Halc

Why should it be for our racing submarine? Like a fish, there's no particular requirement to keeps a certain pressure inside. By all means, pressurize it if it helps. Even humans can take that so long as they don't mind the time it takes to depressurize. Similarly, given the power our sub is going to need, the buoyancy of the thing (be it positive or negative) seems a drop in the bucket compared to the sorts of forces we plan for it. Likewise, an airplane going fast need not worry about the fact that it has greater density than the air it displaces. Propulsion on the other hand is a serious issue, as it was for the early supersonic aircraft. I think fish tails would work better than a propeller, and in the worst case, we can always fall back to our rocket. Nobody said we had to sustain the speed for a long time. The shock wave is also a serious issue as you point out. The thing will be shaped like a needle, splitting the water 'gently' to the side rather than compressing it in a shock wave. Minimum sonic boom. This reduces the problem to one of the increased friction resulting from the greater surface area presented. Imagine an amoeba, injecting its skin into new territory, and then moving all its interior guts into that new expanded volume, closing it in behind. Just do that a lot faster. They have creatures that word this way. No propulsion since the skin is effectively stationary relative to the water and need not even be particularly slippery. All the motion (movement of its center of gravity) takes place internally.

The original statement has multiple interpretations, left ambiguous by your translation. This being a topic concerning translation of English to logic, I approve the translation. The two interpretations seem to generate mutually exclusive statements. Each has implications seemingly opposite of each other. The interpretations break into logical statements involving two of three variables: S(sunny), G, and E for weddings in garden and elsewhere respectively. Note that neither interpretation is a function of all three variables. Both G and E can be false if no weddings are held that day, and can both be true if weddings are held in both places. Interpretation 1: On sunny days, all weddings are held in the garden. S => ~E This implies that if there is an indoor wedding, it is not a sunny day. It does not imply that if it is a sunny day, there is a wedding. Interpretation 2: On sunny days, there are weddings held in the garden. S => G This implies that if there are no weddings today, it is not a sunny day. It does not imply that if there is an indoor wedding, it is not a sunny day.

Are the requirements such that the object moving has to A) maintain its speed, B) be self propelled and air-propelled, and C) survive the event. If A) is a requirement, the shuttle coming in doesn't count since it is slowing down, as is a meteor coming in at mach 100. If B) is a requirement (probably the most interesting question), then it becomes a question of how much thrust you can generate in a medium using that medium for reaction mass. This is reminiscent of my favorite Bonneville event: Fastest wheel-powered vehicle, far more challenging than simply strapping a cockpit and skateboard to a missile. If C) is a requirement, the meteor is out. This seems a reasonable requirement since anybody can just go out in space, come back and skim the atmosphere at 0.5c. Earth might even survive a graze like that. Related question I've posed to the dinner table: How fast must a ping-pong ball come in from space to destroy a submarine at 100m depth? (We have interesting dinner table conversations)

Sure it is. Bob (owner of Bob's solar farms) was known to sink all his profits into expansion of the business, and thus Bob's solar farms increased its energy production capacity at an average rate of 40 megawatts per year over the last decade.

Gravity for a fixed mass object scales with the inverse square law (shown first by Newton, late 1600's), not the square-cube law, which instead relates area to volume, and also orbital periods (Kepler, around 1600). We're not talking about a fixed mass object here, so yes, the square-cube law is relevant, but only when Newton's work is applied. A planet of half the radius would have 4x/8x the gravity (8th the mass, half the radius squared). Ghideon gives a good answer to the OP. A basketball sized Earth would exert as much gravity as a basketball sized rock, which is negligible in comparison to what we're used to.

Not so. I'm functionally a cyclops, and the 3D glasses they give you in the cinema do nothing for me, but I get my 3D vision from motion (moving myself) rather than from binocular vision. It doesn't work as well, but enough that I don't need to resort to groping to perceive depth.

It is not a real paradox since the Theory (SR in this case) predicts nothing different than what happens in reality. It is only when the situation is assessed by a naive person who does not know the theory well that inconsistencies might first appear. If I am figuring out something and I get an inconsistency like that, my instinct is like that of a scientist: I assume I made a mistake somewhere, and do not assume the theory is wrong. The wiki article expresses this: "Therefore, at first sight, it might appear that the thread will not break during acceleration." This already implies somebody who isn't very familiar with SR, since anybody who knows it well knows that a rigid object (one that retains its proper length at all times) undergoes different magnitudes of proper acceleration along its length. The two spaceships undergo identical proper acceleration, therefore the assembly of the two connected ships does not constitute a rigid object and will not retain its proper length. So it is no surprise that the string breaks. It is similar to the twins 'paradox', which seems to always want to use twins because it pushes an intuitive button in humans that twins are always the same age. But this naively makes the assumption that the twins have similar acceleration histories. Once this 'rule' is found to not be a valid rule, the paradox resting on the rule vanishes. A true paradox will falsify a theory. If a theory resting on a set of premises predicts that X is true and that X is not true, that is a true paradox, and it is a general indication that at least one of the premises leading to the paradox is false. In the case of Bell's spaceships, all observers agree that the string should break given what they're doing, so there's no paradox.

The distance between the ships does contract in the S frame, but since the proper distance between the ships increases, it effectively stays the same in the S frame. So say the ships (separated by 1 light hour) accelerate identically (same proper acceleration, commencing simultaneously in S) to 0.6c. In the new inertial frame (T) where they both eventually come to rest, the ships are now 1.25 light hours apart (proper distance), which is length contracted to 1 light hours in frame S. The 1 light hour string is length contracted to 0.8 LH in S, which isn't enough to connect them. It is fully 1 LH in T, which also isn't enough to connect them.

The distance between the ships in frame S remains the same, but the length of the rope contracts in that frame and doesn't span the distance anymore. So it breaks. There is no paradox in this. One rigid ship of significant length does not accelerate at an equal rate along its length, so the pair of ships tied by a string does not behave as a rigid object would. The ships in this example are assumed to be essentially small objects of negligible length. Only the string has non-negligible length. I don't know what you mean by seeing a person running. In the frame of either ship during the acceleration phase, the rear one is going slower and the lead ship is going faster. So again, the string must break. There is no disagreement between the frames, and thus no paradox at all. Your spacetime diagram shows all of this.

This is what I am getting at, and what nobody is answering. I am after the invariant spacetime interval between a pair of events separated in a time-like manner, under GR rules. In flat spacetime, the interval between two events is not a function of any worldline connecting the two events. Nevertheless, the one inertial worldline connecting the two events in question happens to have a proper temporal length equal to that interval (or to the square root of it). It is the worldline that maximizes this proper time between the events. Any other worldline will give a smaller time. This is by definition, and all observer will agree that Fred's clock must measure X time between this event and that. I don't care about this. I'm asking if any one of their time measurements happens to equal the spacetime interval between those two events. The measured times (worldline lengths) are obviously different, and the interval is not different for each observer, so at most only one of them can directly measure it. It's either X, Y, Z, some 4th worldline, or none of them. Z seems the only viable candidate (the one on a ballistic trajectory straight up, falling back just in time for the 2nd event. Z measures the maximal value, just like the one 'correct' observer in the flat scenario. I cannot think of a worldline that can log more time in my gravity scenario. How is a bunch of observers agreeing on the lengths of each other's worldlines in any way relevant to the spacetime interval being invariant? Said interval being invariant means that it isn't dependent on a choice of worldline connecting the two events. Again, this isn't my question. My question is this: If the interval between the two events in my example is 1 (we choose our units so the interval is 1), does any of the three clocks read 1? I can tell you that Z will always trace the longest worldline, using the equivalence principle logic in my prior post. I'm not asking which traces the longest worldline. I'm asking if any of the three is taking a direct measurement of the one worldline-independent interval between those two events. I don't know how to do the tensor computation, else I'd not be asking.

All three observers are present at both events, and they all measure a different time on their clocks, so it can't be as simple as just reading the clock. You picture an orbit too high. One meter altitude is plenty to orbit an ideal sphere. The tower is just there to keep Y from getting belly burns and to give Z some acceleration space. Let's just say the blinks are far enough apart that the Z observer is going to measure the most time. If 90 minutes isn't enough for that, we can make it a month. X just sits at the light, experiencing proper acceleration. Remember, the planet is not spinning and has no air. Y orbits multiple times, but is inertial, and follows a geodesic at constant potential, so I assume he's dilated due to velocity, per H-K. Z goes up, fast at first, but quickly slowing to a near halt for most of the month at an altitude of considerable higher potential, so the +dilation due to potential will be greater than the -dilation due to the motion at either end of the trajectory. Z does his acceleration before the first blink, so the entire duration between the two events is inertial and follows a geodesic, albeit a different one than Y's worldline. This seems a nit-picky objection. Work with me here. I'm not after exact figures. I'm just pointing out that these three observers are going to measure different times between the events, so they can't all have measured the frame-invariant interval between the two events. If one of them is by definition correct, then which? Your equation doesn't immediately shed light on that. You hint that it is X, the one experiencing proper acceleration. This seems to contradict the equivalence principle which I think would have chosen Z. I can have observer X and Z in flat SR space, but in equivalent accelerating Rindler space. In that scenario, the two blinks take place at the same location in some inertial frame for Z and in the Rindler frame for X. In the inertial frame, X goes out and back, and Z is stationary at that location the entire time. In the Rindler frame, X is stationary and Z goes up and down, changing potential along the way, kind of like with Earth above, but different velocity relative to X. I cannot work out a path equivalent to Y in that scenario.

It cannot be done 'all at once' since different parts of the ruler need different levels of proper acceleration. I did a topic once positing the minimum time it would take a 100 light year Born-rigid object to move one light hour in frame X, beginning and ending at rest in that frame. We had to carefully define independent acceleration all along its length to keep it always stationary in its own frame. It can be done, but it is necessarily not all moving at the same speed in any other frame, which doesn't contradict born rigidity. The only rule was that at no point could there be either stress or strain. Keeping it stationary in its own frame wasn't a requirement, but we couldn't find a quicker solution. It takes over 55 days to move the object as measured in any frame since the speed never got very high. The math is pretty trivial until you investigate alternate methods, but none of them yielded a shorter duration.

Of course. I don't know my tensor calculus, but I know that tensors are frame invariant, and any scalar quantity like that must thus also be frame invariant, so that answers that. Thanks. Some specific examples then: OK, it makes more sense for the guy present at both events to have an easier job of determining the interval than the remote guy. Is the interval then just the time on his clock, or is it more complicated than that? I ask because I can have three observers X,Y,Z. The light blinks every ~90 minutes. X is our guy located at the top of a tower on a spherical non-rotating Earth (effectively Schwarzschild spacetime). Y is in low orbit and comes around exactly when it blinks. Z is lauched straight up a somewhat under escape velocity and comes back exactly in time for the second blink. All three observers are present at both events. Y I think will measure the shortest time and Z the greatest. Those two are inertial in that they follow geodesic worldlines. X does not and is under 1g of proper acceleration between the events. My guess is that all three are going to have to do some complicated mathematics to compute the invariant interval between those two events. None of them has the luxury of simply looking at his watch. I also admit not being able to explain concepts like time-killing vector fields. Heard of them, but I'm just a novice at this.

I can think of few frame-invariant things. In fact, the interval seems to be a rare such thing. Yes, anybody can compute what is being measured relative to a different frame. We're all doing it in this forum all the time. That doesn't make the thing being measured frame invariant unless there is somehow a way to determine the objective frame that all can determine independently instead of an arbitrary choice that all decide to use. There is no such frame. In particular, there is no objective measure of gravitation potential. It is always relative. Ideally one would assign 0 gravitational potential to a place infinitely distant from all matter, but of course there is nowhere you can go to get away from it all. I've never seen the gravitational potential of Earth expressed in any absolute way. If you could do that, you could compute the objective dilation of a clock here on Earth and then get a figure for the actual rate of time passage in the universe. It would probably change over time. Again, I've never seen any valid attempt at this, just religious presumptions that we're the standard and the universe is all about us, just like we're also by chance the only stationary thing in the universe, and thus the center of it.

The interval cannot be invariant between observers at different gravitational potentials. Consider a light blinking at the top of a radio tower (on a non-spinning planet if we want to be more precise). We have an observer at the top and one on the ground, each measuring the interval between the blinks. They're both stationary relative to the tower, so the interval is trivially just the time between the blinks, but the two observers will measure different elapsed times on their local clocks because they're at different potentials and don't run in sync. So the spacetime interval being invariant is a property only of flat Minkowskian spacetime.

Dandelions and Homo are genetically similar enough to be connected, but that doesn't mean that either is descended from the other. It just means we share a common ancestor at some point. The more recent that common ancestor, the closer we are related to a thing.

Gifts of silver and gold will vir- ginity tend to put an end to. The tortithe and rabbit rathed theee timeth thith month. The rabbit won twithe and the torithe onthe. Started choking, turning purple, A hardy slap and one good burp'll put you to rights. Quoted from memory. Credit: James Hogan. There was one with orange as well, but it was really reaching, and I don't remember it.

Accelerating an electron to nearly light speed is regularly done in particle accelerators, which then go on to add momentum to said electron perhaps another thousand-fold, and the electron still doesn't go faster than c. I think that is a practical example of verification that's been done. I cannot parse what you have in mind with the M and the little m's. A rock is composed of a bunch of molecules, which is a set of smaller masses making up a larger mass. That fact seems totally irrelevant to the limit the separation rate of a pair of rocks. GR limits separation speed locally to c, but non-locally, this is not the case. Hence there are distant galaxies increasing their proper distance from us at a rate greater than c. No object can have a peculiar velocity greater than c. Peculiar velocity is local in a sense: it is its velocity relative to the center of mass of all matter within some radius (typically the size of the visible universe, or possibly the Hubble radius) centered at the location of the object.

I said no such thing. Either your reading comprehension skills are completely absent or your are deliberately twisting everybody's replies. I suspect the latter. I said that since the 'traveler' considers himself stationary in his own frame (by definition no less), clock 2 rushes towards him from 36 light minutes away, and this takes 45 minute, which is less than SoL. That makes no statement about what he sees while this is going on. 10 years away as measured in a different frame than your own, yes. If I have a fast enough ship, I can get to the far edge of the galaxy in my lifetime, despite it being 70,000 LY away in the frame of the galaxy. That speed is called proper velocity (dx/dt') and there is no limit of c to it. Normal velocity (dx/dt, or distance and time measured in the same frame) is bounded locally by light speed.

That error has already been pointed out. The example shows clock 3 traveling the light hour in 5/4 hours. That's what it means to be moving at 0.8c. Due to time dilation, clock 3 just happens to have logged only 3/4 hours during that trip. Relative to the clock's own frame, only 45 undilated minutes pass, but it is stationary (by definition), which is hardly traveling faster than SoL. In that frame, clock 2 comes to it at 0.8c, and moves 36 light-minutes during those 45 minutes, which is also less than SoL. 2: As seen some observer A, the incoming clock is seen to go from a light hour away to 'here' in 15 minutes, yes. That's the difference between its actual speed in that frame and the speed of the light the observer is observing.

Yes, it takes 1:15 in the frame of 1st/2nd clock for the 3rd clock to get one LH away. In the frame of that 3rd clock, it is stationary, so it isn't traveling anywhere in the 45 minutes it logs, so it isn't moving faster than light.

Because one and a quarter hours is an hour and 15 minutes, not an hour and 25 minutes.

Not going to watch a 6 minute video to look for where he says that. As swansont points out, there is no such thing as 'the gravity'. It needs to be more specific. The gravitational potential above the surface of some mass is -GM/r, which is proportional to M/r, so perhaps by 'the gravity' he means gravitational potential. It can be expressed in the same units as velocity.

Janus is describing the ballistic theory of light, which is actually a much better fit to Galilean relativity which asserts no preferred frame. One cannot 'continue to see B as "frozen"' '. Light from that moment can at best be measured once, not continuously. Oddly enough, if the missile moves at 2c, with either theory, if they look at a 'stationary' Earth from the missile, Earth will appear to run backwards. At 11:00 missile time, it will see Earth at 9:00 given a 10:00 departure. The 9:00 light from Earth gets to the same location as where the missile is at that time. Funny thing is, the image of Earth will appear in front of the missile, not behind it like intuition suggests. Sound works the same way. The post of mine discussed both, but the part to which you responded involved more of a Newtonian preferred-frame ether theory which denies Galilean relativity. Don't get the question. The only measurements you describe are both of them in each other's presence at the time of departure. They both measure 10:00 on both clocks at that time. Not sure what it would mean for those identical observations to 'cancel'.

It depends on what theory Galileo uses. Using the ballistic theory of light, the missile is immediately invisible at A because it is going faster than light, and thus light emitted by it cannot propagate towards A. It can only fall behind. Using a pre-relativity ether theory (Newtonian), and assuming A is stationary in that ether, the missile clock reads 11:00 when it passes C and it takes 2 hours for that light to go back and reach A, so A sees the missile clock ticking at 1/3 rate. In both cases, the missile can see nothing behind it because it is superluminal, similar to how a supersonic aircraft cannot hear any noise from behind. Both theories predict different results than what is actually observed, but the actual theory has no different answer for the problem since it forbids the missile from having a space-like worldline. As you say, relativity forbids that you go there, but the other two theories do not.