Markus Hanke

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)?

I’d say a feeling (or any kind of mind-state in general) is never in itself reprehensible, because it is the result of very many different internal and external causes and conditions that we generally do not choose to put in place. What we can choose though - at least to a degree - is how to act in response to our mind-states. Thus, merely having personal distaste or discomfort over anything is ethically neutral, whereas (eg.) beating someone to a bloody pulp because of such mind-states, is not.

It’s interesting to ponder what this actually means in the case of black holes, because there is no obvious reason why a region of completely empty spacetime (which is what a Schwarzschild BH is, in classical GR) should have finite non-zero entropy associated with it at all. Even if we disregard the issues around singularities etc for the moment, if we consider an ordinary star of 1 solar mass, then its total entropy is on the order \(~10^{57}\). If we let that star collapse gravitationally, and ignore any energy losses during the collapse process, then the resulting 1 solar mass Schwarzschild BH will have entropy on the order \(~10^{77}\), which is 20 orders of magnitude larger. While it is interesting in itself that the total entropy of the system increases by orders of magnitude, what’s really surprising is that the resulting BH has finite, non-zero entropy at all - remember again that classical Schwarzschild spacetime is everywhere empty. Simply and somewhat sloppily put, entropy is a statistical property that reflects the number of ways the microstates of a system can be rearranged without affecting its overall macrostate. But a Schwarzschild BH is just empty spacetime - every point within that manifold is exactly the same as every other point, meaning no small local neighbourhood can be physically distinguished from any other small local neighbourhood. And because this is a classical model, the spacetime manifold is implicitly assumed to be smooth and differentiable everywhere (disregarding the singularity for now), or else the entire formalism of GR makes no sense. Thus we can just pick any arbitrary pair of events within the volume in question, and swap them - doing this will not change anything about the BH at all. Because the manifold is smooth and differentiable, there are infinitely many such pairs in any given volume of empty spacetime, so the total entropy of this system should diverge. But it doesn’t - it’s finite and well defined, and always >0. IOW, there is a finite, well defined number of distinct operations one can perform in such a volume that leaves the overall system unaffected. It would seem to me that this is possible only if spacetime within such a volume is not in fact smooth and continuous everywhere - there needs to be some kind of non-trivial structure present at least in some subregion of the volume enclosed by the event horizon. The mere fact of geodesic incompleteness in and around r=0 doesn’t seem to account for this (a point singularity has no degrees of freedom, and isn’t part of the manifold in any case) - it would take a very non-trivial kind of micro-structure to yield an entropy of the magnitudes mentioned above. So what is this micro-structure? I for one would dearly like to know…

Well put +1 I think it is a crucially important life skill to - in some situations - be able to respect things that we don’t personally like. This isn’t always easy, since we generally tend to equate our own preferences, views, beliefs and opinions with some notion of “truth” about the world. It takes a certain amount of introspective awareness to recognise this dynamic and suspend it, if and when necessary; sadly, not everyone is able to do this.

Definitely. People here generally aren’t gifted with the patience to wade through gibberish either. LaTeX isn’t “rare”, and it’s not found only on this website - it’s the international standard for typesetting documents that contain mathematical notation. Every major science discussion forum uses this to display maths. It’s really not that hard, and there are also many free online LaTeX editors you can use, for example: https://www.mathcha.io Give it a try sometime.

Check part VI, chapter 27.5-6. In my copy that is page 718 onwards, where they talk about the expansion factor and the possible geometries of hypersurfaces of homogeneity. The expressions for the non-zero components of Riemann for FLRW come from my notes, which, if memory serves right, I got from MAPLE at some point. It’s considered the gold standard so far as the theoretical aspects of GR are concerned for a reason. Truly fantastic text, most of my own GR knowledge comes from here. There are even some bits which are not found in any other GR text that I’m aware of.

A lot of people are talking about the ChatGPT project of late, so I decided to try it out for myself, because I was curious. I’ve asked it a lot of technical questions with regards to physics and some other topics, and was pretty impressed how well it was able to answer most of them. However, the AI wasn’t 100% spot-on. At least twice I caught it giving dubious answers - the first one was, with some goodwill, highly misleading; the other one was just flat out wrong. Interestingly, I was able to tell the AI about the wrong answers, and when I asked the same questions again (in a new chat session), it had one of them (not the other one!) corrected. So the moral here is that one should be very careful about using ChatGPT as a source of scientific information - at the very least, the answers it gives should be cross-checked against other sources to ascertain their correctness, and not just accepted at face value. This is an AI, not an online encyclopedia - the system makes mistakes, it learns, and it continuously updates its own internal states. It’s not infallible, so we are well advised to use it with caution. PS. I fear that people referencing AI-generated output from sources such as ChatGPT to backup their arguments is going to become “a thing” before too long on forums such as ours. That’s going to be a problem, for several reasons; most importantly, as mentioned above, AIs make mistakes, and sometimes give wrong answers, so I don’t think we should consider thinks like “ChatGPT said…” to be a valid reference when it comes to backing up arguments. Secondly, I noticed that ChatGPT sometimes answers the same question in different ways, using different wording - so you can’t link to a specific textual answer in a way that others can check and recreate; it’s dynamic output, not static text. That’s problematic when it comes to using it as a reference. What do other people here think? Do you guys feel there might be a need to develop an “AI-Generated Content Policy” for this forum?

Misner/Thorne/Wheeler “Gravitation” describes this in some detail. Consider first the general form of the FLRW metric: \[ds^{2} =-dt^{2} +a( t)^{2} d\Sigma ^{2}\] wherein \(\Sigma\) designates a 3-surface of uniform curvature (which could thus be elliptical, Euclidean or hyperbolic). The full Riemann tensor for 3D+1 spacetime with this type of metric then has six functionally independent components: \[R_{1100} =R_{2200} =R_{3300} =a\ddot{a}\] \[R_{1122} =R_{2233} =R_{3311} =-a^{2}\dot{a}^{2}\] Thus spacetime is never Riemann-flat, unless a(t)=const. On the other hand, the 3D+0 Riemann tensor for a given 3-surface \(d\Sigma\) of space is \[^{3}R_{ijkl} =\frac{k}{a( t)^{2}}( g_{ik} g_{il} -g_{il} g_{jk})\] wherein k is called the curvature parameter, so that for the choice k=0 each 3-surface is Euclidean and flat. MTW motivates the presence of the curvature parameter in this expression by using the following exact solution of the field equations: \[d\tau ^{2} =-dt^{2} +a( t)^{2}\left(\frac{du^{2}}{1-ku^{2}} +u^{2} d\phi ^{2}\right) ;\ u=\frac{r}{a( t)}\] wherein t is the total time recorded on a co-moving “dust clock” since the beginning of the universe.

Ok, that’s fair enough! I freely admit that I’m more versed in the physics of time dilation than I am in the engineering details of the GPS system. The salient point here is though that if you didn’t account for gravitational and kinematic time dilation at all, the system couldn’t work in the way it does now, or at least it would be a lot less accurate.