md65536

"One timeline" is meaningless to me. t' inside t is also meaningless... you're effectively asking if the time measured by one clock is inside the time measured by another clock. But you could say that if Bob leaves the train at 2050 and returns to it at 2100, its world line between those events is within the future light cone of the train in 2050, and the past light cone in 2100, and that's what you want: all observers agree that Bob's trip happened after the 2050 event and before the 2100 event. However, if Bob didn't start/end near the train and was always far enough away, that's not true for all observers. As well, Bob's cousin Babs, who suppose shares Bob's inertial frame and clock for an inertial part of Bob's trip, but was in the Andromeda galaxy during proper time interval t', is completely outside the light cones mentioned, so in that sense t' is not "inside" t, and far away events occurring at proper time t' can be before, during, or after (depending on frame of reference) the t interval from 2050 to 2100. Yes, the block universe has one time dimension, but you can choose it in different ways (as MigL said, "You can have many different 'foliations' of the 4D hypercube, and all are equally valid.") If you have the block universe in coordinates so that the front of the train is always at the same location with one x value, its world line is a straight line aligned with the T axis. But you can rotate (hyperbolicly) the block universe so that Bob's world line is now aligned with the new T' axis, and the train front's world line span different x' values. This doesn't imply multiple universes, blocks, trains etc. any more than someone in Russia having a different vector pointing "up" than someone in Argentina, implies that there are multiple Earths. Say they each have a y-axis that points "up", you have one Earth described using 2 different sets of coordinates, each more appropriate than the other to some observer.

I was thinking about this as an example for another thread but thought it would only add confusion. Say you have a train with proper length 100 m, traveling so fast that it contracts to 1 m according to the tracks' reference frame. Suppose that in the train's frame, lightning strikes the front and back of the train simultaneously. How far apart are the lightning strikes on the track? Can anyone answer this with just a few seconds thought? I can't, I have to figure out the details and calculations, but if I ask a different question... They're the same question, but only one seems intuitively obvious.

There are events on the those same 2 world lines that are 100 m apart to that observer, yes. They're on the same world lines, but they're not the same pair of events that are 1 m apart in Bob's frame. The two observers use different time coordinates, and 2 events at the same t and 100 m apart in the train's frame, don't have the same t' value in Bob's frame. The 1 m length of the train that Bob measures is between 2 events on the respective world lines with the same t' value. This is "relativity of simultaneity." Using world lines instead of lengths isn't going to change the details of the simpler examples you're using them to represent. I suppose we persist because we expect a response like "That's something I don't get, let me try to understand that first" instead of "What if I ignore all that and ask the same question in different words?" Block universe, world lines... it's like expecting to find some aspect of relativity for which the rules of relativity don't apply.

I did mean the same thing, but failed to state it. I should have said I meant the spatial distance between two events at the same time in a particular frame of reference. Did I imply the invariant length of a spacetime interval? ...thus proving my point that you have to specify these details for things to make sense to others!

That doesn't work unless you know the time at the light source, and you can't measure that from the detector. You theoretically can't measure the one-way speed of light, but you don't have to measure it since it is by definition equal to c. You can measure the 2-way timing of light, and find the one-way speed because literally by definition, the time that it takes for the light to go 1 way is the same as the time it takes to go the other way. You can however confirm that light from a distance D takes the same time regardless of the motion of the light source. For example, if you have 2 sources moving in opposite directions, and a signal from each of them when they're at the same location, you can verify that the 2 signals arrive at the same time. You don't have to know what time they're sent at, if all you care about is that they were sent at the same time, and you can make sure that happens by sending them from the same location.

Your line of questions shows you're not going to understand the answers. Why don't you start with simpler concepts first and understand them before talking about world lines? The world line of a particle is made up of all the events in the particle's entire lifetime. You're asking for the distance between two arbitrary lines. I think what you're really asking for is the distance between particular pairs of events on those world lines, but you're not specifying that unambiguously. If you're using coordinates where Bob's world line remains fixed at one x,y,z location and only varies in T, then (based on the situation you've implied), the distance between a given event at time T1 on the world line of the front of the train, and an event at the same time T1 on the world line of the back of the train, will be 1 m apart. If you use other coordinates, you'll get other answers. If you're talking about measurements in the block universe, you should specify what coordinates you're using. If you're talking about observations, you should make it clear what frame of reference you're using. If you're measuring the distance between 2 objects over their entire lifetimes (ie. world line) you should make it clear what time you're talking about. And if you're talking about a single moment across a distance, to even make sense of that requires the set of coordinates or frame of reference! If you want specific answers, your questions have to make sense, and they haven't been.

Could you please stop taking what is said and twisting it to fit your existing misunderstanding? It's been repeated ad nauseam that that there's only one train. No, because those are 2 different coordinate systems. For example, the front of the train can be at rest at x2=0 for the coordinate system of an observer at the front of the train, while Bob might use a coordinate system where his ship is at the origin and the location of the front of the train is x1=several light years and changing. If the front of the train enters a tunnel, that's an event. Everyone agrees that the location of the front of the train coincides with the location of the entrance of the tunnel when it enters, but that could be at x2=0 and x1=several light years. Though it might make more sense with all the other basic stuff about relativity that I'm sure you've read, if you called them x and x' to denote that they're different coordinate systems.

Did none of the rest of what I wrote make sense? The particle has only one world line. The world line is the events (the x and t locations) that the particle passes through over its lifetime. The world line is a fixed set of events, but the x and t values of those events change if you rotate it into other coordinate systems representing other observers. For example, put a stick on a grid with an x and t axis. If you align it with the t axis, it has the same x value at different t, representing the world line of a particle at rest. If you rotate the grid, the stick stays the same but now it has different x at different t, representing a different observer, for which the particle is moving. This is an oversimplified analogy and it uses the wrong rotation, so don't draw too many conclusions from it. If you rotate the stick or grid, the stick's x and t components change, but the length of the stick stays the same. It's the hypotenuse in the Pythagorean theorem r^2 = x^2 + t^2. With the correct hyperbolic rotation, s^2 = x^2 - t^2 stays the same.

Sure, that would be a block of events, and the "strings" are world lines of particles. Imagine you have a block of say wood, and you draw a small line on it to represent a world line. You can rotate the block in your hands, but the shape and length of the line remains invariant. Say that you rotate it through a fixed grid of coordinates in a room. Using just 2 dimensions for simplicity, you can make the line align with the x-axis, or the y-axis, or anything in between. By rotating it, you can change how long the line is in the x-dimension and how long it is in the y-dimension, without changing its length. But the block universe also has a time dimension, so instead of x and y, imagine it's t and x. The line could be a function x(t) representing how far x a particle moves in time t in a particular set of coordinates, and you get different x-lengths in different coordinates just by rotating the block. In this analogy the rotation is a spherical rotation which keeps r^2 = x^2 + y^2 invariant as you rotate it. With spacetime, the rotation is hyperbolic, which instead keeps s^2 = x^2 - t^2 invariant as you rotate it. So you can't just turn it upside down and reverse time etc.

Do you mean worldline of an object? An event is a single point in spacetime (and in the "block") and many worldlines can pass through it. Worldlines still exist in the block, and one's proper time is an invariant length of the world line. Different coordinate times can be defined (including just using T to represent time) and the coordinate times between distant events can be calculated in the block. I don't understand what you mean by it being "devoid of time", since all the measures of time are still there. Some philosophical "flow or time" or whatever might be taken out, but time in SR and GR is the measurement, not the concept. Where's the problem? You're using coordinate time t to be the time in Earth's frame. You're stating that the length of the train is relative, and depends on the observer (ie. reference frame). That agrees with observation. That agrees with relativity. Nothing there implies illusion.

Then S(t) is a hypersurface of events representing a single moment? How do you determine t? It sounds like you're implying it uses dates according to a clock on Earth? What if you have a block universe with all of the same events, but with a t value that corresponds with Bob's clock?

Don't you have to choose your coordinates for such a block universe? Then you can measure the distance between points with the same T value. For example, choose coordinates for the block universe where T corresponds with a clock that is stationary relative to the train. Take a pair of events with the same T value, one at the front of the train and one at the back. You can find the proper length of this train, say 100 m. Or choose coordinates where T' corresponds with Bob's proper time and find 2 different events with the same T' and find that the distance between those 2 events is 1 m. The aliens would also have to choose a set of coordinates, and could measure the length contraction of the train relative to its proper length.

In your aliens' view of the universe, is Bob's ship moving? Is the train moving? If so, how fast? Does this question support the argument that movement is an illusion and not actually happening?

No, you don't need two. You can have the two, but there's nothing that stops working or making sense with just the frame-dependent definition of simultaneity. For order of events, causality is enough, and that doesn't depend on frames of reference or simultaneity. There is no "god's eye view" needed. You can add it in, but it doesn't explain any observations that can't be explained without it. Therefore it's likely more misleading than helpful for explaining. It would be like saying "To understand life and death, we have to first understand ghosts." Nothing requires that ghosts aren't real, but everything observed can be explained without ghosts. For order of events, if two distant events are simultaneous, they're not causally linked (one doesn't cause the other). Their order is frame-dependent, and there's no problem with that because their order has no bearing on causality and on what other events are effected, and thus no bearing on what can be observed (since observations can be treated as a set of events). If the god's eye view is necessary, it's for something other than what's been observed or what's predicted to be, ie. metaphysical. The "universal now" is simply not needed for the universe to function.

I bet reddit.com/r/Whatisthis/ or reddit.com/r/whatisthisthing/ would figure it out within an hour. I figured it might be part of a tool, like a plane depth adjust, or a proportional divider knob. The closest I've seen is on a horological tool, like in the top left: The knob slides and tightens to lock. A string might fit in the groove, and the part is used to set the tension on it. This one looks like it spins freely though.

I wouldn't say that exactly. I'd also call it "inertia" instead of artificial gravity. But objects do tend to have some rotation, and inertia does contribute to forces acting on things (eg. the Earth bulges around the equator, along with a feeling of less pull of gravity than at the poles, at a fixed distance from the centre of the Earth), and for a static object that inertia does not require energy to maintain. My point is that if you choose a force that requires constant use of energy for one observer, and another force that doesn't for another observer, that's not a difference that concerns the equivalence principle. The principle doesn't say that any 2 different things you choose should be the same, and if there's a detectable difference (eg. in energy use) then the principle doesn't hold. Effectively it says if you make everything else the same, gravity is not detectably different from acceleration.

It wouldn't take infinite energy and you wouldn't reach a speed of c relative to anything. If you accelerated away from Earth with a constant proper acceleration for seventy years, the coordinate acceleration of Earth away from you would decrease and approach zero as its speed approached c, because of the velocity addition or composition law of SR. I don't think the difference in energy use is related to localness of effects, or whatever. You could hover over the Earth on a stationary platform that uses energy to create 200 lbs of thrust for 70 years. The equivalence principle still applies. You can't tell whether you're accelerating away or overcoming gravity. You could also simulate gravity by being in a rotating ship in freefall for 70 years, and use no energy, and experience no gravitational acceleration. The equivalence principle might not apply since that would be a rotating frame of reference.

Well, the curvature of the parabola decreases with increasing positive x. There's an analogy for what you're talking about. "Local" effectively means at small enough distances that the effects of distance don't matter, and that's not a fixed size. At small x, you have to zoom in to a smaller area to make the parabola appear flat, and at large x, it can appear flat over a larger range. Analogously, near to a gravitational mass "local" might be very small, but very big much farther away. If you're free-falling near a black hole and being spaghettified, spacetime is still locally Minkowskian but at distances much smaller than a human.

I think it also has intrinsic curvature in the usual setup. Sure, but the gravitational time dilation should depend only on their relative gravitational potential (right?) and not how space between them curves to produce that difference in potential. For example if you have 2 locations at different potentials in a constant gravitational field, you have 2 locally Minkowskian regions and no curvature anywhere, but you still get gravitational time dilation. Yours is an interesting example because the regions are intuitively flat. I guess the Riemann tensor is zero at those locations? But for example a region just outside the hollow shell is also locally Minkowskian in the coordinates of a particle in free fall, and locally measurably "flat" to such a particle (but spacetime is not actually flat there and the Riemann tensor isn't zero). I think I'm too stuck on confusion about the meaning of "locally Minkowskian," and trying to visualize it as horizontal on a diagram showing local coordinates, but then don't know how distant curved space could be shown. Now I remember you mentioned the covariance of tensors to me before, and I looked up the components of the Riemann tensor and couldn't make sense of it (same as now!). I think I need to learn more basics before understanding curvature in different observers' coordinates.

Tying this in with the rubber sheet analogy: The height on the sheet or on Earth corresponds (assuming constant g) with gravitational potential, proportional to how much energy it takes to lift an object to a specific height. The derivative of that, the slope of the sheet or ground, corresponds with gravitational force---how quickly a marble would accelerate as it rolls along that point. The derivative of that represents curvature. If we imagine the rubber sheet to be semi-rigid, and you actually have to bend it into a curved shape, the severity of the bending at a given point corresponds with curvature. You can tilt a sheet, and that corresponds with a constant gravitational force but no curvature. (Just to complete the analogy, curvature corresponds with tidal forces. If you have a toy car where the front and back are separate and connected by a spring, and roll it down a hill with constant slope, the spring doesn't stretch or compress. If you put it on a curve, eg. on top of a ball, the front and back are on different slopes and can pull away from each other.) The "local flatness of spacetime" implies that if you're looking only at a single point, you can't detect curvature, but you also can't detect gravitational force (is that right???). So for the Earth analogy, using it only to show what you can detect locally, and not how it affects the motion of objects, we could take the real Earth and modify it so that "up" is always normal to the surface you're standing on. Like in some cartoons where if you're standing on the side of a mountain, your body is tilted so your feet remain "flat" against the slope. Then if you can only look at the ground beneath your feet, you can't tell if you're on a slope or not. Is that a fair analogy to spacetime, or is it only curvature that is locally undetectable, but not gravitational force? Also I have a feeling I've asked a similar thing but still don't get it: Is the curvature relative, so it actually is locally zero but a different value from a distant? Or does local flatness merely mean, like you suggest, that the local value of the curvature only has measurable meaning across some distance? I think it's the latter??? Can curvature be called a scalar field, and is it invariant in a static universe?

Can you please explain the statement that I bolded? You implied that I should back up assertions with detail, and when I looked closer at your analogy that you claim is a "better one", I find that it makes no sense at all. Can you explain what this is meant to show about GR, especially related to the topic of how curvature in space-time is shown? I don't see any questions asked by swansont. I agree with all the points he made.

I already justified the opinion, your model doesn't show the paths of objects bending in a curved space, and the rubber sheet analogy does. But I don't see what your analogy is even trying to say. What I get from it is you're saying that objects can only travel along gridlines through space? Are the streets representative of dimensions? Are they both spatial, or is one meant to be time? (I guess spatial, since you said "ignoring time", but then I don't understand MigL's comment that it "considers the space-time interval".) You can only travel in one dimension at a time, one "first" and then another? And that should give someone an idea of how gravity works?

It's still much better than your buildings and streets analogy.

1) It is meant to represent the curvature of a space (just not real space), and the paths of objects through that space. What I meant was that I don't think the sheet's curvature is intended to actually model real spacetime curvature. I don't think they curve in the same way. But that's fine, it is an analogy of real spacetime curvature, not a model of it. My problem is that it is not clear how closely it is analogous, or even what properties exactly it is representing. Usually with a physical analogy you can clearly see the differences between the analogy and the real thing, and you don't confuse them. Here, the "fabric of spacetime" is such an elusive abstract thing that people see the sheet as a model of that "fabric". 2) I don't understand. It doesn't represent the real path of an object through spacetime (see (1)), yet a marble initially at rest on the trampoline does follow the line to the COG of the mass. 4) Why can't a 1D line curve? Can't you map one 1D space onto another different 1D space? 5) Again it's an analogy. You can't practically show the curvature of a sheet due to mass alone, the thing that causes the curvature in the sheet is only an analogy to the thing that causes curvature in spacetime.