we are used to thinking that measuring a system does not meaningfully change it. In QM these are not true.
Not if the black hole is big enough!
Unless you thought I said event horizon instead of singularity, then that only makes the problem worse! The kill radius of tidal forces increases as the cube root of mass. The radius of the event horizon grows faster (linear with mass), so for supermassive black holes you may survive well into the event horizon. But there is always some non-zero proper time between getting spaghettified and meeting the singularity. This time is pretty short -- some fraction of a second. But the idea is that gravitation/curvature itself definitely can be felt, or even kill you directly, given the right circumstances. It's basically the situation where the equivalence principle fails, when the curvature is apparent over your reference frame.
What if you were to build a reasonably large container of known dimensions (a spherical container would be nice for its symmetry), and you would have a way to measure the volume of that container somehow, shouldn't you see a slight decrease in the volume as you send the container from Earth's surface to high orbit, like geostationary orbit?
*Thinks*. No, I think you would actually expect the opposite result, or nothing. The volume of an object decreases [edit: with decreasing R], as tidal forces compress it into smaller dimensions, trending toward zero volume as you approach a singularity. In practice this would be really hard to measure for the Earth's field -- solid state physics would be critically important, like rigidity of the container, and that pesky thing that is thermal expansion. You'd have to have a super small expansion coefficient and control temperature extremely sensitively.
Also, I think what might be throwing you off here is that gravitation is not curvature of space. It is curvature of space-time.
Let's look at a different thought experiment and see if that helps. Consider two people on a firing range. They aim to strike the same target, and they fire at the same time. What do the trajectories look like? The bullet is much faster than the arrow [citation needed], so it can follow a faster and more direct path to the target. The arrow in comparison must sweep through a high arc, and it takes longer. So their paths through space are not the same -- how could we explain them as being due to spatial curvature? Their paths through time are different as well (same origin, different destination).
What's going on is that the curvature of the two paths in space-time is the same. They both have a radius of curvature of about 1 light year, if this thought experiment is happening on the surface of the Earth. And that's what the geometric theory of gravity is all about -- gravitational field is manifest as curvature in four-dimensional space-time, not curvature of space or time individually.
What are the best ways to measure curvature of space-time? Check for the separation of nearby geodesics. In other words, check for tidal forces. Consider two ships hovering above the Earth, at the same radius, but different (but constant) angular coordinates. Let them be within line of sight of each other, and they measure the distance between them by laser. Now, let them fall, accelerated only by gravity. They both accelerate toward the center of the Earth, but since they had slightly different angular coordinates to start with, their paths will draw nearer to one another. The distance separating them decreases. There is a tidal force acting on them. It seems like there is an acceleration between them.
Now consider two ships at the same angular coordinate, but different radius. They proceed into freefall. The ship at smaller radius has a greater acceleration toward the Earth, so the distance between them increases. Again, tidal force, except now in the case of objects oriented radially to the acceleration vector.
One of the fundamental pieces of the derivation of general relativity is to produce the curvature tensor from the equation of geodetic deviation, which works just like in the above examples. Tidal forces and curvature are very tightly linked. :)
edit: typose.
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I knew of some articles about quantum mechanical effects that might be apparent to biological systems (I think I remember in birds or something, it's been a while), but I didn't hear about us smelling them.
The research is ongoing and inconclusive, but certainly it appears that at least some animals can do it.
Unless you thought I said event horizon instead of singularity, then that only makes the problem worse!
Yeah, I thought you referred to crossing event horizon instead of falling to singularity (which, I should note, I suspect is a mathematical artefact from the theory rather than a physical object at the "bottom" of the black hole).
*Thinks*. No, I think you would actually expect the opposite result, or nothing. The volume of an object decreases [edit: with decreasing R], as tidal forces compress it into smaller dimensions, trending toward zero volume as you approach a singularity. In practice this would be really hard to measure for the Earth's field -- solid state physics would be critically important, like rigidity of the container, and that pesky thing that is thermal expansion. You'd have to have a super small expansion coefficient and control temperature extremely sensitively.
Also, I think what might be throwing you off here is that gravitation is not curvature of space. It is curvature of space-time.
Let's look at a different thought experiment and see if that helps. Consider two people on a firing range. They aim to strike the same target, and they fire at the same time. What do the trajectories look like? The bullet is much faster than the arrow [citation needed], so it can follow a faster and more direct path to the target. The arrow in comparison must sweep through a high arc, and it takes longer. So their paths through space are not the same -- how could we explain them as being due to spatial curvature? Their paths through time are different as well (same origin, different destination).
What's going on is that the curvature of the two paths in space-time is the same. They both have a radius of curvature of about 1 light year, if this thought experiment is happening on the surface of the Earth. And that's what the geometric theory of gravity is all about -- gravitational field is manifest as curvature in four-dimensional space-time, not curvature of space or time individually.
What are the best ways to measure curvature of space-time? Check for the separation of nearby geodesics. In other words, check for tidal forces. Consider two ships hovering above the Earth, at the same radius, but different (but constant) angular coordinates. Let them be within line of sight of each other, and they measure the distance between them by laser. Now, let them fall, accelerated only by gravity. They both accelerate toward the center of the Earth, but since they had slightly different angular coordinates to start with, their paths will draw nearer to one another. The distance separating them decreases. There is a tidal force acting on them. It seems like there is an acceleration between them.
Now consider two ships at the same angular coordinate, but different radius. They proceed into freefall. The ship at smaller radius has a greater acceleration toward the Earth, so the distance between them increases. Again, tidal force, except now in the case of objects oriented radially to the acceleration vector.
One of the fundamental pieces of the derivation of general relativity is to produce the curvature tensor from the equation of geodetic deviation, which works just like in the above examples. Tidal forces and curvature are very tightly linked.
Hang on now, I'm not following you.
I'm sure you're aware of the gravitational time dilation causing a time difference between surface and satellites, being substantial enough that GPS and other geolocation systems need to be specifically corrected to take it into account.
Doesn't gravity cause time and space to stretch equally, though? I'm imagining a thin, rigid shell that has a certain amount of space within it. The circumference of the shell should be expected to remain constant, because it's made of atoms and molecules that tend to balance at a certain distance from each other.
In other words, going across the surface, counting each interval between atoms and adding them together until I've crossed the surface and arrived back to where I begun, I would expect to arrive at the same result for the circumference of the container both on surface and in orbit.
So... if we have this rigid* shell on Earth's surface and we measure the inner diameter to be s = t*c (the time it takes for light pulse to cross the cavity, multiplied by speed of light of course).
On orbit (further from Earth's mass), we should measure the inner diameter to be s0 = t0*c, again the time it takes for a light pulse to cross the cavity in those conditions.
In my visualization, when space is curved (and, correspondingly, time is stretched), light has a longer way to travel, so it takes a longer time for it to pass through that region of space. Isn't that how gravitational lensing works? If a wavefront of radiation passes next to a massive object, the photons nearer the disturbance have a longer way to travel, which causes the inner part of the wavefront to be delayed (and curved).
I would expect that the result for measuring the diameter of the container should be t>t0 and correspondingly s>s0 - on orbit, light takes a shorter time to travel through the container and therefore the distance is measured slightly shorter than on Earth.
Which to me would suggest that the internal volume of the spherical shell should decrease slightly when it is hoisted from surface to orbit. Of course the measurement would be really challenging to do because the effect is so small, but that's what I would expect.
Using the analogy of different paths: On Earth, the photon travels horizontally through the container on a more curved and therefore longer path. On orbit, the photon's trajectory through the container is flatter and thus more rapid. In perfectly flat, euclidian space (closest would be intergalactic space I guess) the photon travels through container in perfectly linear trajectory, which results in smallest measured internal diameter possible.
I'm not sure if tidal forces have anything to do with this, though. I would expect this to work the same in homogenous gravity fields of different magnitudes.
Of course, rigidity of the sphere is a pretty big assumption, but would there really be any particular distortions that would affect things? Tidal forces of course could make the container asymmetric.
But what if you idealize the situation, and instead ask: What is the volume of a sphere with circumference c in euclidian flat geometry, and what is the volume of a sphere with circumference c in curved space-time? And does the diameter of the sphere measured through the sphere change based on curvature of the space?
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Yeah, I thought you referred to crossing event horizon instead of falling to singularity (which, I should note, I suspect is a mathematical artefact from the theory rather than a physical object at the "bottom" of the black hole).
It is, yeah. General relativity is a classical theory. It treats space-time as continuous and infinitely differentiable. A singularity in GR even has a very precise definition, beyond "infinite density". If you find a place on a space-time manifold where all geodesics (space-like, time-like, and null) and all time-like paths with bound acceleration (any allowed observer), end after finite proper time, and the manifold cannot be smoothly extended beyond that location, then you have found a singularity. Classically, gravitational collapse is guaranteed to produce such a region anytime mass falls within its own Schwarzschild radius and forms a trapped surface, as proved by various singularity theorems (e.g. by Penrose).
Of course, the correct theory of gravitation might not be classical. Perhaps some future theory of quantum gravitation may totally rewrite our understanding of what happens in the depths of a black hole. In practice, however, it makes no difference to the fate of a person falling into one. Tidal forces become lethal before you'd be close enough to discern any difference from the classical description (unless you happen to be a human the size of a subatomic particle). There's something disconcerting about that -- it means not only can you not report your discoveries about the 'singularity' to your fellows outside the horizon, you can't even make those discoveries in the first place. Singularities are very well protected.
There are of course other possibilities -- metrics become pretty weird when you allow for nonzero values of charge/spin. Or that 'cosmology with torsion' idea that was tossed around here a while back. And then there's those theoreticians discussing black hole thermodynamics/holography, some saying that maybe there's this firewall sort of thing just above the event horizon. I don't know, it sounds silly to me, but I sure don't understand it well enough. :)
I'm sure you're aware of the gravitational time dilation causing a time difference between surface and satellites, being substantial enough that GPS and other geolocation systems need to be specifically corrected to take it into account.
Certainly! The gravitational time dilation effect was experimentally verified back in 1959 at Harvard, and it is critically important to consider for the proper functioning of GPS. The system would very rapidly become useless if we didn't! But this isn't 'curvature of time', it is again curvature of 'space time'. We do not explain time dilation with 'time curvature' -- the concept makes no physical sense and does not help us to do calculations.
Your thought experiment examines the ability to measure the change in volume elements due to curvature (based on the idea that gravitational field is positive curvature in 4d-Riemannian geometry). This underlying reasoning is correct -- volume in non-Euclidean geometry is different. What I was trying to explain to you is that in practice, it is a terrible idea to try measuring curvature in this way (there are far better ways, and I gave an example). Even excluding the complications of solid state physics, the effects of tidal forces are going to be overwhelming in comparison. You would need an impossibly rigid structure to avoid this. But yes, in answer to your question of an idealized sphere, so that this is purely a hypothetical thought experiment, you would expect a change in the enclosed volume due to the change in curvature.
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Ok, yeah, that makes sense. In other words the experiment is one of those that are valid on thought level, but in practice would require the arsenal of physics tools similar to massless wire, rigid rod (in this case a rigid shell), frictionless surface, ideal gas, and perfect black body...
So I guess instead of providing basis for an experiment, it can still be an useful way to visualize the effects of positive curvature on space-time, and what the phrase "curved space-time" functionally means. I suppose, fundamentally the problem is that when you build a geometric shape in one curvature, it will inevitably be distorted when it's brought to a different curvature.
But it's quite interesting - to consider that geometric shapes or regions of real space might have more volume than they externally appear to hold, depending on the properties of the space within them. It's basically the same thing as TARDIS... or Bag of Holding, for that matter. Just in much smaller extent in most situations.
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I saw a pretty cool demonstration on youtube of a device that was rigged to demonstrate the effect of curvature on world-lines on a 2D surface. Aha, found it:
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massless wire, rigid rod (in this case a rigid shell), frictionless surface, ideal gas, and perfect black body...
It's okay, as long as they don't take away my spherical cows. They're so cute and lovable and maximally efficient methane farms.
So I guess instead of providing basis for an experiment, it can still be an useful way to visualize the effects of positive curvature on space-time, and what the phrase "curved space-time" functionally means. I suppose, fundamentally the problem is that when you build a geometric shape in one curvature, it will inevitably be distorted when it's brought to a different curvature.
That it is! Another consequence of curvature we can visualize fairly easily is the separation or convergence of initially parallel light rays (this is a good way to test the average density of the universe). Or we can consider the reverse situation to what we were looking at earlier -- instead of moving an object through changes in curvature, have changes of curvature move through the object. Those are gravitational waves! Or just watch what Phantom Hoover posted; that guy got it right. (Thanks Phantom, that was cool!)
I should point out real quick that spatial volume can become a really tricky thing in extreme space-time curvature, even in the theoretical (instead of just practical) sense. For instance, the surface area of an event horizon has a unique interpretation that everyone can agree on. Not so for the volume enclosed by it! (http://arxiv.org/pdf/0801.1734v1.pdf) That depends on the choice of coordinates. One could even say that the volume within an event horizon is zero!
So, 3-volume is not an invariant quantity in GR. There is however a 4-volume, akin to 'hypervolume', which is.
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One particularly bad problem with the standard 'bowling ball on a trampoline' illustration is that the trampoline's curvature is actually negative.
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No it isn't, it's positive. Just check whether initially parallel rays come together or separate, or whether the sum of angles in a triangle is more or less than 180 degrees.
Curvature isn't "whether it bends up or down", like "bending downward is negative curvature". If that was true then you can turn negative curvature into positive curvature just by inverting your perspective. But that would be ridiculous, curvature should not depend on your point of view. The curvature of a surface is an intrinsic property of that surface and does not depend on how it is embedded in a higher dimensional space. That's one of the failings of our minds. It's pretty hard to imagine a surface, or a manifold, without imagining what it is embedded in. I can't visualize a 2D manifold without imagining 3D space around it.
So what does negative curvature look like? Negative curvature is like the region immediately around a saddle point. In one direction, it bends up, in the other direction it bends down. Now try extending that, so that the neighborhood of every point on the surface has that property. I don't know about you, but I can't visualize such a thing. My brain can only extend it so far and then simply gives up. :p
Positive curvature on the other hand is extremely easy to visualize. It looks like the surface of a sphere. In fact, the surface of a sphere has the same positive curvature everywhere. Try to draw two parallel lines on the surface of a sphere. If you start them out parallel, then they will come together and intersect on the opposite side of the sphere. Those are longitude lines! They are also perfectly straight lines. You can walk all the way around one without ever turning your step. What about latitude lines, you might ask? They are parallel everywhere, aren't they? Yes, but they are in general not straight. :) If you follow any longitude latitude line other than the equator, you will have to turn slightly at every step, or very sharply if you're near the pole.
And that's the idea behind geodesics -- locally straight lines. Longitude lines are geodesics on the positively curved surface of the Earth. Latitude lines are not geodesics. You must accelerate (constantly change your direction) to follow one. The secret to general relativity is to recognize that all objects that are in freefall follow geodesics. What gravitation does is make those straight lines appear curved, because the underlying geometry of space-time is curved. That curvature arises because of the presence of some mass nearby. "Mass tells geometry how to curve, curvature tells matter how to move."
The cover art and first chapter of the Gravitation textbook (http://www.goodreads.com/book/show/638371.Gravitation) is all about this idea and is probably the most awesome description of it I've ever seen. Far better than any rubber sheet analogy. The rubber sheet analogy is only good if you recognize it only for its description of the effect of curvature on geodesics. If you think of it as the curving of space, then it fails and only causes more confusion. If you ask what causes objects to follow those paths ("don't you need gravity to explain the paths") then it fails again. Ignore gravity in the rubber sheet analogy. Just ask what happens to straight lines when you introduce that warp. And remember that the rubber sheet is not analogous to space, it is analogous to space-time.
edit: Mixed up lats and longs. I do that sometimes.
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The curvature of the trampoline is definitely negative over a significant part of its area, though not right next to the bowling ball. And you can totally visualise negatively-curved surfaces, what you can't have is a surface that has constant negative curvature and has infinite extent in all directions (http://en.wikipedia.org/wiki/Hilbert's_theorem_(differential_geometry)).
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I'm still not sure I see any place on it with negative curvature, but perhaps it is my visualization of the rubber sheet depression itself that isn't right.
Added: Ah, you are right, I was wrong. :) My problem was thinking of the depression as being well fit by a spherical surface over most of the region. Obviously that's not true everywhere, especially if the depression is more funnel shaped, like a tornado. So it is possible to imagine broad regions of negative curvature, but as you point out with Hilbert's Theorem, not globally with constant negative curvature. That's neat!
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The curvature of a funnel that most accurately simulates Newtonian gravitation would be one where elevation is relative to the square of distance, making it the approximate shape of the gravitational potential. That makes a ball rolling on the surface experience a component of gravitation that is aligned with the gradient of the surface, and that sort of makes the effect analogous to Newtonian gravity. Hence, good way to visualize effects of gravity, but I do agree it has fundamental problems with explaining the nature of gravity in general relativity.
One of which is that the curvature of the 2D shape that reproduces effects that resemble Newtonian gravity best is not necessarily analogous to the curvature of a planar sheet of 4-space.
Visualizing negatively curved surfaces is easy (at least objects with locally negative curvature). Negatively curved space is a bit of a different matter, though...
As said before I tend to visualize the effect of curvature instead of exactly trying to figure out how to make a 2D analogy of it or anything of the sort. Basically what I do is imagine that I can isolate an ideal geometric shape of the space, and then measure its properties, and compare those properties to what that shape should have in flat, euclidian geometry.
In a constant geometry you can use a triangle, measure the angles in the corners, and sum it together. But on a surface, it's sufficient to draw one triangle and that's it. In space, that only measures the curvature in the plane of the triangle, and it only returns the average curvature inside the triangle.
To get better results, you'd need a 3D shape instead of a 2D shape, which would allow you to measure the curvature in all three spatial dimensions at the same time.
Another solution, which works better for me, is to take a geometric shape of 4-space again (sphere works best because symmetry and simple definition), and compare its properties to its ideal, euclidian definition. For example, is the relationship between the diameter and circumference exactly pi?
However that again means we have to measure the diameter of the sphere in at least the three cardinal directions, and likewise the circumference, because we're not exactly using the entire sphere but instead circles aligned on the three spatial dimensions.
What's more elegant in my mind (and which spawned the question about whether making a measurement based on it is practical) is to look at the volume of the sphere.
Ideally, in euclidian space, the volume of any sphere regardless of its size is 4/3 pi r3.
The curvature of space affects that, and moreover the relationship between volume and radius becomes non-constant.
On my mind, I visualize positive curvature of space as something that causes the volume of the sphere to become more than 4/3 pi r3.
By contrast, negative curvature would reduce the volume compared to Euclidian norm.
The problem of trying to do an analogy of negative spatial curvature on the rubber sheet representing a world sheet is that the shape at which a region of rubber sheet has the smallest surface area is always flat.
In order to change the curvature of the sheet to any direction, positive or negative, you have to displace it from being flat. That means you're stretching the sheet, which basically always increases its surface area.
In my opinion, that more or less makes the "stretched" rubber sheet closer to being analogous to positive curvature of space, irrespective of the actual shape of the deformation that a weight causes on the sheet.
To have any hope of creating a visualization of negative curvature on a 2D sheet analogy is if you have some way of contracting the sheet instead of stretching it. You can do this by pre-stretching a sheet of rubber or spandex or whatever flexible material, then draw a co-ordinate grid on it.
You can't really visualize negative curvature as analogous to any deformation of the plane, but what you can do is to create an area where you reduce the stress of the sheet, by for example pinching it and then attaching tape on the underside to keep it on the "un-stretched" position. What you're doing is analogous to negative energy causing the space to "contract" from the euclidian standard.
Then you can basically look at the world lines passing the disturbance, and think of them as the routes that a photon would take when it passes that location.
If world lines curve towards the center of disturbance, you have positive curvature.
If the world line curves away from the disturbance, you have negative curvature. This doesn't necessarily depend on the actual shape of the rubber sheet in 3D space. Just concentrate on the coordinate grid, and see whether it contracts or expands relative to what we define as the "standard".
In this analogy, the "pre-stretched" sheet of rubber represents actually un-disturbed euclidian space with zero energy tensor. The "pinching" (or reducing the stress) is analogous to negative energy tensor, while stretching the sheet is analogous to positive energy tensor. It's still a bad analogy in the sense that we need to pre-stress the sheet in order to be able to reverse it by "negative" energy (pinching), but it seems necessary to avoid crumpling the sheet.
Technically it could be possible to use a fabric that doesn't need to be pre-stretched. A very good visualization tool could be material that expands or contracts significantly based on small temperature changes, which could allow creating a "cold spot" that causes the fabric to contract, and a "hot spot" to cause it to expand, and that would distort the world-lines (or co-ordinates) on the sheet so as to visualize the stress tensor's effects on space.
Oh and by the way, here's a fun little detail. A photon has the same acceleration due to gravity that we measure for everything else - approximately 9.81 m/s2. They just travel so fast that they appear to travel straight, but they do fall. If you're doing very precise optical measurements you have to account for that - and if you're doing vertical beams of significant length, and you're measuring wavelength, you might also need to take red-shift or blue-shift into account (although this is not a practical concern, I can imagine it might be necessary in some experiments requiring extreme accuracy).
But in a falling elevator you would measure a photon traveling perfectly straight. As you would in a spaceship on orbit, or in any kind of free-fall situation.
In other words, curvature of space and time seems different depending on your reference frame. Photons traveling vertically separated would still diverge, though...
And every time I get into thinking this stuff I feel like my head's going to implode. :shaking:
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A photon has the same acceleration due to gravity that we measure for everything else - approximately 9.81 m/s2. They just travel so fast that they appear to travel straight, but they do fall.
This is great for Newtonian thinking, but it actually doesn't work very well in General. Oh, photons are affected by gravity, certainly, and their paths are deflected by it. The equivalence principle teaches us this much. But if you treat a photon as something moving at c but accelerating with g, you won't get the right answers. E.g. for gravitational lensing the Newtonian calculation underestimates the deflection angle by a factor of 2. It also fails spectacularly for describing what happens in extreme curvature. You really need the geometric theory of gravitation to get the correct results. :)
So, for extreme curvature, consider a black hole like always.
If we think of photons in the Newtonian perspective, then a photon emitted radially outward from just inside the horizon will move outward, slow down, turn around, and come right back down. Sounds correct, doesn't it? It's just like the orbit of an object with less than the escape speed. But that's not what happens! If it was within the horizon, then it never gains altitude. It gets dragged farther inward from the instant it was emitted. A photon emitted from just barely inside the horizon does not and cannot make it to the horizon.
Similarly, photons emitted from a singularity don't just go up and come back down. They don't leave the singularity at all. You would not see a singularity even if you could put your nose right up next to it.
If a photon is emitted radially outward right on the horizon, then it just stays there, apparently motionless. (This is actually not quite right either, because a photon is not infinitesimal -- it has wave-like properties and so it gets smeared out. Classically though, treating a photon as a point, then it just hovers there on the horizon.) We can say that the photon is moving at c, but space there is also moving inward at c. Or, we can say that space-time is so strongly curved there that null geodesics directed normal to the horizon appear as points in space. An event horizon is a null surface.
In this sense, an effective visualization of a gravitational well (especially that of a black hole) is to think of it not as being like a warped sheet, but like a waterfall. Susskind gives a great presentation of this analogy in his lecture on The Black Hole War, see here (https://www.youtube.com/watch?v=pf0D8A0jRiY&t=8m00s) about 8 minutes in. Highly recommended.
So the correct description of motion in GR is that the photon, and everything else which is in free-fall, has exactly zero acceleration. (This is probably the most awkward but most important principle in all of GR!) There is no acceleration and so the paths are all geodesics, with the photon being the precise case of a null geodesic (ds2=0) because it moves at the speed of light. The geodesics look curved because the geometry is curved.
What acceleration actually is, is your departure from a geodesic. Sitting or standing here on the surface of the Earth, you are accelerating! Jump off of a cliff, that's when you cease to accelerate (though note quite, because air resistance). Or more happily, go into orbit! GR teaches us to totally rethink what acceleration actually means.
Some more detail on ds2=0:
ds is how we measure distances in space-time, aka a metric.
ds2=0 means the sum of the squares of the space and time distances is zero, which sounds really weird until you remember space-time is Minkowskian instead of Euclidean, so the definition of distance is different. In Euclidian 4D space, distance is a logical extension of Pythagoras' Theorem:
dx2 = dx02 + dx12 + dx22 + dx32,
where dx is the differential distance, and the subscripts 0 through 3 designate the coordinate axes.
But in Minkowskian space, time has opposite sign as space, so the measure of distance is really
dx2 = dx02 - dx12 - dx22 - dx32,
where dx0 represents the time coordinate and we use +--- sign convention (time-like convention). We could write it with the signs the other way (positive for space coordinates and negative for time) for a space-like convention. Doesn't matter which we use as long as we are consistent. We also generally write the metric as ds instead of dx, for space-time interval:
ds2 = c2dt2 - dxr2, with r being a dummy index to sum over the three spatial dimensions.
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A photon has the same acceleration due to gravity that we measure for everything else - approximately 9.81 m/s2. They just travel so fast that they appear to travel straight, but they do fall.
This is great for Newtonian thinking, but it actually doesn't work very well in General. Oh, photons are affected by gravity, certainly, and their paths are deflected by it. The equivalence principle teaches us this much.
Well, yeah. But the equivalency principle is a really, really awesome* thing and worth mentioning every time it's even remotely relevant!
The part about extreme curvatures is quite interesting - space dimensions become time-like inside event horizon and the only way to progress is towards singularity etc - but for practical purposes it's almost always necessary to convert things back into the "Newtonian" thinking, right? Especially in non-extreme cases, like consideration of how photons behave here on the surface of the Earth.
Oh, and I was being a bit inaccurate in my previous post. The curvature of space-time can of course be measured in free-fall by the divergence of parallel light rays, or tidal forces, or frame-dragging. The equivalency principle equalizes acceleration with homogenous gravity field (not that gravity is a field in GR anyway, but just to use the familiar term).
I'm not entirely sure what kind of curvature would be required to produce an ideal, homogenous "gravity field". In fact, technically it should be impossible to detect at all, right?
Although I'm not sure if what you wrote actually means that we would be able to detect an universe-wide homogenous gravity field based on photons acting differently than they should?
*Things that equivalency principle makes awesome: Photons have no (invariant) mass, but light - as a collection of photons - still has lots of properties that are equivalent to mass... :drevil:
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Well, yeah. But the equivalency principle is a really, really awesome* thing and worth mentioning every time it's even remotely relevant!
Oh yes. Equivalence principle is crucial, as we'll discover shortly below. :)
As for reverting back to Newtonian for practical purposes, you definitely do if you're only interested in low approximation or the weak field limit. If the difference in your result due to GR's effects is greater than your intended precision, then you probably want to use GR. Otherwise, using GR is a huge waste of computational time for no practical benefit.
The curvature of space-time can of course be measured in free-fall by the divergence of parallel light rays, or tidal forces, or frame-dragging. The equivalency principle equalizes acceleration with homogenous gravity field (not that gravity is a field in GR anyway, but just to use the familiar term).
Exactly. Equivalence principle assumes that you take a reference frame that is sufficiently limited that the curvature becomes negligible over it. It's the same idea as the surface of a sphere appearing flat if you are sufficiently small / close to it. So the equivalence principle makes a real gravitational field look like a homogenous one, so that tidal forces, etc, vanish.
Why would we care about this? Why would we want to intentionally limit our perspective (I remember Luis had trouble with this idea earlier)? Because it acts as a stepping stone for the development of general relativity from special. It teaches us that we need to rethink what we thought we knew about how to do gravitational physics. Furthermore, it is a very powerful tool for us later on, in distinguishing whether statements in relativity can or cannot be true.
In special we learn how to develop the Lorentz transformations once we recognize the constancy of the speed of light (and more fundamentally, the laws of physics) for all observers. We learn of the Minkowskian nature of space-time and the associated coordinate transformations and tensors. But we have a pretty hard time bringing gravitation into special relativity. In fact, they are not compatible. There's a leap of insight that we need, and that leap is facilitated by recognizing that the effects of a uniform gravitational field are indistinguishable from a uniformly accelerated reference frame (equivalence principle). How can we explain this? What we find is that a geometric interpretation of gravitation naturally provides the answer. Indeed, the effect of gravitational redshift implies curvature! (See Schild, 1960, 62, 67).
We'll now discover the power of the equivalence principle with your question about detection/existence of homogenous fields:
I'm not entirely sure what kind of curvature would be required to produce an ideal, homogenous "gravity field". In fact, technically it should be impossible to detect at all, right?
OK, so what kind of curvature would produce a homogenous gravity field? Remember what we discussed earlier about the development of the curvature tensor from geodetic deviation. Let's suppose we have some region of space-time operating under what we will call a homogenous gravitational field. We will put two test masses in it, let them fall, and measure their separations over time. What will we see? Nothing. No separation. There are no tidal forces. So we conclude that there is no curvature. And no curvature means no gravitational field!
Equivalently, we might notice that a simple coordinate transformation will remove the effect of uniform acceleration. And if we can do that, we can again conclude that there is no gravity.
Does that sound nuts, or no? Consider this instead. Suppose we have a little box in orbit above the Earth. Or in free-fall toward the Earth, doesn't matter, they're equivalent for our purposes. There are two test particles in the box. We watch their motions. We see no geodetic separation. As far as we can tell, the two particles just move in straight lines. So we conclude there's no curvature, thus no gravity? Earth's grav field doesn't exist? No, we must remember what the equivalence principle tells us: if you use a frame of reference so limited that curvature appears flat, then you see no geodetic separation, no tidal forces, no gravity. That's the key! If you observe no separation of geodesics, then you conclude the space-time is flat to within precision of measurement and extent of your reference frame.
So, extend the reference frame. Or improve our precision. Then, for a real field, we'll start to detect those tidal forces. We'll conclude that there is a gravitational field there. Thus we can know that a gravitational field exists if, and only if, we can detect curvature. We cannot detect curvature if things accelerate uniformly -- that's the equivalence principle. Equivalently, we know it exists if we cannot globally remove it by a simple coordinate transformation. I.e. we cannot remove the effects of the Earth's field everywhere just by changing coordinates.
Now, return to the concept of a globally homogenous field. If we suppose the universe is permeated by such a field, then we find that this is fundamentally untestable. There would be absolutely no experiment that could detect its effects. No reference frame large enough, no precision high enough.
[Added] Just in case there's any confusion between this and dark energy, no this does not mean dark energy is untestable. I'm leaving out a little bit of detail, which is the importance of symmetries (in this case, isotropy). Dark energy is homogenous in its energy distribution, but it is also isotropic. It has curvature, and the same curvature everywhere. It therefore affects the expansion rate of the universe. The homogenous field Herra and I speak of above, which can be considered as being produced from a flat, infinitely extended mass distribution, is not isotropic. It does not produce curvature, and it does not affect the expansion rate.
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It is also worthwhile to point out that this homogenous field is inconsistent with the formulation of general relativity. There is no solution to the equations that preserves all the conditions of such a field. (See Section 7.4 of Crowell's text on GR (http://www.lightandmatter.com/html_books/genrel/ch07/ch07.html#Section7.4).)
Although I'm not sure if what you wrote actually means that we would be able to detect an universe-wide homogenous gravity field based on photons acting differently than they should?
"Acting differently than they should?" How are photons supposed to act? In GR, photons follow null geodesics. How would a globally homogenous field cause them to do otherwise?
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Something I recently realised is that the best answer for the whole question of 'if you travel back in time do you end up floating in space because the earth moved away?' is that your time machine just stays in its inertial reference frame and starts orbiting the core of the earth.
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Reading through this, I feel like a first year undergrad again; reading through science papers, recognizing a word or two, needing a dictionary for the rest, and lacking context for it all. So much to learn, so little time....
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I at least know the general basics of some of what's being discussed, but I'm still left pulling a complete Keanu "woooooaaaaah" with most of it.
(Also I'm strongly reminded of why I never went any further with physics. I loved most of the conceptual stuff, but the math that arose as you delved into it just destroyed me.)
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Yeah, the mathematics becomes kind of brickwallish (https://www.youtube.com/watch?v=duVq7cXWcYw) in the difficulty curve, especially once you hit tensors. I'm learning my way through tensors through some texts, but actually getting to the point of being comfortable with using them is a slow, arduous endeavor. I'd like to get there before grad school, but we'll see. :)
@Herra, I was reflecting on what you were asking about the behavior of photons earlier, and wondered if you were thinking about what photons do in a uniform field as used in the equivalence principle in the following sense: In a uniform field like that, we would not see the photons bend, right? The field has no effect that we could measure. But in a uniformly accelerated frame, we do see them bend. Let me know if that's close to right and if this helps clarify or not:
What we mean by a uniformly accelerated frame is like having a uniform field permeating that frame, but it is not the same as a frame which is freely falling within a uniform field. The uniformly accelerated frame is like being in a spaceship which is firing its engines. It is also the same as being in a ship sitting on the surface of a planet. If the acceleration of the ship in scenario (a) is the same as the surface gravity of the planet in scenario (b), then there's no way to distinguish the two.
The other example of the equivalence principle is for freefall within a gravitational field, vs. not being subjected to a field at all. You again can't tell the difference.
But, that's not quite right because there is of course an exception. The exception is if your frame of reference is sufficiently large, and precision of measurement sufficiently fine, that you can detect tidal forces. A real gravitational field produces tidal forces, whereas a uniformly accelerated frame (the rocket firing its engines, or a universally uniform field) does not produce them. This is how you detect gravity. You stop trying to care about what's going on around you. Ignore the temptation to look out the gedanken experiment's window to see if there's a gravitational mass nearby. Don't try to do physics with a global perspective. Physics is simple only when examined locally. Locally, you detect gravitational fields because they produce tidal forces. You cannot distinguish gravity through a uniform acceleration.
(Incidentally, this has a lot to do with why we sometimes call gravity a fictitious force, just like Coriolis or centrifugal force. In the local frame, the effect of these forces can be made to vanish by a coordinate transformation. This happens any time you have a force which is proportional to mass, so that all test particles behave the same way. Thinking of gravity as 'fictitious' is pretty mind bending.)