md65536

Another thing is that its gravitational influence is similar to ordinary matter of the same mass, at a distance, so a black hole of one solar mass wandering into the solar system shouldn't be much more likely or a danger of eating us, as a star of one solar mass doing the same. Similarly a black hole created in the lab would not have enough mass to devour the earth on its own. It would only devour extremely nearby matter (probably closer than atoms are to each other*, depending on its mass of course). A BH made from Earth's matter wouldn't collapse the Earth any easier than the Earth's own mass would collapse on itself, so it would have to eat away at nearby matter until it destroyed the structural integrity of the planet before it would collapse. That said, Ton 618 has "The largest black hole ever found in the known universe" at 66 billion solar masses. "A black hole of this mass has a Schwarzschild radius of 1,300 AU (about 390 billion km in diameter) which is more than 40 times the distance from Neptune to the Sun." But it's about 18.2 billion light years away, so like you say, it's distance that keeps us safe. * Edit: I vastly overestimated that distance. Atom spacing is on the order of Angstroms (10^-10 m). A black hole with a mass of a million tonnes would have a Schwarzschild radius of about 10^-18 m, a thousandth the size of a proton! Does that mean such a BH might not even collapse a proton even if it passed right through it?

I found some ranges where you can create pieces of any area in the range, but none of them overlap at any of the values of A through E. Was that on purpose? Because, I found some other possible areas of pieces, including one that has 2 different answers. So I'll extend the puzzle with these... Cut a cube of side 2 cms into eight identical pieces such that the surface area of each piece is : F: 7 sq cms G: 6 + ✅2 sq cms H: 5 + ✅5 sq cms Then bonus: For which pairs of A through H is it possible to create 8 identical pieces with a surface area anywhere in between the two? Based on that answer, how would you cut one cube into 8 identical pieces of surface area a, and cut another cube into 8 identical pieces also of surface a but with a different shape than those from the first cube? I'm not certain that I haven't made any errors here!

You can change the area continuously for some of these cuts just by rotating the cube, so I still suspect there might be some with multiple answers.

Yes, I agree. Sorry for the confusion, I've ended up discussing what would happen in both the instantaneous and gradual acceleration cases, and the whole train stopping simultaneously only applies to the gradual case. In the instantaneous case, yes the train stretches again, because the parts of the train don't stop simultaneously in S', and the "car 0" end (the back end in S) continues moving for a short time while the other end has stopped, in the S' frame. Well it seems to work, but it's confusing. In this case of gradual acceleration, in the vfinal frame, what is observed is that we start with the train having constant velocity -v and is length contracted. Then to start the acceleration phase, the n=N end of the train begins to decelerate first, and the train begins to stretch very gradually at first. Other cars decelerate in turn until the n=0 car begins decelerating last. The n=0 car has the greatest proper acceleration (which all frames must agree on), but since it started last, it maintains a greater speed (with negative velocity) than the rest of the train in the vfinal frame during the entire acceleration phase. Then finally all cars arrive at the same velocity of 0 simultaneously, the train fully stretched to its full proper length. This does sound exactly like a time reversal of what is seen in the track frame, with everyone agreeing that the n=0 end has the higher proper acceleration. But everything I'm describing has a locally constant proper acceleration during the acceleration phase, and satisfies Born rigidity as far as I understand it. Are we talking about the same thing? Another possible source of confusion is that constant proper acceleration isn't constant coordinate acceleration in any single frame, and I may have mistook one for the other in what you wrote.

I think I figured out the basic idea in the track frame, for Born rigidity or anything else that approximately maintains the proper length of the train with non-instantaneous acceleration. In the track frame, the back of the train always accelerates at a higher rate than the front, so that as velocity increases, the train contracts. Then when acceleration stops, the back of the train stops accelerating first, when it reaches velocity v. The rest of the train is moving slower and continues to accelerate (and contract), with the front of the train being the last to stop accelerating, at which point the whole train is moving at v and is length-contracted by gamma. This is consistent with allowing the whole train to stop accelerating simultaneously in a moving frame (v's frame, if I got it right).

The result I get is that as soon as a car accelerates to its new speed, it is in a different inertial frame in which the car in front has accelerated first (or rather decelerated to rest in this frame), and is already out of the way. I argued before that I thought it wouldn't feel any stress due to causality, but now I'm sure that's wrong. Still, I don't see a result from SR that is theoretically impossible for every type of "train". I mean that there's a solution where any individual part (car) of the train has a constant proper acceleration over time, not that all parts have the same proper acceleration. But yes, if the train is meant to reach a specific velocity relative to the tracks, and then stop accelerating, there should be a solution that uses a set of constant proper accelerations, beginning simultaneously in the momentary rest frame of the train, and stopping simultaneously in another momentary rest frame of the train. Such a rest frame should exist because Born rigidity maintains the proper length of the train. Or another way to look at it, the train stopping at its final velocity should be simultaneous in that frame, because the process should be time-reversible. Ie. when the back car is "already up to speed", all of the cars are. The cars had different proper accelerations for different proper times, because they started simultaneously in one frame, and ended simultaneously in another. However... the more I think about it the less I'm sure I have that figured out correctly, because what I just described doesn't seem to make sense from the tracks frame. Maybe an acceleration phase and a separate deceleration phase are both needed, not just a stop to the acceleration. I'll have to think about that more because 2 different answers make sense in 2 different frames, so for sure I'm wrong somewhere. Yes, the same solution works. The point of Born rigidity is that all of the points or parts of the object are accelerated individually, it never specifies "only two engines" unless the train consists of only those 2 points (such as 2 rockets with an imaginary string between them). Besides, if your solution works for cars of length L_0, then it would work for a train with one engine and a length of L_0. Either your solution works on the train as a whole with certain restrictions, or it fails for cars when there are no restrictions (unless you make n infinitely high and L_0 infinitesimal, but then you're just describing a solid rod where every part of it is accelerated locally). Interesting that the second form is what you posted in the very first post, and the first form I posted later, and they are the same but I didn't realize it. We were talking about the same thing all along.

I don't understand, is this an interpretation of OP's stated problem, or are you rejecting instantaneous acceleration and substituting your own problem? If the latter, why not just apply Born rigidity? Is that what you're trying to do? If so, why approximate, and why not allow constant proper acceleration at each part or car of the train? Are you treating the train differently because it's made up of cars, and if so why would a solution for Born rigidity that works on the train as a whole, not work for the individual cars?

Did you mention you already figured this was wrong? I figure that the correct velocity should be greater than c for all v<c, otherwise the accelerations won't be simultaneous in any frame. As well, when v approaches 0, this velocity should approach infinity, because for vanishing v, the frame in which the accelerations are simultaneous approaches the track's frame.

You're not still using "rapidity" as a synonym of acceleration, are you?

Just to expand the realm of possibility, the rail gun wouldn't have to be in the ground/tracks frame. Both what I described and what OP did, here, describe a set of events that are simultaneous in some frame, when gamma is high like 2 in the example. It's as easy to coordinate the events as it is to synchronize a (maybe large) set of clocks in that frame. Then, the rail gun would only need to be as long as the length-contracted train in that frame (by a factor of less than 2 in this example, because the train's speed is less than v in that frame). When figuring out the rate at which the acceleration events happen along the length of the train, I saw that the proper length of the train gets simplified out of the equation. That means that the rate is the same whether the train is short or long. Whether "long" means 1 m or a billion light years, the same answer applies (with the railgun length proportional to train length, as is the timing in the ground frame). (Then yes, it would be impossible to accelerate a full-size train fast enough, so just make the train out of electrons or something lighter, etc.)

Nothing physical is moving faster than c, it's just the timing of the local accelerations across the length of the train moving faster than c. It's only the phase velocity of this wave that's traveling faster than c. That's fine because the propagation of the wave isn't causal; the parts of the train are accelerating independently of each other. I can avoid calling it a wave to be less confusing. Yes, I've assumed "local engines" all along, that was implied by OP. I'm just saying the engines don't have to be part of the train itself. I'm trying to talk about the effects of SR without getting hung up on the difficulties of building a physical version of a thought experiment. I don't want to debate whether it's possible for 2 events to be simultaneous ie. not causally connected.

As mentioned, this isn't a rigid body and that's fine because it's not being accelerated as a rigid body, rather all the parts of the train are accelerated independently, as if it is modelled as a soup of particles. I also think it's not describing a practical impossibility, where every smallest part of the train needs to be a train engine, because the cause of the acceleration can be external. For example the train could be a metal object accelerated by a rail gun. I think it's possible that the train might not feel any stress from being stretched or compressed, it might only feel the local proper acceleration, temporal distortion, etc. This is because the acceleration "wave" crosses the train at a rate faster than c. For any particle on the train, even though the one "behind" it has started moving first, the particle has accelerated before the one behind has caught up and before any causal effect of the particle behind it has reached it. And even though the particle is moving before the one in front moves, by the time the particle gets to the location of the one in front, the one in front has already accelerated. This is true no matter how small you make the separation of the particles. (I guess this assumes that any field effects that can be felt, like EM field, are also accelerated along with the particles. Is this impossible?) On second thought... after a particle has accelerated it is now in the new inertial frame, in which the particle in front has already accelerated first! It has accelerated to rest. This gives an observer in the middle of the train (and who accelerates along with it) a bizarre account of what happens! According to her, the rear of the train begins to compress forward (not fast enough to see or feel it) while the front of the train remains stationary. Then she feels proper acceleration, then the rear of the train begins to stretch backward (but not faster than can be felt??? It seems this compression would have to be visible) while the front of the train again remains stationary! That seems surprising enough that I wonder what details I'm missing or getting wrong. Edit: Due to delay of light I think she'd see the front of the train appear to be approaching and stretched out (due to aberration of light). These are images of the front of the train before those parts accelerated, seen in the post-acceleration frame. But if it can be seen it can be felt?

https://en.wikipedia.org/wiki/Born_rigidity The "Class A" section contains an entire class of applicable motions. Instantaneous (not generally simultaneous) acceleration using the solution I proposed (different from OP's I think), does not satisfy Born rigidity, because it requires different parts of the train to accelerate at different times, in the 2 rest frames (before and after acceleration) of the train. Therefore the train's length changes in those frames, and its proper length doesn't even seem defined while accelerating. However, it should be the only solution that involves a single instantaneous acceleration at each point of the train, with which the train has the same proper length before and after the acceleration. It would seem not even Born rigidity can be satisfied with an instantaneous acceleration. However, OP's proposal never mentioned anything about a requirement of rigidity. For that matter, they neither required that the proper length of the train is the same before and after, I assumed that. The way I figure it, if you have the back of the train accelerate at time 0, the rest of the train accelerates over time until the front of the train finally accelerates when the length of the train has become L/gamma in the initial frame, where L is its original proper length. By that time, the back of the train has traveled a distance of (L - L/gamma) at velocity v. Therefore the time when the front of the train accelerates would be t=d/v = (L - L/gamma)/v. Then the velocity of the "wave" would have to be d/t = Lv/(L - L/gamma) = v/(1 - 1/gamma) For gamma=2, it seems intuitive that the wave would have to travel at 2v, to reach the front of the train at the same time that the back of the train traveling at v gets halfway there. With v = 3^0.5 c/2, I get a velocity of the wave equal to 3^0.5 c. Like you said this is faster than c, so it's non-causal and needs local forces to accelerate the parts of the train.

This doesn't refute the validity of Born rigidity especially in terms of kinematics. If you understood Born rigidity you'd know this thread is concerned with the "irrotational motions" class of Born rigidity, which is broken if the train turns. So bringing it up is a strawman argument. It sounds like you're trying to argue that Born rigidity can't be satisfied, by bringing up irrelevant counter-examples where it is not satisfied.

I've thought about it more and I still think my first reply is correct. The issue is, if there's an instant change in velocity, does length contraction apply to all of the velocities in between as if it accelerates through them all in an instant, and I say it doesn't. SR doesn't predict the effects of acceleration, rather it is an assumption that acceleration doesn't have an effect, only velocity itself does (the "clock postulate"). SR neither says that the train would survive the acceleration, nor that it wouldn't. It says the train would be contracted at the beginning velocity, and also at the end, but not what strains would happen to the train in the zero length of time in between. So, whether the train can accelerate instantly or not is an assumption we have to make, not one we can derive from SR itself. If the train parts can instantly accelerate from 0 to v, then I think my answer works, and if not then it doesn't, and SR doesn't change that. But yes, if there's any frame in which a train is moving and then all parts of it simultaneously stop, SR says its new proper length is smaller than it was, so it physically must get squished. Going from -u to +u instantaneously doesn't have a moment when it is at rest, so SR does not by itself say it will behave as if at rest in that case.

I agree this is irrelevant nonsense and you're incorrectly applying definitions that have no bearing on the issue.

It's not a problem if we consider only the the kinematics of the system, and treat the train as a set of particles, and consider only as many particles as needed. Usually one at the front of the train and one at the back is enough for most things. The solution to Bell's spaceship paradox is the same if you consider one large metal ship, or 2 small ships with an imaginary string between them. Einstein used trains in thought experiments without problems. Considering only kinematics, the forces on the train and communication between the parts is irrelevant. Things like lengths and speeds are what's important. I think this is wrong but I'm not certain. If it squishes (like, physically crumples) in that one frame, that must happen in all frames. I don't see how that's possible in the moving train's frame, where the train is simply stretching out from length-contracted to rest length. Are you saying that what I described will crush the train with high v, or that I made some other mistake? Also, if the train "loses contraction" only for an instant, the effects of the "squishing" would be causal, and there's no time for the effects to propagate. Also, the "squishing" seems to imply that you can't accelerate an object without either stretching or compression strain on the object (you can only minimize it with gradual acceleration), do you agree? However, Born rigid acceleration is possible without spacial strain. I'll have to think more about this.

Right, the parts of the train aren't accelerated simultaneously in the ground frame, only the in-between frame. The parts are each accelerated instantly but not simultaneously, in other frames. In the ground frame, the train starts at its proper length and ends up length-contracted; the rear of the train must accelerate first. In the post-acceleration train's frame, the train starts length-contracted and ends at rest at its proper length; the front of the train must accelerate first. This isn't a question of practicality, it's mathematical. If an object can be made to accelerate all at once from -u to +u, it doesn't get pulled apart (in any frame) or physically squished, just length-contracted. If you want to debate whether a real train can do this, you should probably find out what v is before deciding if it can handle it, because in practical cases v is small enough that length-contraction is negligible.

You can do it if all the parts of the train can change speed instantly. Say you have a train of proper length L, at rest on some tracks, and instantly accelerate each part of it to velocity v so that it has a proper length of L in its new frame. There's a frame of reference in between those two, where the tracks + rest train are moving in one direction at some velocity -u, and after the train accelerates, it's moving in the opposite direction but same speed +u, so that the length contraction factor before and after are the same. This is the frame in which all parts of the train would accelerate simultaneously. You can find u using the composition of velocity formula, so that v is the composition of u and u. Is that what you're talking about, and does this match your result? I suppose that if you did this repeatedly using small accelerations, in limit form you'd get Born rigid motion?

I'm trying to make sense of this and can't. I think you have it backwards. s^2 is the spacetime interval between 2 events. I don't think that equation can directly represent length-contraction of an object, because the length of an object in a reference frame is the spatial distance between 2 simultaneous (in that frame) events. The spacetime interval is the measure between the same 2 events in different frames, while length contraction compares 2 different sets of events in different frames. I don't think you can put 2 lengths of an object (ie. proper and contracted) into the equation and get the same s^2, or in other words I don't see how you can express "longer t and shorter x" with that equation. The interval being invariant implies that the space and time coordinates actually change in the same way, not opposite. That means that in one frame, if 2 events are a certain time apart and distance apart, in another frame where they're a longer time apart, they're also a greater distance apart. I always confused that idea with the way that length contraction seems to say the opposite of that, so I'll just go through an example to show how relativity of simultaneity clears up the confusion. Say you have a train passing through a tunnel so that in the train's frame, the train is exactly the length of the contracted tunnel. The events are the front of the train passing the exit of the tunnel, and the rear of the train passing the entrance of the tunnel, simultaneously. The spacetime interval has (delta)t=0 and x=the proper length of the train = contracted length of tunnel, so s^2 is negative, a spacelike interval. In the tunnel frame, the tunnel is longer than it is in the train frame, while the train is length-contracted and fits entirely inside the tunnel. s^2 is the same since it's invariant, with x'=proper length of tunnel > x, but now t' is also larger than t because the train's rear enters the tunnel before the front of the train leaves. If I were to confuse things, I might say "let x' be the contracted length of the train, and it is smaller while t' is larger", but those statements are talking about 2 different spacetime intervals.

There are additional replies better than I can give here: https://astronomy.stackexchange.com/questions/32445/head-on-collision-of-two-black-holes If energy is lost from a system (in its CoM frame), the system loses mass. The binary system is losing mass equivalent to the gravitational wave energy radiated away, before the collision. If an individual component of the system doesn't radiate energy itself, that component don't lose rest mass. Radiating gravitational waves can reduce angular momentum, so the remnant BH should be decreasing its angular momentum during merger and ringdown. After they're merged, the system only has the one object in it. Your question is basically asking about the components of the system before the merger, and the system as a whole after, that's the main difference. Also, I think the energy radiated while merging is vastly greater than that radiated during the inspiral phase.

These are inspiral orbiting black holes. Yes, I see the numbers in the wiki add up to zero kinetic energy included. However if you look at the references eg. [4], the numbers don't add up exactly, and the uncertainties in the estimates of the black hole masses are so huge that it's not possible to give an exact figure for other energy in the system from the masses. The equation describing this would be more like, remnant BH mass Mf = m1+m2+everything_else_ignored - radiated_energy/c^2. It's likely that whatever is ignored is small on a scale of solar masses. These aren't two masses falling directly toward each other, they come together because their orbits decay due to energy lost to gravitational waves. The kinetic energy is being radiated away as they approach. When they merge they'd generally have net angular momentum, resulting in a spinning black hole. The rotational energy of a black hole contributes to its mass (https://en.wikipedia.org/wiki/Kerr_metric#Mass_of_rotational_energy). Certainly then, some kinetic energy of the orbiting black holes contributed to the mass of the resulting black hole. In this case, at about 8 solar masses of energy radiated, much much more than any energy not accounted for is lost to gravitational waves.

What are you basing this on? You're saying it should be one thing but is "really" another, where do you get the latter from? If you apply your equations to quarks forming a proton, it really does include the KE. From https://en.wikipedia.org/wiki/Proton "The mass of a proton is about 80–100 times greater than the sum of the rest masses of its three valence quarks, while the gluons have zero rest mass. [...] The rest mass of a proton is, thus, the invariant mass of the system of moving quarks and gluons that make up the particle, and, in such systems, even the energy of massless particles is still measured as part of the rest mass of the system." In the case of a black hole you can't measure or assert any internal motion, but the externally measurable rest mass is still there and the energy is still conserved.

It's easy to explain the hyperbolic rotation relation in terms of other things. If you assume that the speed of light is constant, and consider a path of light c*tau long in one frame, then consider it in another frame (as commonly done with a light-clock at rest and moving, in thought-experiments used to explain time dilation), chosen to form a right triangle with a sides x and c*tau and hypotenuse c*t (right? am I messing up the details?). Then by the Pythagorean Theorem, the value of x^2 - (ct)^2 is constant for a given tau. That equation happens to describe a hyperbola. You don't need to explain why it's described by a hyperbolic rotation, if you instead explain why the speed of light is constant and why Pythagoras' theorem holds. "Hyperbolic rotation" is just an equivalent mathematical description of the relationship. But likewise, a constant speed of light and Pythagorean theorem could also be just descriptions of what's happening, equivalent to some other things. Maybe there's some simplest explanation of what's "really" happening, or maybe they're all just equivalent descriptions of what we observe. I think we have the same point, that you don't need to explain a given description of a phenomenon to explain the phenomenon itself. Likewise, you don't need things to "really" be rotating for a mathematical equation for rotation to be a valid description.