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 Apr 11 comment Is there an energy density limit in GR? @Roy Again, the issue is this quantity is coordinate system dependent. The scientist can use whatever coordinate system he wishes to describe a lab experiment. Coordinate system choices are not "physical" which is the very reason the physical laws are independent of them. So we need to cast the question into invariant terms. You instead seem to be saying $T^{00}$ is already physical since it can be measured with a given coordinate system ... if you take that stance on "physical" then the question is not interesting and already answered. Apr 11 answered What is a virtual ground? Apr 10 comment Is there an energy density limit in GR? @Roy I'm not sure how to answer your question there. If you could clearly define what you mean by "physical energy density", maybe you should ask your question in the last comment as a follow up to John's question. It would likely be much more interesting than discussion on limits of the coordinate dependent $T^{00}$. Apr 10 comment Is there an energy density limit in GR? @Roy In my opinion, it sounds like John is still struggling with the concept that only invariant, not coordinate dependent, quantities have any real meaning. So this leads to sloppy associations like 'large amounts of matter can collapse to a black hole' -> 'E=mc^2' -> 'energy is limited by black hole formation' -> 'if I move to fast I become a blackhole'. This is clearly wrong, but it is a common question/confusion students have. Apr 10 comment Is there an energy density limit in GR? @Roy $E^2 + B^2$ is just $T^{00}$ for an electromagnetic field, and therefore is just as coordinate dependent. The usual way of defining a 'proper density' for an object or fluid is to state the density in its rest frame. We don't have that luxury with a electromagnetic plane wave for example. So I don't think it is meaningful to try to define a proper density in that case. There could be a clever way to redefine the terminology that isn't immediately obvious, but without referring to other vectors like a velocity, the only scalar invariant is the trace $T$ which is 0 for electrodynamics. Apr 9 comment Why do car keys have longer range when held next to your head? I have taken one of these apart before. The antenna was a copper trace running in a spiral around the outside of the PCB. I think it would be foolish if the engineers didn't design the antenna correctly, so that is probably fine. Also, I am very confused about the "pointing the key" comments people are making. I don't think it emits a roughly focussed 'beam' in any direction. So while I don't doubt people are seeing increased range, some of the details seem quite sketchy and anecdotal instead of systematic tests. Apr 9 comment Can the Big Rip really rip apart an atomic nucleus? @anna This is not a matter of personal "view" or opinion. Consider the ratio of two lengths so that we can ignore the units completely. The ratio (bohr radius / separation of distant galaxies) is changing in time due to expansion of the universe. Experimental results, and especially mathematical predictions from theoretical models are NOT a matter of opinion. @Marek is correct here. Apr 9 comment Is there an energy density limit in GR? @Roy Mass definitions and the hoop conjecture are indeed some interesting follow ups to this question. However I'm not entirely sure the MM book is even talking about 'proper' energy density since he uses it to claim a limit on the electric field. Even in SR I don't think it is meaningful to try to define a proper density for an electromagnetic plane wave as there is no inertial frame in which it is at rest. Considering his "derivation" of GR, it is probably best to just ignore the Motion Mountain book whenever it invokes GR. Apr 6 comment Can you see yourself in a mirror when you are riding on top of a light stream? @dbrane I think it is assumed the mirror is not moving relative to you (ie. you are holding the mirror, not travelling at a large relative velocity towards the mirror). So aberration and doppler shift will not cause any effect for the person holding the mirror, no matter how fast they are moving according to an arbitrary choice of coordinate system. Apr 5 revised Are these two quantum systems distinguishable? found mistake Apr 5 revised Are these two quantum systems distinguishable? added 6 characters in body Apr 5 revised Are these two quantum systems distinguishable? typos Apr 5 comment Are these two quantum systems distinguishable? I see you marked an accepted answer. I find this question fascinating and would like to see it discussed further. I really was hoping people would add more analysis. Since no one did, I added what at initial glance (may be wrong) appears to be a counter example to hopefully at least get more discussion going. Apr 5 answered Are these two quantum systems distinguishable? Apr 4 comment Does the foam on top of boiling maple sap affect the rate of evaporation? For a potentially complicated system like this, it seems the best solution would be experiment. Can't you just try it both ways for the sake of curiosity? Apr 3 comment Angular deficit Basically, in the line defect example at least, the angular deficit appears to be a topological quantity and therefore we should be able to define it in a coordinate system independent manner. So if we add other matter to the background of that line defect situation, we can still calculate the angular deficit even though we don't have a nice flat spacetime 'background' anymore. Ultimately it would be nice to have a definition in terms of an integral over an area or some paths, and involving only the metric or curvature. Apr 3 comment Angular deficit @Roy No this isn't specifically about cosmic strings, but they give the only introduction to this that intuitively describes angular defects. But even there, instead of viewing it as a wedge removal, we can scale the coordinates to get back to the usual $0 \leq \theta \leq 2\pi$ and get the metric: $g^{\mu\nu} = dt^2-dz^2-dr^2 - k^2r^2 d\theta^2$ where $k$ is the scaling parameter to map theta back to the usual range. It seems that angular deficit isn't clearly defined unless comparing directly to flat spacetime. If there was any other matter, could we even still define an angular deficit? Apr 3 comment Can GR be derived by postulating a maximum force? @Deepak Thanks for clarifying. Your initial comment seemed to come off as saying you agree with Schiller's claim here, which took me aback. I hope we can both agree the claim is incorrect. I considered writing an answer, but am glad someone else at least posted something. It would be nice to see an answer attempting more analysis of Schiller's actual derivation and conterarguments though. Apr 3 asked Angular deficit Apr 2 revised Why can different batteries with the same voltage send different currents through the same object? removed obvious statement