5 ms·

The separate space and time interval measurements between two events can vary among observers, but they will all measure the same spacetime interval, which in t

by d_tr 6y ago

The separate space and time interval measurements between two events can vary among observers, but they will all measure the same spacetime interval, which in turn determines whether light is "fast enough".

caminocorner 6y ago
Is there a somewhat longer (maybe illustrated) example of this principle somewhere? You've captured my interest and I want to read more
- jfoutz 6y ago
  maybe this one? https://en.wikipedia.org/wiki/Ladder_paradox https://en.wikipedia.org/wiki/Ladder_paradox depending on where you stand, the ladder can fit in the garage, or it can't. If I remember right, it's from the original special relativity paper, which is super accessible. General was always beyond me, but special is just high school math.
- alephu5 6y ago
  https://en.m.wikipedia.org/wiki/Relativity_of_simultaneity https://en.m.wikipedia.org/wiki/Relativity_of_simultaneity
- d_tr 6y ago
  The links provided are nice but I will leave a comment here in case it is useful. If I measure the temporal (dt) and spatial (dx) distance between two events A and B, then I can calculate what another observer would measure (dt' and dx') using the so-called Lorentz transformation, provided that I know his velocity relatively to me (v). The Lorentz transformation is a linear operator, written down as a matrix. Now, the spacetime interval (ds) between A and B, is computed with the formula ds^2 = dx^2 - c^2 dt^2. The interesting property here is that the Lorentz transformation leaves ds^2 unchanged, i.e. dx^2 - c^2 dt^2 = dx'^2 - c^2 dt'^2. So, it also does not change the sign of ds^2, which determines whether light is fast enough to travel a distance dx within time dt. The animation in the Wikipedia link by alephu5 shows the Lorentz transformation in action for a smoothly varying value of relative velocity. The events A B C are all separated by positive ('spacelike', light not fast enough) intervals, which graphically means that the line connecting them has a slope of less than 45 degrees in that graph, and the Lorentz transformation can tilt that line both ways and change the ordering of the events in the t axis. If two of these events on that graph were separated by a negative ('timelike') interval, the line connecting them would have a slope larger than 45 degrees and the Lorentz transformation could not alter their relative ordering in the t axis, meaning that all observers would agree on the ordering.
- shaunpersad 6y ago
  Light cones helped me understand causality from a slightly different perspective: https://en.wikipedia.org/wiki/Light_cone https://en.wikipedia.org/wiki/Light_cone https://www.youtube.com/watch?v=OZv3ycr6Jxg https://www.youtube.com/watch?v=OZv3ycr6Jxg
6gvONxR4sf7o 6y ago
How's this work with multiple paths between things? Like in special relativity, it's all uniquely defined, but how's it work in GR? Shortest spacetime path?
- deleted 6y ago
  [deleted]
- contravariant 6y ago
  In short, yes. To expand on that a bit, this shortest path is known as a geodesic and one of the more important axioms of general relativity is that all laws of physics are preserved locally when traveling along a geodesic. In particular all such observers should measure the same speed of light, since it's a simple property of electrodynamics. Interestingly they won't measure the same CMB, showing that the local part is important.
  - 6gvONxR4sf7o 6y ago
    That's neat, thanks. When you say it's an axiom that it's all preserved locally, does that mean that all the laws only state local properties (i.e. it's all just differential equations over spacetime)? If so, is that why stuff like QM has trouble, where something like entangled things at a distance might be hard to express locally, or is it actually easy to transform spooky stuff into a local statement and QM issues are something else entirely?
    - contravariant 6y ago
      It's a bit too simple to say that that's the problem, it's not that general relativity can't deal with global features of reality, those will simply not look the same to all observers. The reason I mentioned CMB is that space necessarily seems to have a preferred speed (even with general relativity you can't have a distribution of speeds that looks the same from all frames of references), and measuring the blueshift in CMB allows you to measure an absolute speed in some sense (it might still be local to our part of the universe, but that includes pretty much everything we can interact with). The obstacle to combining general relativity and quantum mechanics is, in short, that general relativity is a classical theory of physics (e.g. exact positions, energy, momentum and all that), whereas quantum mechanics expands classical mechanics to get quantum mechanical laws of physics. Now for whatever reason the techniques we used to turn electromagnetism etc. quantum mechanical fail to work on the (classical) theory of general relativity. And IMHO that's about as far as we've gotten, a lot of work's gone into it but it's honestly hard to tell if we've gotten a better grasp on why general relativity refuses to 'quantize'. Personally I blame the fact that the mathematical foundations of quantum mechanics aren't strong enough to support a quantum mechanical description of geometry itself, but I may not be the most qualified person to judge this.

by d_tr 6y ago

caminocorner 6y ago

Is there a somewhat longer (maybe illustrated) example of this principle somewhere? You've captured my interest and I want to read more

jfoutz 6y ago
maybe this one? https://en.wikipedia.org/wiki/Ladder_paradox https://en.wikipedia.org/wiki/Ladder_paradox depending on where you stand, the ladder can fit in the garage, or it can't. If I remember right, it's from the original special relativity paper, which is super accessible. General was always beyond me, but special is just high school math.
alephu5 6y ago
https://en.m.wikipedia.org/wiki/Relativity_of_simultaneity https://en.m.wikipedia.org/wiki/Relativity_of_simultaneity
d_tr 6y ago
The links provided are nice but I will leave a comment here in case it is useful. If I measure the temporal (dt) and spatial (dx) distance between two events A and B, then I can calculate what another observer would measure (dt' and dx') using the so-called Lorentz transformation, provided that I know his velocity relatively to me (v). The Lorentz transformation is a linear operator, written down as a matrix. Now, the spacetime interval (ds) between A and B, is computed with the formula ds^2 = dx^2 - c^2 dt^2. The interesting property here is that the Lorentz transformation leaves ds^2 unchanged, i.e. dx^2 - c^2 dt^2 = dx'^2 - c^2 dt'^2. So, it also does not change the sign of ds^2, which determines whether light is fast enough to travel a distance dx within time dt. The animation in the Wikipedia link by alephu5 shows the Lorentz transformation in action for a smoothly varying value of relative velocity. The events A B C are all separated by positive ('spacelike', light not fast enough) intervals, which graphically means that the line connecting them has a slope of less than 45 degrees in that graph, and the Lorentz transformation can tilt that line both ways and change the ordering of the events in the t axis. If two of these events on that graph were separated by a negative ('timelike') interval, the line connecting them would have a slope larger than 45 degrees and the Lorentz transformation could not alter their relative ordering in the t axis, meaning that all observers would agree on the ordering.
shaunpersad 6y ago
Light cones helped me understand causality from a slightly different perspective: https://en.wikipedia.org/wiki/Light_cone https://en.wikipedia.org/wiki/Light_cone https://www.youtube.com/watch?v=OZv3ycr6Jxg https://www.youtube.com/watch?v=OZv3ycr6Jxg

6gvONxR4sf7o 6y ago

How's this work with multiple paths between things? Like in special relativity, it's all uniquely defined, but how's it work in GR? Shortest spacetime path?

deleted 6y ago
[deleted]
contravariant 6y ago
In short, yes. To expand on that a bit, this shortest path is known as a geodesic and one of the more important axioms of general relativity is that all laws of physics are preserved locally when traveling along a geodesic. In particular all such observers should measure the same speed of light, since it's a simple property of electrodynamics. Interestingly they won't measure the same CMB, showing that the local part is important.
- 6gvONxR4sf7o 6y ago
  That's neat, thanks. When you say it's an axiom that it's all preserved locally, does that mean that all the laws only state local properties (i.e. it's all just differential equations over spacetime)? If so, is that why stuff like QM has trouble, where something like entangled things at a distance might be hard to express locally, or is it actually easy to transform spooky stuff into a local statement and QM issues are something else entirely?
  - contravariant 6y ago
    It's a bit too simple to say that that's the problem, it's not that general relativity can't deal with global features of reality, those will simply not look the same to all observers. The reason I mentioned CMB is that space necessarily seems to have a preferred speed (even with general relativity you can't have a distribution of speeds that looks the same from all frames of references), and measuring the blueshift in CMB allows you to measure an absolute speed in some sense (it might still be local to our part of the universe, but that includes pretty much everything we can interact with). The obstacle to combining general relativity and quantum mechanics is, in short, that general relativity is a classical theory of physics (e.g. exact positions, energy, momentum and all that), whereas quantum mechanics expands classical mechanics to get quantum mechanical laws of physics. Now for whatever reason the techniques we used to turn electromagnetism etc. quantum mechanical fail to work on the (classical) theory of general relativity. And IMHO that's about as far as we've gotten, a lot of work's gone into it but it's honestly hard to tell if we've gotten a better grasp on why general relativity refuses to 'quantize'. Personally I blame the fact that the mathematical foundations of quantum mechanics aren't strong enough to support a quantum mechanical description of geometry itself, but I may not be the most qualified person to judge this.