Markus Hanke

No. What I mean is that the metric of spacetime itself becomes time-dependent here, in such a way that it appears to a far-away stationary observer as if the EH was oscillating with a quadrupole moment, even though the “position” of the horizon remains constant locally in a small neighbourhood. So what oscillates here is the relative separation between events, but not the coordinate position of the events themselves. This is somewhat similar to what’s called the “ring-down” phase at the end of a BH merger, where energy-momentum is dissipated away via gravitational radiation.

I think this would depend. In principle both EM radiation and G radiation should propagate along the same geodesics, so for “ordinary” objects like stars etc and “ordinary” G waves the deflection angle should be nearly the same since any non-linear effects are negligible. However, if the wave length of the G radiation becomes very large, and/or the background field is very strong, I would imagine there might be situations were such effects cannot be neglected, and the deflection angles differ. I’m not sure though. This may be relevant: https://arxiv.org/abs/2001.01710

Same here…welcome to my world

Oh yes, they are. The Lorentz transformation leaves the metric invariant: \[g_{\mu \nu } =\Lambda _{\sigma }^{\mu } \Lambda _{\rho }^{\nu } =g_{\sigma \rho }\] Rewrite this in matrix notation: \[\Lambda ^{T} g\Lambda =g\] Take determinant on both sides: \[det\left( \Lambda ^{T}\right) det( g) det( \Lambda ) =det( g)\] Since none of these determinants is ever zero, and since the determinant of the transpose equals the determinant of the original matrix, you get: \[det( \Lambda )^{2} =1 \] which implies that \(\Lambda\) is always invertible. Thus, inertial frames related via Lorentz transformations are always symmetric. ! Moderator Note It is against the rules of this forum to post personal theories into the main physics section, let alone onto an existing thread. If you wish to discuss this, you must open your own thread under “Speculations” and explain your thoughts there (don’t just give links).

Yes, they do “carry” energy-momentum. That’s an interesting question (I never really considered sources of directional g-radiation), the answer to which I am not sure about. There are two main issues here that make this situation somewhat different from Newtonian mechanics: 1. The energy-momentum in a gravitational wave cannot be localised, and 2. There is no global law of energy-momentum conservation in curved spacetimes of this sort. Point (2) is to say that yes, energy-momentum is conserved everywhere locally in small neighbourhoods that can be considered approximately flat; but there is no such conservation law for larger, curved regions. Take careful note here of the difference between a conservation law being violated, and no such law existing in the first place. In the absence of meaningful energy-momentum conservation (in the Newtonian sense), it is not at all clear to me whether or not a source of directional g-radiation would receive an impulse in the opposite direction. I’d say before even considering this issue one what have to first ascertain whether such a thing as a directional beam of g-radiation can even exist, which is in itself not clear. I’m very careful to commit to any definite answer on this, as this is one of those scenarios where ordinary Newtonian intuition can very quickly lead one down the wrong road. The best I can say is that it would be necessary to actually do the maths, which would be highly non-trivial.

I am autistic indeed, and in the very fortunate position that my own personal autistic profile is such that for me the advantages of being on the spectrum far outweigh the challenges (which exist, but may not be obvious here). Unfortunately this is not true for many other - perhaps even the majority of - autistic people, a large proportion of whom suffer significantly from their autistic traits and common comorbidities, and above all from the failure of the wider neurotypical world to understand, respect, accept and accommodate these traits. Ultimately whether or not the “D” belongs in there is very much a matter of personal circumstance and experience - for me personally, I do not consider my autism to be a “disorder” in any way, and wouldn’t choose to change it even if given the opportunity; however, other autistic people may think about their own situation very differently, and this absolutely needs to be respected too. Everything considered, I am not a very “typical” representative of the autistic community. And for those of you who aren’t aware - English is also not my native language; my real-world vocation has nothing to do with science; and I am not university-educated. The foundations of physics are simply a matter of personal interest to me, so it’s all self-taught. And rest assured that if I’m among real experts on these subjects matters (e.g. among some of the regulars over on PhysicsForums), I’m also left feeling ignorant and dumb Which is why I’m mostly just a silent reader there. But for me this is rarely a negative experience, since I consider ignorance to be an opportunity to acquire new understanding, which is never a bad thing. PS. To give perhaps a better insight into the subject matter of this thread - when I respond to posts, my entire focus is always 100% on understanding better how the world works. That means when I see a statement that doesn’t gel very well with the current scientific consensus on the matter at hand, I’ll simply say so - social considerations never come into it for me at all. I don’t set out to intentionally hurt or belittle people, but neither do I go out of my way to mollycoddle others’ feelings (unless they are obviously vulnerable in some way). The social aspect just simply isn’t on my radar at all. A verbal or mathematical statement is either a good description of some aspect of the world, or it isn’t - that’s all there is to it for me. This is also how I roll in the real world - I am very focused on concepts, insights, and values, and have little to no interest in social or cultural conventions. That get’s me in trouble sometimes, since most other people appear to be reifying socio-cultural conventions into some sort of universal truths or standards. I’m just not everyone’s cup of tea I guess

There is no known physical mechanism that can “shield” gravity in this way, so no reflection - in the sense that term is used for light e.g. - can happen. It is, however, possible to deflect gravitational waves, i.e. change their principle direction of propagation via interaction with background curvature, or other gravitational waves. Yes they can, and in full theory of GR that is a highly non-linear process (but linear approximations can sometimes be used to describe this). Yes, the Einstein equations emit solutions - both in vacuum and in so-called null dusts - which can be physically interpreted as the equivalent of standing waves. I don’t know about the amplification bit, since this is a non-linear situation, so one would have to run the maths on it. This should theoretically be possible I think, though again, one would have to do the maths (which wouldn’t be trivial at all) to be absolutely sure. I don’t understand this question…can you explain further?

Good point, this never occurred to me +1 Very different indeed. In fact, based on GR in 2D, the only gravity that could exist in such a world would be found in the interior of mass-energy distributions - there could be no gravity at all in vacuum.

Does the forum software here support the ability to number posts within a thread? If so, wouldn’t it make sense to turn that function on? More than once now have I had to refer back to what someone said earlier on a thread, in a way that just can’t be done easily using the quote function. There are just situations where it is easier to simply refer to a post number (“You made this claim in post #…”). Thoughts?

Yes, precisely +1. The basic twin scenario is simple, I don’t understand why people feel the need to add so many extra complications to it that do nothing to illuminate the underlying physics. This seems to be a problem with SR in general.

Ah ok…I can see how someone might be tempted to look at it in this way. That didn’t even occur to me. Thanks for pointing it out. Obviously though, since there is exactly one geodesic (=extremal) path connecting any pair of events in spacetime, there must always be at least one local section of the journey where the two travellers are not related via a Lorentz transformation. Yes, perhaps you’re right and it’s that simple.

For me personally, it’s fair to say that I’ve been concerning myself with the theory of relativity for quite some time, and while I’m not an expert by any reasonable metric, I still dare say that my knowledge of the subject is above the average one would find in a group of randomly-chosen members of a (reasonable well-educated) public. Yet, even after all these years, I am still failing to understand why so many people consider these scenarios “paradoxical”? I get that the outcome of such experiments can appear surprising at first glance, but that’s not the same as calling it “paradoxical”. I’m not being condescending, sarcastic or whatever else, it’s a genuine question. I just don’t get it. Analogy(!!!): It’s a little bit like flying from, say, New York to London - you can fly eastbound, and follow a suitable geodesic (i.e. a great circle segment, ignoring vertical motion for simplicity now, all other factors being equal), and get to London in a certain amount of time. Or you can choose to fly westbound, and likewise follow a geodesic, just in the opposite direction. Or you can choose some other route that isn’t a geodesic at all, so long as they all start at the same place and terminate at the same place. No one would be surprised by the fact that for these three cases, the onboard clocks read different elapsed times - I mean, it’s rather obvious that there is exactly one, and only one, route that minimises the total in-flight time. Any route that diverges from that flight path will necessarily be of different duration, assuming all other factors remain equal. It’s no different for a path in spacetime, that is traced out by different travellers between the same two events. In topologically trivial Minkowski spacetime (which is the stage against which this is set), there is precisely one - and only one - path between any pair of given events that extremises (minimises or maximises, depending on sign convention) its own length, which is by definition equal to the proper time recorded on a clock that physically travels this path through spacetime. Any path that varies from this one choice must necessarily be of different length, ie a physical clock travelling along it will record something other than the extremal value. If you set this up as a variational calculus problem, you unsurprisingly end up with the geodesic equation - the extremal path between any pair of events here is a geodesic of Minkowski spacetime, which physically corresponds to a traveller moving inertially. The reverse of this statement is just as true - in singly-connected Minkowski spacetime, any path between given events, the length of which differs from the extremal value obtained from the variational calculus problem, is necessarily non-inertial at least within some small region along it, since the extremal path is a unique solution to the equation. Why is this considered “paradoxical”? I struggle to even consider it “surprising”, since it seems entirely obvious to me that this is what will happen, just like in the analogy of the planes above. You can’t - in general - take two different routes between the same points and reasonably expect them to have the same lengths (unless you cheat by introducing non-trivial topologies etc). Note that this isn’t about why SR is what it is (ie hypothetical underlying mechanisms etc), but simply about why this result should be considered surprising or paradoxical. To me it isn’t, unless I am missing something so basic that I can’t even see it - in which case I’d be grateful if someone could point it out to me. My other issue is that I’ve seen the original scenario amended such that the travellers involved do not actually connect the same pair of events. What meaningful physical conclusion - in terms of elapsed times - would one hope to arrive at by comparing paths that don’t connect the same events? I don’t get this either. It seems even less surprising that - again in general - you get different results if you compare paths between different events.

A physical paradox can arise only if there is an inconsistency in transformations between frames. Here’s what I mean by this. Say you perform a Lorentz transform to get you from frame A to A’ - by doing so, lengths and times in A’s choice of coordinates will change, since a Lorentz transform is nothing other than a combination of hyperbolic rotations and boosts. A linear transformation, in other words. But what happens if you perform a second Lorentz transformation, identical to the first one, only with a negative argument (-v instead of v)? The already transformed lengths and times associated with A’ will transform again - since this new transform is the same one as the original one, only in the opposite relative direction, one should recover the original frame A. If one doesn’t, the situation is not internally self-consistent. So, consistency in this context means (L denotes Lorentz transform): L(v)A -> A’ L(-v)A’=L(-v)L(v)A -> A A applies L, and arrives at A’. Likewise, A’ applies L, and arrives at A. There is perfect symmetry between these frames. In a situation where such consistency holds, it is not possible to construct any physical paradoxes based on this linear transformation alone, because any pair of inertial frames will always agree on how they are related to one another. So, in order to show that one cannot construct physical paradoxes in Minkowski spacetime based on the axioms of SR, it is sufficient to show that every Lorentz transformation has a unique and well defined inverse, such that \(L L^{-1}=I\), wherein I is the identity element. Suffice to say that this is indeed the case, and if you want I can present the formal proof here (or you can just Google it yourself).

Wouldn’t it be easier then to just write down a general proof that there can be no physical paradoxes based on any pair of inertial frames related via Lorentz transformations? Because that’s easy enough to do, and it would apply to all possible such scenarios.

Ok, fair enough, that’s a valid motivation. So do you feel that the conundrum, as you call it, is due to not having analysed the specific scenario correctly, or do you feel that SR as a model is not internally self-consistent? The latter is easy to address in a very general manner; whereas disentangling the former may be tedious and time-consuming, and (IMHO) not very illuminating so far as the actual underlying physics are concerned. Not cynical so much as sceptical. For two reasons, mainly: 1. I’ve been participating on various science forums for a long time, and sadly it’s a very common tactic for people to try and covertly smuggle “anti-relativity” type of sentiments onto the main physics sections (where such things aren’t supposed to go) by dressing them up as specific scenarios that purport to show some sort of paradox or contradiction. Once other posters clarify the mistake that was made in analysing the scenario - which lead to the apparent contradiction in the first place -, the OP then refuses to take on board anything that is explained to them, and will stick hand tooth and nail to their contention that relativity is self-contradictory and thus wrong. Usually such threads end up locked or abandoned. Sorry to say, but this is an exceedingly common modus operandi. 2. The general impression you gave on your other thread was not one of someone having genuinely come here to learn; it also had some of the hallmarks mentioned under (1). Perhaps I’ve gotten the wrong impression somehow, but as an uninvolved reader of said thread I’ve got to tell you that the vibes weren’t good. But maybe I got you all wrong, and maybe you are right in that I have become cynical by having spent too many years on science forums…if so, I’ll take responsibility for that. So let’s see how this thread goes

So let me ask you this straight - what is the purpose of these threads? Are you seeking to genuinely understand how to correctly analyse situations such as these, or are you just trying to show that relativity must be wrong by presenting “conundrums”? I’m asking this because thus far you haven’t shown yourself receptive to anything that was explained to you, and I therefore see no reason to assume that this thread will go any differently than your last one. Based on what I have seen so far, I personally suspect it is the latter of the two options - but please do correct me if I misjudged. Thus, instead of getting endlessly entangled in highly specific scenarios that do nothing to illuminate the actual issue, I suggest it would be far better if you were to simply state clearly and directly what your contention with regards to Special Relativity actually is, and we can all save ourselves the needless beating around the bush and cut straight to the chase instead.

Yes, it’s indeed not different from EM waves. What is different though is the way gravitational waves interact with one another, and with any background curvature that might be present.

Nice summary! +1

I think both your discomfort, as well as your reaction of politely declining, are perfectly ok here - those are not examples of homophobia.

I wasn’t around yet at the time of the moon landing itself, but I was born and grew up in a former East Bloc country myself. So ditto. A lot of mud was slung both ways across the Iron Curtain, but no one doubted the scientific and technological capabilities of the other side, for obvious reasons. I don’t know for sure, but I would also bet that the USSR remotely tracked every step of the moon mission, and intercepted each and every communication, including telemetry. Fun fact: the Soviet Luna-15 mission (a robotic attempt to collect samples of moon rock and dust) crash landed right as Apollo 11 was on the moon, and just a few hundred miles away. Interestingly, what the USSR did deny was that there was a race to the moon going on between the two nations at all. But they never denied that the Apollo mission was successful.

What difference does it make what anyone “believes”? I always felt that the best response one can give to MLHs isn’t scientific at all, but political - the USSR and Maoist China believed it to be real, and that’s to say an awful lot given the global political, military and intelligence situation back in the day. Had this been fake, you can be absolutely sure that the communist bloc would have found out about it, and oh boy would they have had a field day with that

Lifting anything to a point of higher gravitational potential requires you to put energy into the system in the first place; likewise, producing magnets of substantial strength also requires lots of energy. Whatever motor device you then construct based on these, you will never get out any more energy than you originally put in. You simply can’t cheat nature. So I really don’t get the point of all this? We already have hydropower, and we already have photovoltaic systems - these aren’t new inventions. You really don’t want any more unnecessary mechanical parts such as moving iron balls etc, since these just reduce the overall efficiency of the system.

Because that violates the singularity theorems. Once you have an event horizon, the formation of a gravitational singularity within the region enclosed by it is always inevitable, at least in classical GR. But that notwithstanding, I was specifically referring to a classical Schwarzschild BH, where we have \(T_{\mu \nu}=0\) everywhere by definition - there are no distributions of energy-momentum of any kind, anywhere in this scenario. The event horizon in this type of spacetime encloses a region that is completely empty (at least classically), and yet you can still associate thermodynamic entropy with this black hole. So the question arises: thermodynamics of what, exactly? There are no constituents or “states” to this system at all, in the classical picture.

That’s a really good question, actually. In GR texts (which is what I am mostly familiar with) this is never really made any more explicit than connecting event horizon area to entropy - but the entire subject is generally treated under the heading “black hole thermodynamics”, and it is directly linked to the “temperature” of a black hole, so it stands to reason that it is the thermodynamic type of entropy that is in question here. Which again raises the issue just in what sense a region of completely empty spacetime should exhibit a property such as temperature…

Well, perhaps there’s a gap in my own understanding then. So how would you physically interpret the notion of “entropy” associated with a region of spacetime (ie a region on a semi-Riemannian manifold)?