The horizon problem for faster than light travel(eriklentzphd.blogspot.com)
eriklentzphd.blogspot.com
The horizon problem for faster than light travel
https://eriklentzphd.blogspot.com/2022/01/the-horizon-problem-for-faster-than.html
4 comments
> “the horizon problem”
It looks like the author was referring to this: “In relativity theory, a horizon is also a boundary on the region which an observer can see. Horizons pose an issue for warp drives as one cannot control the warp bubble if one see or communicate with it.”
And for more context, the author seems to note that faster than light travel is possible under Einstein’s physics: https://www.sciencealert.com/faster-than-light-travel-is-pos...
It looks like the author was referring to this: “In relativity theory, a horizon is also a boundary on the region which an observer can see. Horizons pose an issue for warp drives as one cannot control the warp bubble if one see or communicate with it.”
And for more context, the author seems to note that faster than light travel is possible under Einstein’s physics: https://www.sciencealert.com/faster-than-light-travel-is-pos...
Sure, but only for certain uses of the term ‘possible’. There are solutions to Einstein’s equations that imply faster than light travel or communication, but that doesn’t mean such solutions correspond to physical reality. For example such solutions can include terms in the equations that correspond to negative energy densities, or infinite masses, or trajectories around infinitely extended one dimensional massive objects. Such solutions somewhat stretch the meaning of the word ‘possible’.
Hasn't the Casimir effect given possible real life negative energy densities?
No. All that happens in the Casimir effect is that the vacuum state between a pair of uncharged plates is slightly different from the vacuum state in the absence of all matter whatsoever. There is no "negative energy" anywhere in the Casimir effect, despite pop science articles that appear to claim the contrary.
I'm probably wrong but I thought the argument was being made that you could generate the same effect?
https://hal.archives-ouvertes.fr/hal-00981257/document#:~:te....
https://hal.archives-ouvertes.fr/hal-00981257/document#:~:te....
This paper is claiming that the Casimir Effect could be used to generate a "Natario warp drive spacetime", but the validity of that claim is questionable. (As far as I can tell, this paper has not been peer-reviewed so its claims have not been checked by other experts in the field.)
Also, even if the claim about the Casimir Effect is correct, since the Casimir Effect occurs between a pair of uncharged plates, the "warp bubble" would have to be between the plates--but that means the warp bubble can't move the plates, so using it as a "warp drive" would not be possible.
Also, even if the claim about the Casimir Effect is correct, since the Casimir Effect occurs between a pair of uncharged plates, the "warp bubble" would have to be between the plates--but that means the warp bubble can't move the plates, so using it as a "warp drive" would not be possible.
Again I suspect there is more that I don't understand but this paper goes into geometries that could possibly lead to being able to generate the required negative energy.https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-02...
That said I think you still need such huge energies that it's impractical.
That said I think you still need such huge energies that it's impractical.
> this paper goes into geometries that could possibly lead to being able to generate the required negative energy
The geometries in question don't generate negative energy, they are generated by negative energy (which actually isn't a very good term for the kind of stress-energy that is involved). This paper is just talking about different possible geometries that could exist between Casimir plates, based on different possible configurations of "vacuum energy" that could exist between the plates. Even if one of these geometries turned out to be a "warp bubble", it still wouldn't change the issue that I described before, that all of this is taking place between uncharged plates, and the plates are outside the "warp bubble" so the bubble can't move them, so this is not a viable option for an actual "warp drive".
The geometries in question don't generate negative energy, they are generated by negative energy (which actually isn't a very good term for the kind of stress-energy that is involved). This paper is just talking about different possible geometries that could exist between Casimir plates, based on different possible configurations of "vacuum energy" that could exist between the plates. Even if one of these geometries turned out to be a "warp bubble", it still wouldn't change the issue that I described before, that all of this is taking place between uncharged plates, and the plates are outside the "warp bubble" so the bubble can't move them, so this is not a viable option for an actual "warp drive".
Good to know thanks for the explanation.
> the author seems to note that faster than light travel is possible under Einstein’s physics
He's talking about a highly speculative form of "warp drive" he has proposed that supposedly does not require negative energy, as all previous "warp drive" solutions (such as the Alcubierre drive) did. His speculations have not been accepted as valid by other physicists working in the field. (Also, calling such travel "faster than light" is problematic, since a light beam emitted from inside a "warp bubble" would still outrun the bubble itself. These are really just highly counterintuitive spacetime geometries that are most likely not physically possible anyway because the kinds of materials that would be needed to create them don't exist and can't be manufactured.)
He's talking about a highly speculative form of "warp drive" he has proposed that supposedly does not require negative energy, as all previous "warp drive" solutions (such as the Alcubierre drive) did. His speculations have not been accepted as valid by other physicists working in the field. (Also, calling such travel "faster than light" is problematic, since a light beam emitted from inside a "warp bubble" would still outrun the bubble itself. These are really just highly counterintuitive spacetime geometries that are most likely not physically possible anyway because the kinds of materials that would be needed to create them don't exist and can't be manufactured.)
The quoted sentence has dropped the word "can't" I suspect.
Reworded: "Horizons pose an issue for warp drives as one cannot control the warp bubble if one can't see or communicate with it."
The problem, then, is that warp drives as posited create KVF horizons that would seem to make them impossible to control, enter, or leave. That would mean warp drive is not permitted in the universe.
Some KVF horizons, like the outer ergosphere of a black hole can be detected, entered, and exited. Conclusion, warp drives are not ruled out on the grounds that they involve horizons because we don't know what kind of horizon they might have.
tl;dr Warp drives aren't ruled out... yet.
Reworded: "Horizons pose an issue for warp drives as one cannot control the warp bubble if one can't see or communicate with it."
The problem, then, is that warp drives as posited create KVF horizons that would seem to make them impossible to control, enter, or leave. That would mean warp drive is not permitted in the universe.
Some KVF horizons, like the outer ergosphere of a black hole can be detected, entered, and exited. Conclusion, warp drives are not ruled out on the grounds that they involve horizons because we don't know what kind of horizon they might have.
tl;dr Warp drives aren't ruled out... yet.
Faster than light travel and time travel are both fundamentally impossible in our universe and can't be achieved. I'm sure we'll learn all sorts of cool things looking for ways in. The reality is these will sadly live only in science fiction. There are plenty of things that are fundamentally impossible in our universe, faster than light travel being one of them.
"Heavier-than-air flying machines are impossible." -- Lord William Thomson Kelvin, British mathematician and physicist, president of the British Royal Society, 1895.
Edit: See https://news.ycombinator.com/item?id=27816225 for a discussion of the paper that launched this whole topic.
Edit 2: The actual paper you want is this one by the author of OP, the first one is referring back to this: https://arxiv.org/abs/2006.07125
Edit: See https://news.ycombinator.com/item?id=27816225 for a discussion of the paper that launched this whole topic.
Edit 2: The actual paper you want is this one by the author of OP, the first one is referring back to this: https://arxiv.org/abs/2006.07125
Scientifically speaking, the fact that someone was wrong in the past, doesn't mean someone else is wrong now, even if they use a similar sentence structure.
PS Yes, I am fun at parties, if I ever get invited to one again!
PS Yes, I am fun at parties, if I ever get invited to one again!
You're technically correct but missing the point. Past hubris is a warning that we are capable of similar hubris.
What we can say is our current model of the universe does not allow for any practical travel beyond the speed of light.
What we can say is our current model of the universe does not allow for any practical travel beyond the speed of light.
We can also say that our current model of the universe is orders of magnitude more accurate, so it's more likely that we're correct about the speed of light - something which has been experimentally tested and verified by hundreds if not thousands of people - than the opinion of one particular Lord Kelvin regarding the viability of aircraft.
It's just as hubristic to assume that science is as ignorant now as it always has been.
It's just as hubristic to assume that science is as ignorant now as it always has been.
Yes but we do have theories that suggest getting from point A to point B faster than light could be possible. We just have to do some things that appear impractical such as bend spacetime.
If FTL were possible, don’t you think there’d already be evidence of the time travel? First thing I’d do with FTL is travel back to the earliest habitable epoch in the Universe and start building galaxy-sized Dyson spheres.
Can you elaborate please.
Why does FTL produce time travel?
Why go back and build Dyson spheres?
Why does FTL produce time travel?
Why go back and build Dyson spheres?
To answer the second question, so that I can be the bad guys from the end of that Cixin Liu trilogy. Exponential growth is infinite and resources are not, so take any available advantage.
Short answer to the first, because the speed of light is the top speed at which information can propagate in the Universe.
Short answer to the first, because the speed of light is the top speed at which information can propagate in the Universe.
I don't think a gold standard is a viable pyramid scheme for a multiplanetary species. There's too much of it in space. You need something that is rare even after traveling for hundreds of years. A lot of wars were simply fought over resources. It's even better if your society uses the resource as money. The only thing I can imagine is using habitable planets as currency, this would require your galactic empire to have a dozen planets though because a single planet does not make a currency and the debt trap of borrowing 1 planet at 3% interest would fall apart because colonizing the next planet may take too long to pay off the interest. I.e. you need some planets to lose before you take that loan.
Edit: Great filter hypothesis to explain Fermis paradox. The pyramid scheme needed to enslave the galaxy is also powerful enough to destroy a planet before it develops space faring technology.
Edit: Great filter hypothesis to explain Fermis paradox. The pyramid scheme needed to enslave the galaxy is also powerful enough to destroy a planet before it develops space faring technology.
Just to point out obvious, you are applying a human logic to an theoretical alien species of theoretically any type of behaviour/value/technological/political angle.
Who knows what they are like and what they value.
Who knows what they are like and what they value.
Thank you another book/trilogy to add to the reading list.
I suggest you also add The Wandering Earth by the same author to your list, which is a collection eleven of his short fiction stories. I'd even go as far as say it's better to read it first before the trilogy.
You’re welcome! Incidentally, there’s no FTL in the story.
In special relativity, faster-than-light travel or communication is equivalent to time travel.
But, it doesn't enable time travel. It just makes calculation easier, like «bending» of space-time array.
If you switch from relativity to regular physics with medium instead of pure vacuum (10 years passed since 2012 already, so relativity is busted at this point), then c is the speed of light in medium, so we don't need to bend anything or use negative t to calculate outcomes.
If you switch from relativity to regular physics with medium instead of pure vacuum (10 years passed since 2012 already, so relativity is busted at this point), then c is the speed of light in medium, so we don't need to bend anything or use negative t to calculate outcomes.
> so relativity is busted at this point
Excuse me?
Excuse me?
Yes but that is only if you are traveling through space faster than light. If you simply skip some of the distance with something like an Alcubierre drive there is no time dilation to deal with.
> simply skip some of the distance
My understanding as a lay person of the physics that allow for things like this, is that they require starting conditions that wouldn't produce a universe anything like what we actually live in, or require energy levels around the output of galaxy superclusters to be focused into planck lengths or such. "Technically not impossible" is different for physicists than to normal people ;-)
My understanding as a lay person of the physics that allow for things like this, is that they require starting conditions that wouldn't produce a universe anything like what we actually live in, or require energy levels around the output of galaxy superclusters to be focused into planck lengths or such. "Technically not impossible" is different for physicists than to normal people ;-)
No, it still results in time travel.
> To answer your question: any FTL drive can be used to create closed time-like loops. With these you can travel back in time, but you get stuck in a loop.
https://physics.stackexchange.com/a/247048
> To answer your question: any FTL drive can be used to create closed time-like loops. With these you can travel back in time, but you get stuck in a loop.
https://physics.stackexchange.com/a/247048
Okay, I think there is something I don't understand here. That link references a book I might have to read "Time Travel and Warp Drives".
Thanks!
Thanks!
Those theories would still result in time travel becoming possible and all the assorted paradoxes that come with it.
[deleted]
Valid warnings also don't mean anything.
Which ignored the obvious and prevalent heavier-than-air flying birds. We have no such examples to point at with FTL travel.
"Inertia is a property of matter" - fundamental unchangeable property of the universe
...so far.
What about perpetual motion machines though? :)
> Faster than light travel and time travel are both fundamentally impossible in our universe and can't be achieved.
That may very well be, and true in our lifetime, but given how little we know about a lot of things, and that in many cases we don't even know what we don't know, I would hesitate in making an absolute statement.
That may very well be, and true in our lifetime, but given how little we know about a lot of things, and that in many cases we don't even know what we don't know, I would hesitate in making an absolute statement.
It's been proven rather thoroughly though. Look at Kip Thorne's body of work. His book: Black Holes and Time Warps is accessible.
c0 is the speed of light only incidentally. It is the speed of causality first. You can warp spacetime but you can't change the speed limit.
You might think there would be a hypothetical way to cheat it from some unknown phenomenon, but then any occurrence of that phenomenon would destroy itself in zero time.
c0 is the speed of light only incidentally. It is the speed of causality first. You can warp spacetime but you can't change the speed limit.
You might think there would be a hypothetical way to cheat it from some unknown phenomenon, but then any occurrence of that phenomenon would destroy itself in zero time.
[deleted]
Plenty of current physics allows for the possibility - even special relativity allows for it.
Personally, I think that the “speed of light” is a misleading misnomer. It’s the speed of information propagation in flat space - and even that’s a misnomer, as speed is a very human concept, as is time, as is space.
For all we know we’re waddling around in flatland, thinking that what we perceive is what is. More likely than not, most everything we know isn’t even wrong.
To borrow Haldane - “Nature is not only queerer than we suppose, it is queerer than we can suppose.”
Personally, I think that the “speed of light” is a misleading misnomer. It’s the speed of information propagation in flat space - and even that’s a misnomer, as speed is a very human concept, as is time, as is space.
For all we know we’re waddling around in flatland, thinking that what we perceive is what is. More likely than not, most everything we know isn’t even wrong.
To borrow Haldane - “Nature is not only queerer than we suppose, it is queerer than we can suppose.”
Doesn't the concept of Wormholes, folding space or Warp engines bypass FTL blocks?
You're essentially relocating from one location to another by bypassing the distance... Leaving this universe long enough to travel a comparatively smaller distance/time in another then pop back into our universe a long long way but short amount of local time later.
Sure you dont travel FTL but you do arrive in the location Faster than light would take to travel in mundane space.
You're essentially relocating from one location to another by bypassing the distance... Leaving this universe long enough to travel a comparatively smaller distance/time in another then pop back into our universe a long long way but short amount of local time later.
Sure you dont travel FTL but you do arrive in the location Faster than light would take to travel in mundane space.
I believe all of those violate causality, at least if you have more than one of them (more than one pair in the case of wormholes).
Causality might very well be locked to this universe/space.
I mean its all just sci-fi at the moment. But lots of things we take advantage of today was at one time in the past as well.
I prefer to be optimistic about potential breakthroughs.
I mean its all just sci-fi at the moment. But lots of things we take advantage of today was at one time in the past as well.
I prefer to be optimistic about potential breakthroughs.
I am not familiar with this field but extraordinary claims require extraordinary evidence and I am not aware of any evidence absolutely ruling out the possibility. From my understanding we are still unsure about very fundamental aspects of physics and the universe so there are certainly unknown unknowns left which could dramatically change our understanding of everything.
> extraordinary claims require extraordinary evidence
I think you've got the sign on your skepticism reversed. The burden of proof is very much on those who claim that FTL travel is possible.
From Wikipedia:
I think you've got the sign on your skepticism reversed. The burden of proof is very much on those who claim that FTL travel is possible.
From Wikipedia:
The special theory of relativity implies that only particles with zero rest mass (i.e., photons) may travel at the speed of light, and that nothing may travel faster.
As of the 21st century, according to current scientific theories, matter is required to travel at slower-than-light (also STL or subluminal) speed with respect to the locally distorted spacetime region. Apparent FTL is not excluded by general relativity; however, any apparent FTL physical plausibility is currently speculative.
The burden of proof for people making that claim is on the person making that claim. The burden of proof for making the claim that it’s “fundamentally impossible” is likewise on the person making that claim.
For all we know, our understanding of the universe could be at a stage of analogous to “there are four elements: earth, air, fire, and water”. Many things that we enjoy daily were once “fundamentally impossible”.
For all we know, our understanding of the universe could be at a stage of analogous to “there are four elements: earth, air, fire, and water”. Many things that we enjoy daily were once “fundamentally impossible”.
> Many things that we enjoy daily were once “fundamentally impossible”.
Not so. They may have been conceptually impossible but did not run into the "fundamental" laws of physics. FTL travel breaks physics. A lot abiut our observable universe would be different if it were possible.
The evidence that the universe obeys these laws is significant and, currently, not counter indicated.
Not so. They may have been conceptually impossible but did not run into the "fundamental" laws of physics. FTL travel breaks physics. A lot abiut our observable universe would be different if it were possible.
The evidence that the universe obeys these laws is significant and, currently, not counter indicated.
> but did not run into the "fundamental" laws of physics
Those "fundamental" laws of physics didn't exist several centuries ago.
Those "fundamental" laws of physics didn't exist several centuries ago.
Yes they did, we just hadn't figured them out yet. If we were very wrong in them, we'd have to be wrong in a way that doesn't affect all the technology using the wrong theory - which is part of why we refined them, because shit didn't work.
That's what it means to be contradicting fundamental physics. It implies all of our other knowledge is wrong, but we are just lucky enough that our machines work anyway. Which, unless we see a phenomenon that we can't fit into our existing theories, seems unlikely
That's what it means to be contradicting fundamental physics. It implies all of our other knowledge is wrong, but we are just lucky enough that our machines work anyway. Which, unless we see a phenomenon that we can't fit into our existing theories, seems unlikely
Let's invent new laws of physics and hope the universe adopts them.
> FTL travel breaks physics
Like quantum mechanics broke physics?
Like quantum mechanics broke physics?
Quantum mechanics _fixed_ physics. Newtonian physics predicts that an electron will spiral into the nucleus of an atom within a few nanoseconds of the atom’s creation.
We observed quantum effects before the theory. We haven't seen any phenomena so far which require physics that could lead to FTL
By this standard, there is literally nothing that can be called impossible. Perpetual motion machines can't be said to be impossible because, who knows, maybe there is a God and tomorrow He could flip the switch for free energy from off to on.
It's simply the difference between claiming something is "impossible based on our current understanding/model of physics" and "impossible regardless of what we don't understand". Science gives us the first one, but the original commentator unambiguously claimed the second. Our models of the universe being wrong happens all the time, sometimes in significant ways, and even now there are plenty of unexplained things in our universe that have the potential to change our understanding. It seems pretty optimistic to assume our current models are guaranteed correct in every circumstance we haven't yet observed even when they can't explain everything we have actually observed.
So yes, perpetual motion could start working tomorrow, or could already exist for all we know. That doesn't mean it's wrong to say physics _currently_ says it's impossible and any claim better have some pretty good evidence to dispute it (which none currently do). But that also doesn't mean that if we're around centuries/millennia from now we won't have found a special case where our current understanding breaks down.
So yes, perpetual motion could start working tomorrow, or could already exist for all we know. That doesn't mean it's wrong to say physics _currently_ says it's impossible and any claim better have some pretty good evidence to dispute it (which none currently do). But that also doesn't mean that if we're around centuries/millennia from now we won't have found a special case where our current understanding breaks down.
I don’t think that anyone made the claim that FTL travel is possible; they merely said that there’s still so much we don’t know about the universe, so definitely ruling something out is a bit rash.
With our current understanding of the universe, yes, indeed, FTL travel is not possible. But it’s naïve to think that our understanding of universe won’t change.
With our current understanding of the universe, yes, indeed, FTL travel is not possible. But it’s naïve to think that our understanding of universe won’t change.
After developing some intuition for relativity you realize that asking whether FTL is possible is more akin to asking whether you can remain inside a circle after traveling radius + 1 from the center, than it is to asking whether a car can be modded to exceed its top speed. It is nonsensical, not logically coherent within all currently known physics and it goes to the fundamental nature of motion itself.
It can't be ruled out, but neither can Russell's Teapot.
It can't be ruled out, but neither can Russell's Teapot.
"FTL is impossible" doesn't seem like an extraordinary claim at all, but actually quite an ordinary one. The natural world seems to obey a light-speed limit every time it's been tested.
Sure, it might be wrong, but as far as claims go it has a lot going for it, both experimentally and theoretically.
Sure, it might be wrong, but as far as claims go it has a lot going for it, both experimentally and theoretically.
Well, technically time travel forward is easy.
And to say that backwards time travel isn't possible at this point doesn't make much sense. We don't even have a clear understanding of what time is, or why it goes forwards.
And the singularity should be around the corner, and that might present a computerized solution to emulate reverse time travel.
And to say that backwards time travel isn't possible at this point doesn't make much sense. We don't even have a clear understanding of what time is, or why it goes forwards.
And the singularity should be around the corner, and that might present a computerized solution to emulate reverse time travel.
Not sure if it matters too much from an individualistic perspective. Near light-speed travel will allow a person to go anywhere instantaneously, they'd just have to contend with leaving the time and space they know behind them.
Only for sufficiently small values of anywhere. Even near-light-speed communication in fiber optics has detectable and significant delay just in crossing a continent. A trip made physically by humans every day.
Near-light-speed travel to the Moon would take a few seconds. Near-light-speed travel to Mars would take longer than my drive to the grocery store. These are not "instantaneously."
Near-light-speed travel to the Moon would take a few seconds. Near-light-speed travel to Mars would take longer than my drive to the grocery store. These are not "instantaneously."
I'm talking about time dilation. From the perspective of the pilot of a near light speed ship a trip across the galaxy would be instantaneous.
It's instantaneous at 100% of light speed (impossible for massive objects). At "near" light speed, you need seventeen nines (0.99999999999999999c) to bring a trip across the galaxy down to a subjective day. I'm not sure how near your idea of "near" is. You also need instantaneous acceleration at both ends, which is quite a requirement.
OK. Only Pluto in a few hours, Mars in a few minutes, and the moon in less time it would take for me to reach the toilet from my living room.
Not literally instantaneous, but still practically instantaneous at the scale of the solar system.
Not literally instantaneous, but still practically instantaneous at the scale of the solar system.
> Near light-speed travel will allow a person to go anywhere instantaneously
Only if you ignore the enormous acceleration/deceleration forces needed for this.
Only if you ignore the enormous acceleration/deceleration forces needed for this.
Surprisingly it's not that much. At a constant 1G acceleration you could travel across the galaxy (100k light years) in 12 years (from your reference frame)[0].
I definitely agree that such travel would require enormous amounts of energy.
[0]: https://en.wikipedia.org/wiki/Space_travel_under_constant_ac...
I definitely agree that such travel would require enormous amounts of energy.
[0]: https://en.wikipedia.org/wiki/Space_travel_under_constant_ac...
also, "instantaneously" is a stretch on a galactic scale. 299792000 m/s isn't as fast as it sounds if you need to cross the entire galaxy.
Time dilation doesn't matter here.
Time dilation doesn't matter here.
[deleted]
[deleted]
While faster-than-light (FTL) travel is impossible through spacetime as PhD physicists know it (i.e. as "Relativity-confirming experiments reveal it"), there are forms of apparent FTL that have mainstream PhD acceptance, because they don't allow information to travel from point A to point B faster than light and thus do not violate causality.
The expansion of the observable universe itself being apparently FTL is perhaps the most interesting. If we point the Hubble telescope (and soon JWST) towards opposite ends of the observable universe, we observe extreme redshift galaxies receding away from each other in opposite directions greater than twice lightspeed if we measure according to "distance in light years" from earth. But no photons or information from a galaxy at one side of our observable universe can be sent to a galaxy at the opposite end: they are receding away from each other too rapidly, these two galaxies exist Beyond the Cosmological Event Horizon relative to each other and are forever unknowable to each other, and thus causality is not violated.
An advanced alien race in galaxy A might be able to beam a message to earth before galaxy A vanishes Beyond the Cosmological Event Horizon within the next 100 million years earth-time, and likewise galaxy B might beam a message to earth within the next 100 million years earth-time, but earth can't relay the message from A to B or B to A because they have already forever receded from each other and will soon both likewise forever recede away from us.
The expansion of the observable universe itself being apparently FTL is perhaps the most interesting. If we point the Hubble telescope (and soon JWST) towards opposite ends of the observable universe, we observe extreme redshift galaxies receding away from each other in opposite directions greater than twice lightspeed if we measure according to "distance in light years" from earth. But no photons or information from a galaxy at one side of our observable universe can be sent to a galaxy at the opposite end: they are receding away from each other too rapidly, these two galaxies exist Beyond the Cosmological Event Horizon relative to each other and are forever unknowable to each other, and thus causality is not violated.
An advanced alien race in galaxy A might be able to beam a message to earth before galaxy A vanishes Beyond the Cosmological Event Horizon within the next 100 million years earth-time, and likewise galaxy B might beam a message to earth within the next 100 million years earth-time, but earth can't relay the message from A to B or B to A because they have already forever receded from each other and will soon both likewise forever recede away from us.
There is no restriction on relative velocities between objects from a given observer as long as all objects have velocities less than the speed of light. You can point laser pointers in opposite directions and the "front" of the laser pointer will recede at "2x" C from the opposite pointer.
In the case of the expanding universe, our standard "flat" space time represented by minkowski space is warped into de-sitter space. As distances increase apparent acceleration increases up to and beyond the speed of light due to the expansion of space time.
In the case of the expanding universe, our standard "flat" space time represented by minkowski space is warped into de-sitter space. As distances increase apparent acceleration increases up to and beyond the speed of light due to the expansion of space time.
Nice! Great solution to a common inconvenience.
Granted this article is largely above my reading/comprehensive level, but after describing multiple types of horizons and then saying a wrap drive horizon largely resembles a black hole horizon, I can't see where the author actually defines "the horizon problem".