Markus Hanke

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.

I just read through this entire thread, but I’m confused as to what your actual question really is. I think studiot provided a very good explanation above, though. Could you perhaps explain again what the confusion is?

No, all observers agree that the frequency shift happens in this scenario. Both observers agree that light emitted at A will be redshifted once it arrives at B. I don’t know what you mean by this, but it’s a real, measurable effect that isn’t just a visual ‘artefact’ of some kind. Yes, but the effect is pretty small. If there is relative motion of the star with respect to us, then the different effects will combine - there is redshift due to the star’s own gravity, blueshift due to Earth’s gravity, and either blue or redshift due to relative motion. What the overall net frequency shift will be then depends on the relative magnitude of each effect. Yes, that’s cosmological redshift. For large distances, this effect will be much larger than any local effects, so light from far-away sources is overwhelmingly redshifted. By the relativistic energy momentum relation: \[E=\sqrt{p^2 c^2 + m^2 c^4}\] which is just the magnitude of the energy-momentum vector, being part of the energy-momentum tensor, which follows as a conserved quantity from Noether’s theorem for time-translation invariance. So the idea behind it is symmetries of spacetime. For m=0 this gives p=E/c.

Massless particles still carry momentum - you can exert a measureable force on an object simply by exposing it to light in the right way. So in that sense photons definitely do “push”. The same holds true for gluons, though due to the nature of QCD the situation here is more complicated. While it is true that geodesics don’t depend on the nature of the free-falling object (they depend only on the geometry of spacetime, plus boundary conditions), it is highly misleading to say that “gravity doesn’t affect photons”. After all, if you send a photon through any kind of gravitational gradient, it will experience frequency shift, ie it will either red- or blue-shift.

Electrodynamics doesn’t claim this. What it says is that magnetic field lines do not end anywhere, meaning they must form closed loops through a bar magnet. It’s important to remember then that this is a vector field, meaning it has both magnitude and direction at each point. Since field lines form closed loops, the direction of the field (which is its tangent) is opposite at the two ends of the magnet - but we are still dealing with the same field. So when we are talking of N and S poles, these are just arbitrary conventions to indicate relative orientations; they are not actual physical entities in the sense of ‘magnetic charges’. If you cut a bar magnet through the middle, you don’t get an isolated N and an isolated S pole; you get two new bar magnets, each of them with two opposite poles.

I think it’s also important to mention here that we don’t actually live in a Newtonian universe, except as an approximation in the low-energy, low-velocity regime. What this means is that Noether’s theorem actually applies to an action within at least a 4D Minkowski spacetime (ignoring gravity for now) - in which case the conserved quantity associated with time-translation invariance is not just ‘energy’, but the full stress-energy-momentum tensor. This should be obvious, since ‘energy’ on its own depends on the observer, and is thus conserved only within a given frame.

I can see what you mean, but I’m not sure if this is in fact true. Look at the animal kingdom, eg ants - I don’t think ants share any kind of ethics with us, and neither do they have any cultural mechanisms to limit expansion and consumption (other than an equilibrium with their environment). And yet as a species they have been very viable for an amazingly long time. Of course an individual ant isn’t intelligent, and even its degree of sentience is debatable - but then, who’s to say that an alien intelligence isn’t distributed, akin to ant colonies? This is, in fact, a much more resilient form of intelligence than having everything centred in a relatively small number of highly complex, but intrinsically vulnerable individual brains. Of course, this is all speculation.

Actually, the Dark Forest conjecture assumes only this: 1. The primary goal of any civilisation is survival, before any other consideration 2. Civilisations develop, evolve and expand, but local resources available to them do not - they remain finite 3. Any possible communication between distant civilisations is subject to the constraints of the laws of physics, in particular the finite speed of light. This means they can’t be sure about one another’s psychologies and ethics (communication takes longer than cycles of technological evolution), leading to a ‘chain of suspicion’ The rest is simply a straightforward application of game theory - it’s called a sequential game with incomplete information. You can work out the possible evolution of such ‘games’, and the results aren’t pretty. A strong case can indeed be made for the most rational course of action being either to remain hidden, or to initiate a preemptive strike and destroy all other known races. This sounds dark and terrible, but I’m afraid it is logically and mathematically sound. Of course you can add ethics into the game - but then you need to make an additional assumption of all (!) other civilisations being somehow self-bound by some code of ethics that modifies or qualifies point (1) above. That’s a pretty heavy assumption to make, given that you have no way of knowing anything about another race’s psychological make-up and ethical motivations, and the results are disastrous should you get it wrong - if just one civilisation chooses to disregard that code of ethics and goes rogue, everyone else ends up being wiped out. Here’s a good article with a few more details: https://towardsdatascience.com/aliens-the-fermi-paradox-and-the-dark-forest-theory-e288718a808