Joined: 3/2/2010
Posts: 2178
Location: A little to the left of nowhere (Sweden)
I agree that the idea isn't unfounded, the question became a debate over if there is enough air in the tires of a car to accomplish what the movie did. I was just lacking the necessary skills in physics to produce proofs of it (not) working, which Ferret Warlord helped me with above.
While there's more parameters to it than what Ferret Warlord used in his estimation, I don't see how those parameters can affect the end result enough to change the outcome. I think it's fairly safe to say that the stunt is not possible.
Thanks for the help!
Still it's way closer to being possible than the situation with the floating house in Up. I read somewhere that it would require roughly ten thousand the number of balloons depicted onscreen to get a house to float in air in real life.
Joined: 10/27/2004
Posts: 1978
Location: Making an escape
thatguy wrote:
Still it's way closer to being possible than the situation with the floating house in Up. I read somewhere that it would require roughly ten thousand the number of balloons depicted onscreen to get a house to float in air in real life.
Not to mention the strings would have to be evenly spread across the whole foundation, and not just the iron grate in the fire place that holds the burning logs. Whatever that part is called.
A hundred years from now, they will gaze upon my work and marvel at my skills but never know my name. And that will be good enough for me.
This has probably been discussed already a million times all over the internet, but nevertheless I think it would be an interesting question.
At the end of the first Superman movie (1978) there's a ridiculous deus-ex-machina about Superman flying around the Earth and reversing time. Never mind the ridiculousness, I was thinking what actually would happen in this situation?
Assume that Superman were to fly in circles around the Earth at, say, 0.999c. How does this affect his time with respect to the time on the surface of the Earth?
(This ought to have some effect. After all, flying in circles at that speed means he's in constant acceleration. In fact, it's quite an enormous amount of acceleration, but he's Superman so he can withstand it.)
Actually, the Silver Age Superman (upon which the 1978 film was based) could fly forwards and backwards in time at will; flying around the Earth and reversing time was the way the filmmakers decided to use to explain what was going on (and given how many people misunderstood the scene, they clearly failed in this regard...). So the only ridiculous part is the fact that he had this power in the comics to begin with, not the fact that he used it in the film.
Anyway: assuming that, on the Earth's reference frame, Superman is moving in circular orbits of constant speed around the equator (supplying whatever acceleration to keep moving in this orbit by his own power), and ignoring the effect of gravity for the sake of simplicity, we can compute how much proper time elapses for Superman. Using spherical coordinates, his motion as seen from Earth's reference frame is given by
r(t) = constant
theta(t) = pi/2
phi(t) = v * t / r
Plugging that into the line element for spherical coordinates in space-time, we have that
[c * d(tau)/dt]^2 = c^2 - (dr/dt)^2 - [r * d(tetha)/dt]^2 - [r * sin(theta) * d(phi)/dt]^2 = c^2 - v^2
where tau is Superman's proper time. Rearranging, we get the normal "gamma" factor:
d(tau)/dt = sqrt[1 - (v/c)^2]
So without gravity, he would just get his normal time dilation for speed. Which is expected: acceleration does not directly affect proper time in SR, it only indirectly affects it because it affects velocity; but in circular motion with constant speed, the velocity only changes direction.
The next question would be: if you factor in gravity, how are things changed? Ignoring Earth's rotation (I can deal with it later), the line element would end up being
[c * d(tau)/dt]^2 = (1 - rs / r) * c^2 - [r * d(phi)/dt]^2 = (1 - rs / r) * c^2 - v^2
or
d(tau)/dt = sqrt[1 - (v/c)^2 - rs / r]
where rs = 2 * G * M / c^2 is the Schwarzschild radius of the Earth; that is, rs = 0.00887 m. You still have to divide it by the radius of Superman's orbit, which will be greater than 6400 km -- meaning the effect of gravity will be tiny given the high speed you game for Superman. The terms involving Earth's angular momentum would be about an order of magnitude or more smaller than the term with the Schwarzschild radius, so we don't need to pay much attention to them.
I can give however many details are desired for the derivation of the equations above; I just went more for the results in this post.
So without gravity, he would just get his normal time dilation for speed. Which is expected: acceleration does not directly affect proper time in SR, it only indirectly affects it because it affects velocity; but in circular motion with constant speed, the velocity only changes direction.
That's strange. Since he's in constant acceleration, I would have thought that his frame of reference is constantly changing (even in SR), which would cause an effect similar to what happens eg. with the so-called twin paradox (which is also based on changing frames of reference due to acceleration).
Warp may (or may not-- I don't really know) be touching upon a subject known as Ehrenfest's paradox. I was interested in it back in high school, but since I never became comfortable with general relativity, I don't have any insights to offer. I can't even properly interpret Wikipedia's resolutions of the paradox.
Superman's reference frame will indeed be constantly changing; but this is irrelevant since I performed the calculation in an inertial frame. SincenI computed the rate of change of Superman's clock compared to that of an inertial frame (Superman's proper time versus Earth's proper time) the result is an invariant – it does not depend on which reference frame is used to compute this value. Had I tried to work on Superman's reference frame, I would have to deal with countless issues – for example, Superman's frame will undergo Thomas precession, and he will experience a breakdown in simultaneity (see Bobo's post) – which would make the problem much harder.
But the ultimate conclusion would be the same – his observed "time dilation" and "space contraction" will not be different than if he were moving with constant velocity.
Had I tried to work on Superman's reference frame, I would have to deal with countless issues – for example, Superman's frame will undergo Thomas precession, and he will experience a breakdown in simultaneity (see Bobo's post) – which would make the problem much harder.
Doesn't look much harder. You just have to put phi-w*t instead of phi here:
marzojr wrote:
dt^2 gets the familiar c^2-v^2 term, but now the line element is not diagonal, there's a term multiplying dt*d(phi), but since superman is at rest in his own frame d(phi) = 0, so he feels the effects of a diagonal line element, just as he would if he wasn't spinning. But, for example, if superman had small oscillations in the phi direction, he would measure fictitious forces that wouldn't happen in an inertial frame.
I think that's the source of Warp's confusion, even though his proper time is the same as in a movement in a straight line, there must be some physical effect that's different in the rotating frame. How could Superman measure his own centripetal acceleration?
I don't know even nearly enough of GR to understand whether an object in acceleration (which is what Superman in this scenario is, given that he needs to constantly change his flying direction, and thus he's experiencing measurable acceleration) experiences the same passing of time as a non-accelerating object moving in a straight line.
I don't know how time works in accelerating frames of reference.
I don't know even nearly enough of GR to understand whether an object in acceleration (which is what Superman in this scenario is, given that he needs to constantly change his flying direction, and thus he's experiencing measurable acceleration) experiences the same passing of time as a non-accelerating object moving in a straight line.
No knowledge of GR is needed. Check the formula for the time dilation, it only depends on the magnitude of the velocity, not on its direction. So, if everything the acceleration does is change the direction, the factor of time dilation stays the same.
If it's not clear yet, take the example of a frame with linear acceleration on the x direction. It's a non-inertial frame, but the y and z axis, being perpendicular to the motion, are completely unaware of this acceleration. The same thing happens in the rotating frame. Here the frame is non-inertial, but the t axis doesn't care because it's in some sense "perpendicular" to the acceleration.
Warp wrote:
I don't know how time works in accelerating frames of reference.
It's almost the same as in classical mechanics. You can't apply Newton's laws of motion in non-inertial frames, but if you insist, you can get correct results by adding fictitious forces to the system. They're called fictitious because they don't correspond to actual physical interaction, are simply artifacts introduced to solve the problem.
So, in SR you can do the same. Solve the problem in an inertial frame, change coordinates to non-inertial ones and evaluate the fictitious forces that arise (more technically, we usually evaluate the line element, since calculating forces gets a bit messy in SR). Then when you want to solve a more complicated system you just solve the problem in the accelerating frame inserting the fictitious forces. These "forces" might affect the time dilation of a system, or might not.
GR is constructed in a way that its equations can be computed in any frame, inertial or not, so we don't have to do the previous procedure (although in practice the equations are much harder to solve). This was actually a requirement of gravity. You can find thought experiments by Einstein that prove that a theory that restricts its laws to inertial frames gets absurd in the presence of a gravitational field, but that's another story.
Doesn't look much harder. You just have to put phi-w*t instead of phi here:
That is indeed a popular way of doind it; it also happens to be wrong: it does not result in Superman's frame, far from it. You can Google for "Trocheris-Takeno transformation" and "modified Trocheris-Takeno transformation" for two alternative ways of moving to a rotating frame, as well as for references and critiques of the transformation you give. Trust me, correctly handling rotating reference frames is still an open research question, it is not simple.
Quantum entanglement, the double-slit experiment and things like the delayed-choice quantum eraser experiment have always been rather esoteric and have always had a feeling of been very far removed from actual everyday life and visible applications (other than possibly some rather non-descript-to-the-layman interference patterns on a photographic film).
However, researchers have now taken a step further and have actually demonstrated quantum entanglement in action: Taking a "photograph" of an object with photons that never touched nor reflected from said object. Instead, photons emitted by a light source are split in a manner that they get entangled, then one stream hits the object and the other stream hits the camera. Thus the photons that hit the camera never came into contact with the object being "photographed" this way. Yet the image appears on the film. The only reason for this is quantum entanglement.
http://www.eurekalert.org/pub_releases/2014-08/uov-qpe082814.php
The universe is generally described as four-dimensional, with the fourth dimension being time. However, does the time axis really exist?
If I have understood the history of special relativity correctly, Einstein himself didn't describe spacetime as four-dimensional in his original paper (published in 1905). It wasn't until 1907 that Hermann Minkowski came with the idea that the theory could be most conveniently formulated in four-dimensional space. In fact, his formulation is described like this (emphasis mine):
"In mathematical physics, Minkowski space or Minkowski spacetime (named after the mathematician Hermann Minkowski) is the mathematical space setting in which Einstein's theory of special relativity is most conveniently formulated."
In other words, thinking about the universe as four-dimensional spacetime seems to be just a convenient mathematical tool to make calculations and understanding easier, rather than describing actual reality.
Or does it describe actual reality (thus being what could be called serendipity)?
In the context of relativity, especially general relativity, time acquires a character no less real than that of space dimensions: relativistic physics becomes geometry in a 4-dimensional non-Euclidean space, and this space is curved. What is more, if you consider only curvature of space, instead of space-time, you get several results wrong: for example, you get half the value of light deflection by gravity that you observe with modern experiments. So in this sense it can be said to exist.
Moreover, traditional "classic" quantum mechanics has long since been replaced by quantum field theory, which basically can be summed up as quantum mechanics in the spacetime of special relativity (although this is a gross simplification). Since quantum field theory can give some pretty accurate results in general, one can infer that time as an axis exists here as well.
Fact of the matter is that, if you really want to, you can describe all of physics without time as a "true" axis. Your equations will be hideously complex, much more so than the alternative; and so will interpretations of what you come out with, because you will have to add a load of novel effects to explain the appearance of relativistic effects. In the end, it is simpler to just use relativity. Scientists, being ultimately pragmatic, takes this approach and assume that the theory means time exists as a separate axis.
But you ought to know that the creator of any theory or hypothesis in science is not the ultimate arbiter about it -- physics has moved on from Einstein's original views quite a bit, so what he believed about relativity is essentially irrelevant today except for a historical perspective. Physicists today work with time as a separate axis as a matter of fact.
It is worth noticing that time is not quite the same as the other spacetime dimensions for one reason - you can not travel backwards in time (in special relativity anyway). This is the property that conserves the uniqueness of the time axis within the spacetime it sits in. Otherwise you could pick any axis and label it time, much like you can rotate your paper however you want and still draw x- and y-axes on it.
It is worth noticing that time is not quite the same as the other spacetime dimensions for one reason - you can not travel backwards in time (in special relativity anyway). This is the property that conserves the uniqueness of the time axis within the spacetime it sits in.
Since Marzo mentioned quantum field theories (and I've been wanting to talk about QFTs for a while :P), I think it should be mentioned that the picture changes radically when you introduce quantum mechanics to relativity. In a classical relativistic field theory (like the electromagnetic one in Maxwell's equations) you have to systematically hunt down and eliminate all solutions that travel backwards in time in order to maintain causality in the theory. So, causality has to be added as an extra postulate.
In QFT, however, it's the opposite. Interpreting the equations naively will tell you that the probability of a particle propagating faster than light is small, but positive. But if you understand relativity in the correct way, the famous equation E=mc^2 allows the particle to trade some of the mass for energy and split into many other particles via radioactive decay, and quantum mechanics requires that you sum over all possible trajectories of the system. Therefore, it's logically inconsistent to consider a system of a single particle, because in nature they'll always be under influence of a field generated by all the particles it can transform to.
In this setting, the probability of a path is a complex number, and when you calculate the probability of faster than light propagation, you get 0, because every path is canceled exactly by another path going the same direction, but backwards in time. The surprising result is that QFT maintains causality because of the time-reversed paths, not in spite of them :D
The time-reversed solutions are the antiparticles, since you can understand the positron as an electron carrying its negative charge backwards in time, so it reverses its interactions in our forward-time frame, and is thus indistinguishable from a particle with the same rest mass of the electron, but opposite charge.
I have a question. In more complicated QFTs, we encounter ghost particles, which must have their interactions summed into the formulas for every physical process. They have unphysical properties, but that doesn't matter, because every time you calculate the probability of observing them directly, it turns out to be zero, so ghosts don't introduce a contradiction.
That leads to my (somewhat philosophical) question. If we understand that the particles and their interactions form the fundamental physical laws, do ghost particles "exist" (even in the sense of spontaneously appearing and then disappearing before you can detect them)? For one side, their interactions have to be considered for the theory to make any sense, and a theory without ghosts can't be asymptotically free, as detected in experiments, so their existence is detected experimentally. However, if you try to produce/observe them, you'll always fail, and that suggests they're just a mathematical trick to solve the problem.
If someone asks you "Do ghost particles exist? Yes or No?", what's your answer and why?
EDIT: Happy 500th post for me! \(^o^)/
Re: time travelling backwards in time in relativity: technically, you can do it. Well, not you: it is possible that some things do travel backwards in time, the equations of relativity (special or general) don't prevent that. What they do prevent is for something that was going one direction in time to turn around and go the other direction (although some of the weirder geometries in general relativity allow just that by moving in a specific way).
Re: ghost particles: personally, I think that the usual interpretation is a load of crap, an attempt to make sense of a mathematical trick that gives the correct results. Especially since gravity is nowhere to be found in QFTs because no one figured out how to do it yet*. Yes, it gives correct results; for all we know, it might be because QFTs give a way of approximating the effects of gravity in the quantum scale.
* And yelling "but gravity is too weak!" is a lame excuse -- general relativity is nonlinear, meaning gravity can become exceedingly strong in short range depending on the distribution of stress-energy-momentum, or even cause some other weird stuff such as flipping the sign of eletromagnetic fields at close range, as happens for a spinning charged blackhole.
I think there was a solution to a particle orbiting two rotating black holes (inside their ergospheres) to "catch up" with itself, iow. encounter itself.
I think that it has something to do with the fact that distances inside the ergosphere of a rotating black hole increase faster than c (due to frame dragging). While nothing can travel between two points faster than c, the distance between two points can increase faster than c (which is a different thing). Thus if you curve the path in a manner that it creates a loop, you can have a particle that "travels faster than c" to meet itself, basically traveling back in time. And this is exactly what would hypothetically happen under certain configurations of rotating black holes.
If this were to happen in real life, I don't even dare to guess what happens if a particle collides with itself. At the very least it ought to cause some kind of paradox. (I think some scientists postulate that such paradoxes are impossible because there's some fundamental property of existence that forbids them from existing...)
There is Novikov's self-consistency principle/conjecture, which he formulated after trying (and failing) to set up a theoretical particle version of the grandfather paradox and always coming up with no paradox. The Wikipedia page sums it up nicely. So that particle hitting itself would probably just deflect its past self in such a manner that, when it bumped into itself in the future, would cause the same deflection it had experienced.
So it would be like "if we shoot this particle at this closed timelike curve exactly in the right direction with an exact momentum, it will collide with itself, setting itself in a path to collide with itself" and if they do it, it will happen, but only then.
It seems still paradoxical, though. If the collision never happens, it would just continue on its normal path (which may not result in any collision with itself). Why would the collision happen in the first place? Seems like some kind of ontological paradox.
The original idea was this: if you want the particle to bump into its past self in such a way that its trajectory will be changed enough that it won't bump into itself in the future (thus creating a paradox), you find out that you can't -- if the particle bumped into its past self, it will do so in a manner that will deflect it to bump into its future self in exactly that way (thus preventing a paradox).
In a sense, yes, it would be an ontological paradox; Novikov's principle is exactly that: the only paradoxes you can get in a time travel scenario are non-contradictory ontological paradoxes.
Re: ghost particles: personally, I think that the usual interpretation is a load of crap, an attempt to make sense of a mathematical trick that gives the correct results. Especially since gravity is nowhere to be found in QFTs because no one figured out how to do it yet*. Yes, it gives correct results; for all we know, it might be because QFTs give a way of approximating the effects of gravity in the quantum scale.
* And yelling "but gravity is too weak!" is a lame excuse -- general relativity is nonlinear, meaning gravity can become exceedingly strong in short range depending on the distribution of stress-energy-momentum, or even cause some other weird stuff such as flipping the sign of eletromagnetic fields at close range, as happens for a spinning charged blackhole.
I agree that the usual interpretation is not good. It's said the ghosts are just the gauge being fixed in a strange way, so you shouldn't worry about them. Well, the photon also appears by fixing the gauge and you rarely see people saying photons are unphysical and just mathematical artifacts. For me, if you're gonna use exotic gauges in QFTs, at least treat the particles you introduce seriously. I guess people just find it fun to couple fields to QFTs without worrying about what they're doing. It's a smart way to derive the theories, because there are few gauge groups and you can brute force them all until you find one that works. The problem is that the standard model ends up being a horrible mess that frightens small children.
Your remark about gravity made me curious. What's your interpretation of asymptotic freedom? Is there a good reason to believe gravity alone can introduce it at the QCD scale?
That is not what I said; but then again, I said it in a very inaccurate way. What I meant to say was this: for all we know, the sum-over-all-possibilities aspect of QFTs could just be a way of approximating the effects of gravity at the quantum scale, and a full theory of quantum gravity would explain why this process works (and why renormalization is needed in QFTs), and give a whole different interpretation of the whole process. There has been some research on that front, but it is not yet "mainstream" physics.
But don't pay too much attention to me in this; particle physics and QFTs are not my field, I studied GR far more. That is, before I decided to leave physics for computer science.
It is worth noticing that time is not quite the same as the other spacetime dimensions for one reason - you can not travel backwards in time (in special relativity anyway).
How do you even define "traveling in time"? Movement is a change of position over time, so how does that work when you're talking about the position in the time dimension?
p4wn3r wrote:
If someone asks you "Do ghost particles exist? Yes or No?", what's your answer and why?
I don't read that question as a question about ghost particles, since the answer would not yield a better understanding of ghost particles, nor would it predict any new testable properties.
Instead, I read that question as a philosophical question about the concept of "existence": How do we define existence in such a way that the concept is useful even in weird corner cases like these? Cases that were unthinkable back when humankind first discovered the concept of existence, when existence were just the things we could see or touch?
Since this is a philosophical question, I refuse to attempt an answer.
Of course the scientific question is: "Does this set of formulas, which includes unobservable particles, accurately describe the experimental results we obtained?"
Same for warp's question. We all agree that time exist. Great. We also agree that formulas using a four-dimensional spacetime agree with experimental results and are thus a good model for the world we live in. Also great.
Does that mean that we can split our formulas into small parts, and for each part find a direct physical representation in the real world? Well.. try hitting the universe with a hammer until it breaks, sift through the pieces until you find time and ghost particles, then look at them in isolation. If you can't, you'll have a hard time answering the original question.
What you can ask is: "does time as it exists in the universe have dimension-like properties?". Considering that the simplest way to model time is as a dimension, I think you can answer that.
Oh, and don't forget that Novikov's self-consistency conjecture is just that: a conjecture, a sentence starting with "Wouldn't it be nice if ...?" Of course paradoxes are a problem, but they don't go away if you claim they don't exist, no matter how reasonable the conjecture sounds.
Warp wrote:
If this were to happen in real life, I don't even dare to guess what happens if a particle collides with itself. At the very least it ought to cause some kind of paradox.
Just to be clear, particles don't have a unique identity, so from the particle's point of view, it doesn't collide with itself, but just with another particle of the same kind. The SciFi-notion that terrible things happen if you see or touch your past self, but you're safe as long as you're just blindfolded in the same room, those are just SciFi.
The paradox would only arise from the causality loop, but that doesn't require both particles to be the same. Particle A could emit a photon B which [ travels around the black hole / falls into the worm hole / jumps into the Tardis ], thus arriving in A's past and hitting it.
