Markus Hanke

Well put +1 I think it is a crucially important life skill to - in some situations - be able to respect things that we don’t personally like. This isn’t always easy, since we generally tend to equate our own preferences, views, beliefs and opinions with some notion of “truth” about the world. It takes a certain amount of introspective awareness to recognise this dynamic and suspend it, if and when necessary; sadly, not everyone is able to do this.

Definitely. People here generally aren’t gifted with the patience to wade through gibberish either. LaTeX isn’t “rare”, and it’s not found only on this website - it’s the international standard for typesetting documents that contain mathematical notation. Every major science discussion forum uses this to display maths. It’s really not that hard, and there are also many free online LaTeX editors you can use, for example: https://www.mathcha.io Give it a try sometime.

Check part VI, chapter 27.5-6. In my copy that is page 718 onwards, where they talk about the expansion factor and the possible geometries of hypersurfaces of homogeneity. The expressions for the non-zero components of Riemann for FLRW come from my notes, which, if memory serves right, I got from MAPLE at some point. It’s considered the gold standard so far as the theoretical aspects of GR are concerned for a reason. Truly fantastic text, most of my own GR knowledge comes from here. There are even some bits which are not found in any other GR text that I’m aware of.

A lot of people are talking about the ChatGPT project of late, so I decided to try it out for myself, because I was curious. I’ve asked it a lot of technical questions with regards to physics and some other topics, and was pretty impressed how well it was able to answer most of them. However, the AI wasn’t 100% spot-on. At least twice I caught it giving dubious answers - the first one was, with some goodwill, highly misleading; the other one was just flat out wrong. Interestingly, I was able to tell the AI about the wrong answers, and when I asked the same questions again (in a new chat session), it had one of them (not the other one!) corrected. So the moral here is that one should be very careful about using ChatGPT as a source of scientific information - at the very least, the answers it gives should be cross-checked against other sources to ascertain their correctness, and not just accepted at face value. This is an AI, not an online encyclopedia - the system makes mistakes, it learns, and it continuously updates its own internal states. It’s not infallible, so we are well advised to use it with caution. PS. I fear that people referencing AI-generated output from sources such as ChatGPT to backup their arguments is going to become “a thing” before too long on forums such as ours. That’s going to be a problem, for several reasons; most importantly, as mentioned above, AIs make mistakes, and sometimes give wrong answers, so I don’t think we should consider thinks like “ChatGPT said…” to be a valid reference when it comes to backing up arguments. Secondly, I noticed that ChatGPT sometimes answers the same question in different ways, using different wording - so you can’t link to a specific textual answer in a way that others can check and recreate; it’s dynamic output, not static text. That’s problematic when it comes to using it as a reference. What do other people here think? Do you guys feel there might be a need to develop an “AI-Generated Content Policy” for this forum?

Misner/Thorne/Wheeler “Gravitation” describes this in some detail. Consider first the general form of the FLRW metric: \[ds^{2} =-dt^{2} +a( t)^{2} d\Sigma ^{2}\] wherein \(\Sigma\) designates a 3-surface of uniform curvature (which could thus be elliptical, Euclidean or hyperbolic). The full Riemann tensor for 3D+1 spacetime with this type of metric then has six functionally independent components: \[R_{1100} =R_{2200} =R_{3300} =a\ddot{a}\] \[R_{1122} =R_{2233} =R_{3311} =-a^{2}\dot{a}^{2}\] Thus spacetime is never Riemann-flat, unless a(t)=const. On the other hand, the 3D+0 Riemann tensor for a given 3-surface \(d\Sigma\) of space is \[^{3}R_{ijkl} =\frac{k}{a( t)^{2}}( g_{ik} g_{il} -g_{il} g_{jk})\] wherein k is called the curvature parameter, so that for the choice k=0 each 3-surface is Euclidean and flat. MTW motivates the presence of the curvature parameter in this expression by using the following exact solution of the field equations: \[d\tau ^{2} =-dt^{2} +a( t)^{2}\left(\frac{du^{2}}{1-ku^{2}} +u^{2} d\phi ^{2}\right) ;\ u=\frac{r}{a( t)}\] wherein t is the total time recorded on a co-moving “dust clock” since the beginning of the universe.

Ok, that’s fair enough! I freely admit that I’m more versed in the physics of time dilation than I am in the engineering details of the GPS system. The salient point here is though that if you didn’t account for gravitational and kinematic time dilation at all, the system couldn’t work in the way it does now, or at least it would be a lot less accurate.

Calculating the total time dilation when there is a combination of differences in gravitational potential, as well as relative motion, is a standard exercise that’s done in most undergrad courses on GR. It’s not that hard so long as you can use the Schwarzschild metric, which is highly symmetric. Usually it’s done in the context of GPS satellites, because the software on your GPS receivers has to explicitly account for both these effects. At an orbital height of ~20000km and a relative speed of ~3.9km/s wrt to the ground observer, the gravitational contribution works out to about \(38 \mu s \), and the kinematic contribution is roughly \(7 \mu s \). Because of the symmetries of the Schwarzschild metric, you can just add these together to get total time dilation. That’s like saying we have never been on the surface of the sun, so we don’t really know it’s hot. It’s a silly argument. We have had a large number of crafts of different kinds both on the surface of the moon, and in orbit around it. We have also bounced lasers and radar signals off the moon’s surface. All of these things explicitly take into account time dilation - it affects orbital mechanics, it affects light travel times, and it affects frequency shifts. No discrepancies with expected physics have ever been observed in that regard. True, we haven’t done that specific experiment you are suggesting - but we have done many, many others where time dilation plays a role too, so if anything unexpected was going on, we would have seen it long ago. Why is the movement of a planet/star/moon different from the movement of a satellite? Then why do you insist that this experiment must be conducted on the moon? It’s not the total amount that’s the problem, but the cost/benefit ratio. All the above were designed to observe specific phenomena, which our models indicated should be there, so these are direct tests of specific predictions. On the other hand, what you are suggesting is a wild goose chase - there is nothing whatsoever to suggest that anything out of the ordinary will happen if we perform that experiment of yours, because motion and gravity isn’t any different on the moon than it is in Earth orbit. Research funding in fundamental physics is a very limited resource, so we are careful where we invest it. Like I said, if you want to test time dilation on the moon’s surface with respect to an Earth observer, then bounce a laser or a radar echo off it, and compare propagation times and frequency shifts to what our models say they should be. It’s a much easier test that addresses the same issue of time dilation, and it’s been done many times since 1946 - Earth-Moon-Earth communications is in fact an entire sub-discipline of aeronautical engineering. Sure. I’m not opposed to performing such ann experiment, if anyone wants to provide the necessary funds. The more tests of GR/SR the better, so far as I am concerned. I just think it would be a waste of money, since there are much easier ways to test the physics in question here. Wrong, see above. The radar echo “Moon bounce” is a direct test of this (radar echo goes Earth-Moon-Earth), because any discrepancies in predicted time dilations would show up as anomalous frequency shifts in the reflected signals. Needless to say, no such thing as ever been observed. Yes, I’m denying that, because it’s a pop-sci misconception. Look at Newtonian gravity - there are any number of experiments that are in direct contradiction to what this model predicts. And yet, we are still using it very successfully, and we are even teaching it to our kids in school. The point here is that all models have a domain of applicability - a set of circumstances which they are able to model very well. So long as you stay within that domain, the model will continue to work for your purposes, just as Newtonian gravity continues to work for us within its domain of applicability, even though it’s been experimentally “disproven” in many different ways. And so it is with GR - we have already know for a long time that its domain of applicability is limited; we’re just not entirely sure precisely where those limits are. So if some experiment comes along that contradicts GR, then in the first instance we will tighten those limits. But it won’t ever be abandoned - that’s never going to happen, because it has already proven far too accurate and useful. That’s true. But it’s also dangerous to become obsessed with a single tree, and forgetting the rest of the forest - which is what you seem to be doing here.

The FLRW metric is an interior solution to the Einstein equations - it represents the geometry of spacetime in the interior of a homogenous, isotropic distribution of dust. Since this “dust” is certainly a gravitational source, spacetime here cannot be flat. When it comes to questions of geometry, one must carefully distinguish between space and spacetime. These are not the same things at all. In the FLRW scenario, spacetime is always curved in a particular way; but, for the right choice of parameters, each 3D hypersurface of space within that spacetime can be flat. Careful - time translation symmetry accounts for energy-momentum conservation; the conserved quantity associated with this symmetry in Noether’s theorem is the energy-momentum tensor, not energy. The trouble is that, once you are working in a curved (as opposed to flat) spacetime, the mathematical expression that represents sources and sinks of energy-momentum no longer identically vanishes, so in general there is no law of energy-momentum conservation for regions of curved spacetime. The pesky extra terms that stop it from happening are related to the curvature of spacetime - energy-momentum isn’t conserved because curved spacetime itself contributes to the total energy-momentum. Unfortunately the energy-momentum of gravity cannot be localised, and cannot be written in a form that all observers agree on (in isn’t a tensorial expression). It is, however, possible to fix this by forming a certain combination of energy-momentum from sources, and energy-momentum from spacetime curvature; the overall expression is then again covariant. So to make a long story short, if you want to write down a law of energy-momentum conservation in curved spacetime, you need to account not only for gravitational sources, but also for the contribution that comes from gravitational self-interactions, ie spacetime curvature itself.

I’m still not sure what you are actually suggesting. Do you mean you want to have two clocks, one stationary on the moon and one stationary on the Earth, and then compare them? If so, then yes, you’d get a gravitational and a kinematic component to the total time dilation - though the kinematic contribution will be very small. This is in no way different from the GPS receiver in your car - the software installed on it will correct for both the gravitational as well as kinematic time dilation between yourself and the satellite. If it didn’t, your position would be off by a whopping 10km per day! Honestly, I don’t see the point in all this? We have known these principles for a long time, and are using them every day in some commonly available technologies. There’s no mystery in it. Why would you expect things to be any different on the moon, as opposed to a satellite? It’s not like either gravity or motion function any differently there than they do here. There’s nothing wrong in principle with performing such an experiment, but I don’t think anyone will throw loads of money at this, since we have no reason whatsoever to expect that anything special would happen. If there was some physical difference between gravity on Earth and elsewhere in the solar system, then we’d have already noticed this in other ways.

Motion with respect to what, exactly? Motion is not an intrinsic feature of a physical system, it’s simply a relationship to some external point of reference. As such, the local geometry of spacetime around an isolated gravitational source does not depend on whether that source happens to be moving with respect to some external reference point, unless this source is itself embedded in a region of curved spacetime (eg a binary system of some sort). As I’ve said before, in binary systems you cannot neatly separate the effects of the two bodies, because their gravity combines in highly non-linear ways. Of course we have. If you combine ordinary Schwarzschild geometry with relative motion, you get the Aichlburg-Sexl ultraboost. This is the same geometry as Schwarzschild, but expressed in different coordinates to reflect relative motion. If you want to see the effects of a binary system, then this can also be done, but only numerically on a computer (it’s too complicated to write down and solve on a piece of paper).

I’m confused now, because to my understanding this discussion is about time as a dimension. So if you speak about time being equal to change in the present, is your macroscopic universe 3D or 4D? Because that makes a huge difference to the physics! We’ve previously brought up an example - unstable elementary particles, such as muons for example. They are created, and then they exist for a period of time without any changes taking place in their rest frame at all, before they decay (which is an example of change). This notwithstanding, I don’t think anyone here argues that change doesn’t exist. I would just say that change is always defined relative to something, and that something isn’t necessarily time. This is most clearly seen in mathematics. If you have a function of both time and space, such as f(x,t), then this function “changes” in time at a rate of \[\frac{\partial f( x,t)}{\partial t}\] But it also “changes” in space at a rate of \[\frac{\partial f( x,t)}{\partial x}\] and its total “amount of change” is the sum of these. So by the above two examples, you can see that one can have passage of time without change, and one can have change without reference to time. While in ordinary everyday life we usually equate these two, based on how our senses and minds experiences the world, if you think about it more carefully you’ll find that they are really not the same thing at all.

I think it’s important to remember that QM is a completely deterministic model - given the initial state of any quantum system (plus boundary conditions), it tells us exactly how this system will evolve, in a deterministic way. It’s just straightforward linear algebra and differential equations. It’s only the observables that are probabilistic - each eigenvalue (measurement outcome) of these operators appears with a certain well-defined probability. You can’t - in general - predict the actual outcome of an observation, but you can predict the probability for each possible outcome with certainty. There’s a difference between determinism and predictability - the evolution of quantum systems is deterministic, and at the same time the outcome of individual measurements is not predictable. It is also possible for a system to be both deterministic and predictable, but still not computable by any conceivable real-world computation device, because there are too many variables or degrees of freedom involved. So this whole issue isn’t quite as straightforward as often assumed - one sometimes has to consider the entirety of the determinism-predictability-computability triad.

…which is funny if you think about it, since without time there wouldn’t be any such thing as vision, since EM radiation wouldn’t propagate, eyes wouldn’t perceive, and brains wouldn’t process. So when you visually see an extended object, this feat requires space and time. I think this discussion is going off on a lot of unnecessary tangents. Really, one only needs to ask how many pieces of information are required to uniquely specify an event in our universe - this is very much an everyday, direct experience kind of question, and requires no philosophical or mathematical acumen. If you want to set up a meeting with someone, you have to give them a place: “Meet me at the statue on Trafalgar Square” (which is a particular location on a 3D grid). But if you leave it at just this - a location on a 3D grid -, the meeting isn’t likely to ever happen, because this is not unique. It applies to Trafalgar Square on 6th June 1967, and it equally applies to Trafalgar Square tomorrow morning at 8am. So to uniquely specify the event in a way that admits no ambiguity, you have to specify an instant in time as well. This can be done in many ways - by reading on a shared clock, as an arbitrary parameter on the statue’s world line, or as a detailed description of a serious of changes starting from some agreed point in the past, or some other way. The main point is that this information is extraneous to the 3D grid, it cannot be reduced to any combination of purely spatial information - to put it simply, spatial information alone is simply not enough to uniquely specify an event in our universe. There’s a reason why we all use maps and calendars in our daily lives, and physics does much the same, and for the same reasons.

No. It’s not. You are asking about relativistic motion through a gravity well other than the Earth’s, and electromagnetic radiation propagating to/from Venus, Mercury, as well as various probes we have sent out, provides a perfect example for that.

This thread is just bizarre, to be honest. I am autistic, and have been involved in the neurodivergence community for quite some time, both on a local and an international level. Aphantasia is a relatively common comorbidity for people on the spectrum; I personally know 3 (maybe 4) fellow autistic people who have this, and my social circle is by no means large. Even among the general neurotypical population, this isn’t rare - there has been a study conducted on this only a few months ago: https://www.sciencedirect.com/science/article/abs/pii/S1053810021001690 Evidently, aphantasia and its opposite - hyperphantasia - are very much real. Claiming otherwise isn’t just ignoring the evidence, but - and that’s much worse - it is dismissing and invalidating the suffering and challenges of those who have it, because this has a very real impact on people’s lives.

You are also assuming a ruler when you draw out your 3D coordinate system. Yes, it is different - that’s what I’ve been attempting to demonstrate in that graphic I provided earlier. A clock provides an additional degree of freedom that uniquely specifies the evolution of the system. It’s extraneous information that removes the ambiguity inherent in a 3D-only graph of your system. Mathematically speaking, a time axis is orthogonal to each of the three space axes, so time “information” is linearly independent from the spatial hypersurface. You do not seriously propose that we can just do away with time in all our physics models and expect it to still work out correctly, do you? Surely you can see that this doesn’t work. If all the available dimensions are spatial in nature, and there is no other external information, how will there be “change”?

Have a look at the following situation: this is a plot of a 3-cube that has evolved in time, but instead of using a time axis, the evolution is all plotted on the same 3D volume to just reflect “movement” (as you suggest), without reference to an external “time” at all. The movement/change here is a combination of rotations (angle not necessarily constant, and rotations not necessarily in the same direction), and a change in colour. Without being given any information other than points within that the same 3D volume (ie only the above picture), can you tell me what the initial state of this system is, and how it evolves? What in here corresponds to past, present, future? You can’t do this, unless additional information is provided that is not itself an element of this same 3D volume. I think you can see the issues. And this is an idealised evolution in just three discrete steps - real-world systems, especially classical ones, feature continuous evolutions, with rotations around all three axes. Try to plot that into a single 3D volume, and what you’d get in the continuum limit is a solid ball - you couldn’t even tell the original shape any more. On the other hand, if you were to plot the evolution of the above system on a clearly labelled time axis with separate 3D plots at t=1,2,3, then there are no ambiguities at all - you can tell exactly what the original state was, and how it evolved in time.

I don’t see how this is different from the Shapiro delay, which has been tested extensively in a variety of ways.

This is true, though I meant something different - my argument was that eliminating time as a dimension will reduce the dimensionality of the spacetime manifold to 3D+0, so that the indices in the field equation now only run over a range \(\mu,\nu=1…3\). If you do this, then the Weyl tensor vanishes identically, and, because the vacuum field equations also imply \(R_{\mu\nu}=0\), you are left with both the trace and the trace-free part of the Riemann tensor vanishing. In other words, spacetime must be Riemann-flat everywhere in vacuum. So in 3D+0, assuming validity of GR implies that there is no gravity between massive bodies in vacuum.

Ok, point noted - though I did make it quite clear that this refers to general transformations between frames, and not just composition of velocities. They were just meant to be general Lorentz transformations, and I left the parameter unspecified. I don’t think it matters what kind of parameter you choose - speed, angle, gamma factor etc -, since the transformation itself always remains linear; it acts on 4-vectors and general tensors, not the parameter itself.

Actually, I formulated this badly - GR does still work, even if you take out time as a dimension. The problem is just that the kind of gravity you get then is nothing like the gravity we actually see in our universe.

If U is a 4-vector, then the formula for a Lorentz transformation from the original frame to some primed frame is written as \[U’=\Lambda(v)U\] where \(\Lambda\) is a square matrix, the Lorentz transformation matrix, and v is some parameter of the transformation (not necessarily speed!). I invite you to verify yourself that \[\Lambda(v+u)=\Lambda(v)+\Lambda(u)\] and (c=const.) \[\Lambda(cv)=c\Lambda(v)\] by whatever means you find most convenient. The above two relations define the property of linearity in the context of matrix transformations. So yes, the Lorentz transformations are indeed very much linear - as of course they have to be, since they map lines into lines, ie inertial frames into inertial frames. This is pretty trivial tbh. P.S. Cross posted with joigus, studiot and Grenady! Had my reply open on screen some time before hitting “Submit”.

Yes, unfortunately you are right. I see a lot of problems with the way physics is presented in various media. Sadly this appears to be true across the board, including the other sciences too. I think they meant explaining the model itself, rather than any underlying ontology. Well, relativistic effects are always relationships between reference frames, so this is not really a surprise. But the true power and beauty lies in the exact opposite - that relativity allows us to write the laws of physics such that they do not in any way depend on which reference frame is chosen. It’s about the fact that nature appears to be generally covariant (within the classical domain) and thus does not care about observers at all. That’s a powerful symmetry.

Yes, and this is true for all clocks - including the distant stationary one! Now think about this - if the distant stationary clock measures purely its own time (“behaviour”), as you correctly say, then why do you expect it to be able to accurately measure any process that does not, in fact, take place within its own frame? Can you see the issue? The reverse situation is just as true - the in-falling clock cannot, based on its own readings, expect a distant stationary clock to tick out the same amount of time. Exactly! This is just the basic principle in GR - the laws of physics are covariant, so all observers agree on them. In this case, all observers agree that the in-fall world line does in fact intersect the horizon, including the distant Schwarzschild observer. He just doesn’t physically measure this on his own clocks and rulers, because those are inextricably linked to his distant, stationary frame, and thus unable to measure anything about the in-falling particle. They can only measure things in their own local frame. There are two entirely separate concepts to consider here - there is the manifold, which is spacetime itself, ie the set of all points in space at all instances of time (“events”); and then there is the coordinate chart that covers the manifold, which is simply a map that assigns a unique label to each event. The manifold is like the physical set of streets that makes up a city, whereas the chart is the choice of names we assign to those streets. It should be obvious straight away that the choice of street names is entirely arbitrary (so long as they are unique) - we can erase and re-write all street names, without affecting any of the physical layout and geometry of our city. You can also have different people employing different names for the same street; there’s potential for confusion when you do this, but so long as both sets of names are unique, there will be no problems or contradictions; you can map them into each other 1-on-1 in a unique way. The time it takes you to drive from one address within the city to another is not affected by the way the streets are named in any way. Essentially, the street names have no physical significance so far as the layout of the city is concerned. And so it is with spacetime - you’ve got the spacetime manifold and its geometry, which is given by the distribution of gravitational sources (“the city). This is entirely separate from the coordinate chart which you choose to label each event on that manifold (“the street names”). How long it takes to inertially free-fall from one event to another as measured on a co-moving clock is likewise not affected in any way by what kind of coordinate chart you choose to use. Just as is the case for the city, the choice of coordinate chart has no physical significance whatever so far as the geometry of spacetime is concerned. What happens at the event horizon is that the Schwarzschild coordinate chart becomes singular, in the same way as spherical coordinates become singular at the poles on Earth. So the question then becomes whether this is purely a coordinate singularity, where only the coordinate chart fails due to the way it is defined, but the manifold itself remains perfectly regular; or whether this is a curvature singularity, where both the coordinate chart fails and the manifold ceases to be smooth and regular. The former has no physical significance, it’s just an artefact of the way we choose to label our events; whereas the latter means that the manifold is geodesically incomplete, ie we cannot physically extend free-fall geodesics past that region. The simplest way to distinguish between them is to try and cover our spacetime with a different coordinate chart (remember that this choice is arbitrary and has no physical significance!), and see what happens at the horizon. Instead of Schwarzschild coordinates we can use (e.g.) Gullstrand-Painleve coordinates, Novikov coordinates, Eddington-Finkelstein coordinates, Kruszkal-Szekeres coordinates, or any other convenient choice. If we can find even only one coordinate chart that remains smooth and regular at the horizon, then we know that the original singularity was of the coordinate kind, and thus has no physical significance so far as the manifold is concerned. And that’s indeed the case here - in Schwarzschild spacetime, there are many coordinate choices that remain smooth and continuous even at the horizon. To be absolutely sure, we can also check in a more direct way, by considering a covariant quantity that does not depend on coordinate choices, such as the curvature tensors. More specifically, one looks at the principle invariants of the Riemann tensor and the Weyl tensor, which indicates how the curvature of spacetime behaves at the region in question. There are altogether five of such invariants. When we calculate them at the horizon (using any coordinate chart of our choice), we find that they are all finite and well defined, indicating that spacetime is smooth and regular there. This is in contrast to the central singularity - no matter what coordinate chart we choose, the central singularity is always singular; we cannot eliminate this by choosing different coordinates. Also, the curvature invariants all diverge there. This indicates that the central singularity is a physical one - a region of true geodesic incompleteness. There are other ways to distinguish these singularity types, but you get the idea - the event horizon is a coordinate singularity (no physical significance), whereas the central one is physical.

No one knows the answer to this. There is also the far less intuitive possibility of the “tower” being quite finite, while at the same time lacking any irreducible ontology. Rovelli’s relational interpretation of QM would be an example of this. I don’t think so either - though of course we can’t be sure. I don’t think the principle of relativity can be derived from any fundamental axioms - it’s an empirical observation about how the world works. We simply don’t see any variations in the laws of physics between observers, at least not within the constraints of our experimental abilities. You can never “absolutely” prove any model of physics. We can, however, but upper bounds on the magnitude of any possible Lorentz-violating effects, and these bounds are very stringent indeed. The laws of acoustics are just a special application of the laws of fluid dynamics - and those can be written in fully covariant form using the energy-momentum tensor, so they don’t depend on the observer. Based on that you could, if you wanted to, write a model of relativistic acoustics that is observer-independent. Yes, there are very many tests of Lorentz violations, both historical and modern, and none of them has ever found any hint of such a thing in fact existing. Like I said, this places very stringent upper bounds on such violations. Yes In the rest frame of those very same ultra-high-energy cosmic protons you just mentioned, my computer does in fact operate at those very speeds. What do you mean you “don’t buy it”? Do you doubt that the mathematics provide the correct answer when you run the numbers? It’s rather easy to show that they do in fact work out, in a fully self-consistent way. Once again, Minkowski spacetime here is a descriptive model, the purpose of which is to provide a framework to make predictions for real-world scenarios. And it evidently does this really well. Of course, it has no explanatory power as to why this model - as opposed to some other description - works so well. Here’s where we come back to the question as to how fundamental (or not) spacetime is, and what, if anything, underlies it. These questions don’t as yet have an answer. You appear to be using a different definition for the term “geometry” than mathematics do. Intuitiveness is not a required feature for any aspect of mathematics or physics, or any other science for that matter. It just needs to work, and be internally self-consistent. Euclidean geometry seems nice and intuitive to you only because as being human you happen to experience a domain of the universe that is roughly Euclidean in nature; this does not afford it any physically privileged status, however. Non-Euclidean geometries are equally well formulated and understood, and are equally self-consistent. Besides, intuitiveness is highly subjective - to me, for example, Minkowski space seems perfectly natural, and very well suited for the task at hand.