md65536

So that would be... time according to a set of eternally inertial physical clocks that are infinitely far away from mass (or far enough that acceleration is negligible, since the fundamental observers are never accelerated), or time according to a nearby (generally passing by) eternally inertial clock that ticks as if there were no mass in the universe (essentially an abstract clock)? Would those be the fastest possible clocks? Any clock that is accelerated or in a gravity well would tick relatively slower? Very deep in a gravity well, you may have "cosmic time" ticking by very fast. I suspect it might not even be defined, with extreme-enough "lumpiness". If that's right, the "cosmic time" could not be represented with a Cauchy surface; the latter would follow around the curvature of "lumps" while the former would cut right through them. This is another notion of a common "now" in the universe. This would be no different than choosing any other clock and calling it a universal clock and counting all of time in the universe according to the chosen clock while ignoring local clocks, except that by choosing an inertial clock away from matter, you can have "identical" clocks all over, all marking the same time.

Do you mean that if there was a BBQ planned, you would expect someone to mow the lawn in a thunderstorm? If you mean it as a nonsense example, there's no point in dismissing the general case with a specific example that doesn't make sense. It might technically be called "circular reasoning" if there's nothing more specific???

It seems to be a "vicious circle", in this case formed by removing the reasoning that got you into (or can take you out of) the circle in the first case. Defendant offers no plea only due to lack of evidence. Defendant rejects evidence only because of lack of plea. I haven't found any more specific term for it.

Restating what's been said, so it's clear there are no contradictions: 1. The answer is "no", there is not a common moment throughout the universe that could be considered "now" by all observers who experience it. 2. Our locally defined 'now' could not be used as a global now, because for one thing it is not even a definite moment in some locations. 3. We could arbitrarily define a moment using a Cauchy surface, and that would globally (across the surface) be experienced as a moment and consistently separate a past and future everywhere, but it would not meaningfully represent 'now'. The answer is still no, but if you wanted to invent your own meaningless coordinate system it's possible to define 'now' consistently everywhere, just like you could choose a center of the universe or a middle of the surface of the Earth... meaningless outside of your invented coordinate system.

How so? The magnet must have 2 poles. No matter how they're oriented you should be able to detect varying field strength??? (Unless the magnet was too weak, which would ruin the puzzle.)

Do the general arguments include these assumptions? 1. The universe cannot be closed (must be globally hyperbolic or flat). 2. It can't have local singularities like those at the center of black holes. I think these are accepted mainstream assumptions, but they're not proven. Black hole singularities are not expected to be real, but it is probably not answerable without a quantum gravity theory. I'm guessing they would not permit a Cauchy surface??? Edit: I googled and came up with this: "if you look at the Penrose diagram for an astrophysical black hole, there clearly are Cauchy surfaces. The singularity is spacelike, so Cauchy surfaces don't have to intersect it. – Ben Crowell" So it would seem I'm wrong... black hole singularities don't interfere with the ability to define a "now" everywhere. If I understand that right, it's kind of interesting... You can define a "now" for the universe, but it won't include the centers of black holes! At first I think "that's not the whole universe then", but it is... the singularities are completely in the future of any such "now" that we're talking about. They could exist in 4d spacetime, just not "now". (Not so weird if you think of spacelike intervals to an outside observer being timelike inside a BH horizon.) Edit: Just thinking some more... we can speak of a BH's singularity at a specific time in our coordinates. We can define from here its existence in a moment of time "now" (here), but there or near enough, our defined moment isn't even a moment. The usual concept of "now" that humans use, essentially assigns a local (ie. Earth) time to all events in the universe, which is not a common moment throughout the universe, and is not a Cauchy surface. Thinking about black holes and how spacelike curves can be timelike elsewhere, it becomes pretty obvious pretty quick why we don't try to define a common now in the universe, and instead work in local coordinates. If we chose to define a "now" using a Cauchy surface, everyone would agree that the surface is a moment, with a consistent clear separation of past and future, but no one (not even us who get to choose how to define the surface) would say that far distant points on that surface agree with local coordinates. I think.

I agree with xyzt. I think you've asked the same questions in various threads over the years, and now you're adding a misuse of "foliation" into your arguments. A point on a spacetime diagram represents an event. An observer is better represented by its worldline, a more-vertical-than-not path on the diagrams, and the events on the x-axis will eventually be observed if the observer sticks around long enough. A short-lived observer could be represented by a point, and it wouldn't observe those events you're talking about, but so what? Is something not real if you can contrive a particular observer that can't observe it? I've seen this argument before and it doesn't seem to have progressed. Forget foliations, look up world lines and space-like intervals, and why not devote one thread (not this) to figuring out your idea?

'Unphysical' isn't the right word, because it has a physical meaning, and it can be measured. Eg. send a light signal to two receivers the same distance from you, and they'll receive them simultaneously, even though the two reception events are not directly causally connected to each other. The best you could say is that simultaneity is 'conventional'. http://plato.stanford.edu/entries/spacetime-convensimul/gives an overview. It is widely accepted (though not settled) that simultaneity is indeed conventional. To relate that back to this thread... simultaneity is meaningfully defined in flat spacetime, using Einstein's definition, whether or not that is merely a convention. It would seem there's no such meaningful definition for curved spacetime, even though you could arbitrarily make one (eg with a foliation of Cauchy surfaces) if the universe fits the requirements (if it's globally hyperbolic or whatever). You could for example take all events in the universe, and construct a Cauchy surface that means "this is all the events that I consider to be happening 'now'". That too would be a convention, but I think it would be physically meaningless, because you would probably have no way of unambiguously measuring the simultaneity of events no matter how you choose to define it. In flat spacetime Einstein's definition of simultaneity defines an unambiguous measurement of simultaneity.

Fair enough. To get back on track, I'll stress what I didn't say clearly enough: Spacetime diagrams aka Minkowski diagrams only represent flat spacetime and can't fully represent the foliations or Cauchy surfaces being talked about, except in a simplified way (with straight parallel lines, instead of curved surfaces). You can use the diagrams and your maths to examine the topic with a simpler subset of spacetime, but you can't generalize the simplified case into an answer for the thread.

The thread is talking about the universe, not a flat spacetime. Your post is off-topic. Why not start a new thread? It's misleading to post maths that relate to a new topic without mentioning it. Some people might think you're talking about the question that everyone else is discussing.

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.