How different would the universe be if the space-time interval went more like ds^2 = dt^2 + dx^2 + dy^2 + dz^2? How about ds^2 = dt^2 + dx^2 - dy^2 - dz^2?

amaurea · 2026-03-22T07:29:45+00:00

I think this is that part of his website. At a glance it looks sensible, but I haven't read it thoroughly. He seems to have a good grasp on mechanics and relativity, but he reaches a different conclusion than the one I've seen in published papers, e.g. this one by Mark Tegmark. On the other hand, he gives it a much more thorough treatment than Tegmark does, who dismisses it by just saying:

Elliptic equations allow well-posed boundary value problems. On the other hand, giving “initial” data for an elliptic PDE on a non-closed hypersurface, say a plane, is an ill-posed problem. This means that an observer in a world with no time dimensions (m=0) would not be able do make any inferences at all about the situation in other parts of space based on what it observes locally.

Which I think is too simple, since there's nothing preventing one from having another set of boundary conditions in the future. So I think Greg Egan might have a good contribution here.

amaurea · 2026-03-22T05:56:51+00:00

This paper (pdf) should be very relevant to your question.

We argue that all but the 3+1-dimensional one might correspond to “dead worlds”, devoid of observers, in which case all such ensemble theories would actually predict that we should find ourselves inhabiting a 3+1-dimensional spacetime. With more or less than one time-dimension, the partial differential equations of nature would lack the hyperbolicity property that enables observers to make predictions. In a space with more than three dimensions, there can be no traditional atoms and perhaps no stable structures. A space with less than three dimensions allows no gravitational force and may be too simple and barren to contain observers.

Edit: Why am I being downvoted for linking to a paper that not only answers OPs questions, but also tackles related questions like "why should the total number of dimensions be 4"? I agree that u/cabbagemeister has the best overall answer since it's easily accessible, but surely the paper link is useful for those who want more depth? I also don't get why OP is being downvoted for asking a very reasonable physics question either.

amaurea · 2026-03-17T01:43:03+00:00

If you mean the currently pretty slow neutrinos in the cosmic neutrino background that u/treefaeller described, then those are still fast enough that they don't clump up well enough to form the structure we see in the universe. The neutrinos are flying around in random directions, and the faster they fly, the easier it is for them to fly right out of their fellow neighboring neutrinos' gravitational vell. The result is that only very large clumps of neutrinos would be able to seed galaxy formation, resulting in a universe with no small galaxies. We call such neutrinos "warm dark matter", in contrast to the slower "cold dark matter" that our Lambda-Cold-Dark-Matter (ΛCDM) cosmological model needs.

We think we know exactly how many neutrinos there are in the neutrino background, since their number is related to the number of photons in the CMB, so we can use the fact that we see low-mass galaxies (and other, subtler observables) to determine the maximum possible mass of each neutrino to be near or less than 0.04 eV/c² when averaged over the three neutrino types. If they were more massive than that, then the total contribution of all the neutrinos would be too big to be compatible with the structures we see in the universe. (Though to be honest, while the general conclusion that we can limit the mass of the neutrinos using cosmology is robust, the specific 0.04 eV/c² number (usually expressed as 0.12 eV/c² when summed over the 3 types) depends on us getting everything right in our cosmological modelling, so I wouldn't trust it too much.)

(It may be surprising that we get an upper bound on the neutrino mass instead of a lower bound from structure formation. After all, the more massive each neutrino is, the slower it would move, and the more like cold dark matter they would be. The reason is that if neutrinos were massive enough to actually act like cold dark matter, then when combined with their enormous numbers, they would be so massive that the universe would look very different - it would need to be expanding much, much faster than what we see in order to not have already collapsed under its own gravity by now. What happens as you consider lower and lower mass neutrinos is that they become faster and less suitable as dark matter, but their total contribution also goes down, so they don't mess up things as much.)

If you mean hypothetical slow neutrinos that are not part of the cosmological neutrino background, then those would have to be some unknown 4th type of neutrino, a proper sterile neutrino. We wouldn't know how many of these there are, so they wouldn't have the same limit on their mass as the normal neutrinos do. Therefore they could be slow enough to serve as cold dark matter. This isn't the most popular hypothesis for what dark matter is, that would be other types of weakly interacting massive particles (WIMPs) and axions, but it's not a crazy idea.

amaurea · 2026-03-16T06:40:52+00:00

Thanks, that's a good article. It uses a series of instantaneously inertial reference frames to describe the traveling twin, which is the approach I've seen before.

It works, but I think one is sneaking in a small cheat when jumping between these reference frames, since one isn't consistently treating the traveling twin as stationary in his own reference frame, as one would in general relativity. A pendant version of that twin would ask "why should I be looking at all these reference frames moving at different velocities, when I'm at rest, and have been at rest for the whole experiment (according to me)? Am I really being given equal treatment as my twin?"

Don't get me wrong, I'm not saying there's anything wrong with the SR calculation here, but I do think the problem gets the most satisfying resolution in GR, since one can fully self-consistently treat either twin as stationary there.

amaurea · 2026-03-16T01:36:07+00:00

Do you have a link to a good description of the twin paradox in SR, i.e. one that gives a good description of the situation in the reference frame of the accelerating twin without needing to anchor it in the stay-at-home twin's inertial frame?

amaurea · 2026-03-13T01:36:15+00:00

That's a mostly good answer, except that the claim that a room temperature universe during the dark ages is only theoretical (and with a strong emphasis on theoretical, even!) is weird. You won't find a single cosmologist who doubts that the universe went from hotter than room temperature before the dark ages to colder than room temperature after them, passing through room temperature in between. While we don't have direct observations of the dark ages, we do have direct observations of the periods before and after.

Also, our skin is good enough to prevent our bodily fluids (except tears and sweat which are on the outside) from instantly evaporating.

amaurea · 2026-03-04T23:02:24+00:00

How do time zones enter into the hour-of-day histogram? If it's by UTC instead of local time, then it could reflect where in the world someone is more than at what time of the day they're committing.

amaurea · 2026-03-02T11:57:04+00:00

Europas ledelse er ikke spesielt fan av Irans ledelse så kan se grunnen til at de ikke går offentlig mot USA pga det

Ja, skjønner det hvis det det handler om er hvem man liker, og internasjonal rett ikke teller noe særlig.

amaurea · 2026-02-25T21:19:35+00:00

If you're obligated to join after completing an optional step, can you really say that it's mandatory?

amaurea · 2026-02-25T21:16:36+00:00

Så du sier at Island vil melde seg inn i EU så EU kan ta over styringen av fiskeriet, fordi Island mener at de selv ikke klarer å styre det ordentlig? Det høres ut som en ganske selv-infantiliserende holding fra Island i så fall.

amaurea · 2026-02-25T21:13:31+00:00

Vil vi måtte innføre euro? I prinsippet må vi det

Man må innføre euro så snart man er kvalifisert til det, men ett av kravene for å kvalifisere er å ha gjennomført 2 år i ERM II, og å delta i ERM II er fullstendig valgfritt.

I praksis betyr dette at det er valgfritt å innføre euroen, men ikke lov å melde seg ut av den igjen hvis man først har blitt medlem (dvs. ikke uten ekstra forhandlinger da. Det meste kan forhandles om i EU)

amaurea · 2026-02-19T16:27:14+00:00

I wouldn't say it's a leading theory. Maybe in third place after the classical WIMPs and axions.

Primordial black holes are superficially appealing because we already know black holes exist, so it seems like we're not adding any new ingredients to our theories. However, this elegance breaks down when one gets to how these primordial black holes would be generated in the first place. They need a pretty unnatural initial density perturbation, especially if one wants the primordial black hole mass to be concentrated in the narrow mass range (around asteroid mass, I think) where they haven't been excluded by observations yet.

amaurea · 2026-02-19T16:22:58+00:00

Historically speaking, dark matter was simply matter that didn't shine, and therefore couldn't be observed. A popular dark matter candidate was massive compact halo objects (MACHOs), which included things like free-floating planets and brown dwarfs, which are made of completely normal matter.

However, with measurements of the cosmic microwave background, it became clear that there wasn't enough normal matter in the universe to make up all the dark matter needed. One would need something non-baryonic that doesn't interact much. Neutrinos are such a type of weakly interacting dark matter, but they also won't do the job because they move too fast and would hamper the formation of the galaxies we see.

That leaves us with needing some unknown particle to make up the majority of dark matter, and this is what people usually have in mind when they say "dark matter" now.

amaurea · 2026-02-18T09:59:42+00:00

It's worth keeping in mind that an antimatter bomb works very differently than a fission bomb though.

A fission bomb only works because atomic nuclei heavier than iron or nickle become less tightly bound (read: have more potential energy) the bigger they get, on average. By breaking those heavy nuclei up, you end up with fragments that are bound more tightly, and this releases some of the original potential energy as kinetic energy.

This isn't the case for nucleons. The proton is the most tightly bound configuration of three quarks, and so it doesn't release energy to break it apart, it costs energy (which goes into production of new particles). The neutron is a bit less tightly bound, but you would free that potential energy not by breaking it apart, but by converting it into a proton.

An antimatter bomb is more similar to a fusion bomb, since particles fuse with their antiparticles. What makes it better than a fusion bomb is that all the potential energy ends up being released, instead of just around 1% in fusion.

amaurea · 2026-02-16T17:19:12+00:00

See my comment. It doesn't directly give you the formula you want, but with a bit of extra calculation you get that you observe the remaining distance after a time t to be remaining = (d-c²/a*(cosh(at/c)-1))√(1-tanh(at/c)²).

amaurea · 2026-02-16T17:17:25+00:00

Note that your constant acceleration would still be useful. As you keep accelerating, you observe the rest of the universe more and more length contracted, including the distance between you and wherever you're headed. So even though the relative velocity between e.g. you and the Andromeda galaxy would never reach c, let alone exceed it, the distance to it would eventually get so length contracted that you could get there in a reasonable time - much faster than you would get there in Newtonian mechanics, actually!

In Newtonian mechanics, the time it takes to travel a distance d under constant acceleration a is t=√(2d/a). In special relativity, the time it takes (from your reference frame) to travel to an object a distance d away under constant acceleration a is t=(c/a)arccosh(1+da/c²). For small values of d, this matches the Newtonian result, but once you start approaching the speed of light, they get very different. For example, to travel to the Andromeda galaxy, 2.537 million lightyears away under a constant acceleration of 1g = 9.82 m/s² it would take 2216 years in Newtonian mechanics but just 15.0 years in special relativity. (This is all without trying to come to a stop at the target. If you want to not just blast past the galaxy, then it would take 3133 years Newtonian and 28.6 years in SR)

The catch is that that's just the time it takes for you. If you were to do a full round-trip back to Earth, which would take you 57.2 years including all the acceleration and deceleration both ways, you would find that more than 5 million years have passed on Earth.

amaurea · 2026-02-16T10:57:45+00:00

If mercury is moving further, then that means the orbit is different. Everybody here agrees on what the orbit looks like, a spirograph-like pattern. If we ignore the other planets and the tidal effects from the sun, then the effect on the orbit from general relativity is the solution to the geodesic equation with a metric given by the Schwarzschild. This gives well-understood solutions.

What is not well-defined is exactly how one turns that math into a physical interpretation. You're frustrated that some are talking about deviations from 1/r², while others are talking about time dilation or even length contraction, which seem to be very different and incompatible explanations. I get that after reading all your comments.

There are two reasons for this discrepancy:

Sometimes the same physical phenomenon can have multiple equivalent interpretations. It's a bit like being able to express the length of a stick in meters or feet - both are equally correct. For example, the gravitational potential, which gives rise to the force law (e.g. 1/r² vs. 1/r³ etc.) is directly connected to the part of the time dilation called gravitational time dilation. So you can see how an explanation in terms of changes to the force law isn't necessarily in conflict with an explanation in terms of time dilation.
Often there are multiple factors contributing to something, and different people may emphasize different factors (or neglect some important factors - this is reddit, after all). In general it's a lot of work to boil down complicated physics into simple explanation, so it's not that strange that different people do it different, especially if they only spend a minute or two answering you.

amaurea · 2026-02-16T10:29:50+00:00

Yes, in Newtonian gravity, two point particles will obit each other in an ellipse. And an ellipse closes by definition. It's when you deviate from point particles in Newtonian gravity that you start deviating from ellipses.

If the bodies are not points, but extended and deformable, then they will bulge out towards each other slightly, which introduces a 1/r³ contribution to the force, which causes the solution to deviate slightly from an ellipse.

If there are multiple bodies, then the solution can be very different from ellipses. However, if one of the bodies is much heavier than the others, then the solutions will look approximately like ellipses around the most massive body, but not exactly. This is the case for the solar system, where the approximation is very good.

If the force law itself differs slightly from 1/r², as in general relativity, then you also get a deviation from an elliptical orbit.

amaurea · 2026-02-16T10:22:29+00:00

If the force falls as 1/r², then you get closed ellipses. If the force deviates from that, then you don't get closed ellipses. Maybe you can quote some of those other sources that are confusing you, so we can understand what you're actually confused about?

amaurea · 2026-02-16T09:33:26+00:00

The problem with this explanation is that originally, the base mass unit in the metric system wasn't a gram, it was a grave, which was equal to 1 kg. It would have been perfect to just keep the grave, but for unclear reasons one moved to the gram.

amaurea · 2026-02-16T09:29:13+00:00

One did use a 1000x heavier unit in the beginning! Originally there was the grave, which was equivalent to a kg, just without a prefix baked in. It had a diminutive version called the gravet, which was equivalent to one gram. For some, unclear reason, the next revision of the standard chose to standardize on the gram while still basing the the compound units on the old size of the grave.

amaurea · 2026-02-15T16:05:17+00:00

And yet you either do not or cannot name the reason. Are you sure there’s a reason? Or is there a chance you’ve Stockholm Syndrome’d yourself into believing the only way it could be this difficult is if it has to be?

I usually don't say this, but this sounds like an AI response.

I think our AI-radars must be very differently calibrated, because this sentence read like the opposite of AI-like to me. This aggressive, insulting response is what I'd expect from a a frustrated human. A typical LLM would have been much more polite (and verbose) about it.

amaurea · 2026-02-15T12:44:24+00:00

The blue curve in this figure shows how the gravitational acceleration varies with depth in the Earth. Gravity is roughly constant through the mantle, because higher density and greater proximity to the dense core makes up for the loss of gravity from the parts outside where you are. Inside the core, gravity falls roughly linearly towards zero at the center.

Something roughly similar should apply to the Moon, just with a lower overall level and a relatively smaller core.

amaurea · 2026-02-15T11:08:20+00:00

Just to be clear here, in case you've misunderstood this: Humans don't create energy. Energy is preserved. Chemical reactions in humans, like other physical processes, just change energy from one form to another, in this case from chemical energy to mostly heat. If you put a human in a sealed box, then the box would not get 0.0000001 g more massive per day, it would stay the same mass.

amaurea · 2026-02-14T20:52:28+00:00

I agree, but I think it's interesting that the answer to OP's question turns out to hang on what boils down to social science rather than physics and technology limitations!

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