Markus Hanke

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

I presume you are talking about the Alcubierre metric, being a solution of the Einstein equations. There are numerous problems with this concept, to the degree that I would nearly call it “unphysical”. But assuming for now that the creation of such a ‘bubble’ were somehow possible, it would be a completely impractical - and highly dangerous - means of travelling. The pivotal issue is that once the bubble has been created, there would be no physical means of accelerating it, meaning you could neither steer nor stop it. There also wouldn’t be any means to enter or exit it without getting ripped to shreds by tidal forces. Furthermore, if the bubble enters the gravitational influence of another body (star etc), it would begin to distort and undergo other non-linear interactions with the background curvature, which would be very bad news for whatever is in the interior. If you want a concept that is at least remotely plausible, you should look into the Krasnikov tube. Not as flashy as a warp drive, but more practical if you could somehow manage to create one.

Put yourself in their shoes. We’re intelligent, but also largely still driven by ancient instincts that were designed to give us a survival advantage in a largely hostile environment. Most people’s thought patterns are overwhelmingly ego-centric, along the lines of “what can I get out of this?”. People spend their entire lives chasing sense pleasure and running away from discomfort, putting them at the mercy of external circumstance. Our societies institutionalise greed (economic systems), hatred (militarism and nationalism), and mass delusion (corporate media). At any given time there are at least half a dozen active war zones around the world; we can’t even get along with our own species, and our own natural environment - never even mind with aliens. All these things were indispensable survival tools in the distant past, but now our technology has outpaced our ethical and psychological evolution - a very dangerous situation. We’re animals waving thermonuclear warheads around. Would you really want to get involved with such a species? I sure wouldn’t - I’d recognise their potential, and perhaps would watch from a distance, but otherwise would choose to wait until they’ve outgrown their first set of teeth at least. And as a species we’re nowhere even near that point, IMHO. Besides, if there are several spacefaring species in our galaxy competing for finite resources, then there’s quite a lot to be said for remaining silent and invisible, as a general rule (‘Dark Forest Theory’): https://bigthink.com/surprising-science/the-dark-forest-theory-a-terrifying-explanation-of-why-we-havent-heard-from-aliens-yet/ Apologies for being dark and pessimistic. Personally I see a lot of beauty and potential in Homo sapiens sapiens, but for now I see us as being little better than animals with a knack for technology. So I’m not in the least surprised that no one else has made contact.

Looks correct to me, this is what I got too 👍

1. Write down the transformation functions x=..., y=..., z=... for spherical coordinates 2. Calculate the Jacobian matrix from these 3. Calculate the determinant of the Jacobian matrix, in order to get dV=... 4. Determine the integration limits in your new coordinates - radius and equatorial angle are trivial, but for the polar angle you need to find the intersection of the ball and the cone (hint: eliminate z from the equations in the OP) 5. Write down the volume integral using your limits and volume form, and evaluate

Janus beat me to it! Maybe just as well, since I’m actually somewhat confused now - my understanding of the original scenario was that the distance A-B was to be 1 light-minute as seen from frame C, and that clocks were synchronised (taking into account light travel times) to all be seen as showing zero at the instant when C passes A, again as seen from C. My previous answers were based on that. But reading the last few posts, I may not have understood correctly what the OP meant, as the scenario seems to be different now...?

It’s not that simple. In frame C, the instantaneous tick rate of B is dilated, whereas the distance A-B appears longer for frame B than frame C. We have thus far only concerned ourselves with the final result of the experiment (C accumulating less time than B in total) - if you want to analyse what C visually sees on clock B at every moment of the journey, then things become complicated, because C and B do not share a common notion of simultaneity while there is a spatial separation between them. So you would have to account for relativity of simultaneity as well, and the analysis will lead you to the relativistic Doppler effect. In actual fact, what you visually see is that the hands on the distant clock advance faster as compared to your own clock - that’s because these intervals “tick out” a total spatial distance that’s different from yours. The key issue here is that there’s a difference between instantaneous tick rate, and total accumulated time - the instantaneous tick rate of B is dilated with respect to C, but you’re not sharing the same notion of simultaneity, so if you integrate those small infinitesimals, you end up with longer intervals, and thus more overall time passed as seen by you. It’s like the distant clock projects pictures of itself at you at a steady rate of 1 frame per time unit - but because you are moving towards the clock at high speed, you are encountering each picture at less than one time unit in your own frame, and there will be more of those pictures in total. This is why it visually looks like it’s running fast. But if you were to compare each picture individually to your own clock, using some appropriate concept of relativity of simultaneity, you’d find the instantaneous readings to be dilated with respect to you. The overall times are different, because the overall distance A-B is also different between these frames. I’m sorry I don’t know how to explain it better - this is neither intuitive nor particularly simple, despite the quite basic scenario. The devil is in the details. Nonetheless, once the maths are done correctly (also not as simple as it might at first seem!), they are found to correspond to what we actually find in the real world.

No, C will show less elapsed time at the moment of reading his clock as it passes B, since it has been in constant relative motion with respect to B throughout (we assume there was no starting or stopping). Yes, correct. That’s because observers C and A/B don’t agree on the distance between A and B - a phenomenon called length contraction, which goes hand in hand with time dilation. Also, there’s a difference between total elapsed time, and instantaneous “tick rate”. The train C reckons that the distant clocks A/B run slower, but he also reckons that the distance A-B is shorter as measured in his frame, so it takes less time on his own train clock (C) to reach B, than it does on B’s clock. So what this means is that yes, B is dilated as reckoned in frame C (just as C is dilated in B) - but the distance A-B is also longer in frame B than it is as seen from C (because A is stationary with respect to B, but neither A nor B are stationary with respect to C). So, when the train arrives, then, as seen from the train C, clock B had to have been measuring out a longer distance at a slower tick rate (remember that speed of light is the same in both frames!), meaning it has to have accumulated more total time than C - even though it was seen as dilated at each instant, just as C was dilated as seen from B. A proper analysis of this will need to reference relativity of simultaneity at each point of the journey, but rest assured that everything works out as self-consistent. In the end, they both agree that total time in C is less than B. In other words, both observers C and B agree that at the moment of C’s passing by B, the clock C shows less total elapsed time - but they disagree on the reason why that is so. B will say it’s because clock C ran slower, whereas C says it’s because the distance A-B was shorter and thus it took less time to cross it. But both agree on the physical outcome - which is the crucial bit. Take careful note of the highlighted bit - comparing the clocks needs to happen at a single instant, when the clocks pass by one another at negligible spatial distance, or else you’d need to account for light travel time as well. A real-world example of just such a scenario is atmospheric muons. These are short-lived, very fast moving particles that are produced just outside the earth’s atmosphere. Without relativity they don’t live long enough to have sufficient time to reach the Earth’s surface. But they do, we detect them. Why? Because of relativity. In the frame of the earth, the muon moves very fast, so it’s associated clock ticks slower (=it lives for longer), and thus reaches the surface in time before it decays. In the frame of the muon, earth’s atmosphere is length-contracted, so it has to cross a shorter distance, and thus lives long enough to reach the surface. Both observers agree on the outcome (=no paradoxes), but they disagree on the reason. This is exactly analogous to your train experiment. Ok, agreed 👍

There are a whole lot of technicalities involved, but fundamentally it boils down to two basic issues - background independence, and time. GR is background-independent, in the sense that it does not make reference to any underlying frame or system of coordinates; what’s more, it is the metric (system to measure space and time) itself that is a dynamic variable here. Gravity is geometry of spacetime. QM on the other hands is formulated as wave functions that explicitly depend on space and time, so it is background-dependent. Reconciling this is proving to be very difficult. In GR, time is treated on equal footing as space, and thus forms spacetime. It’s a geometrical dimension, and a fundamental part of the world. Time in QM, however, is just another parameter appearing in the theory, and is not the same as space. Whether we’ll ever bring these together into a fully self-consistent model of quantum gravity that has the proper limits, is anyone’s guess. There are various attempts and candidate models at present, but none of them is universally accepted as being the ‘right’ one. It’s an area of active research, so we’ll see what happens. On a side note, you can formulate quantum field theories in curved spacetime, but that’s not the same as quantum gravity.