Markus Hanke

That’s not the case though, not even in the most symmetric spacetimes. Consider a freely falling particle in Schwarzschild spacetime - a faraway observer will see the particle recede, and then slowing down and getting dimmer and redder as it approaches the horizon. Another observer falling in parallel to the particle (at not too great a distance) will not see this effect, for him the particle appears the same the entire time, and will actually get closer to him as they fall. So visual appearance in curved spacetime does depend on the observer, because different events are linked by different sets of null geodesics.

A point in 3-space can be considered as the limit of the volume of a sphere with r-> 0. You can formally set up the volume as an integral function, and then take the limit.

I remember having gone through Young/Freedman many years ago, an older edition. From what I remember, you will only really need (1)-(5), as well as (8) and (9) of the topics in your maths text. I would recommend going through the whole thing though!

What do you mean by “observe”? You mean visually?

Actually, distances generally become longer as compared to a far-away reference ruler, though it depends on the type of spacetime you are in. And this is precisely the physical meaning of mathematical curvature - that relationships between events (ie measurements of space and time) now depend on where and when you make them, relative to a reference point. Emphasis here being on the word ‘relationships’ - these aren’t effects that locally happen to clocks and rulers; it is how two or more such measurements are related in spacetime, which is why you can only detect them by comparing measurements. Locally all clocks tick at 1 second per second, and all rulers measure 1 meter per meter.

Good point. Example: Hyperion Cantos tetralogy by Dan Simmons. Very much recommended to anyone who hasn’t read it yet. Another one is Solaris by Stanislaw Lem. And for the brave ones, there’s always Roadside Picnic by the Strugatski brothers.

I completely agree with this, also because philosophy tackles some questions that aren’t readily amenable to the scientific method. What irks me a bit about philosophy in general though is that rarely - if ever - is there any consensus reached on anything. At least it seems to me like that. There just seem to be individuals with differing opinions, often diametrically opposed. But maybe that’s just me (and I do have an interest in philosophy)...

I don’t think it is very meaningful to try and define G across a larger region of curved spacetime; it’s numerical value would then depend on the distribution of matter-energy in that region, and also on the observer whose clocks and rulers are used. It wouldn’t be an invariant.

Think about this a little more. So you have G=kT, but now allow the constant to become a function (what does it even mean to multiply a non-covariant function with a covariant object such as a tensor? How do you define this consistently?). First thing you’ll notice is that for T=0 you still get G=0, and thus R=0, which is the exact same as ordinary GR - meaning there’s no change whatsoever to vacuum solutions, including Schwarzschild. You still get event horizons and everything else. The problems begin in the interior of mass distributions, because now your equation is no longer covariant, meaning the geometry of interior spacetime depends on the observer - which is totally nonsensical, and certainly not what we see in the real world. Furthermore, because of observer dependence you cannot, in general, match the interior metric to the exterior vacuum one while maintaining smoothness and differentiability at the boundary - so it’s not just meaningless, but also mathematically inconsistent. Lastly, the actual interior solutions themselves would now be completely different; using known equations of state for fluids would not result in anything even remotely resembling real-world astrophysical objects such as stars, directly contradicting observation. So, the idea is nonsensical, mathematically inconsistent, and contradicts real-world observation. And this is just the obvious issues. For example, there’s also deeper reasons why the equations have the specific form they do, but I think the above is already enough to show why this can’t work. Indeed. And which means you can’t simply change the field equation like that, and expect things to still work.

This is meaningless. The GR field equation, being a tensor equation, is a purely local constraint on the metric; thus there’s no time dilation involved. Besides, if you allow G to vary, the overall equation is no longer covariant, and will have completely different solutions. An event horizon is not in general the same as a singularity in your chosen coordinate system. These happen to coincide for Schwarzschild coordinates, but that’s mostly just because of the way these coordinates have been defined. Physically there is no infinite time dilation at the EH - any test particle in free fall will reach the horizon, fall through it, and continue you on down, all in a finite, well defined proper time. It’s just that in-fall time diverges for a stationary far-away observer on his own clock; here again it is crucially important to recognise the difference of global vs local. A Schwarzschild coordinate system is true and valid locally for a stationary distant observer, but tells you nothing about what physically happens at the horizon itself.

Mass, angular momentum, and charge of a BH are not entirely free parameters, but they are linked via the concept of ‘irreducible mass’ - there’s an equation linking these, which I won’t typeset here. The point is that for a given mass, there’s an upper limit to angular momentum; rendering the singularity naked would need to exceed this limit. The EH (herein aka ‘outer horizon’) is the next horizon inside the ergosphere, and these two coincide at a point on the rotation axis, above the pole. Spacetime is stable most of the way to the Cauchy horizon (‘inner horizon’), but then becomes highly unstable. In Kerr spacetime there’s only mass and angular momentum. If you add charge as well, then you are in Kerr-Newman spacetime. It’s a causality boundary - above the Cauchy horizon ordinary causality always holds, but below it causal structure becomes non-trivial; this is the region where you find closed timelike curves.

Yes, that’s right. Just before that happens, it will wrap itself around the ring, like a donut. Note that in Kerr spacetime there are a number of additional important surfaces, other than the event horizon. Also, the region below the EH is highly unstable, so any deviation from true vacuum will make this decay into a different type of spacetime. It’s an idealised solution. But there’s a consolation: if recent results in quantum gravity (ref Netta Engelhardt etc) prove correct, then BHs do have hair after all; in fact they would sport some thick, luxurious fur! So it might (proof outstanding) be possible to reconstruct your identity from information encoded in Hawking radiation, after your in-fall (the concept is called ‘Python’s lunch’). Not only would that allow the world to know your fate, but it also preserves unitarity that way, and thus resolves the information paradox. The caveat is that the decoding algorithm requires full knowledge of all quantum gravity DoF, and may even then not be computable with any conceivable real-world computer. All of this is work in progress at the moment.

You’d need a situation where there’s enough angular momentum so that the event horizon sort of wraps around the ring singularity, like a donut, leaving a hole in the middle. This is theoretically possible, but I don’t know if any real-world BH ever gets to that point (I’d say not). I also don’t know how tidal forces would behave in such a scenario. My feeling is that you’d have to cross precisely through the Center of the ring, or else you’d get ripped apart and pulled into the horizon. This is a very unstable situation, and probably not realistically doable.

Well, depending on the total mass of the BH you’d still get ripped to shreds long before you get anywhere near the singularity. It’s worse here even, since there’s also an angular component to the tidal forces - you’d be elongated lengthwise and pulled sideways 😳 This is a cosmic meat grinder!

A pretty horrible way to kick the bucket. One can only hope that the BH isn’t very massive, so that it’s over quickly.

Well spotted 😄 Never consciously registered with me. What did register though was the fact that all drive systems appear to be facing rigidly backwards...kind of hard to steer or decelerate that way 😏 Same issue in Star Wars and many other movies as well. The other issue of course being that it is very difficult for screenwriters to even conceive of something that is truly alien in that sense, because we are trapped in our own frame of reference. Most sci-fi creatures are just variations on various animal/plant life, or on mythological archetypes such as ghosts etc. It’s very hard to imagine something that is scientifically feasible (at least in principle), and at the same time entirely alien. PS. As a young kid I used to be captivated by Space:1999 and The Tripods. Nowadays these seem rather old fashioned and poor quality, but back in the day they were great! Star Trek and Star Wars as well, needless to say.

Correct - but with the caveat that the concept of ‘gravitational potential’ can only be meaningfully defined in certain highly symmetric spacetimes, such as Schwarzschild. It is not a generally applicable concept. Indeed. It vanishes locally in those regions, but not globally. Yes, correct. Think back to your math lessons in high school - remember how you drew simple graphs such as y=x^2. No question that the graph is globally curved. But now imagine you were to choose some point (eg x=2), and zoom into the graph there. What happens? The more you zoom in, the flatter it will begin to look. It’s just like that. This takes a while to really get your head around. I’m sorry I won’t try to offer a proper answer here, as typing LaTeX code on an on-screen phone keyboard is just too cumbersome and time consuming. What I will say though is don’t focus on the components, but on what objects you pass to the tensor, and what you get out as a result. Rough outline: Imagine you have two test particles, whose world lines are initially parallel. Now choose a point on one of these world lines - take the unit tangent vector at that point (which is physically just that particle’s 4-velocity). Then, still at that same point, take the perpendicular separation vector that connects it to the other particle’s world line. Now imagine the Riemann tensor as a machine with four slots (the four indices). Input the tangent vector into slots 2 & 4, and the separation vector into slot 3; leave the first slot empty. The output of the machine then is a new vector (because we left one index open) - it tells you how fast the separation between the test particles begins to change, and in what direction (relative acceleration between the test particles). Writing this down in math notation immediately gives you the geodesic deviation equation, bearing in mind that we need to use a covariant, not ordinary, second derivative. So the Riemann tensor is a machine that takes the tangent vector on one world line, and the separation vector between them as input; and produces as a result an acceleration vector that tells you how that separation between particles changes over time (geodesic deviation). This deviation can be a combination of any space and time direction, and can be complicated - you can get the world lines twisting around one another in a helix configuration, and all kinds of fancy stuff like that. This is really most of what there is to it in a GR context - Riemann has other uses as well (eg one can calculate tidal forces from it), but I won’t get into this here. It does in fact reflect all possible degrees of freedom of gravity. The individual components of the tensor represent tidal effects between various combinations of directions, but I really don’t think it’s helpful to try to look at it this way - it won’t help you understand. Better to think of it as a machine with slots that take an input, and produces an output; with the indices of the tensor being those slots. Any tensor can be conceptualised in that way - eg the Ricci tensor takes a future-pointing time-like unit vector into both slots, and produces a real number that is the rate at which a small volume changes when in free fall. In vacuum R(u,v)=0, so in vacuum a small volume in free fall is conserved (but its shape will get distorted, which is described by a different tensor). Inside a matter distribution, neither volume nor shape would be preserved. (Note carefully that this geometric interpretation only holds so long as there is no expansion, shear, or vorticity, which is true for most simple spacetimes) Hopefully this makes any kind of sense to you.

Yes, that’s right. As pointed out earlier in the thread, spacetime is only locally flat. If the accelerometer is too large, it will begin to measure tidal effects.

No, this one is new to me. Thanks for bringing it up. Having skimmed through the link, my first impression is that this formalism is not nearly as elegant and intuitive as the standard one (and full equivalence with GR is yet to be shown). I kind of fail to see the advantage, though the point about substructure is interesting. See studiot’s comments on intrinsic vs extrinsic to begin with. Furthermore, there is not really any force involved in gravity - when you have initially parallel test particles in free fall, and attach an accelerometer to them, it will always read exactly zero, so no forces; nonetheless in the presence of gravity their geodesics will begin to deviate. Good question! This point is a bit subtle, and really the answer should be “both of the above, depending on context”. The physical manifestation of curvature is geodesic deviation - meaning that initially parallel world lines will begin to deviate as they extend into the future. It is thus necessary for world lines to have at least some extension in spacetime before “parallel” and “deviate” even make sense - you can’t speak of parallelism at a single event. Thus curvature has measurable meaning only across some distance. I’m highlighting the word ‘measurable’ because counterintuitively the mathematical object describing curvature (Riemann tensor) nonetheless is a local object, like all tensors. For clarification on this point, refer back to the example about calculus in my previous post. However, there are also scenarios where the effects of gravity are in some sense ‘relative’. Consider a hollow shell of matter, like a planet that has somehow been hollowed out (not very physical of course, but I’m just demonstrating a principle here). Birkhoffs Theorem tells us that spacetime everywhere in the interior cavity is perfectly flat, ie locally Minkowski. There’s no geodesic deviation inside the cavity. Now let’s place a clock into the cavity, and another reference clock very far way on the outside, so both clocks are locally in flat Minkowski spacetime. What happens? Even though both clocks are locally in flat spacetime (no gravity), the one inside the cavity is still gravitationally dilated with respect to the far way one! This is because while both local patches are flat, spacetime in between them is curved - if you were to draw an embedding diagram, you’d get a gravitational well with a ‘Mesa mountain’ at the bottom; and the flat top of that mountain sits at a lower level than the far away clock, thus the time dilation. So in this particular case one could reasonably say that gravitation effects are ‘relative’ between local patches. Or you can put it like this: both regions are Minkowski, but one is more Minkowski than the other The isn’t very intuitive, but mathematically perfectly consistent - if you look at the world lines of the clocks, you’ll find that while they appear parallel in space (they’re simply at rest wrt to one another), they deviate in spacetime. In GR it is crucially important that one fully understands local vs global, or else there’ll be no end to misunderstandings and problems. This point is where most, if not all, apparent ‘paradoxes’ in GR arise. In general, no, it’s not a scalar - it’s a rank-4 tensor field, the Riemann tensor. However, you can choose to look at only certain aspects of curvature, such as how volumes change (rank-2 Ricci tensor), or how areas differ from Euclidean counterparts (rank-0 Ricci scalar), or the average Gaussian curvature of a small region of space (rank-2 Einstein tensor). But to capture all aspects, you need the full rank-4 tensor with 20 independent components. Tensors are not invariant, but covariant - meaning their individual components do vary in just the right ways so that the relationships between the components remain, hence the overall object is the same for all observers. Remember a tensor is all about the relationships between its components.

Do you mean Euclidean geometry by ‘old geometry’? If so, then yes - Euclidean geometry cannot guarantee that the spacetime interval is the same for all observers; physically, this means that all observers experience the same physics, irrespective of their states of relative motion. That’s what we see happening in the real world, so we need any good model to reflect that. Minkowski geometry (and Riemann geometry in GR) do this natively and very elegantly.

There isn’t really any kind of ‘action’ in the mechanistic sense of the word. It’s just that test particles and their world lines are themselves part of spacetime, so they cannot do anything other than follow its underlying geometry. There’s no duality of any kind. See below analogy for clarity. As I’ve mentioned in my last post, there is information, in the form of the metric which determines the relationship between points. So it isn’t a ‘zero set’. This is true even very far from any sources - even spacetime without gravity has geometric structure that is different from that of Euclidean space. This is (eg) why you can’t accelerate to the speed of light - the fundamental reason for this is geometric, so geometry has real measurable consequences. It’s exactly like the calculus you learned at school - the derivative of a function is defined at a single point, yet gives you information about the slope of the entire function. That’s because what it really does is tell you about the relationship between neighbouring points on the graph of the function - how it changes from point to point. If you’re given just the (local) derivative, plus boundary conditions, you can reconstruct the entire function, even though any one single point of the function is just an (x,y) pair. To give an analogy (!!!) - suppose you have two people starting out on different points along the equator, and flying north simultaneously at a constant altitude. When they start out on the equator, let them be - say - 1000miles apart. What happens? The further north they get, the smaller the distance between them becomes. Eventually they’ll meet at the pole. Why? There is no detectable ‘action’ or force between the two planes. Each plane starts off at 90 degree angle from the equator (so their trajectories are initially parallel), and they always fly straight (there’s never any detectable change in direction from their initial trajectory). Yet they approach one another. That’s because they are both confined to the surface of the Earth, which is a sphere; so they must follow its intrinsic geometry. The metric governing this has real, detectable consequences. There is no detectable information about this at any one point on the Earth’s surface. This is because the geometry concerns relationships between points, so what you do is take measurements of path lengths, areas, or angles. For example, you’ll find that the sum of the angles in a triangle on Earth’s surface is no longer exactly 180 degrees - it’s possible to directly measure this deviation. But you can’t do it at a single point, you need to measure across some distance. That’s because the effects of a non-flat metric are accumulative - mathematically, you integrate components of the metric to obtain path lengths. To put it differently, the metric defines an inner product of tangent vectors, so it’s a local object, but with global effects across the manifold. Similar principles are true for curved spacetime as well. You can measure path lengths through spacetime pretty much directly (Shapiro delay, Pound-Rebka, gravitational wave detectors,...) and find that they differ from what you’d expect in a flat geometry. You can also directly measure angular distortions in the geometry, ie gyroscopic precessions, frame dragging etc. Gravitational light deflection is in effect a demonstration of the angle sum in a large triangle being different from 180 degrees close to a massive body. And so on.

I can only second that +1

Yes - but you can see how test particles in free fall (ie only gravity acts on them) move in space as they age into the future. In particular, you can see what happens over time when test particles initially move in parallel. That’s because in a small enough local area, spacetime appears flat, just like the surface of the Earth looks flat if you only look at a small patch of it. Global curvature emerges from the way many of such small local patches are assembled. To put it differently, a single point contains no information; but there’s information in how such points are related to one another. The physics are in the relationships, not the points themselves.

That’s an interesting perspective, I’ve never looked at it quite in that way. Thank you for bringing that in here. Given this, how would you characterise the relationship between QM/EM that governs the interaction between individual H2O molecules, and Navier-Stokes that describes the dynamics of very large ensembles of such molecules (ie flows of liquid water, for example)? The dynamics at play are remarkably different, it seems to me. I don’t quite get this example, since the scale this process (molecular vibrations) happens on is the same both here and far away. They’re just separated in space.

Yes you are right - but I disregarded this (and other effects) here, for clarity and simplicity. I think current models suggest it will be Mercury and Venus only.