Markus Hanke

The salient point though is that there is no local experiment you can perform which distinguishes these cases. The equivalence principle is always a purely local statement. Measuring force and distance over long periods of time gives you a region of spacetime that is too large to be considered ‘local’, so of course there is no equivalence. But if you put an astronaut into a windowless box and ask him to tell whether the downward force is due to gravity or due to acceleration, without any outside references, then there isn’t anything he can do to tell the difference.

It’s quite alright to ponder these issues. That’s how understanding is made. But here’s another thing - in the GR field equations, Lambda is not directly identified with any kind of energy. Regardless of which side of the equation you choose to put it, what happens is that, in vacuum, it stops the Einstein tensor from being zero. The physical meaning of this tensor is that, once a future time direction is chosen, it gives the average of scalar curvature in the spatial directions. This means that, taken at face value, Lambda is best understood as a background curvature that is always present, even in the absence of all gravitational sources. It’s purely a geometric entity that modifies the global geometry of the manifold. Identifying this background curvature with Dark Energy is one possible hypothesis - but there’s no principle that requires us to posit this. There are other options.

I second that +1

You would be moving pretty fast relative to your starting point, but not faster than c. Remember the laws of Special Relativity!

Yes. Also, remember previous discussions we had about degrees of freedom. Scalar curvature (I presume you mean the Ricci scalar) only measures one particular aspect of curvature; it can’t account for all necessary degrees of freedom.

So does it require agency? Does it presuppose an independent agent who ‘has’ free will, or is it more akin to a self-regulating complex system with multiple feedback loops? Could you explain further? If (in my specific example) the action is initiated before the motivation and decision making process becomes conscious, then how was the decision ‘free’? Are you familiar with Carlo Rovelli’s ‘relational QM’? It resolves the issue rather elegantly.

True, it is, at least in some sense. But even if you disregard inflation completely, you can only extend the Lambda-CDM model as far back as ~10^(-32)s or so - and you’d struggle to explain the large-structure of the universe without inflation. But under no circumstances can you extend it to the Planck epoch, that’s outside its domain of applicability, since GR is a purely classical model.

It’s far more complex than that, because determinism does not imply predictability; and determinism+predictability don’t imply computability. That notwithstanding, it has been known for some time that consciously intentional actions (eg moving your arm) are preceded by motor neuron activity in the brain by ~10ms. That means the brain physically initiates motor action before you ever become aware of any decision-making to act. Where does that leave free will? I personally think free will is a silly concept, as you need to make a ‘causal gap’ argument to postulate it. It also requires independent agency, which is also an extremely dubious concept, to say the least.

The Lambda-CDM model is a model based on classical General Relativity, so it only describes the evolution of spacetime after the inflationary epoch. You cannot extend it further back than that.

What do you mean exactly by ‘radius of curvature of spacetime’? Again, I don’t know what you mean by this, since the tidal curvature of spacetime certainly does not follow a linear relationship like this. For a radial infall such as the one I spoke about the relevant quantity is the (t,r,t,r) component of the Riemann tensor - which is an inverse cube law.

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.