Thanks again, Mike, for your help in this thread (and the many others)!!
I've tried to understand your previous post, re: providing the event times in Andromedian frames. To simplify (or de-anthropomorphize) the paradox, I drew an illustration for a GRB, where the beginning of the burst can be the 'speeches and invasion plan', and some point midway along the light-curve (say after it peaks in intensity) would correspond to the 'invasion force deployment'.
(click thumbnail for larger image)
The placement of the red and blue detectors (counter-orbiting satellites) is a bit arbitrary. I could have placed them back a bit, in their respective orbits, such that the blue arrow passes through the red detector, on its way to the blue detector. The red detector shouldn't detect the GRB event, from that distance, until the red arrow arrives (after the relativistic phase between red and blue inertial frames elapses). Hard to believe that's correct, certainly doesn't sound possible....
Thanks again, Mike, for your help in this thread (and the many others)!!
A pleasure indeed, and I do hope/yearn/fear correction should I lead you astray. :-)
I've written a long reply today, I'm afraid!!
Quote:
I've tried to understand your previous post, re: providing the event times in Andromedian frames. To simplify (or de-anthropomorphize) the paradox, I drew an illustration for a GRB, where the beginning of the burst can be the 'speeches and invasion plan', and some point midway along the light-curve (say after it peaks in intensity) would correspond to the 'invasion force deployment'.
(click thumbnail for larger image)
The placement of the red and blue detectors (counter-orbiting satellites) is a bit arbitrary. I could have placed them back a bit, in their respective orbits, such that the blue arrow passes through the red detector, on its way to the blue detector. The red detector shouldn't detect the GRB event, from that distance, until the red arrow arrives (after the relativistic phase between red and blue inertial frames elapses). Hard to believe that's correct, certainly doesn't sound possible....
Darn!! But that's because it's not... :-)
A very nicely done diagram! Some points of interest:
- There is no 'stationary' frame of reference. Prior to Einstein most, if not all, believed in some ( perhaps hard to find ) universal rest frame. The Michelson-Morley experiment buried that, and much discomfort that then caused. This is a key point in relativity.
- What you do have are inertial frames and non-inertial frames. An inertial frame is one where the Law of Inertia holds. The Law of Inertia states that a body subject to no forces continues it's current path without deflection. This includes the case where it may be stationary with respect to the frame. If then there are no forces, it'll then stay put.
- This de-references the issue to one of 'what are forces?' or 'how do you know if there is a force about?' Here it can get a bit squirrelly, if done without some care. Now take some torch, shine it in a direction ( let's do this is in a vacuum for the present ). Whatever path it takes we'll call that a 'geodesic', or a 'straight line' in ordinary terms. If your body cannot be seen to continue along any ( suitably aligned ) beam of light then it is deemed to be subject to a force. So now the classification of frames depends upon the comparison of material body motion vs. photon motion.
- An inertial frame is otherwise termed a non-accelerated frame. Special Relativity deals with such inertial frames. Really there are no absolutely exact inertial frames, as we've yet to find a region of space with no influences. But we can still use the concept with exceptional precision anyway.
- Non-inertial frames are where the Law of Inertia fails. So the path of a body within it will always deviate from that of any light beam. So no matter how I try to arrange the beam, the body will never 'stay in the bulleye' or whatever trick I use to determine it's conformance to the beam. ( In my mind's eye I am thinking of some body moving away from me with a target painted on it's back, and a sniper's red laser spot.... ).
- A non-inertial frame is otherwise known as an accelerated frame. This means that bodies within it that seem motionless are, when viewed from an inertial frame, exhibiting acceleration ie. changes in the velocity vector with time - speed and/or direction. I can calculate from such measurements in the inertial frame and come up with a vector quantity which contains the magnitude and direction of that acceleration. A good example would be a pencil floating in 'mid-air' on the International Space Station - apparently static from within the station, but orbiting the Earth ( acceleration, change in velocity vector with time.... ) when seen from afar.
- Suppose I note that some behaviour in a non-inertial frame is proportional to the mass, with all other things being equal. Then I can equally consider that frame to be one where a gravitational field is present. This is the Principle of Equivalence. You can pass from descriptions of things (i) as an accelerated frame to/from (ii) a gravity field with strength equal to the acceleration but oppositely directed. Hence you go up in a lift and feel heavier, go down in the lift and feel lighter. Jump off the roof and you will feel briefly weightless, extend that time a bit by skydiving, and even go to the ISS with your pencil and fall endlessly ie. orbit. Be careful as this is only 'best true' if you don't go too far in time or space. A local rule really.
- Now you can go the whole hog to General Relativity and state that because gravity is universal, it's not really a 'force' but a condition of space & time. Light paths are still geodesics. Masses still move but are not 'accelerated by gravity' at all. They follow a path in space and time that would be labelled as 'warped' from a purely inertial frame perspective. So a light beam ( or Mercury too ) moving nearby our Sun seem to be deflected by a force when observed distantly & inertially. But you wouldn't observe it that way within the relevant local non-inertial frames nearby the Sun - it would be a 'free-fall' experience. Weird huh??
So now I've sunk your boat: as we can't legitimately model your satellites using Special Relativity, because they are in fact orbiting/accelerated.
Now back to our earthly friends awaiting signals from Andromeda. The guy in the car is said to have several days headstart on the information from Andromeda compared to the pedestrian. Then his phase difference - (v*x)/(c*c) - implies that at car type speeds ( relative to the other guy ) he has to travel an absolutely enormous distance to achieve that. This is because his 'v' in the numerator is low, and 'c' squared in the denominator is really, really big. You'd need a humungous 'x' to bring the whole phase up to those several days advantage as quoted.
For example, at 1 m/sec ( a 'reasonable' car speed ), 100000 seconds ( about a day ), speed of light at 3000000000 m/sec then x = 90000000000 metres.
Thus to earn his headstart over the pedestrian in time, he must have started quite a long time ago ( in a galaxy far far away? ), to enable him to arrive at the moment when the origins of their reference frames co-incide. At that point he can toss a note out the window to his pedestrian mate and tell him of the impending doom. If it was Nelson from The Simpsons we'd get a 'haha'. :-)
Of course the origins won't actually co-incide and other imprecisions abound but the gist is that both observers will, at about the same point in spacetime, report quite differently about light received from Andromeda. This is highly anti-intuitive and hence the 'paradox' label. I can't see any internal inconsistency though, provided one can accept that Relativity demands the death of a Newtonian universal time. Such is the price ....
Even if your GRB detectors were well out from the Solar System at the moment they passed each other - how are they going to 'run up' to that moment? Remember the big 'x'. If you pull one or the other up, and send them around for another go at it, then you've performed accelerations. That alas will wipe the slate and the time advantage is lost.
My apologies for the length.
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Boat? What boat? You mean that froth from my fast and furious treading in the science-waters? I've been in over my head for so long, I'm happy to just break the surface :)) I figured somewhere in the depths is the answer for how to walk on the stuff, but lacking it in the meantime, a boat makes a lot more sense! :)))
I'm afraid I still don't see the distinction between when to use GR instead of SR, or how that affects calculations on the measurements from one inertial frame to another. I understand that the satellites are in free-fall around the warped space near a massive object. The question is how events from a distance appear to the satellites. I see many frames of reference: the individual inertial frames of each satellite, the inertial frame of the planet with the satellites going round it, the inertial frame of the solar system with the planet itself in free-fall around a larger massive object... topping it off, why would SR be required for the Global Positioning System, but not to ascertain and evaluate the simultaneity of events at a (much farther) distance, as they appear in one inertial frame compared to another?
It's a bit of a stretch to consider that an observation of a distant object means that you're seeing it as it was in a different time, and not as it currently is. When you look at a distant object, you're looking back into the past. Doesn't makes sense to talk about distance without considering time. And if you're in motion, and moving in a direction different from another observer, then only with synchronized clocks, GR, and SR will your measurements agree.
It's a bit more of a stretch that simultaneity is therefore also relative, and that the rods and clocks in one inertial frame will have different lengths and times when measured from a different inertial frame. Observers in both frames are still looking into the past when observing distant objects, but they're observing failure of simultaneity at a distance. Observations are in agreement, however, when measurements are adjusted by taking into account the relativistic phase, that is, the difference in the amount of distance/time experienced by observers with different sets of rods and clocks (each set contracted and dilated as they are along the path of their respective trajectories).
Maybe this is where I'm having trouble applying the paradox. I'm making it out to be a case of the tail wagging the dog. The time and distance of the entire universe isn't going to instantaneously wax and wane just because the inertial frame I'm in is going round and round. But my question then becomes: isn't that how SR says things will appear, even though it isn't how things currently are?
Don't worry about me drowning; I've found I can still breathe the stuff even if I don't fully comprehend it. :)
Boat? What boat? You mean that froth from my fast and furious treading in the science-waters? I've been in over my head for so long, I'm happy to just break the surface :)) I figured somewhere in the depths is the answer for how to walk on the stuff, but lacking it in the meantime, a boat makes a lot more sense! :)))
Well done, and it is terrific when it crystallises though.... so hang in for the 'Eureka' moments in life. :-)
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I'm afraid I still don't see the distinction between when to use GR instead of SR, or how that affects calculations on the measurements from one inertial frame to another. I understand that the satellites are in free-fall around the warped space near a massive object. The question is how events from a distance appear to the satellites. I see many frames of reference: the individual inertial frames of each satellite, the inertial frame of the planet with the satellites going round it, the inertial frame of the solar system with the planet itself in free-fall around a larger massive object .... topping it off, why would SR be required for the Global Positioning System, but not to ascertain and evaluate the simultaneity of events at a (much farther) distance, as they appear in one inertial frame compared to another?
Fair points. They are calculational models, so it's a question of the level of approximation of description that's desired. That depends then on the purpose - real experiment vs. gedanken etc... I recall ( late 70's? ) a trip was taken ( on the Concorde? ) across the Atlantic with a pre-synchronised 'portable' atomic clock. On landing it differed from it's 'stationary' twin back in the UK by an amount consistent with SR ie. so only the velocity of the plane was used. Strictly speaking though they were all accelerating, in a gravity field etc. Nowadays GPS accounts very well using a GR model, and locational accuracy would drift by kilometers per day if it were not used. This is mainly due to 'time warping' rather than 'space warping' effects. The accelerations used, and gravity field strength, hereabouts are all pretty mild on the cosmic scale of things. Which is good for our health!! We really do orbit about a quiet unregarded yellow sun ( as Douglas Adams would have put it ).
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It's a bit of a stretch to consider that an observation of a distant object means that you're seeing it as it was in a different time, and not as it currently is. When you look at a distant object, you're looking back into the past. Doesn't makes sense to talk about distance without considering time. And if you're in motion, and moving in a direction different from another observer, then only with synchronized clocks, GR, and SR will your measurements agree.
It's a bit more of a stretch that simultaneity is therefore also relative, and that the rods and clocks in one inertial frame will have different lengths and times when measured from a different inertial frame. Observers in both frames are still looking into the past when observing distant objects, but they're observing failure of simultaneity at a distance. Observations are in agreement, however, when measurements are adjusted by taking into account the relativistic phase, that is, the difference in the amount of distance/time experienced by observers with different sets of rods and clocks (each set contracted and dilated as they are along the path of their respective trajectories).
Well put! Yeah, a stretch it is indeed - from our low speed, low energy, near space existence which gives a certain expectation of things, to rather non-evident behaviour. The runs are on the board though as SR, GR and quantum mechanics for that matter are spectacularly successful theories.
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Maybe this is where I'm having trouble applying the paradox.
Well generally paradoxes are not a good thing to apply!! You see the next question: does the scenario as quoted, aka. Andromeda, actually apply to this universe? Which is what my hint was about finding out about their habits over there in Andromeda. I think the Penrose proposition is good in that it very much highlights what we mean about reality, and thus whether our theories, as they are, actually match experience. I'm not aware of any experimental verification of it's predictions. I think it's one of those clever gedankens ( like Schroedinger's Cat ) that points the way to refinements. Science is always a work in progress and theory perpetually runs the risk of invalidation due to it's exposure to experimental results.
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I'm making it out to be a case of the tail wagging the dog. The time and distance of the entire universe isn't going to instantaneously wax and wane just because the inertial frame I'm in is going round and round.
Yes... that is the intuitive view, hence my earlier mention of the 'above frame' view. We can have a mental model, a God's eye view or whatever, but the desire of science is to measure reality so: theory must always point back to a frame for measurement. What does wax and wane is a given perspective/frame view of the universe. We don't need to worry too much about anthropic stuff either, because what is implied when the word 'observer' is used is really 'any measurable spacetime event'. So if say, some motion produces a red shift of emitted photons, then an atom will behave in a measurably different fashion due to that frequency shift of received radiation. This all happens night and day regardless of human views and habits. I certainly find superfluous the view that the world stops just because I do, or if I sleep, or if I look the other way etc..... all that light inside the fridge, and trees falling in forests stuff.
Alas 'relativity' has been misunderstood as implying a sort of 'anything goes' philosophy, when in fact it simply RELATES measurements in different frames in the specified fashion. Einstein actually proposed calling it 'Invariant Theory' to emphasise the constancy of light speed when measured in all frames; and the constancy of the form of physical laws when expressed in said frames.
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But my question then becomes: isn't that how SR says things will appear, even though it isn't how things currently are?
Ahhh, but you're now talking of a 'reality' behind the measuring now ... which is trickier. Not bad, just trickier these 'above frame' views. ( A real cracker is how does a single photon passing through a device interfere with itself .... but we won't go there )
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Don't worry about me drowning; I've found I can still breathe the stuff even if I don't fully comprehend it. :)
Amphibious? Superb :-)
Cheers, Mike.
( edit ) See this as regards the airline/clock thing.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
...
- This de-references the issue to one of 'what are forces?' or 'how do you know if there is a force about?' Here it can get a bit squirrelly, if done without some care. Now take some torch, shine it in a direction ( let's do this is in a vacuum for the present ). Whatever path it takes we'll call that a 'geodesic', or a 'straight line' in ordinary terms. If your body cannot be seen to continue along any ( suitably aligned ) beam of light then it is deemed to be subject to a force. So now the classification of frames depends upon the comparison of material body motion vs. photon motion.
...
So, if light is bent due to gravitational lensing, does that mean that your inertial reference frame is bent as well?
Quote:
...
A GPS receiver pretty well does this ( with the more satellites the better ), as it finds itself on the overlapping portions of wave fronts radiated by the GPS satellites. The GPS signal not only times pulses but encodes the detail of who/what/where sent them, and all is corrected by known relativistic effects ( big topic ). LIGO uses GPS which in turn relies upon the best Caesium clocks.
...
Just an addition: although GPS uses very accurate clocks, there is always the issue of precision as two clock are never aligned exactly. Therefore, a fourth GPS satellite is required such that the exact time can be included as a variable.
BTW: "the more satellites the better" should be read as "a sufficient number of satellites with a good spread over the sky". If you would have, say, a hundred satellites in almost the same position, the position estimation would still be inaccurate.
Does LIGO really use GPS, or does it actually use differential GPS?
So, if light is bent due to gravitational lensing, does that mean that your inertial reference frame is bent as well?
Ooohhhh.... I'd put it that the light path is bent in an accelerated frame compared to an inertial one. The 1919 eclipse observations by Eddington et al did exactly that: Sun present vs. Sun absent. The classic example is the enclosed lift accelerating upwards ( roofwards ) with a light beam shot from one side to the other. The beam will hit the opposite wall at a lower ( more floorward ) point than it would have if it wasn't accelerating. This is basically because during the finite time during which the light travels across the lift, the lift moves more upwards than it would if not accelerating. Now use the Equivalence Principle to then equate this to behaviour in a gravity field ie. lensing. I don't know about calling that frame bending though, not sure.... I think there is some subtlety applied to talking of 'reference frames' vs 'co-ordinate systems'. I think the purpose there is to separate the ideas of intrinsic curvature/metric vs. a specific choice of origins and axis orientation. ( That is, one needs to specify base vectors when quoting matrix elements ).
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Just an addition: although GPS uses very accurate clocks, there is always the issue of precision as two clock are never aligned exactly. Therefore, a fourth GPS satellite is required such that the exact time can be included as a variable.
BTW: "the more satellites the better" should be read as "a sufficient number of satellites with a good spread over the sky". If you would have, say, a hundred satellites in almost the same position, the position estimation would still be inaccurate.
Absolutely right. I was alluding to the greater number of distinct wavefront surfaces you can define your intersection with the better.
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Does LIGO really use GPS, or does it actually use differential GPS?
Just 'plain' GPS I think, so as to ensure a correct application of simultaneity between the two LIGO's, being ~ 1800 miles apart. Differential GPS suggests some sort of local surveying role, which I don't think is required as we are seeking a differential path 'length' between the two arms. It's the laser beams that do that.
Aside: The wavelength of the waves that are sought detection of are presumed to be of much greater extent than the LIGO dimensions. This means that during a light beam round trip of the facility only a small segment of the wave will be sampled. Like a small boat on a long ocean swell. Otherwise the wave phase may change sign during the circuit traversal and the summation of phase changes ( accumulating from point to point by integration of the metric ) may in fact be subtractions. Like a big boat's hull on little ripples. The effect would be to diminish the signal at higher frequencies.
Cheers, Mike.
( edit ) Note: Suppose that I try to punt an object with mass across the lift along the same path as the light beam. I'll fail because I can't achieve light speed with non-zero mass bodies, thus it'll take longer to get across, the lift will go upwards further and I'll hit the wall at a lower point than the beam did. Hence I'll know I'm in a non-inertial frame.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Just 'plain' GPS I think, so as to ensure a correct application of simultaneity between the two LIGO's, being ~ 1800 miles apart. Differential GPS suggests some sort of local surveying role, which I don't think is required as we are seeking a differential path 'length' between the two arms. It's the laser beams that do that.
Aside: The wavelength of the waves that are sought detection of are presumed to be of much greater extent than the LIGO dimensions. This means that during a light beam round trip of the facility only a small segment of the wave will be sampled. Like a small boat on a long ocean swell. Otherwise the wave phase may change sign during the circuit traversal and the summation of phase changes ( accumulating from point to point by integration of the metric ) may in fact be subtractions. Like a big boat's hull on little ripples. The effect would be to diminish the signal at higher frequencies.
Cheers, Mike.
Thanks Mike!
I had the idea that GPS would be used to correct for misaligment in the LIGO arms due to earthquakes. I can somehow image that dGPS could also have advantage in timing problems, but the ambiquity problem needs to be solved as well at that range. I doubt that the scientist would want to rely on such a method. OTOH, I wouldn't depend on GPS only, but use it as a periodic calibration means for my own clocks. Or leave it out completely since it's the difference of time that matters.
Do you have a clue on the required timing accuracy?
I had the idea that GPS would be used to correct for misaligment in the LIGO arms due to earthquakes. I can somehow image that dGPS could also have advantage in timing problems, but the ambiquity problem needs to be solved as well at that range. I doubt that the scientist would want to rely on such a method. OTOH, I wouldn't depend on GPS only, but use it as a periodic calibration means for my own clocks. Or leave it out completely since it's the difference of time that matters.
Yep!! There are discussion papers in the LIGO archives that discuss timing and alternatives too: here, this, that, and the other.
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Do you have a clue on the required timing accuracy?
From memory: about ten microseconds is desired. The Hanford and Livingston are at ~ 10 milliseconds apart. My apologies as I quoted ten nanonseconds in the Detector Watch II thread.
Cheers, Mike.
( edit ) Imagine the following scenario which is entirely plasuible as things currently stand:
- some nearby supernova cracks off.
- the 'visual' astronomers detect it.
- a neutrino observatory, or two, notes a blip in arrivals ( this happened in '87 ).
- several of the gamma ray detectors also light up.
- the LIGO's find in the data near co-incident wave arrivals.
So while these are all 'nearly' co-incident, we really, really like to be as precise as possible here. Because:
- we get a blow by blow description of some stars death ( and something's birth ). These are heavily modelled situations and we get to whittle the pack, refine the thinking.
- we get the sequencing of low energy photons, high energy photons, neutrinos and gravitons. Do they travel at the same speed, who's got mass? etc.
- the waveforms of all of the above.
... so we need to be sure of timing hereabouts if we are going to unravel events thereabouts. Nice.... :-)
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
After reading and re-reading the posts I have come to the conclusion that I may never understand. But I sure am going to sound smart to those that understand even less than I.
The described Andromeda Paradox (as well as Twin and other paradoxes) is only paradox for those who do not understand what they are talking about. It is a paradox only as long as you dont understand the notions involved.
To illustarate how a bad understanding of a notion (the notion of mass in this case) can lead to confusion here is the "paradox": if mass increase with the speed increase then why ordinary objects no matter how small mass they have - why dont they become black-holes when accelerated to high enough speed ?
Now back to Andromeda paradox:
what standing still observer thinks is "there-now" in Andromeda and what moving in the car observer thinks is "there-now" in Andromeda are indeed two different things (different dots in space-time).
But neither of them knows what that "there-now" is. They will only know about what it is when light will come from there. In other words there-now still does not exist. The world simply does not exist that way. If u see the Sun right now u think of it as being there-now (8 light-minutes away from you) but you cannot state for sure that the Sun exists there now (it may have exploded already but the light still travels to inform you about it). So there-now there-now dots of space-time you think form your plain 3D world that is simultanious with you they simply dont exist yet in real world. What in reality exists is the light that has traveled to you informing about past events from some there-in-the-past dots
So i see no paradox at all. Some people simply got confused thinking about there-now as something that observer see. in reality the both observers see the same past(events that happened long before 1.5mil years ago if we talk about Andromeda) and they each can only speculate about their "there-now" projections of the world
Thanks again, Mike, for your
)
Thanks again, Mike, for your help in this thread (and the many others)!!
I've tried to understand your previous post, re: providing the event times in Andromedian frames. To simplify (or de-anthropomorphize) the paradox, I drew an illustration for a GRB, where the beginning of the burst can be the 'speeches and invasion plan', and some point midway along the light-curve (say after it peaks in intensity) would correspond to the 'invasion force deployment'.
(click thumbnail for larger image)
The placement of the red and blue detectors (counter-orbiting satellites) is a bit arbitrary. I could have placed them back a bit, in their respective orbits, such that the blue arrow passes through the red detector, on its way to the blue detector. The red detector shouldn't detect the GRB event, from that distance, until the red arrow arrives (after the relativistic phase between red and blue inertial frames elapses). Hard to believe that's correct, certainly doesn't sound possible....
RE: Thanks again, Mike, for
)
A pleasure indeed, and I do hope/yearn/fear correction should I lead you astray. :-)
I've written a long reply today, I'm afraid!!
Darn!! But that's because it's not... :-)
A very nicely done diagram! Some points of interest:
- There is no 'stationary' frame of reference. Prior to Einstein most, if not all, believed in some ( perhaps hard to find ) universal rest frame. The Michelson-Morley experiment buried that, and much discomfort that then caused. This is a key point in relativity.
- What you do have are inertial frames and non-inertial frames. An inertial frame is one where the Law of Inertia holds. The Law of Inertia states that a body subject to no forces continues it's current path without deflection. This includes the case where it may be stationary with respect to the frame. If then there are no forces, it'll then stay put.
- This de-references the issue to one of 'what are forces?' or 'how do you know if there is a force about?' Here it can get a bit squirrelly, if done without some care. Now take some torch, shine it in a direction ( let's do this is in a vacuum for the present ). Whatever path it takes we'll call that a 'geodesic', or a 'straight line' in ordinary terms. If your body cannot be seen to continue along any ( suitably aligned ) beam of light then it is deemed to be subject to a force. So now the classification of frames depends upon the comparison of material body motion vs. photon motion.
- An inertial frame is otherwise termed a non-accelerated frame. Special Relativity deals with such inertial frames. Really there are no absolutely exact inertial frames, as we've yet to find a region of space with no influences. But we can still use the concept with exceptional precision anyway.
- Non-inertial frames are where the Law of Inertia fails. So the path of a body within it will always deviate from that of any light beam. So no matter how I try to arrange the beam, the body will never 'stay in the bulleye' or whatever trick I use to determine it's conformance to the beam. ( In my mind's eye I am thinking of some body moving away from me with a target painted on it's back, and a sniper's red laser spot.... ).
- A non-inertial frame is otherwise known as an accelerated frame. This means that bodies within it that seem motionless are, when viewed from an inertial frame, exhibiting acceleration ie. changes in the velocity vector with time - speed and/or direction. I can calculate from such measurements in the inertial frame and come up with a vector quantity which contains the magnitude and direction of that acceleration. A good example would be a pencil floating in 'mid-air' on the International Space Station - apparently static from within the station, but orbiting the Earth ( acceleration, change in velocity vector with time.... ) when seen from afar.
- Suppose I note that some behaviour in a non-inertial frame is proportional to the mass, with all other things being equal. Then I can equally consider that frame to be one where a gravitational field is present. This is the Principle of Equivalence. You can pass from descriptions of things (i) as an accelerated frame to/from (ii) a gravity field with strength equal to the acceleration but oppositely directed. Hence you go up in a lift and feel heavier, go down in the lift and feel lighter. Jump off the roof and you will feel briefly weightless, extend that time a bit by skydiving, and even go to the ISS with your pencil and fall endlessly ie. orbit. Be careful as this is only 'best true' if you don't go too far in time or space. A local rule really.
- Now you can go the whole hog to General Relativity and state that because gravity is universal, it's not really a 'force' but a condition of space & time. Light paths are still geodesics. Masses still move but are not 'accelerated by gravity' at all. They follow a path in space and time that would be labelled as 'warped' from a purely inertial frame perspective. So a light beam ( or Mercury too ) moving nearby our Sun seem to be deflected by a force when observed distantly & inertially. But you wouldn't observe it that way within the relevant local non-inertial frames nearby the Sun - it would be a 'free-fall' experience. Weird huh??
So now I've sunk your boat: as we can't legitimately model your satellites using Special Relativity, because they are in fact orbiting/accelerated.
Now back to our earthly friends awaiting signals from Andromeda. The guy in the car is said to have several days headstart on the information from Andromeda compared to the pedestrian. Then his phase difference - (v*x)/(c*c) - implies that at car type speeds ( relative to the other guy ) he has to travel an absolutely enormous distance to achieve that. This is because his 'v' in the numerator is low, and 'c' squared in the denominator is really, really big. You'd need a humungous 'x' to bring the whole phase up to those several days advantage as quoted.
For example, at 1 m/sec ( a 'reasonable' car speed ), 100000 seconds ( about a day ), speed of light at 3000000000 m/sec then x = 90000000000 metres.
Thus to earn his headstart over the pedestrian in time, he must have started quite a long time ago ( in a galaxy far far away? ), to enable him to arrive at the moment when the origins of their reference frames co-incide. At that point he can toss a note out the window to his pedestrian mate and tell him of the impending doom. If it was Nelson from The Simpsons we'd get a 'haha'. :-)
Of course the origins won't actually co-incide and other imprecisions abound but the gist is that both observers will, at about the same point in spacetime, report quite differently about light received from Andromeda. This is highly anti-intuitive and hence the 'paradox' label. I can't see any internal inconsistency though, provided one can accept that Relativity demands the death of a Newtonian universal time. Such is the price ....
Even if your GRB detectors were well out from the Solar System at the moment they passed each other - how are they going to 'run up' to that moment? Remember the big 'x'. If you pull one or the other up, and send them around for another go at it, then you've performed accelerations. That alas will wipe the slate and the time advantage is lost.
My apologies for the length.
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Boat? What boat? You mean
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Boat? What boat? You mean that froth from my fast and furious treading in the science-waters? I've been in over my head for so long, I'm happy to just break the surface :)) I figured somewhere in the depths is the answer for how to walk on the stuff, but lacking it in the meantime, a boat makes a lot more sense! :)))
I'm afraid I still don't see the distinction between when to use GR instead of SR, or how that affects calculations on the measurements from one inertial frame to another. I understand that the satellites are in free-fall around the warped space near a massive object. The question is how events from a distance appear to the satellites. I see many frames of reference: the individual inertial frames of each satellite, the inertial frame of the planet with the satellites going round it, the inertial frame of the solar system with the planet itself in free-fall around a larger massive object... topping it off, why would SR be required for the Global Positioning System, but not to ascertain and evaluate the simultaneity of events at a (much farther) distance, as they appear in one inertial frame compared to another?
It's a bit of a stretch to consider that an observation of a distant object means that you're seeing it as it was in a different time, and not as it currently is. When you look at a distant object, you're looking back into the past. Doesn't makes sense to talk about distance without considering time. And if you're in motion, and moving in a direction different from another observer, then only with synchronized clocks, GR, and SR will your measurements agree.
It's a bit more of a stretch that simultaneity is therefore also relative, and that the rods and clocks in one inertial frame will have different lengths and times when measured from a different inertial frame. Observers in both frames are still looking into the past when observing distant objects, but they're observing failure of simultaneity at a distance. Observations are in agreement, however, when measurements are adjusted by taking into account the relativistic phase, that is, the difference in the amount of distance/time experienced by observers with different sets of rods and clocks (each set contracted and dilated as they are along the path of their respective trajectories).
Maybe this is where I'm having trouble applying the paradox. I'm making it out to be a case of the tail wagging the dog. The time and distance of the entire universe isn't going to instantaneously wax and wane just because the inertial frame I'm in is going round and round. But my question then becomes: isn't that how SR says things will appear, even though it isn't how things currently are?
Don't worry about me drowning; I've found I can still breathe the stuff even if I don't fully comprehend it. :)
RE: Boat? What boat? You
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Well done, and it is terrific when it crystallises though.... so hang in for the 'Eureka' moments in life. :-)
Fair points. They are calculational models, so it's a question of the level of approximation of description that's desired. That depends then on the purpose - real experiment vs. gedanken etc... I recall ( late 70's? ) a trip was taken ( on the Concorde? ) across the Atlantic with a pre-synchronised 'portable' atomic clock. On landing it differed from it's 'stationary' twin back in the UK by an amount consistent with SR ie. so only the velocity of the plane was used. Strictly speaking though they were all accelerating, in a gravity field etc. Nowadays GPS accounts very well using a GR model, and locational accuracy would drift by kilometers per day if it were not used. This is mainly due to 'time warping' rather than 'space warping' effects. The accelerations used, and gravity field strength, hereabouts are all pretty mild on the cosmic scale of things. Which is good for our health!! We really do orbit about a quiet unregarded yellow sun ( as Douglas Adams would have put it ).
Well put! Yeah, a stretch it is indeed - from our low speed, low energy, near space existence which gives a certain expectation of things, to rather non-evident behaviour. The runs are on the board though as SR, GR and quantum mechanics for that matter are spectacularly successful theories.
Well generally paradoxes are not a good thing to apply!! You see the next question: does the scenario as quoted, aka. Andromeda, actually apply to this universe? Which is what my hint was about finding out about their habits over there in Andromeda. I think the Penrose proposition is good in that it very much highlights what we mean about reality, and thus whether our theories, as they are, actually match experience. I'm not aware of any experimental verification of it's predictions. I think it's one of those clever gedankens ( like Schroedinger's Cat ) that points the way to refinements. Science is always a work in progress and theory perpetually runs the risk of invalidation due to it's exposure to experimental results.
Yes... that is the intuitive view, hence my earlier mention of the 'above frame' view. We can have a mental model, a God's eye view or whatever, but the desire of science is to measure reality so: theory must always point back to a frame for measurement. What does wax and wane is a given perspective/frame view of the universe. We don't need to worry too much about anthropic stuff either, because what is implied when the word 'observer' is used is really 'any measurable spacetime event'. So if say, some motion produces a red shift of emitted photons, then an atom will behave in a measurably different fashion due to that frequency shift of received radiation. This all happens night and day regardless of human views and habits. I certainly find superfluous the view that the world stops just because I do, or if I sleep, or if I look the other way etc..... all that light inside the fridge, and trees falling in forests stuff.
Alas 'relativity' has been misunderstood as implying a sort of 'anything goes' philosophy, when in fact it simply RELATES measurements in different frames in the specified fashion. Einstein actually proposed calling it 'Invariant Theory' to emphasise the constancy of light speed when measured in all frames; and the constancy of the form of physical laws when expressed in said frames.
Ahhh, but you're now talking of a 'reality' behind the measuring now ... which is trickier. Not bad, just trickier these 'above frame' views. ( A real cracker is how does a single photon passing through a device interfere with itself .... but we won't go there )
Amphibious? Superb :-)
Cheers, Mike.
( edit ) See this as regards the airline/clock thing.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
RE: ... - This
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So, if light is bent due to gravitational lensing, does that mean that your inertial reference frame is bent as well?
Just an addition: although GPS uses very accurate clocks, there is always the issue of precision as two clock are never aligned exactly. Therefore, a fourth GPS satellite is required such that the exact time can be included as a variable.
BTW: "the more satellites the better" should be read as "a sufficient number of satellites with a good spread over the sky". If you would have, say, a hundred satellites in almost the same position, the position estimation would still be inaccurate.
Does LIGO really use GPS, or does it actually use differential GPS?
Regards,
Bert
Somnio ergo sum
RE: So, if light is bent
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Ooohhhh.... I'd put it that the light path is bent in an accelerated frame compared to an inertial one. The 1919 eclipse observations by Eddington et al did exactly that: Sun present vs. Sun absent. The classic example is the enclosed lift accelerating upwards ( roofwards ) with a light beam shot from one side to the other. The beam will hit the opposite wall at a lower ( more floorward ) point than it would have if it wasn't accelerating. This is basically because during the finite time during which the light travels across the lift, the lift moves more upwards than it would if not accelerating. Now use the Equivalence Principle to then equate this to behaviour in a gravity field ie. lensing. I don't know about calling that frame bending though, not sure.... I think there is some subtlety applied to talking of 'reference frames' vs 'co-ordinate systems'. I think the purpose there is to separate the ideas of intrinsic curvature/metric vs. a specific choice of origins and axis orientation. ( That is, one needs to specify base vectors when quoting matrix elements ).
Absolutely right. I was alluding to the greater number of distinct wavefront surfaces you can define your intersection with the better.
Just 'plain' GPS I think, so as to ensure a correct application of simultaneity between the two LIGO's, being ~ 1800 miles apart. Differential GPS suggests some sort of local surveying role, which I don't think is required as we are seeking a differential path 'length' between the two arms. It's the laser beams that do that.
Aside: The wavelength of the waves that are sought detection of are presumed to be of much greater extent than the LIGO dimensions. This means that during a light beam round trip of the facility only a small segment of the wave will be sampled. Like a small boat on a long ocean swell. Otherwise the wave phase may change sign during the circuit traversal and the summation of phase changes ( accumulating from point to point by integration of the metric ) may in fact be subtractions. Like a big boat's hull on little ripples. The effect would be to diminish the signal at higher frequencies.
Cheers, Mike.
( edit ) Note: Suppose that I try to punt an object with mass across the lift along the same path as the light beam. I'll fail because I can't achieve light speed with non-zero mass bodies, thus it'll take longer to get across, the lift will go upwards further and I'll hit the wall at a lower point than the beam did. Hence I'll know I'm in a non-inertial frame.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
RE: Just 'plain' GPS I
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Thanks Mike!
I had the idea that GPS would be used to correct for misaligment in the LIGO arms due to earthquakes. I can somehow image that dGPS could also have advantage in timing problems, but the ambiquity problem needs to be solved as well at that range. I doubt that the scientist would want to rely on such a method. OTOH, I wouldn't depend on GPS only, but use it as a periodic calibration means for my own clocks. Or leave it out completely since it's the difference of time that matters.
Do you have a clue on the required timing accuracy?
Regards,
Bert
Somnio ergo sum
RE: I had the idea that GPS
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Yep!! There are discussion papers in the LIGO archives that discuss timing and alternatives too: here, this,
that, and the other.
From memory: about ten microseconds is desired. The Hanford and Livingston are at ~ 10 milliseconds apart. My apologies as I quoted ten nanonseconds in the Detector Watch II thread.
Cheers, Mike.
( edit ) Imagine the following scenario which is entirely plasuible as things currently stand:
- some nearby supernova cracks off.
- the 'visual' astronomers detect it.
- a neutrino observatory, or two, notes a blip in arrivals ( this happened in '87 ).
- several of the gamma ray detectors also light up.
- the LIGO's find in the data near co-incident wave arrivals.
So while these are all 'nearly' co-incident, we really, really like to be as precise as possible here. Because:
- we get a blow by blow description of some stars death ( and something's birth ). These are heavily modelled situations and we get to whittle the pack, refine the thinking.
- we get the sequencing of low energy photons, high energy photons, neutrinos and gravitons. Do they travel at the same speed, who's got mass? etc.
- the waveforms of all of the above.
... so we need to be sure of timing hereabouts if we are going to unravel events thereabouts. Nice.... :-)
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
After reading and re-reading
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After reading and re-reading the posts I have come to the conclusion that I may never understand. But I sure am going to sound smart to those that understand even less than I.
Thanks for the insight
The described Andromeda
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The described Andromeda Paradox (as well as Twin and other paradoxes) is only paradox for those who do not understand what they are talking about. It is a paradox only as long as you dont understand the notions involved.
To illustarate how a bad understanding of a notion (the notion of mass in this case) can lead to confusion here is the "paradox": if mass increase with the speed increase then why ordinary objects no matter how small mass they have - why dont they become black-holes when accelerated to high enough speed ?
Now back to Andromeda paradox:
what standing still observer thinks is "there-now" in Andromeda and what moving in the car observer thinks is "there-now" in Andromeda are indeed two different things (different dots in space-time).
But neither of them knows what that "there-now" is. They will only know about what it is when light will come from there. In other words there-now still does not exist. The world simply does not exist that way. If u see the Sun right now u think of it as being there-now (8 light-minutes away from you) but you cannot state for sure that the Sun exists there now (it may have exploded already but the light still travels to inform you about it). So there-now there-now dots of space-time you think form your plain 3D world that is simultanious with you they simply dont exist yet in real world. What in reality exists is the light that has traveled to you informing about past events from some there-in-the-past dots
So i see no paradox at all. Some people simply got confused thinking about there-now as something that observer see. in reality the both observers see the same past(events that happened long before 1.5mil years ago if we talk about Andromeda) and they each can only speculate about their "there-now" projections of the world