5 ms·

> and allows for new metrics without the most serious issues present in the Alcubierre solution. ...like violating causality?

geuis 4y ago
Subliminal, which is mentioned multiple times.
- texaslonghorn5 4y ago
  ^ this, and I'm guessing it was autocorrected (because that's what almost happened to me) but the term is subluminal
  - deleted 4y ago
    [deleted]
usrbinbash 4y ago
In case someone wants to read a really good explanation about this; http://www.physicsmatt.com/blog/2016/8/25/why-ftl-implies-time-travel http://www.physicsmatt.com/blog/2016/8/25/why-ftl-implies-ti... Generally speaking, all three of FTL (travel/communication), causality and relativity cannot all be true at the same time. If you can FTL-travel/communicate (they are the same really), then you can come up with a scenario in which an observer can see an event happening, and then see the cause of that event after it happened. Extending that logic, if the observer can also move at superluminal speeds, he could prevent the cause of the event after seeing the event happen, leading to a paradox.
- codethief 4y ago
  > Extending that logic, if the observer can also move at superluminal speeds, he could prevent the cause of the event after seeing the event happen, leading to a paradox This is a guess, i.e. one possible outcome physicists are considering. People have proposed alternative outcomes of FTL like the (in my opinion much more sensible) Novikov consistency principle, which roughly proposes that spacetime and the entities it contains (e.g. an observer's wordline) should be looked upon as a whole, in the sense that they need to be self-consistent. Spacetime is not time-dependent and does not evolve, so it does not make much sense to say "something something leads to a [spacetime] paradox".
  - codethief 4y ago
    Addendum: Self-consistency basically amounts to periodic boundary conditions – not really surprising when you have (almost) closed time-like curves.
    - zachf 4y ago
      Periodicity in time is unusual to think about in physics, partly because you start to get wacky results. If you could establish a CTC in nature, it would allow for computers that can efficiently (i.e. in polynomial time) solve not just NP problems but actually even all of PSPACE [0]. You can interpret this in two ways. There’s the hopeful way: “we should spend a boatload of money trying to find CTCs in our universe since they’ll let us create super-ultra computers”. And then there’s the pessimistic way, that nature probably isn’t going to give us a free lunch like that. Sadly, I’m in the pessimist camp on this one! [0] https://www.scottaaronson.com/papers/ctc.pdf https://www.scottaaronson.com/papers/ctc.pdf
- yellowapple 4y ago
  This explanation, like previous explanations for "FTL implies time travel" I've read, presupposes that the signal is actually moving at superluminal speeds, i.e. actually covering a distance greater than 299,792,458 meters in a single second from the signal's perspective. This would not be relevant for Alcubierre drives or wormholes or anything else that warps spacetime, since the whole point of such things is to stretch/contract space such that the thing is traveling a much shorter distance per second. In short, if I were to instantaneously poof out of existence here on Earth, poof into existence on Mars a second later, grab a rock, poof out of existence again, and poof back into existence back on Earth again another second later and hand you that rock, it doesn't seem reasonable to assert that I traveled backward in time when I clearly traveled forward by two seconds. The whole lecture on that external observer is irrelevant, since there's nothing to observe unless the observer happens to be in my bubble/wormhole/whatever - and even then, one'd only be observing subluminal actions/signals within that region of spacetime.
  - xenadu02 4y ago
    The problem is that if FTL travel exists then our existing theories about spacetime are wrong in some way (despite making lots of great predictions) or time paradoxes are possible in which case we have no idea what the consequences could be. Your example assumes there is some underlying rate at which time advances for the universe (or at least Earth and Mars) and that spacetime as we know it (including relativity and time dilation) are just some kind of modifier on top. But theory and experiment so far point to that not being the case. There is no "pop out of existence here and pop in over there" without time travel (as best as we can tell). The whole light cone / worldline explanations are more formal explanations of that. Now you can magic this problem away by proposing any number of schemes... like saying the entire universe's worldline is exists within a metaworldline and time travel actually resets the state of the universe as it was in the past then re-runs the universe... but all of that always proceeds forward in the metaworldline. In other words all past histories existed in a causal order, changing the past just adds "new commits" to the universe but history is never really rewritten. Bam! Our magic theory solves all paradox problems without requiring billions of parallel universes and allows time travel! But it's not a theory we can test or make predictions with so it isn't a useful scientific theory. It might as well be literal magic. Similarly you could propose that GR is wrong... but your new theory is gonna need to match GR's predictions that have proven true while making some new ones we can test, while also avoiding or explaining causality and paradoxes.
tsimionescu 4y ago
Space-time can expand at faster-than-light speeds - this is known for sure, since we live in a 93-billion light-years wide universe that's only 13 billion years old. As far as I understand, the current conjecture is that an Alcubierre drive could move at faster than light speeds (if negative mass/energy to build it existed), but that it it tried to move to its own past, it would either destroy itself because of some conjectured quantum gravity phenomenon - this is called the "chronology protection conjecture" and Alcubierre himself talked about it: > The conjecture has not been proven (it wouldn't be a conjecture if it had), but there are good arguments in its favor based on quantum field theory. The conjecture does not prohibit faster-than-light travel. It just states that if a method to travel faster than light exists, and one tries to use it to build a time machine, something will go wrong: the energy accumulated will explode, or it will create a black hole. [0] https://web.archive.org/web/20160318223348/http://ccrg.rit.edu/files/FasterThanLight.pdf https://web.archive.org/web/20160318223348/http://ccrg.rit.e... (last three slides touch on this area)
- raattgift 4y ago
  > Space-time can expand at faster-than-light speeds - this is known for sure Not exactly. In the standard cosmology, in an equatorial slicing (cf. slicing a cone with each plane perpendicular to the axis, giving circles) of the expanding universe, space expands very slightly at each point in space as the cosmological time ticks by. If we choose a point (p,t) and some coarsening procedure to reduce the count of points immediately around (p,t) to a finite number, then very soon after at (p,t+\epsilon), and using the same coarsening procedure, there will be more points immediately around p. So, for example, for something (p=const,t_0) we might count seven points in which we could find that something at t_1: {(p,t_1),(p x+1,t_1),(p x-1,t_1), (p y+1, t_1), ...}. But there might be, say, twelve points immediately around (p,t_n), twenty-four around (p,t_{n+n}), etc. The same thinking applies at every point in space with the same value of t. As the universe ages, the numbers of points in space is already enormous and at each of those points in space we add more points.[1] To that last paragraph we add a system of coordinates where there are cosmological observers[2] "stuck" at a particular spatial coordinate like (x,y,z) at all times, observing more space appearing between them, requiring coordinates in between to label that space. Physically these observers, if already distant, see each other's image becoming smaller, dimmer, and redder, as if they were accelerating away from each other. Physically neither observer detects such an acceleration -- they are in perfect free-fall. The metric expansion, and accelerated expansion, can be (and is usually treated as) purely local. Nothing interacts superluminally, and there is no need from observation to have large nonuniformities in local expansion in the known universe. That is, at large enough distance scales, the local expansion at any point is well-modelled as constant at all points and at all times: the cosmological constant. (At smaller distance scales, in regions dominated by gravitationally collapsing matter (including dark matter), the local expansion is zero. "Manhattan is not expanding", nor is the rest of the solar system or anything in our galactic cluster as far as we can tell). In the unknown early universe, various approaches to cosmic inflation essentially generatesa lot more new points around old points than expansion does, and the difference in inflation around any point can be large (loosely, the number-of-points-generated-at-point-p gap between "expanding" and "not expanding" is much narrower than the gap between "inflating" and "inflating less" let alone "not inflating"). > 93-billion light-years wide universe that's only 13 billion years old There are a lot more points between galaxy clusters in an expanding universe than there is in a non-expanding universe. (Inflation already stopped making new points in space long before the first protons formed, let alone galaxies). For the most part part galaxy clusters tend to become more compact over time; the matter in them is thick and trending thicker. Expansion doesn't arrest that trend at all. Galaxy clusters shine across the electromagnetic spectrum, and also expel neutrinos and some amount of hot dust and gas. That thin expelled matter does not block expansion very close to it, and so thin matter gets smeared across new points as they appear in its immediate neighbourhood. In the cosmological frame this means stretching their wavelengths or equivalently reducing their kinetic energy or equivalently reducing their temperature adiabatically. However, the expansion is not strong enough to break molecular bonds, so molecules will cool; they won't snap apart because of standard expansion. (Likewise expansion doesn't ionize atoms or fission nuclei; it barely distorts clouds or streams of thick-enough molecular gas that float out of galaxy clusters. The key to turning expansion off is thickness of matter, which may be generated by electromagnetic interactions or even just gravitation). Now, let's talk about "actual" FTL. There is "local" FTL in which a massive object and a pulse of light starting at the same point ends up with the massive object winning a short race ("short" being e.g. micrometers to kilometres). Our notions of causality arise from the structure of spacetime ("a Lorentzian manifold" gives us a particular <https://en.wikipedia.org/wiki/Causal_structure https://en.wikipedia.org/wiki/Causal_structure>), and (if not interfered with, i.e., in vacuum) massless objects (like a pulse of light) move at a particular speed that no massive object can exactly reach. A local FTL event is incompatible with a Lorentzian manifold, but if we see local FTL, out goes the objection that "everything we do strongly supports our idea that the universe is Lorentzian". (<https://en.wikipedia.org/wiki/Modern_searches_for_Lorentz_violation https://en.wikipedia.org/wiki/Modern_searches_for_Lorentz_vi...>) While reliably observing a local FTL event might seem inconvenient for the theories, relativists could certainly cope with what they would see as a breaking of the global hyperbolicity condition of our universe (which we get from having local Lorentz Invariance everywhere). Alcubierre-like ideas do not break local Lorentz invariance; there is no local FTL, so it is less disturbing in some ways than tachyons or whatever. Indeed, Alcubierre neatly packaged up his idea into a "bump function" on a perfectly normal Lorentzian spacetime. The Alcubierre idea essentially just breaks the constant cosmic expansion at points immediately around the "ship", making a lot lot lot more points behind the ship and destroying (or shrinking) a lot of points in front of it. It's a local effect confined to the region around the thin shell of the "warp bubble". The hardest thing is to make the space around the ship relax back to something close to what it was before the bubble zipped through it, and that's the source of many of the objections rooted in properties of as-yet-unobserved matter. However, if we assume that this is all workable and ship or computer memory chip or whatever goes from A to B faster than light can, our global causal structure cannot be the strongest ones we can get by having local Lorentz invariance everywhere. In principle we are probably ok climbing down to nearly any rung of the <https://en.wikipedia.org/wiki/Causality_conditions https://en.wikipedia.org/wiki/Causality_conditions> however for most of those we can't "just" take a set of initial values and evolve them forward, which is everyone's preferred approach to problems in general relativity. The intial value formulation of general relativity came decades after exact solutions like Schwarzschild's and Lemaître, Tolman and Bondi, and perturbations upon all those, so this is a luxury that was not always available to relativists, and at least if future relativists lose global hyperbolicity they will have computers with lots of fast memory. Examining a warp bubble spacetime that is on a lower rung of the causality ladder essentially requires completely specifying the values of all the fields at all the points in a large region as an "exception" sandwiched between two time-separated "initial" values fields. This requirement was part of Alcubierre's motivation to think about a warp bubble -- he already had an interest in the initial values approach and where it becomes hard to use and where it breaks down (indeed he wrote a textbook that deals with that, <https://academic.oup.com/book/9640) https://academic.oup.com/book/9640)>). As Alcubierre suggests in the quote you found, global hyperbolicity is what we appear to have in our universe, and there is nothing obviously "enforcing" it. So why isn't there obvious FTL in many places (or even everywhere)? Who knows. There is no "right" answer to that, and it might end up that it's just a feature of our universe like its three spatial and one timelike dimension. (cf. <https://en.wikipedia.org/wiki/Globally_hyperbolic_manifold https://en.wikipedia.org/wiki/Globally_hyperbolic_manifold>). - -- [1] This invites a Zeno's paradox view of the metric expansion. At later times there are more steps that one must take between "0" and "1" (and at the same time even more steps between "1" and "2" on the same ruler). One has to visit every mark on the ruler between origin and destination, but the number of marks doubles and re-doubles and re-re-doubles over the course of one's travel. [2] Experts will recognize that HN is a place for informal descriptions, and that here I flick (without explicit warning) between thinking in terms of comoving coordinates (more space between each coordinate, or more time for a pulse of light to move between two coordinates) and, without saying so, thinking in terms of Fermi coordinates (e.g. <https://link.springer.com/article/10.1007/s00023-011-0080-9 https://link.springer.com/article/10.1007/s00023-011-0080-9>) when taking a more, um, expansive view of Raychaudhri's equations. I apologize if that makes this harder to follow for people familiar with the mathematical details of physical cosmology.

geuis 4y ago

Subliminal, which is mentioned multiple times.

texaslonghorn5 4y ago
^ this, and I'm guessing it was autocorrected (because that's what almost happened to me) but the term is subluminal
- deleted 4y ago
  [deleted]

usrbinbash 4y ago

In case someone wants to read a really good explanation about this; http://www.physicsmatt.com/blog/2016/8/25/why-ftl-implies-time-travel http://www.physicsmatt.com/blog/2016/8/25/why-ftl-implies-ti... Generally speaking, all three of FTL (travel/communication), causality and relativity cannot all be true at the same time. If you can FTL-travel/communicate (they are the same really), then you can come up with a scenario in which an observer can see an event happening, and then see the cause of that event after it happened. Extending that logic, if the observer can also move at superluminal speeds, he could prevent the cause of the event after seeing the event happen, leading to a paradox.

codethief 4y ago
> Extending that logic, if the observer can also move at superluminal speeds, he could prevent the cause of the event after seeing the event happen, leading to a paradox This is a guess, i.e. one possible outcome physicists are considering. People have proposed alternative outcomes of FTL like the (in my opinion much more sensible) Novikov consistency principle, which roughly proposes that spacetime and the entities it contains (e.g. an observer's wordline) should be looked upon as a whole, in the sense that they need to be self-consistent. Spacetime is not time-dependent and does not evolve, so it does not make much sense to say "something something leads to a [spacetime] paradox".
- codethief 4y ago
  Addendum: Self-consistency basically amounts to periodic boundary conditions – not really surprising when you have (almost) closed time-like curves.
  - zachf 4y ago
    Periodicity in time is unusual to think about in physics, partly because you start to get wacky results. If you could establish a CTC in nature, it would allow for computers that can efficiently (i.e. in polynomial time) solve not just NP problems but actually even all of PSPACE [0]. You can interpret this in two ways. There’s the hopeful way: “we should spend a boatload of money trying to find CTCs in our universe since they’ll let us create super-ultra computers”. And then there’s the pessimistic way, that nature probably isn’t going to give us a free lunch like that. Sadly, I’m in the pessimist camp on this one! [0] https://www.scottaaronson.com/papers/ctc.pdf https://www.scottaaronson.com/papers/ctc.pdf
yellowapple 4y ago
This explanation, like previous explanations for "FTL implies time travel" I've read, presupposes that the signal is actually moving at superluminal speeds, i.e. actually covering a distance greater than 299,792,458 meters in a single second from the signal's perspective. This would not be relevant for Alcubierre drives or wormholes or anything else that warps spacetime, since the whole point of such things is to stretch/contract space such that the thing is traveling a much shorter distance per second. In short, if I were to instantaneously poof out of existence here on Earth, poof into existence on Mars a second later, grab a rock, poof out of existence again, and poof back into existence back on Earth again another second later and hand you that rock, it doesn't seem reasonable to assert that I traveled backward in time when I clearly traveled forward by two seconds. The whole lecture on that external observer is irrelevant, since there's nothing to observe unless the observer happens to be in my bubble/wormhole/whatever - and even then, one'd only be observing subluminal actions/signals within that region of spacetime.
- xenadu02 4y ago
  The problem is that if FTL travel exists then our existing theories about spacetime are wrong in some way (despite making lots of great predictions) or time paradoxes are possible in which case we have no idea what the consequences could be. Your example assumes there is some underlying rate at which time advances for the universe (or at least Earth and Mars) and that spacetime as we know it (including relativity and time dilation) are just some kind of modifier on top. But theory and experiment so far point to that not being the case. There is no "pop out of existence here and pop in over there" without time travel (as best as we can tell). The whole light cone / worldline explanations are more formal explanations of that. Now you can magic this problem away by proposing any number of schemes... like saying the entire universe's worldline is exists within a metaworldline and time travel actually resets the state of the universe as it was in the past then re-runs the universe... but all of that always proceeds forward in the metaworldline. In other words all past histories existed in a causal order, changing the past just adds "new commits" to the universe but history is never really rewritten. Bam! Our magic theory solves all paradox problems without requiring billions of parallel universes and allows time travel! But it's not a theory we can test or make predictions with so it isn't a useful scientific theory. It might as well be literal magic. Similarly you could propose that GR is wrong... but your new theory is gonna need to match GR's predictions that have proven true while making some new ones we can test, while also avoiding or explaining causality and paradoxes.

tsimionescu 4y ago

Space-time can expand at faster-than-light speeds - this is known for sure, since we live in a 93-billion light-years wide universe that's only 13 billion years old. As far as I understand, the current conjecture is that an Alcubierre drive could move at faster than light speeds (if negative mass/energy to build it existed), but that it it tried to move to its own past, it would either destroy itself because of some conjectured quantum gravity phenomenon - this is called the "chronology protection conjecture" and Alcubierre himself talked about it: > The conjecture has not been proven (it wouldn't be a conjecture if it had), but there are good arguments in its favor based on quantum field theory. The conjecture does not prohibit faster-than-light travel. It just states that if a method to travel faster than light exists, and one tries to use it to build a time machine, something will go wrong: the energy accumulated will explode, or it will create a black hole. [0] https://web.archive.org/web/20160318223348/http://ccrg.rit.edu/files/FasterThanLight.pdf https://web.archive.org/web/20160318223348/http://ccrg.rit.e... (last three slides touch on this area)

raattgift 4y ago
> Space-time can expand at faster-than-light speeds - this is known for sure Not exactly. In the standard cosmology, in an equatorial slicing (cf. slicing a cone with each plane perpendicular to the axis, giving circles) of the expanding universe, space expands very slightly at each point in space as the cosmological time ticks by. If we choose a point (p,t) and some coarsening procedure to reduce the count of points immediately around (p,t) to a finite number, then very soon after at (p,t+\epsilon), and using the same coarsening procedure, there will be more points immediately around p. So, for example, for something (p=const,t_0) we might count seven points in which we could find that something at t_1: {(p,t_1),(p x+1,t_1),(p x-1,t_1), (p y+1, t_1), ...}. But there might be, say, twelve points immediately around (p,t_n), twenty-four around (p,t_{n+n}), etc. The same thinking applies at every point in space with the same value of t. As the universe ages, the numbers of points in space is already enormous and at each of those points in space we add more points.[1] To that last paragraph we add a system of coordinates where there are cosmological observers[2] "stuck" at a particular spatial coordinate like (x,y,z) at all times, observing more space appearing between them, requiring coordinates in between to label that space. Physically these observers, if already distant, see each other's image becoming smaller, dimmer, and redder, as if they were accelerating away from each other. Physically neither observer detects such an acceleration -- they are in perfect free-fall. The metric expansion, and accelerated expansion, can be (and is usually treated as) purely local. Nothing interacts superluminally, and there is no need from observation to have large nonuniformities in local expansion in the known universe. That is, at large enough distance scales, the local expansion at any point is well-modelled as constant at all points and at all times: the cosmological constant. (At smaller distance scales, in regions dominated by gravitationally collapsing matter (including dark matter), the local expansion is zero. "Manhattan is not expanding", nor is the rest of the solar system or anything in our galactic cluster as far as we can tell). In the unknown early universe, various approaches to cosmic inflation essentially generatesa lot more new points around old points than expansion does, and the difference in inflation around any point can be large (loosely, the number-of-points-generated-at-point-p gap between "expanding" and "not expanding" is much narrower than the gap between "inflating" and "inflating less" let alone "not inflating"). > 93-billion light-years wide universe that's only 13 billion years old There are a lot more points between galaxy clusters in an expanding universe than there is in a non-expanding universe. (Inflation already stopped making new points in space long before the first protons formed, let alone galaxies). For the most part part galaxy clusters tend to become more compact over time; the matter in them is thick and trending thicker. Expansion doesn't arrest that trend at all. Galaxy clusters shine across the electromagnetic spectrum, and also expel neutrinos and some amount of hot dust and gas. That thin expelled matter does not block expansion very close to it, and so thin matter gets smeared across new points as they appear in its immediate neighbourhood. In the cosmological frame this means stretching their wavelengths or equivalently reducing their kinetic energy or equivalently reducing their temperature adiabatically. However, the expansion is not strong enough to break molecular bonds, so molecules will cool; they won't snap apart because of standard expansion. (Likewise expansion doesn't ionize atoms or fission nuclei; it barely distorts clouds or streams of thick-enough molecular gas that float out of galaxy clusters. The key to turning expansion off is thickness of matter, which may be generated by electromagnetic interactions or even just gravitation). Now, let's talk about "actual" FTL. There is "local" FTL in which a massive object and a pulse of light starting at the same point ends up with the massive object winning a short race ("short" being e.g. micrometers to kilometres). Our notions of causality arise from the structure of spacetime ("a Lorentzian manifold" gives us a particular <https://en.wikipedia.org/wiki/Causal_structure https://en.wikipedia.org/wiki/Causal_structure>), and (if not interfered with, i.e., in vacuum) massless objects (like a pulse of light) move at a particular speed that no massive object can exactly reach. A local FTL event is incompatible with a Lorentzian manifold, but if we see local FTL, out goes the objection that "everything we do strongly supports our idea that the universe is Lorentzian". (<https://en.wikipedia.org/wiki/Modern_searches_for_Lorentz_violation https://en.wikipedia.org/wiki/Modern_searches_for_Lorentz_vi...>) While reliably observing a local FTL event might seem inconvenient for the theories, relativists could certainly cope with what they would see as a breaking of the global hyperbolicity condition of our universe (which we get from having local Lorentz Invariance everywhere). Alcubierre-like ideas do not break local Lorentz invariance; there is no local FTL, so it is less disturbing in some ways than tachyons or whatever. Indeed, Alcubierre neatly packaged up his idea into a "bump function" on a perfectly normal Lorentzian spacetime. The Alcubierre idea essentially just breaks the constant cosmic expansion at points immediately around the "ship", making a lot lot lot more points behind the ship and destroying (or shrinking) a lot of points in front of it. It's a local effect confined to the region around the thin shell of the "warp bubble". The hardest thing is to make the space around the ship relax back to something close to what it was before the bubble zipped through it, and that's the source of many of the objections rooted in properties of as-yet-unobserved matter. However, if we assume that this is all workable and ship or computer memory chip or whatever goes from A to B faster than light can, our global causal structure cannot be the strongest ones we can get by having local Lorentz invariance everywhere. In principle we are probably ok climbing down to nearly any rung of the <https://en.wikipedia.org/wiki/Causality_conditions https://en.wikipedia.org/wiki/Causality_conditions> however for most of those we can't "just" take a set of initial values and evolve them forward, which is everyone's preferred approach to problems in general relativity. The intial value formulation of general relativity came decades after exact solutions like Schwarzschild's and Lemaître, Tolman and Bondi, and perturbations upon all those, so this is a luxury that was not always available to relativists, and at least if future relativists lose global hyperbolicity they will have computers with lots of fast memory. Examining a warp bubble spacetime that is on a lower rung of the causality ladder essentially requires completely specifying the values of all the fields at all the points in a large region as an "exception" sandwiched between two time-separated "initial" values fields. This requirement was part of Alcubierre's motivation to think about a warp bubble -- he already had an interest in the initial values approach and where it becomes hard to use and where it breaks down (indeed he wrote a textbook that deals with that, <https://academic.oup.com/book/9640) https://academic.oup.com/book/9640)>). As Alcubierre suggests in the quote you found, global hyperbolicity is what we appear to have in our universe, and there is nothing obviously "enforcing" it. So why isn't there obvious FTL in many places (or even everywhere)? Who knows. There is no "right" answer to that, and it might end up that it's just a feature of our universe like its three spatial and one timelike dimension. (cf. <https://en.wikipedia.org/wiki/Globally_hyperbolic_manifold https://en.wikipedia.org/wiki/Globally_hyperbolic_manifold>). - -- [1] This invites a Zeno's paradox view of the metric expansion. At later times there are more steps that one must take between "0" and "1" (and at the same time even more steps between "1" and "2" on the same ruler). One has to visit every mark on the ruler between origin and destination, but the number of marks doubles and re-doubles and re-re-doubles over the course of one's travel. [2] Experts will recognize that HN is a place for informal descriptions, and that here I flick (without explicit warning) between thinking in terms of comoving coordinates (more space between each coordinate, or more time for a pulse of light to move between two coordinates) and, without saying so, thinking in terms of Fermi coordinates (e.g. <https://link.springer.com/article/10.1007/s00023-011-0080-9 https://link.springer.com/article/10.1007/s00023-011-0080-9>) when taking a more, um, expansive view of Raychaudhri's equations. I apologize if that makes this harder to follow for people familiar with the mathematical details of physical cosmology.