md65536

No it doesn't. The Cauchy surface is a representation of 'now'. The foliation is the decomposition of all of spacetime into a set of all moments, not just one 'now' but all of them. It's like mixing up what is a tree and what is a leaf. The universe can't be fully represented by a Minkowski diagram because spacetime isn't flat. The thread is talking about a common moment throughout the universe, that includes in gravity wells etc. If you want to explain the maths, start with this:

Those are Minkowski diagrams, which represent one spatial dimension and time in flat spacetime. Time is on the vertical axis, and a horizontal line represents one moment for the observer at rest in the diagram's reference frame. A set of horizontal lines covering the diagram would be a foliation into Cauchy surfaces for that observer. If you allow an infinite number of instants of time, you could allow an infinite number of lines, completely filling the diagram. A set of straight lines at an angle, up to approaching 45 degrees, would represent a foliation of that flat spacetime according to other inertial observers. In curved spacetime the lines wouldn't be flat. Add another spatial dimension and it would be a curved sheet instead of a line. Consider all 3 spatial dimensions and you get a 3d surface in a 4d volume. I feel like it would take pages to describe the details of the analogy, and all the ways that it is not like the math. And still it would not be as precise as the one line of maths that it represents. Also I think I have a terrible habit of trying to base conclusions off analyzing an analogy instead of the maths. Perhaps an analogy is a useful way of thinking about what the maths represent, but the details will only be found in the maths! Eg. in the definition of a foliation, etc.

Dumbing it down to try to figure it out... A foliation of spacetime is by analogy like a book of pages. Each page represents a Cauchy surface ie. one moment of time throughout the universe, in that particular foliation. The book can be twisted and curved, maybe stretched, I dunno, however none of the pages can intersect (the moments are well-ordered) and there are no "gaps" between pages anywhere (the foliation covers all of spacetime). Edit: Note the pages/Cauchy surfaces are 3d and as big as the universe at that moment. The book analogy drops one spatial dimension to represent space as a 2d page. However there is not a unique foliation. Different variations of books, distorted (and oriented?) in different ways, can cover the same spacetime (though not every foliation of spacetime would satisfy the homomorphism? Eg. rotate the book enough or twist it enough and a page can no longer represent a moment). What this means is that you could choose a particular foliation and say that its Cauchy surfaces each represent a common "now" shared throughout the universe, however the choice would be arbitrary and it wouldn't be easy to get everyone to agree on it. In very different frames of references, the "common now" would not be experienced meaningfully as a single moment. This would be like arbitrarily choosing an inertial frame of reference and defining a universal time based on it (Lorentz ether theory?), which wouldn't describe local time very well in other frames, though everyone would agree with the arbitrary choice's ordering of events and their causes. Is this explanation accurate?

Yyyyup. I like to think things through using common sense so I'll try to explain it as I understand it: Imagine that you have a ship with enough fuel to accelerate to half the speed of light. Suppose after that, you meet up with a refueling ship that is traveling at that same speed (you're now relatively at rest with it). You refuel, and can accelerate to half the speed of light relative to that ship. Suppose you can repeat this indefinitely, and there is always a refueling ship ready at whatever speed you get to. Can you reach the speed of light by repeating this a finite number of times? Remember that the speed of light is equal to c in all reference frames. If you are traveling at the speed of light, you are traveling at that speed relative to all inertial observers. But after every acceleration, you end up in a new rest frame, and now have to accelerate all the way from v=0 to v=c relative to that new observer. In each rest frame light is still faster than you are by a speed of c. You can never catch up... It's as if you've not made any progress at all. Does that make sense? Even if it does, you're still repetitively moving faster relative to an observer in that first rest frame, so why do you never reach a speed of c relative to it? It is because relative velocities aren't additive. Instead, using the "composition of velocities" formula, you can repeatedly compose 0.5c (or anything less than c) and the result will still be less than c after any finite number of repetitions. In other words after each acceleration phase, you've greatly changed your velocity relative to an observer you were recently at rest with, but only slightly changed your velocity relative to an observer that was already traveling at near-c relative to you. That's what the maths will describe.

In other words, you will effectively have to lift an entire column of water to make room for the object at the bottom. The water then falls bit by bit as the object rises.

Yes, the speed of light is c in all frames is a postulate of SR. And given the postulates and other assumptions, relativity of simultaneity is necessary. So right there is a "good reason" for it. If you want to argue it is superfluous, you must show that the postulates can be assumed and still lead to a different result, without relativity of simultaneity (in other words that there is a mistake in the derivation of SR, which everyone has missed for 110 years). Or you could argue that the postulates themselves are flawed (you could find evidence for something better, which everyone has missed for hundreds of years). It makes no sense to argue that relativity of simultaneity isn't necessary without even addressing the reasons why it is necessary in SR. Unfortunately that means that most of your original post can be thrown out in revision. You're attacking imagined pink elephants that no one else is imagining. To say relativity of simultaneity doesn't make sense, you'd better understand very well the details of why it makes sense to others, and find a flaw in that, rather than just supposing it's all an assumption of something ridiculous.

http://en.wikipedia.org/wiki/Time_dilation#Muon_lifetimeis a counter example I think. A particle traveling through the atmosphere at relativistic speeds will be time-dilated according to Earth observers (and vice versa). This is true for inertial particles traveling through space, passing through the Earth unaccelerated.

I think it's more likely a mixup between frames, but I can't be sure because the explanation makes no sense to me and no concrete example was given. I can come up with an example using the Andromeda paradox. Suppose somewhere in Andromeda a cat knocks a glass off a table (event A) causing it to shatter on the floor (event B). On Earth, an observer O- walking towards Andromeda has that event A and B already happened last week, while an observer O+ walking away has that they will happen next week. (This is just Andromeda paradox, according to standard simultaneity). An observer P can walk toward Andromeda, and turn around and walk away, and the coordinate time that elapses on Andromeda according to P is negative. P can walk alongside O- and say that effect B has already happened, and then alongside O+ and say that effect A has not happened yet. This is the closest I can imagine that the explanations of an effect preceding a cause are describing. And it's all true, but it's a trick. The problem is that events were selectively transformed from one frame to the other, and event B in the O+ frame was compared to event A in O- frame. Of course all sorts of impossible things can be derived from mixing frames. Yet, at no point can P say that B precedes A. As P accelerates, both events A and B must be transformed to compare them. While alongside O-, P agrees that event B happened last week, but so did event A, earlier. While alongside O+, P agrees that event A happens next week, but so will event B, later. CasualKilla's statement that is in contention, "If event A causes event B, then event A occurs before or simultaneously with event B in any reference frame," is still true, for any given frame even in the example. I know it's not fair to set up an argument for the opposition and then knock it down, but in absence of a better example from them it is the best I can figure. One could describe a space-like interval between A and B where the order actually is reversed, or something close to light-like that requires calculation to make sense of it, but if A and B are time-like separated, their order won't be reversed by choosing a different reference frame.

Sorry for vexing you, but thank you for being respectful and willing to help and learn from others, who are still learning too. I do not understand this yet, but I do not want to be one of those who gives up trying to learn about relativity just because someone on a forum tells them they won't get it. Unfortunately every other source I've read says that SR does not change the order of causally connected events. The maths do not allow it without speeds exceeding c. I am having trouble making sense of your counter claim. Your two other cases in full are: I do not yet understand how less elapsed time reverses the order of cause A and effect B. The elapsed time is still positive. You say this is an explanation, but it is not clear enough to me. This does not show that event B (which is caused by event A) precedes event A, as you claim it does. I appreciate any help in understanding this. It is also (x',y',z',t'), etc. An event doesn't occur in only one frame. But I empathize with you.

Thanks! Would it be fair to point to this example in the future as demonstrative of your level of understanding of the equations you transcribe? You have event A happening at time 0 and event B happening at time t, according to A's clock. Time t is after time 0. You have event A happening at time T and B happening at time T+t, according to B's clock. Time T+t is after time T. Nowhere is demonstrated B preceding A. Events don't have single clocks, so I would have called the events and the clocks by different names. A and B are different clocks, I wouldn't have used the same variable t. Letting B happen at time T+tau allows for all of the different situations you described. Certainly, t can be less than, greater than, or equal to tau. However neither t nor tau is negative. By any of these clocks, event B occurs after event A.

Please continue to the example where event A causes event B, but B precedes A. I'm not seeing where you're going (unless you think that the order of events depends on the coordination of clocks, and that an event happens earlier if you set a nearby clock to read earlier).

That's wrong. You've written out maths to cover all cases, including space-like intervals which don't have a unique order, but which cannot describe causal relationships. Do you understand that? Or do you have an example where event A causes event B, but B precedes A?

The statement is about causal relations. There is no "general case" in which CasualKilla's statement is false.

The sign can change only for space-like intervals. Causal relations are time-like or light-like, they can't be space-like. CasualKilla is correct. If event A causes event B, there is no frame of reference in which event B precedes event A. Do you have a counter-example, of an event that precedes its cause? The only exception I can think of is faster-than-light particles, which are only speculative.

The link is a .cgi, which means "common gateway interface", which means that it is a program on the web server that dynamically generates the web page that you get. I assume the program checks the input and gives you a "correct" or "incorrect" message depending. Nowadays, most websites contain dynamic content so you can't assume that a webpage that you get is the only one that you'll ever get. However, I've come up with the same 73-digit answer twice, which the website says is wrong. Wolfram alpha solved it for me , but I might be making a mistake. There are always bugs possible, but I'd bet on a bug in the puzzle website before a bug in wolfram alpha's solution. Can you split that into 2 equations of the form "x mod d = r"? Wolfram alpha will solve that. Note that x in your input means something different.

20740059257 considers only the first few equations in the puzzle. The full answer I have is 73 digits long. I believe it's right (go wolfram alpha!) but if it's not, the answer will be smaller than that. I suspect that OP says the answer is 80 digits because the web page input allows up to 80 characters. However, brute force calculations and hardware-precision numbers aren't going to work. Note that considering ONLY the last equation, you know that x mod 800009 = 438462, so the smallest possible answer is 438462, and the next smallest is 800009+438462, so it is a waste to check every number in between. Even so, with optimizations like that I think you'll end up running a brute force algorithm for many magnitudes of time greater than the expected age of the universe.

Wolfram has no trouble with these equations and 80-digit numbers. On second thought I don't think it matters that 2 and 600008 have a common factor. "x mod 600008 = 318753" already implies that "x mod 2 = 1". They're consistent and the latter is superfluous. I don't see how its inclusion could screw up the answer.

It should only take a few minutes of computation. You'll need to do "multiple precision" or "big int" calculations, eg. gmp c library. However you can also offload the calculations to wolframalpha! I came across CRT, I think that's essentially the way I tried... Does the CRT apply here, since 2 and 600008 are not coprime? Is that okay, or perhaps 600008 is a typo?

This was posted in Homework Help... is it an assignment? Why do you want just the answer? If it's for homework, please explain what you've got so far. How do you know it's 80 numbers long? I came up with an answer but it's fewer than 80 digits and the website says its wrong...

The video is nothing but pseudo science that uses some science jargon but doesn't actually explain anything. 2:30 some physical phenomena are describes as "mysteries" that "don't fit" in the clueless scientists' view of the world. 2:50 Equations are only a vague attempt to explain stuff. Throwing out old assumptions and constructing "a new blueprint for reality" is a bold step that should be taken. 3:20 "Scientists believe" ... that "nature just doesn't make sense. Hmm..." 5:00 "We need to assume that space is literally and physically quantized" -- why? -- "That it's made of interactive pieces." Oh god, this is another "Space is made out of tiny particles" theory... 7:50 Space is now a medium. 8:05 Curvature is "explained" in terms of the density of these quanta. Denser quanta = less resonating = "they experience less time". 9:30 it somehow explains quantum tunnelling, 10:22 and "where the constants of nature come from" (pi) 12:20 and of course it explains dark matter and dark energy. 12:40 scientists can't explain it, but 13:40 a change in the density of the imaginary particles of space "is going to cause a gravitational field" explains it. So it is just another "scientists don't know anything (as far as I understand)", "my theory explains everything, by analogy, if you just use your imagination", and "I can fit it into some scientific jargon you may have heard, like 11 dimensions, quantum mechanics, Einstein curvature!, dark matter and dark energy!" attempt to sound legitimate.

A person would experience passing the event horizon, only there wouldn't be any nearby effects that indicate its happening. By analogy, if you walk between two markers, you experience passing the halfway point, but you wouldn't know you're there except through calculation or observations of distant locations. "Infinitely dilated" is only relative, according to a distant observer. A local clock still ticks at 1 second per second as the observer falls past the event horizon. Except for gravitational gradients ("spaghettification") in practical cases, the in-falling observer doesn't notice anything weird nearby, even while passing the horizon. You're speaking of relativistic mass, and that's just the mass equivalence of the total energy of a thing. Mass is "rest mass". The Schwarzschild radius event horizon is based on mass, not total energy, and it doesn't change with relative motion of the observer.

Interesting idea but I think you have to state your assumptions, and there are a few, including: - A part of the system can't be predicted without simulating the whole system (everything is connected, and there is no way to begin from a known intermediate state). - The universe is perfectly deterministic to computable precision (which contradicts current theory as others mentioned), but also cannot be computed in parallel. Computing "more" requires "faster", which might not hold true. You would need to specify the limitations of your imaginary simulator, not just assume it is the simplest of systems. Why would the computer have to simulate the results of its own calculations? In your calculator example, the calculator calculates that 2+2=4. It would not need to simulate the workings of itself to know that the simulated calculator will also produce 4, since it already knows it calculated that. Or to put it another way... you're saying that suppose a machine is able to predict a future state X, you want to show that this will derive a contradiction. Suppose state Y is the state of the present after that prediction has been made, and we assume that X depends on Y (simulating the results of the simulation is required). Well that gives a solution to the paradox right there: We must also assume that the simulator is inside the universe and is much smaller than the universe (or otherwise there's no point). Thus to simulate the simulator should require much less processing than to simulate the rest of the universe. So all you need to do is make the simulator more powerful than it needs to be to simulate up to the present. For example say that 90% of the simulator is simulating the rest of the universe, and 5% is simulating the simulator, and the rest is idle, then if could set it up so that the idle part of the simulator has only an easily predicted effect, then that part doesn't need to be simulated. However I think that there are too many assumptions that are left to choice. Since you're not describing the mechanism for simulating the universe, it doesn't make sense to both claim that it can be done and make arbitrary restrictions. You could as easily make up a rule that the idle parts of any simulator all affect the future as much as anything else does and needs to be simulated, but then this is all just a bunch of made-up rules about what's possible and what's not possible. Edit: Trying to clear my brain fog, I think perhaps you might be right. In my example, even if "simulating the simulator" can be done efficiently, it still assumes that in simulating state X, it first simulates the state Y, at which point the prediction of X is already known in the simulation. So it doesn't make sense that the simulator doesn't yet know state X, but the virtual simulator does. If you also assume that no state can be simulated without knowing what that state is, then I think you're right. On the other hand if you suppose that it's possible to simulate the simulation of state X, and to know all possible effects of that without having to know what X actually is! (OMG confusing), then it could be conceivable to simulate the effects of simulating X on the virtual computer, and then continue on to calculate X on the real computer. Well at the least I think you've proposed a good mind bender of a thought experiment.

This is no longer completely on-topic but is a misunderstanding of a related idea. No it doesn't take energy proportional to distance, to move sideways in a gravitational field (or along the width of the accelerating train). Suppose you roll a heavy sphere along a smooth frictionless horizontal surface. It will take energy to get it moving, but then the ball will keep rolling due to inertia (not requiring extra energy to keep moving). The energy put into its momentum is kinetic energy. Meanwhile if you lift the ball, the energy goes into gravitational potential energy. It won't keep moving. Each meter you lift it requires additional energy. If you roll a ball up an incline (or forward in an accelerating train), each meter moved will require additional energy. They don't, in the inertial frame where "exact same time" is determined. From the track's perspective, the clocks could be coordinated as you say, and remain in sync. This is not the case being discussed in other recent posts. The accelerating train is not an inertial frame, and "at the exact same time" will not mean the same thing for the differently located accelerating observers (they won't agree on simultaneity of events). After the train has completed accelerating, it has an inertial frame, and the differently located observers would from then agree on simultaneity of events, and could arrange for their clocks to be synchronized.

I don't think that's correct. I think by the equivalence principle you could have the clocks in a uniform gravitational field but at different heights (ie. different gravitational potentials) corresponding to their different x positions on the train. Therefore if you do set up the train so that it accelerates uniformly for some time, you still have the clocks ticking at different rates according to GR and can end up out of sync depending on the rest of the setup. As studiot mentioned, this might not describe the original setup. Edit: Just thinking to myself... I understand the instinct to put the clocks at the same height around a massive body in order to get the same field strength, but that's the wrong way to think about it. A uniformly accelerating train is like a uniform gravitational field. It will still take effort to "climb" your way from the back of the accelerating train to the front. The effort is proportional to distance climbed. Equivalently it takes effort proportional to distance climbed, to climb out of a gravity well with a uniform field. The location on the train and in the equivalent gravitational field matters.

I'm trying but I'm hopelessly lost, I don't think I have the basic math skills :/ Where I'm going with it: Am I heading in the right direction? I'm stuck and might not work on this more.