Markus Hanke

You are right in that it constrains the set of all possible global topologies, but what I attempted to point out is that it doesn’t uniquely determine it, at least not in 4D. More precisely, globally different topologies can be compatible with the same local geometry, so local measurements of curvature alone don’t necessarily give this information. The reverse is also true - manifolds can have the same global topology, yet different geometries. So this issue is subtle. The interesting exception is in 2D - here, the Ricci scalar is also the Euler characteristic, so gravity is entirely topological. A bit more info here: https://en.m.wikipedia.org/wiki/Shape_of_the_universe

This is true of course. It should be noted though that if both GR and QFT are valid theories, at least near the event horizon, then the existence of Hawking radiation is inevitable. If it turns out to not exist, then one or both of these models don’t apply in that region.

They didn’t know whether light has mass or not, it was an unanswered question at the time. But since Newtonian gravity can only handle test particles with mass, they had to assume that it did in order to derive any kind of prediction at all. The deflection angle doesn’t depend on the exact value of the mass - it just can’t be zero in Newton’s theory. You have to assume this to derive a deflection angle from Newton’s theory, which was the only theory available back then - it can’t handle massless test particles. You are absolutely right of course - this is one of the areas where Newtonian theory fails, and GR is needed. One must remember though that this wasn’t well understood prior to Einstein. Actually, it doesn’t - they are separate concepts. GR determines only local geometry, but not global topology. For example, the maximally extended Schwarzschild metric could describe both two separate, singly-connected regions of spacetime, or a single multiply-connected spacetime. Geometry is the same in both cases, but the global topology isn’t.

Newtonian gravity has nothing to say about massless particles, so strictly speaking it makes no prediction here. However, if one assumes that photons have a very small but finite mass, then one can use Newtonian gravity to work out how they are deflected around massive bodies. Turns out that deflection angle doesn’t depend on the exact mass of the photon, so long as it is much smaller than that of the central body. The result you get is off by a factor of 2 compared to actual observations - to get the correct angle, one must use GR.

Yes, this is possible; this is in fact one of the standard ways to (in principle) build GW detectors: https://arxiv.org/abs/1501.00996 You need an extended array of clocks for this, since what you are measuring is the dilation between clocks at different positions within a passing GW wavefront. Let’s just say it’s equally hard You’d need an extended array of very precise and perfectly synchronised atomic clocks. This is doable at least in principle with current technology.

No it’s not, because measurements of space and time are inherently observer-dependent concepts - there is no absolute frame at all, so there cannot be a contradiction. What all observers agree on is the spacetime interval. This is a rather silly statement, since whatever device you have used to create this post is based on a relativistic theory - the Standard Model, especially the part of it dealing with electromagnetism. Obviously, your computer isn’t an optical illusion, and using relativity to construct it has resulted in quite a useful machine. Have you ever used a microwave? An old-style CRT Monitor? Had an MRI scan? Seen a thermometer field with mercury? Used the GPS on your phone? Used electricity generated in a nuclear power station? Etc. All of these are things that inherently rely on relativistic effects to work.

By “geometry” I mean how exactly the region of spacetime in question is curved. There’s mainly two considerations that are of relevance in this context here - does spacetime become approximately flat if we go far enough away (asymptotic flatness), or are there distant sources of gravity that need to be accounted for? And secondly - what symmetries does this spacetime have? Can we translate each point within this region a short distance along space and/or time, without affecting any of the physics? For example, each point in ordinary Schwarzschild spacetime can be shifted along a vector pointing - say - 1 second into the future, without changing anything about the physics of the system - it thus is said to admit a time-like Killing field, which is to say it is a stationary spacetime. Killing fields are one way to speak about symmetries in spacetime. And since continuous symmetries are by Noether’s theorem associated with conserved quantities, this will have an impact on how we define concepts such as energy-momentum across an extended region. Different definitions are available for different types of spacetime exhibiting different symmetries; and if you’re in a spacetime that’s complicated enough so that it has very few or no symmetries, then it may not even be possible to define its global mass in a meaningful way at all. Does this make more sense?

No. What it means is that the answer to the question of “how much mass/energy is in a region of spacetime” depends on what kind of geometry that region has. Depending on considerations such as symmetries (Killing fields), asymptotic flatness etc a certain definition may apply, while other definitions may not work. So one has to be very careful which one is to be used. Note also that being in relative motion wrt to a gravitational source does not change the geometry of spacetime, it only changes the coordinate description of it.

It has similar challenges Try for example the word “Eichhörnchennachwuchs“ (=offspring of squirrels), where you have three entirely different sounds for the combination “ch”, all in a single word. You get the idea. Fair play! German is quite tough grammar-wise, whereas Chinese is simple in terms of structure, but requires lots of time and effort in memorising characters. I lived and worked in China for a year when I was younger, and immensely enjoyed learning it…but never used it afterwards, and now, 25 years later, I’ve forgotten pretty much everything 😕 You loose what you don’t use. Couldn’t agree more! I love teaching myself languages - I find it a very useful way to keep the old brain in shape, and it also helps to break up old deeply ingrained habits of structuring information, as you very correctly say.

If you think the phonetics of English are complicated, you should try German Or one of the many tonal languages, such as Thai, or Chinese…

Well, one must remember that Newtonian gravity is only a very simplified approximation that disregards all non-linearities, so it is perhaps not so surprising that some of its concepts turn out to be less general than we take them to be.

What Genady means is that the universe is a curved spacetime, as described by General Relativity. The thing now is that some concepts we are used to from old Newtonian physics do not straightforwardly translate to GR - and “mass” is one of them. The question of “what is the mass associated with a given region of spacetime” has no simple answer; there are in fact several different notions of mass that apply to different sets of circumstances, so it really depends. The underlying reasons for this is that the gravitational field in GR is self-coupling and thus itself a source of gravity (unlike in Newtonian gravity); but this type of energy cannot be localised, and is frame-dependent, so it is difficult to account for in an observer-independent way.

This is just combining two sources of gravity to obtain a new spacetime geometry - gravity nonetheless remains attractive in nature, as it always does for ordinary sources. But for the Alcubierre drive you need actual anti-gravity, which is a completely different thing - it can be shown in a general way that the Alcubierre metric in its original form requires exotic matter; unfortunately you cannot “cheat your way there” just by cleverly arranging ordinary sources. Also, even if you could construct an Alcubierre bubble, I think it would be completely unusable as a propulsion method, since it has some pretty nasty side effects and problems. Just a word of warning - these diagrams depicting spacetime curvature are just visual aids to understanding the basic concept, they are not accurate depictions of actual geometry. Spacetime geometry is intrinsic to the manifold, so there is no actual “direction” to curvature, since spacetime is not embedded into any kind of higher-dimensional space. The crucial concept to understand here is the distinction extrinsic vs intrinsic geometry; this is very important if you want to understand GR.

I disagree. While I know that Icelandic distinguishes these sounds in phonology and orthography, English effectively doesn’t, so there’s no point in this at all. Furthermore the vast majority of people here on this forum presumably will use English keyboards, so these letters are not straightforwardly accessible to them, making this not at all efficient to most of us. There are good reasons why English orthography is largely standardised, so deviating from this is unwise, really comes across as silly, and makes it hard for some to read your text. Trust me, this isn’t making a good impression. My advice to you, if you wish to engage in a proper discussion of your ideas and be taken seriously while doing so, is to stick to standard English orthography. You don’t have to agree with it, you just need to use it.

I disagree, I don’t think you have a bad reputation here. You’re still young, so there are a lot of things about modern physics which you haven’t encountered and learned about yet. We all understand this, so there’s no problem. The most important thing is to keep learning, and keep your mind open - don’t allow yourself to fall into the “I’ve figured this all out” trap. Trust me, the universe is far richer than you can even imagine right now. Keep learning

No, not all quantities in nature are relative in the same way as eg speed is. A good counter-example is proper acceleration - it can be measured locally using an accelerometer, and all observers agree on this measurement. It is not relative to anything. Gravity is similar - if you have an ordinary source of gravity, its effect is always attractive for all observers, never repulsive, irrespective of orientations, states of motion etc. True anti-gravity would require exotic matter to generate, which we have reason to believe does not exist. To put it more technical - “gravity” is a tensorial quantity, and tensors are covariant objects; all observers agree on them.

I suspect what you’re looking for does not really exist, since, as others have pointed out, the Einstein tensor is actually quite a complicated function of the metric and its derivatives, so there’s no simple intuitive way to physically interpret any one of its components in isolation. But taken in its entirety, the tensor measures the extent by which geodesics will on average deviate around a small neighbourhood. It represents average Gaussian curvature in space once a time direction has been chosen. The Einstein equations mean just this - average local curvature (of a very particular type) is precisely equivalent to local energy-momentum, up to a proportionality constant. Note that this tensor only captures some aspects of total curvature, but it doesn’t provide a complete description (which is given by the Riemann tensor) - IOW, when you have G=0 then that does not necessarily mean you’re in a flat spacetime! In practice this tensor is seldom directly used in practical calculations - the Riemann tensor, Ricci tensor and Ricci scalar are more useful objects for practical purposes, and they have easier to understand physical meanings.

I’m not aware of any such concept, at least not under this name. Can you explain what you mean by this? I’m afraid that’s not how spacetime curvature works. So long as you start with positive energy-momentum - irrespective of how this is distributed or oriented in space -, you’ll always end up with ordinary attractive gravity. This is not a question of position or orientation in space.

I think this discussion on Physicsforums, particularly post #12, might shed some light, though I need to go through it more carefully myself.

It’s possible to consider discrete models of fluid dynamics, but generally speaking these don’t take the form of neat, closed formulas that can be easily worked on paper - rather, you’ll be dealing with numerical models on powerful computers. An example of this is CFD-DEM. The standard Navier-Stokes equations are then just the continuum limit of such a model.

Are you familiar with the topological arguments put forward in MTW with regards to why the EFE has the form it does? Not sure if one can call it a “derivation” exactly, but it’s nonetheless very interesting.

Yes, but then you’d end up with a different theory of gravity. In GR, the tensor we’re looking for can only contain first and second derivatives of the metric, must be linear in the latter, symmetric, and divergence-free. Lovelock’s theorem guarantees that the only tensor that fulfills these criteria is the Einstein tensor.

It’s best to look at the Einstein tensor as a kind of machine with two slots - if you feed into both slots a unit vector that points into the future time direction, it will give you as result a real number that represents the average Gaussian curvature in a small spatial region around the event you are working at: \[G_{\mu\nu}t^{\mu}t^{\nu}=k\] So the Einstein tensor is a kind of average curvature around some event. I don’t think it is helpful to try and find specific “meanings” for specific components.

Are you sure this is right? Admittedly I don’t know so much about the history of Christianity, but I’m pretty sure there weren’t any churches at the time these texts were written…

! Moderator Note Moved to Speculations, as this is completely at odds with mainstream astrophysics.