Markus Hanke

That’s right, but the SS isn’t a massive body - it’s a multi-body system. Thus, if you are somewhere close but outside the SS, there will be small variations as the various planets go about their orbits. However these would be tiny, since almost all of the total mass is in the sun. Once you go far enough away, the SS will behave like a single body, since these variations will be too small to be detectable by any reasonable means.

It doesn’t really matter much whether or not the frames are perfectly inertial - non-inertial frames experience time dilation, too. The difference is just that the relationship between such frames is more complicated than a simple Lorentz transformation, but Special Relativity handles that just fine. For practical applications - such as particle accelerators - the deviation from perfect inertiality is usually negligible. If you do want a perfectly inertial frame, you can use clocks in a satellite or on the ISS as your reference; they are in free fall and thus locally inertial.

Except that’s not what happens - in fact, the opposite is true. Kinematic time dilation in inertial frames is symmetric; ‘we’ see the receding clock slow down, yet from the frame of the clock it’s ‘us’ who’s seen to be time dilated. That’s because time dilation is a relationship between frames in spacetime; it is not a physical property of any one frame. And since that relationship is the same irrespective of which of the two inertial frames you are in, you see the same thing from either vantage point.

I also do not think that Dark Matter exists in the way it is generally conceived of, ie as a particulate substance made from hitherto undiscovered particles. However, neither do I believe that any of the currently existing alternatives provide a better solution than standard cosmology does. Furthermore, some of the assertions made in this article are concerning, eg the claim that (paraphrase) “all predictions made by MOND have been verified”. This is quite simply wrong (some of its predictions are in fact in direct contradiction to observation), and I am very surprised that a qualified astrophysicist would say something like this.

I will suggest one other axiom, then: 1) The opening post of this thread is meaningless word salad This being an axiom, no further proof or discussion will be necessary - its veracity is self-evident.

I have followed the debates about the nature of the ‘dark sector’ for many years now, and have looked at the mathematical formalisms of all the various candidate models and ideas, some of them in detail. So I’m drawing from a diverse range of sources, not just a single paper or author. If you look at the bigger picture, you’ll find that many of the alternative models may be better at explaining specific phenomena - but at the cost of failing miserably with other observational data. Furthermore, very many of these alternatives require extra fields or extra dimensions, or make ad-hoc assumptions that aren’t based on any known physics - so they try to explain one unknown by proposing other unknowns, which is kind of useless. For example, the paper you quote assumes the existence of sterile neutrinos below a certain critical mass limit in order to match observations. Other known problems with MOND are never addressed at all. On a meta level, taking into account all available observational data at this point in time, standard GR still provides the best fit. Im aware of the problems in standard cosmology of course, but I don’t think any of the currently existing alternatives provides a good enough solution. That includes MOND and its relativistic generalisations.

What force would counteract gravity in this case, in order to keep the object stable and stop it from collapsing?

The trouble is that MOND is a non-relativistic theory, so comparing it to standard cosmology is kind of useless. At a minimum you’d have to use one of its relativistic generalisations - TeVeS being the most common and popular. And here’s where the issues start, because TeVeS has some serious problems, both so far as observational data is concerned, and in terms of mathematical consistency. And even if you could get it to work properly, you end up with various extra vector and scalar fields that are needed in the model - for which of course there’s no experimental evidence whatsoever. So in the end you just replace Dark Matter and Dark Energy with a bunch of extra unknown fields. It really doesn’t solve anything, on a conceptual level.

Yes, light from outside would be able to reach you, but your visual field would be heavily distorted. No. The global geometry of spacetime used to model the Big Bang is very different from that of a black hole.

The event horizon surface area is a function of mass, charge, and angular momentum. The 3-volume enclosed by this surface depends on the observer, so that can’t be answered in general (the actual calculation itself would also be quite cumbersome). Bear in mind also that the EH is now no longer a sphere, so talking about “radius” will depend on the latitude.

Yes, sure. Local physics in the interior follow the same laws as anywhere else (singularity aside). Not sure what you mean by this... The mass is encoded in the sense that the surface area of the EH is a function of said mass. But it doesn’t mean that the mass is concentrated into a shell of some kind. The EH is a region of empty and regular spacetime, so you can fall through it. Schwarzschild black holes. The same principles hold for all the Kerr-Schild metrics (which are electro-vacuum solutions), in that all parameters in the metric are global properties of the entire spacetime, and not localisable. This is thus true also for charge and angular momentum.

If you restrict your attention to some small region away from the source, through some limited period of time, then you could speak of a causal relationship in a purely local sense - something changed at the source, and awhile later my originally flat patch contains waves. Globally though, across all of spacetime, it’s still an equivalence - a time-dependent source distribution is equivalent to a time-dependent Einstein tensor. The global metric actually doesn’t change at all here, in the sense that its covariant derivative always vanishes. This is pretty subtle stuff. You see, the issue here is that there is no mass “inside” that somehow affects spacetime “outside”. In fact, in ordinary Schwarzschild spacetime there is no mass anywhere - it’s a vacuum solution that’s everywhere empty. It’s thus meaningless to point at any single point or region and say that’s where the mass is. The black hole is actually the entire spacetime; so mass is a global property, not local.

This isn’t true in general, but in the case of an ordinary Schwarzschild black hole, it is fitting to some degree - the mass-energy of the original object is no longer ‘there’ after the collapse; instead you now have a particular configuration of (empty!) spacetime that we call black hole. Actually, it’s not that simple - it’s much more accurate to say that there is an equivalence relationship between (Einstein, not Riemann!) curvature and energy-momentum. These two things differ only by a proportionality constant that fixes up the units - it’s not like one precedes the other causally. To say there’s a region of spacetime with non-zero Einstein curvature is exactly equivalent to saying that region contains energy-momentum that’s distributed in certain ways, and vice versa. Interestingly, this relationship only constraints the quantities in question, but does not uniquely determine them until you impose some boundary conditions. In GR, it’s really the boundary conditions where a lot of the ‘magic’ happens; people often don’t realise this.

The answer to this is that a black hole’s mass isn’t localised anywhere, in particular not “at the singularity”, as one might naively assume. Instead, it is a global property of the entire spacetime, so no issues of causality arise. To be even more precise, the metrics describing black hole spacetimes are actually families of metrics, indexed by up to three parameters. For simple Schwarzschild black holes there is only one parameter, denoted “M”, and it comes from boundary conditions when solving the field equations - it turns out that it physically coincides with the total mass of whatever object initially formed the black hole via gravitational collapse. Thus we interpret it as “the mass of the black hole”, but that’s actually pretty sloppy (and physically meaningless) terminology.

True. Also, at least in purely classical gravity, whatever happens beyond the event horizon cannot have any causal effect on the rest of the universe - which, on a high and global level, precludes any possibility of somehow using a black hole to send spaceships someplace else at superluminal speeds, irrespective of the precise mechanism.

The trouble with these things is that many of them violate more general principles that aren’t specific to just gravity - such as unitarity, causality, locality, various conservation laws etc. At least in the classical realm (spaceships etc) I think it is thus very unlikely that such loopholes exist.

Well, not really. The difference is that these are distant frames in a curved (as opposed to flat) spacetime, so the relationship between them isn’t a Lorentz transform, but something more complicated. They are also not symmetric in the same way a pair of inertial frames in flat spacetime would be.

No, this is a common misconception. The thing here is that the region close to the event horizon does not share any notion of simultaneity with a distant stationary observer (‘Schwarzschild observer’). As a result of this, a distant and stationary clock would measure an infinite amount of time for anything to fall to the horizon - meaning the horizon is never reached as measured in that distant frame only. On the other hand, if you consider a clock that actually travels itself to the horizon, you’ll find that it measures a finite and well defined amount of time; there’s nothing special about spacetime at the horizon at all. The clock just falls through and onwards to the singularity. It does not stop and freeze at the horizon. Time in GR is a purely local phenomenon, so you have to consider clocks that are actually there, and not distant observers.

Minkowski spacetime does not contain any sources of gravitation. If you add such sources, you will get spacetime geometries other than Minkowski.

No. Minkowski spacetime is (3+1)D, and it is perfectly flat.

That’s not true - a Euclidean spacetime would have the same sign for the space and time parts of the metric; for Minkowski spacetime these are opposite. In Euclidean spacetime there wouldn’t be any relativistic effects, since the speed of light can’t be invariant. You need the hyperbolic geometry of Minkowski for that.

I love this +1

The problem with this is that the Big Bang represents a boundary to spacetime itself. Without space or time, there is no causal structure, so speaking about “before” or asking “what caused” the BB is entirely meaningless.

The medium does exist - it is the electromagnetic field (in classical physics). This field extends through all of space and time, and EM waves are excitations of this. Going further, into quantum field theory, there are quantum fields that correspond to the elementary particle types - there’s an electron field, a photon field etc etc. At the moment this is our most fundamental description of reality - which is not to say that there mightn’t be something even more fundamental. There almost certainly is. This may be of interest - if you delineate a volume of space, and then ask “how many particles are in this volume?”, it turns out that the answer depends on the observer! Where one observer sees an empty vacuum, another observer might see a thermal bath of many particles - within the same volume, and all other conditions remaining equal. So the question as to existence and nature of particles isn’t as straightforward as one might think - thinking of them as ‘little balls of matter’ is quite meaningless.

Great post! +1 I would add here that GR is a description of, rather than an explanation for, gravity - in the sense that it deals only with the dynamics of the metric, but does not suggest an underlying mechanism as to why the Einstein tensor is precisely proportional to the energy-momentum tensor, as given in the Einstein equations. In other words, at present we don’t know yet why the concept of Einsteinian spacetime is such a good description of observable reality. This question falls outside the remit of GR, and would require a model with a wider domain of applicability. If you change the distribution of gravitational sources, then the geometry of spacetime will change accordingly, along with it. To be more precise, the changes in geometry will propagate outwards and away from the original position - either as regular gravitational radiation, or simply as unordered wave fronts. These propagate at most at the speed of light, but may propagate at less than c due to non-linear interactions with itself and any background curvature. In other words, the curvature that was there remains in existence, it just gets distributed differently. You cannot ‘unbend’ curvature, you can only shift it to somewhere else - this is why (eg) you cannot smooth out a sphere into a flat sheet, no matter what you do to it. So asking why spacetime returns into its unbent state is meaningless, simply because that’s not what happens.