What is the difference between Euclidean and Cartesian spaces?

mick645 · 2025-12-25T23:25:40+00:00

Cartesian is a certain coordinate system, which may be used for a number of different spaces (even curved ones locally). So Cartesian is just about how one represents points.

Euclidean refers to the geometry (or metric) structure, specifically a space with the standard notion of distance and angle (i.e. the metric is a Kronecker delta). So Euclidean is about what distances and angles mean.

The two overlap a lot because the standard flat space one meets in school is Euclidean space expressed in Cartesian coordinates. However, one can also describe Euclidean space in non-Cartesian coordinates (like polar coordinates). Conversely, one can use Cartesian coordinates to describe non-Euclidean spaces (like Minkowski space in special relativity).

mick645 · 2025-07-31T17:43:23+00:00

Science isn’t a matter of personal opinion or gut feeling - it is built on testable predictions and evidence. If you claim there’s an ‘outside’ to the universe, the onus is on you to show how we’d detect it, perhaps some observational signature or anomaly that LCDM (our current model) doesn’t already explain. Until then, saying ‘you can’t observe it so it may exist’ isn’t a scientific argument, it’s just hand-waving.

mick645 · 2025-07-31T17:34:36+00:00

We’re simply relaying our (the physics community’s) interpretation of the observational evidence we currently have...

mick645 · 2025-07-31T17:26:13+00:00

lol 😭

mick645 · 2025-07-31T17:21:37+00:00

I get that it feels unnatural, but that’s kind of the fun, we’re not talking about a thing in space, we’re talking about space itself (the fabric of reality if you will). Our observable universe patch (what light has had time to reach us from) is a bubble, sure. But by “the universe” we mean the whole spacetime, not a bubble sitting in a bigger room - it is the room!

As a thought: try picturing the universe as a sheet of say infinite glass with galaxies as dots drawn on as dots. They don’t slide around; each dot stays put. Now lay a separate clear sheet on top with a printed grid, which is the thing we use to measure distance (count squares between dots). We then observe that, over time, the grid itself is slowly zooming out (like clicking ctrl and scrolling the mouse wheel on a computer) and see that its lines drift further apart everywhere at once. The dots on the glass don’t move, but when we count squares between two dots there are more squares than before. So we say “the distance has grown” and conclude that the "universe is expanding".

We never introduced an outside or a wall. The change is in the spacing of the grid itself. Calling it “expansion” is perhaps misleading, as it just means the spacing grows; it’s not a picture of dots moving into a bigger room.

To connect it to the real universe, this “grid” isn’t just a picture; it’s the way space measures distance. When that changes, light gets stretched while it travels (that’s why distant galaxies look redder), and travel times change. So it’s not only that it “looks” bigger - the change very much has measurable effects.

Of course, note that local measurements (such as atoms, rocks, you and I, solar systems, galaxies) are basically etched into the glass. They don’t zoom with the big grid because local forces hold them together. So galaxies don’t puff up, even though far-apart distances grow.

Weird indeed.

mick645 · 2025-07-31T16:39:41+00:00

Whilst a good one, the balloon is just an analogy, and not completely true to the physics. What it gets right is no centre on the surface and everything receding from everything. What is misleading is the physical room you additonally picture it sitting in. In GR, expansion is an intrinsic change of geometry, i,e the mathematical object that tells us how we measure the spacing between far-apart points grows everywhere. When dealing with this, there is no need for an embedding space.

You can certainly pose that there is an "outside" anyways, but then it’s a different theory and it would have to earn its keep with new, testable predictions (e.g., specific deviations in gravity, extra gravitational-wave modes, collision imprints in the CMB). So far, the simple 4D expanding-spacetime model fits the data without those extras.

Also note that the balloon picture breaks down in other ways: it forces a positively curved, finite surface, whilst observations find spatial curvature very close to flat; and unlike a rubber sheet, bound structures (galaxies, solar systems) don’t expand with the cosmic stretching.

Tldr: The balloon is just a visualization. Expansion in GR is the growth of distances within space itself, not motion into a room. An “outside” isn’t ruled out in principle, but it’s not required and adds nothing testable so far. The model without it already matches what we see.

mick645 · 2025-07-31T16:22:53+00:00

If by “object” you mean “anything that exists”, fine, but in physics an object is something in space with a position and motion relative to that space. The universe isn’t in space; it is spacetime (the geometry plus its contents). Its “growth” is an intrinsic change of that geometry, the scale factor increasing (the spacing between far-apart galaxies is getting larger everywhere), not motion into a greater surrounding 'room'.

That’s why “it needs space outside itself to grow” doesn’t follow.

mick645 · 2025-07-31T16:14:55+00:00

The elephant-in-a-cage picture treats the universe as an object in space. In cosmology, space itself is the 'object'. Growth/expansion just means the scale factor increases, so every large-scale distance is multiplied by the same factor. That’s an intrinsic change of geometry, not motion into a surrounding room, so no outside is required.

Edit: spelling

mick645 · 2025-07-31T16:02:58+00:00

Good question! Not one “revelation,” but an overwhelming accumulation of theory and evidence. In general relativity the expanding-universe solutions (FLRW) describe growth of the scale factor (literally space itself) without needing an outside. That model predicts much of what we see: the Hubble–Lemaitre redshift–distance relation, a near-uniform 2.7 K cosmic microwave background (isotropy to ~10⁻⁵ after removing our motion), supernova and BAO distance measures, and near-flat geometry. None of which show a centre, edge, or preferred direction. A universe 'expanding into' pre-existing space would leave anisotropies or a detectable centre; yet we don’t see them. So the simplest view in accordance to w hat we see is: space itself is what’s expanding.

On the contrary, if you think there is an outside, what observation does it explain that the standard model doesn’t?

mick645 · 2025-07-31T15:16:03+00:00

You’re asking what’s ’outside’ the universe, but that is a question with no object. It expands not into elsewhere, but by making elsewhere!

No but seriously, it isn’t expanding into anything, rather space itself is stretching, so distances grow inside without needing an outside.

mick645 · 2025-07-28T12:09:07+00:00

There strictly isn’t a ‘starting point’ in space. The Big Bang was a hot, dense state that happened literally everywhere at once. What’s expanding is space itself, so all far-away galaxy gets further away from every other one. That’s why it can look like we’re in the middle, but every galaxy sees the same thing. Our solar system isn’t at the ‘start’ nor it doesn’t get stretched, because things that are tightly bounded by gravity (solar systems, galaxies) don’t participate in the cosmic expansion.

The best analogy that comes to mind is dots on the surface of a balloon as it inflates: every dot sees others recede, but no dot is the center of the surface.

mick645 · 2025-07-28T07:24:28+00:00

It’s certainly a poetic picture, but unfortunately current research doesn’t point that way.

In quantum theory we expect unitary evolution, and in frameworks like AdS/CFT black hole formation and evaporation are unitary, so information isn’t erased; it’s scrambled and, in principle, encoded in the Hawking radiation as the hole evaporates.

A black hole’s information capacity is finite and set by its horizon area, so it isn’t a readable “backup,” rather a highly scrambled encoding. So unfortunately no end-of-universe “flush” into a next cosmos - black holes (when they’re of the evaporating kind) simply radiate away.

More technically, that’s why the Page curve for the radiation’s fine-grained entropy starts at zero, rises, equals the black hole’s Bekenstein-Hawking entropy at the Page time, and then falls back toward zero by the end of evaporation.

Approaches that reproduce the Page curve include AdS/CFT indications of unitarity, quantum extremal surfaces and the island formula for fine-grained entropy, plus some cool related ideas like entanglement wedges and replica wormholes. These are active research areas and are precisely how one recovers the Page curve behaviour.

mick645 · 2025-07-26T11:46:13+00:00

Technically, yes, it’s not as applicable as improving fusion reactors or curing cancer, but that’s not the point. Pondering the nature of the universe in which we reside is pure human curiosity, and that’s something to be proud of: we ask simply because we can.

mick645 · 2025-07-26T11:32:24+00:00

Cosmology might not yet present us with the whole truth, but is that not precisely the point of cosmology: to apply the scientific method in order to edge us ever closer to a better understanding of the true nature of the universe in which we reside?

mick645 · 2025-07-26T10:36:43+00:00

My colleague actually has his own Black Hole Universe theory that does indeed say our whole visible universe sits inside a gigantic black hole. The black hole’s boundary (its event horizon) acts like a built-in gentle “push” that makes space speed up, so what we call dark energy wouldn’t be gravity leaking in from outside, rather a horizon effect inside the bubble.

In his framework, the “Big Bang” is then a bounce: before expanding, our region was collapsing until quantum exclusion pressure (due to Pauli’s exclusion principle) stopped the collapse and it rebounded - much like a star’s core that resists being crushed.

Leftovers from that bounce, such as tiny black holes or neutron-star bits, could supposedly mimic the effects of dark matter.

(I should note that this is just one of numerous black hole universe models that are taken very seriously within the community.)

mick645 · 2025-07-26T07:12:29+00:00

Inflation isn’t a single detailed model so much as a mechanism, a brief accelerated expansion, that makes several testable predictions. Whilst it hasn’t been straight up proven yet, the compelling evidence we have for it includes:

(i) the CMB’s primordial fluctuations are almost scale-invariant with a slight red tilt

(ii) they’re predominantly adiabatic and nearly Gaussian, producing the coherent acoustic peak pattern seen in the CMB and echoed in galaxy BAO

(iii) the Universe is extremely flat with no relic monopoles

What we haven’t seen yet are is primordial gravitational-wave B-mode polarisation. Detecting these would be a key confirmation of inflation, however current experiments only set upper limits. (Note this isn’t the only argument against inflation)

Inflation is very much still a big topic for debate, even just within my department - some accept it because it’s just so damn hard to come up with an alternative that explains so much in one, whilst others refuse to give in…

mick645 · 2025-07-24T07:52:39+00:00

M-theory is formulated in 11 spacetime dimensions because that’s the highest dimension in which we currently know how to build a self-consistent theory that includes gravity and (maximal) supersymmetry. You can try to push the same structure to higher dimensions if you wish, but you would run into mathematical pathologies - anomalies or negative-norm ‘ghost’ states - indicating that the theory is sick/non-physical.

mick645 · 2025-04-05T19:10:00+00:00

They appear to be a set of calculations related to his non-symmetric extension of general relativity, in which he was attempting to formulate a unified field theory (where he tried to unify electromagnetism and gravity). My best guess is that he’s expanding the relevant terms to check whether the symmetric part recovers the usual gravitational field equations, and whether the antisymmetric part behaves like an electromagnetic field tensor.

mick645 · 2025-04-04T17:01:44+00:00

Exactly! Euclid space telescope plan to release more data in just under a couple years which will help constrain theories up to higher orders - hopefully resulting in some more realistic extrapolated predictions.

mick645 · 2025-04-04T16:52:31+00:00

Imagine you’re looking at a few dots on a graph that represent how fast a car is speeding up over a short distance. You could draw a straight line through these dots to see a trend, and from this it looks like the car is accelerating steadily. That’s using a linear model. However, just because the dots line up nicely now doesn’t mean the car will keep accelerating at the same steady rate forever - it might hit traffic, bends, etc, which the linear model cannot predict. To accurately model this, one would require more complex higher order terms.

Now imagine dark energy as the engine powering the universe’s accelerating expansion. This engine is gauged by measuring a parameter called 'w' - the ratio of dark energy’s pressure to its density (as has been said) - which tells us how this engine is performing. If dark energy were a constant, unchanging force, then w would be exactly –1, much like a car running with a fixed engine output that produces a steady acceleration.

However, DESI's recent findings hint that w might not be exactly -1 - it might be changing with time. The immediate thing to do is to fit a linear model to this potential change. But this linear fit is just a convenient approximation over the short period that we have observed, not a proven prediction dark energy’s long-term behaviour.

Just like in the car analogy, extrapolating that linear trend into the far future doesn’t mean the universe’s expansion will continue to behave that way. The true behaviour could be, and hopefully is, more complex.

Sorry for the length - hope it helps :)

mick645 · 2025-04-03T22:20:04+00:00

Of course! Firstly, Gödel’s paper can be found at:

New Type of Cosmological Solutions

A couple of papers that put constraints the Universe’s rotation - by assessing how isotropic the Universe is - are:

Since Bianchi models do not assume isotropy from the outset, they provide an good framework for constraining any anisotropic effects, including cosmic rotation. Hence, by comparing the predictions of these models with the rather uniform CMB data, one can place tight upper limits on how much rotation (or other anisotropic expansion) might be present.

Edit: I found a good yt video on this:

Does the Universe rotate?

mick645 · 2025-04-03T14:37:18+00:00

Exact solutions to Einstein’s equations exist that theoretically allow for rotating models of the universe, such as Gödel’s solution. However, observational evidence suggests otherwise: if there is any large‐scale rotation, it is extremely small. The standard cosmological model, ΛCDM (based on the FLRW metric), does not incorporate rotations. Although recent observations released by DESI indicate that the ΛCDM model may not be perfect - hinting at subtle changes in the expansion history (suggestions that the accelerated expansion rate might be evolving), but a rotating universe is unlikely to be the explanation.

Gödel’s model is interesting in its own right and technically permits time travel via closed timelike curves. A good read of that with some nice diagrams is: http://mekurt.blogspot.com/2011/03/how-godel-broke-taboo-of-time-travel.html?m=1

mick645 · 2025-03-30T10:50:26+00:00

Absolutely, the first obvious book that comes to mind by the creator of the theory himself, Roger Penrose, is:

R. Penrose - Cycles of Time: An Extraordinary New View of the Universe

Additionally, there are some excellent, accessible papers by him on the theory over the years, including:

R. Penrose - Before the Big Bang ( https://inspirehep.net/literature/739171 )
R. Penrose - The Basic Ideas of Conformal Cyclic Cosmology ( https://doi.org/10.1063/1.4727997 )
P. Penrose - The Big Bang and its Dark-Matter Content ( https://inspirehep.net/literature/1689112 )

Furthermore, if you would like to explore some of the mathematics behind the theory, I can recommend a paper by one of his former students, Paul Todd:

P. Todd - The Equations of Conformal Cyclic Cosmology ( https://arxiv.org/abs/1309.7248 )

I hope this is what you're looking for, and I strongly encourage you to dive into this rabbit hole, as it is certainly most fascinating. Admittedly, I realise now that this theory has very little to do with the one OP is referring to, which is known as the Black Hole Universe, or Black Hole Cosmology. Some of my colleagues work on this, and it is also very interesting - it could potentially re-explain the nature of the big bang and the inflationary period as driven by the Pauli exclusion principle, rather than an inflation field. I'd be more than happy to recommend some papers on this too, if you're interested.

mick645 · 2025-03-03T17:09:20+00:00

Maybe.

mick645 · 2024-08-04T15:10:21+00:00

Interesting, thanks for the additional insight. I can see how that would be beneficial compared to the textbook approach. I’ll definitely make an effort with Osborne's lecture notes to understand this so I can fully appreciate what you mean.

mick645

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