mr_homm
Hi all,

I've been away for quite some time, but had a little thought about Special Relativity last week, and I thought some here might enjoy playing with it. BTW, I know the complete answer to the following questions, so I'm not asking for help here, just offering a puzzle for your enjoyment. I'll return every couple of days to check progress and give the answer to each part.

The puzzle is this: you are in a disabled interstellar spacecraft, and you wish to broadcast a message to a rescue craft (let's ignore the fact that rescue will take decades to reach you and you will probably be dead -- this is just a puzzle). You do not know the velocities or locations of potential rescuers.

Part 1: Can you (i.e. is it theoretically possible to) send a general distress message that will contain enough information for a rescue craft to find your present location? (Remember that, since their velocity relative to you is unknown, you cannot simply send your coordinates, as they will be unable to Lorentz transform them into their own frames. So the question is really, can you express your position in a way that DOES NOT require the use of a Lorentz transformation in order for the receiver to find it?)

I'll answer Part 1 myself: Yes. Because if the answer were "no," there would be no more parts to the puzzle!

Part 2: You must of course refer somehow to external events in order to place yourself. Let's call them beacon events. How many beacon events are necessary, and precisely what information about them should you broadcast?

Part 4: Why will you never be rescued if all you send is the minimal set of information that satisfies the criteria of Parts 1 and 2? How much and what additional information must you send to enable a rescuer to actually rescue you?

I'll post my solution to Part 2 in a couple of days. Enjoy!

--Stuart Anderson
Harry Ballsonya
I would create a swarm of muons and send them out in all directions.
mr_homm
QUOTE (Harry Ballsonya+Mar 26 2012, 03:55 PM)
I would create a swarm of muons and send them out in all directions.

This is certainly ONE way to solve the puzzle. But why muons instead of something else? Muons decay, and so your signal would disappear just a few kilometers from your location, unless you gave them such enormous energies that time dilation would preserve them. But then they would be nearly indistinguishable from cosmic rays, so how could the rescuers pick them out from the background radiation? Also, at these energies, it would be more efficient to send a huge light pulse.

Anyway, you are basically correct that a sufficiently bright pulse of just about anything capable of crossing space would do, as a sort of signal flare. So let's count this as an alternate solution to the "official" one. What I'm looking for is more along the lines of an information signal. In other words, the signal should contain a stream of words and numbers, much like a post here, and the information content of the signal, rather than its physical properties, should be decodable by the rescuer.

To make this concrete, let's say that someone else has intercepted the signal and handed the rescuer the message printed on a sheet of paper. What does your text need to SAY in order for the rescuer to find your location?

Still open for other suggestions!
Harry Ballsonya
QUOTE (mr_homm+Mar 27 2012, 01:00 AM)

This is certainly ONE way to solve the puzzle. But why muons instead of something else? Muons decay, and so your signal would disappear just a few kilometers from your location, unless you gave them such enormous energies that time dilation would preserve them. But then they would be nearly indistinguishable from cosmic rays, so how could the rescuers pick them out from the background radiation? Also, at these energies, it would be more efficient to send a huge light pulse.

Anyway, you are basically correct that a sufficiently bright pulse of just about anything capable of crossing space would do, as a sort of signal flare. So let's count this as an alternate solution to the "official" one. What I'm looking for is more along the lines of an information signal. In other words, the signal should contain a stream of words and numbers, much like a post here, and the information content of the signal, rather than its physical properties, should be decodable by the rescuer.

To make this concrete, let's say that someone else has intercepted the signal and handed the rescuer the message printed on a sheet of paper. What does your text need to SAY in order for the rescuer to find your location?

Still open for other suggestions!

Shoot, I thought the muons could speak for me by using their decays to provide a map of spacetime events that would be all the same interval from which I created them. Oh well, back to the drawing board.
flyingbuttressman
So, maybe I'm under-thinking this, but why couldn't you just send out radio messages at regular intervals? Assuming that you don't run out of power, any rescuer could home in on the signal via triangulation.
mr_homm
QUOTE (Harry Ballsonya+ Mar 26 2012, 05:30 PM)
Shoot, I thought the muons could speak for me by using their decays to provide a map of spacetime events that would be all the same interval from which I created them.  Oh well, back to the drawing board.

It's not inherently a bad idea, but would require a truly enormous expenditure of energy to create enough high-energy muons to be detected across interstellar distances. Also, since particle decay is a "memoryless" process, an observer would have no way of knowing whether the observed muon flux was due to a large initial batch that had mostly decayed (implying a distant source) or a small batch that was mostly intact (implying a nearby but weaker source).

You could, I suppose, look at the strength of the muon flux at several different locations and infer the direction of the source from their relative strengths (but this information is available more readily by just noting their direction of travel) and you cannot in any way get the distance to the source of the muons from their flux strengths at several points. Decay is an exponentially distributed process, and as such does not contain the information you would need.

Of course, you could observe the muons at several locations displaced laterally and triangulate on their source. This would work except that muons are charged, and the galaxy has a rather strong magnetic field, which you must map in detail before you can reliably trace the muons to their source. So, all in all, this is a method which would theoretically work, but would require vast resources to carry out. What I have in mind is much more practical, requiring only a single message, relatively minimal energy at the sender and at the receiver end, nothing but observations that can be carried out by a single receiver without moving through space or wating for time to pass.

QUOTE (flyingbuttressman+ Mar 26 2012, 5:50 PM)

So, maybe I'm under-thinking this, but why couldn't you just send out radio messages at regular intervals? Assuming that you don't run out of power, any rescuer could home in on the signal via triangulation.

OK, that would work too, but there is a method where you need only send out a single message. Again, your location should be deducible from the CONTENT of the message, rather than by physical measurements carried out on the message itself. That rules out things like triangulation, muon pulses, and many other things. Regularly spaced radio message are a good practical idea, but the core of the puzzle is what the content of the message should be, in order that the text of the message, all by itself, will allow the rescuer to determine the location from which it was sent. For purposes of the puzzle, we could pretend that you have only enough power to send out one or maybe two messages, and you have to decide what to say to maximize your chances of being rescued (eventually). Also, of course, your ship is disabled, so you can't simply set a course and say "meet me at Barnard's Star" or something of the sort -- in that case, you wouldn't need rescue!

QUOTE (investigator+ Mar 26 2012, 5:52 PM)
Thankyou for this interesting game sir. Answer can be to send image of absorption spectrums of surrounding stars which appear on common star map? Do this many times over standard time intervals so ship speed and direction can be deduced by changes in absorption spectrum pattern. This change pattern will tell what speed and direction stranded ship going and rescuers triangulate from changing pattern to tell which stars are used by stranded ship? The correction by rescuers of absorption spectrums of known stars will help also to tell current relative speed and direction of rescuers compare to stranded ship? They go to where ship will be calculated by these things? Am close or not? Thankyou.

This is the best answer yet. You are correct that you must somehow identify external beacon stars which the rescuer can also observe. Giving their spectral characteristics is an excellent way to do this. However, these patterns would change over time only very slowly, as the disabled ship is moving in free fall, so it has constant velocity relative to these stars over time spans of several decades. Therefore, it would take a very long time before the angle betwen the light from a given star and the direction of the ship's travel changes enough to notice. It is also necessary to measure very many stars this way so that the rescuer can have enough information to identify them.

You are also correct that the rescuer could deduce the how his own velocity relative to these stars differed from the sender's. This would assist in determing a point where the rescuer could rendezvous with the disabled ship. However, your scheme would require several decades of measurements of several thousand stars, which makes it somewhat less practical. Nevertheless, it would technically work, so this is another "alternative solution."

I have in mind something easier to do, which makes use of certain facts about Special Relativity and some fairly common knowledge about certain special types of stars.

I will post a further hint for Part 2 tomorrow, and the my solution to Part 2 the day after, unless of course, someone hits on it before then.

Have fun with it!

--Stuart Anderson
Confused1
Might these things be useful?
-C2.

synthsin75
I would suppose you'd send your observed red/blue shift for several standard candles.
Confused1
Back to pulsars. If all parties have a catalogue of pulsars they can be identified by their period (and visibility .. I suspect they have a narrow 'beam'). Using several of them and sending period and angles should (hopefully) enable a rescuer to home in on your location.
-C2.
(Nice to hear from Mr Homm again)
mr_homm
QUOTE (Confused1+Mar 27 2012, 05:15 AM)
Might these things be useful?
-C2.

Yes, this would be very useful indeed. In fact, this is certainly a part of the solution I was thinking of. The essential point is that there should be some identifiable objects which both the disabled ship and the rescuer can observe. These do nicely for that, as they are easily and uniquely identifiable by their periods.

QUOTE
Back to pulsars. If all parties have a catalogue of pulsars they can be identified by their period (and visibility .. I suspect they have a narrow 'beam'). Using several of them and sending period and angles should (hopefully) enable a rescuer to home in on your location.
-C2.
(Nice to hear from Mr Homm again)

Period to identify them, and angles as seen by the disabled ship... yes, this would let the rescuer locate the ship. I had in mind a slightly different set of information, but yours would also work. I think the analysis at the receiver end would be more complicated for your scheme, but it's certainly doable. So let's pursue your solution AND mine, and see how they turn out. Now, the next question in Part 2 is, precisely how many pulsars will you need (at a minimum) to fix your location?

And yes, it's nice to be back!

QUOTE (->
 QUOTE Back to pulsars. If all parties have a catalogue of pulsars they can be identified by their period (and visibility .. I suspect they have a narrow 'beam'). Using several of them and sending period and angles should (hopefully) enable a rescuer to home in on your location.-C2.(Nice to hear from Mr Homm again)

Period to identify them, and angles as seen by the disabled ship... yes, this would let the rescuer locate the ship. I had in mind a slightly different set of information, but yours would also work. I think the analysis at the receiver end would be more complicated for your scheme, but it's certainly doable. So let's pursue your solution AND mine, and see how they turn out. Now, the next question in Part 2 is, precisely how many pulsars will you need (at a minimum) to fix your location?

And yes, it's nice to be back!

I would suppose you'd send your observed red/blue shift for several standard candles.

Yes, this would also work. You could use known stars with identifiable spectra, or pulsars as Confused1 suggested. Given your observed redshifts as the data to be transmitted, what is the minimum number of standard candles you would need in order to fix the ship's location? (BTW, since you are working with redshifts, it seems that you would also necessarily be able to fix the ship's velocity as well. Of course this is necessary if there is any delay in the rescue attempt.)

Later today I will post my own solution for Part 2 and you can all see how the various solutions compare.

--Stuart Anderson
synthsin75
QUOTE (mr_homm+Mar 27 2012, 09:55 AM)
Yes, this would also work. You could use known stars with identifiable spectra, or pulsars as Confused1 suggested. Given your observed redshifts as the data to be transmitted, what is the minimum number of standard candles you would need in order to fix the ship's location? (BTW, since you are working with redshifts, it seems that you would also necessarily be able to fix the ship's velocity as well. Of course this is necessary if there is any delay in the rescue attempt.)

Yes, I assume the rescuers may be far enough removed so as to not recognize the same patterns of star (or this should at least be accounted for), so a significant delay is to be expected.

I assume I'd need three such beacons to fix my location. This once again assumes the possibility of a far removed rescuer where cosmological redshift would require the intersection of three spheres.
Confused1
I'm assuming I can't get any range data from my pulsars.
Choosing a 'good' three. Treat them as lying on a plane. Taking the angles between pairs of pulsars - I think two of these angles should be sufficient (but I can't work out quite how (so far)), failing that three should do it (but I can't work that out either (so far)). It does look like a small error in the angles could mean a lot of error in the position.
A fun problem, thank you.
-C2.
Confused1
While waiting (hoping) for Mr Homm to post..

If the sides of the plane on which the selected pulsars sit is A,B,C (known from the catalogue) and the distances to the pulsars (which we want to know) is p,q,r and the angles observed to A,B,C are a,b,c then (cosine rule)

A^2 = r^2 + p^2 - 2rpcos( a )
B^2 = p^2 + q^2 - 2pqcos( b )
C^2 = q^2 + r^2 - 2qrcos( c )

From memory three equations and three unknowns ought to be soluble .. I just hope the rescuers can do it because I'm not sure I can (in finite time).

-C2.

Assuming the rescuers can deal with part 3 .. is the extra information required (part 4) .. heeeeeeeeeeellllllllllppppppppp ?
EMPulse
Beacon events with time stamp and output power.

first transmission contains: current local time and interval of transmission or time of next transmission and the output power transmitting at.

second transmission contains: current local time and interval of transmission or time of next transmission and the output power transmitting at.

The rescuer can use the signal strength (varies inversely with the distance) to calculate your distance and the time stamps to calculate your velocity.

Confused1
Another nibble..
Can we assume the existence of a universal coordinate system with agreed units? If "yes" then (I suspect) the problem becomes soluble within my available (maths) technology.
-C2.
mr_homm
@ Confused1:

You're on a good track here, although there are a couple of points still to think through. First, 3 beacons will not do the job. Think about a lower dimensional case to see why: on a LINE, if you gave your distance from a single point, you could be on either side of it, so your position is not pinned down (BTW, a "1-dimensional sphere," a.k.a a "1-sphere" is just a pair of points, because "sphere" in general is defined as the set of all points equally distant from a given point. On a line there will be just two such points, so these constitute the "1-sphere.) If you gave distances from two points, you would be on the intersection of 2 spheres, which would of course only overlap at one point, so 2 beacons let you locate yourself on a line.

On a plane, your distance from two beacons would place you on the intersection of two "2-spheres" (2 dimensional spheres are ordinary circles), but two 2-spheres intersect in a pair of points (i.e. a 1-sphere), so you'd need a third beacon whose distance would pin down your location to one or the other of these points. Similarly in 3 dimensional space, two 3-spheres intersect in a 2-sphere, and a 2-sphere and another 3-sphere intersect in a 1 sphere (which is still 2 points), so you need a fourth 3-sphere.

You can see that in general, an n-sphere and an m-sphere intersect in a k-sphere, where k is minimum(n,m) - 1. Therefore, to locate yourself in 3 dimensional space, you need 4 beacons. The math you give in the next post would work, but you will have lots more equations, since in 3-d you have to think about the triangle formed by you and each pair of beacons, which is 6 triangles. Then you have 6 equations in 3 unknowns. Remember that the equations are quadratic, so each has two solutions, and that's why you need 6 equations, so you can select the right answer from each pair of solutions for overall consistency. This is annoying to do by hand, but most fictional spaceships have pretty good computers!

However, this should set off some alarm bells! We're talking relativity here, and your 3-d space is just a slice through 4-d space-time, and your rescuer doesn't know what slice you're on, since that's determined by your velocity relative to him. So the rescuer will be unable to interpret this information as a position in an unambiguous way. If you send Doppler shift information as well, the rescuer could then interpret your data, but the decoding problem at the rescuer's end is harder.

About the universal coordinate system: that's a no-go for relativity. Different frames will necessarily have different coordinate systems related by the Lorentz transformation. You CAN have everyone use the same units, though -- this isn't intended to be a problem about communicating with space aliens, so you can assume a human rescuer who uses the standard metric system.

And yes, "heeeelllllppppp!!!" would be a good addition to the message.

@ synthsin75:

See above comments to Confused1 about the number of beacons. You don't have to worry about cosmological red shift, because that only comes into play across millions of light-years of distance. You can assume the rescuer is in the same galaxy as the disabled ship, perhaps a few dozen or a few hundred light years away. That means you can use local stellar beacons, known black holes, pulsars, Cepheid variables, flare stars, etc. within the galaxy.

@ EMpulse:

This is a good idea, too. You can get distance from signal strength, as you say, however, you also need to get direction. That's easy, since you can just observe the direction from which the signal came. Velocity is trickier, Doppler shift depends on both the speed and direction of travel. For a given Doppler shift, there is a variety of speeds and directions that will produce it, so you can't get the ship's course in this way.

Remember though, the goal is to use the information content of the signal, not the physical signal itself, to locate the ship. If all you have is a sheet of paper with the text of the transmission, you should be able to deduce the ship's location and velocity. That disqualifies part of your solution, since you would have to measure the difference in arrival times of the two transmissions in order to get the Doppler shift, which is a physical measurement of the signal, not just a reading of its contents. That's what makes this a rather harder puzzle. But in a practical sense, I think your solution could be made to work, you'd just need to figure out what additional information to send to pin down the ship's velocity.

I'll give my version of the solution to this part of the puzzle in my next post.

Thanks for taking an interest in this puzzle!

--Stuart Anderson
Confused1
@Mr Homm,

QUOTE (Mr Homm+)
First, 3 beacons will not do the job.

IF we agree the angles are given in (say) a clockwise direction then isn't the location unique? From the 'other side' the angles would be the same but the other way round (maybe)? Solve for (my) p,q,r distances to beacons and then see which way round gives angles a,b,c instead of a,c,b.

This goes back a while, instead of transforming directly from (say) frame K to M can we not go via some arbitrary frame L without loss of data? If 'yes' then could we not nominate (say) Cape Canaveral (CC) as our agreed arbitrary frame and work from there? From the actual measurements of angles we transform into CC frame and this is the data we transmit. If we transmit our x,y,z velocity (wrt to CC) then our smart rescuers should be able to find us.
Maybe 'heeellpp' sums it up.
Thanks again,
C2.
Edit.. can we not also maintain CC time? With time of transmission, location and velocity that should be enough.
Harry Ballsonya
Cartan–Karlhede algorithm?
mr_homm
QUOTE (Confused1+Mar 29 2012, 11:49 AM)
@Mr Homm,

QUOTE (Mr Homm+)
First, 3 beacons will not do the job.

IF we agree the angles are given in (say) a clockwise direction then isn't the location unique? From the 'other side' the angles would be the same but the other way round (maybe)? Solve for (my) p,q,r distances to beacons and then see which way round gives angles a,b,c instead of a,c,b.

This goes back a while, instead of transforming directly from (say) frame K to M can we not go via some arbitrary frame L without loss of data? If 'yes' then could we not nominate (say) Cape Canaveral (CC) as our agreed arbitrary frame and work from there? From the actual measurements of angles we transform into CC frame and this is the data we transmit. If we transmit our x,y,z velocity (wrt to CC) then our smart rescuers should be able to find us.
Maybe 'heeellpp' sums it up.
Thanks again,
C2.
Edit.. can we not also maintain CC time? With time of transmission, location and velocity that should be enough.

You're right for 3 dimensions here. With 3 beacons at known locations and with the angles to each, you can get your position except that you might be at either of two mirror image points. Using the clockwise/counterclockwise data provides the last bit of information to pin your location exactly. But (there's always a but) ships traveling at different velocities will see these angles differently, even if they pass through the same observation point, because of the Lorentz transformation. So you have to give velocity information as well, in order for your position information to be decoded correctly. You could give the red-shifts/blue-shifts you observe for each beacon, which would be sufficient for the receiver to solve for your x y and z velocity components in his own frame, after which he can interpret your angle data correctly.

So this looks like a full solution to Part 2, although Part 3 will be rather tough, since decoding this set of information might be a bit more complicated.

As to CC time, this is also workable, provided the ship has maintained a record of its course and speed so that you can determine your space-time coordinates relative to the CC origin of coordinates. Although, recall that the original question was about whether the communication could occur without anyone having to do a Lorentz transformation, and the receiver would certainly have to do one from the CC frame to his own in order to navigate to your location. So this is another valid way to get rescued, but it wouldn't count as a solution to the original question.

Confused1
@Mr Homm,

Hmm. I hadn't seen the 4th angle as solving the velocity problem. I rather liked the three points on a plane .. four points is a bit more difficult.
Just musing:
New notation: A[n,m] meaning the the angle between pulsar n and pulsar m

From the lost ship we see:
CODE

call this beast [A]
0,0  1,0  2,0 3,0
0,1  1,1  2,1 3,1
0,2  1,2  2,2 3,2
0,3  1,3  2,3 3,3

From the rescue ship we see:
CODE

call this beast [B]
0,0  1,0  2,0 3,0
0,1  1,1  2,1 3,1
0,2  1,2  2,2 3,2
0,3  1,3  2,3 3,3

The relationship between [A] and [B] hopefully contains all the information required. (or not). Not? Another set of angles from the lost ship would tell the rescuer how [A] evolves over time - giving a new beast ([A1]). Given [A],[A1] and [B] it should be trivial to get [C] which is the answer (in some shape or form). Any hints would be most welcome (especially "you're barking up the wrong tree")
Thanks - great fun is being had here.
-C2.
Confused1
Today (BBC):
Dead stars 'to guide spacecraft' ( http://www.bbc.co.uk/news/science-environment-17557581 )
!
C2.
mr_homm
QUOTE (Investigator+Mar 29 2012, 01:53 PM)
Thankyou Mr homm. One try more? If all have common pan-galactic navigation chart and system why not stranded ship send information on what standard grid references they passed and period between since lose control? Such all this one lot data set information sent in one signal is easy for rescuers to graph line and times over standard grid references to extrapolate present speed and heading of stranded ship? Rescuers just compute interception using extrapolation of all in one message information sent by stranded ship? Thankyou sir for good game.

OK, this is very close to what I was thinking, although you must give a detectable event which the rescuer can see for each grid point you use. Otherwise, the rescuer cannot tell when you passed it, because your clock is not the same as his. If you do give such events, then the rescuer can determine your position and velocity, and hence can compensate for your time dilation. Therefore, the rescuer can find you. So this would work.

On the other hand, it is not absolutely necessary to have such a galactic grid set up. You can still solve the puzzle without such a grid, as I will show when I post my version of the solution.

Thanks for taking an interest in this puzzle!

--Stuart Anderson
mr_homm
QUOTE (Harry Ballsonya+Mar 29 2012, 02:36 PM)
Cartan–Karlhede algorithm?

Not really necessary here. This applies in GENERAL relativity, and its purpose is to compare two spaces (or two regions of space). In other words, it discusses the global properties of the space that the ship is in, rather than locating that ship within the space. Also, in special relativity, space is flat anyway, so there is nothing for that particular algorithm to do. Applying it would basically tell you that 0=0, which isn't very informative.

Disclaimer: I was not familiar with this algorithm, so I looked it up in order to give you an answer.

--Stuart Anderson
mr_homm
QUOTE (Confused1+Mar 30 2012, 08:56 AM)
@Mr Homm,

Hmm. I hadn't seen the 4th angle as solving the velocity problem. I rather liked the three points on a plane .. four points is a bit more difficult.
Just musing:
New notation: A[n,m] meaning the the angle between pulsar n and pulsar m

From the lost ship we see:
CODE

call this beast [A]
0,0  1,0  2,0 3,0
0,1  1,1  2,1 3,1
0,2  1,2  2,2 3,2
0,3  1,3  2,3 3,3

From the rescue ship we see:
CODE

call this beast [B]
0,0  1,0  2,0 3,0
0,1  1,1  2,1 3,1
0,2  1,2  2,2 3,2
0,3  1,3  2,3 3,3

The relationship between [A] and [B] hopefully contains all the information required. (or not). Not? Another set of angles from the lost ship would tell the rescuer how [A] evolves over time - giving a new beast ([A1]). Given [A],[A1] and [B] it should be trivial to get [C] which is the answer (in some shape or form). Any hints would be most welcome (especially "you're barking up the wrong tree")
Thanks - great fun is being had here.
-C2.

This is good. The rescuer can make the observations of B and the sender can make the observations of A and then transmit them to the rescuer. Comparing A and B, the rescuer can deduce the origin of the sender's coordinate system and using A1 as well, he can find the sender's velocity relative to his own. Now you have to actually do the math!

...Or not, if you don't feel like it.

BTW, these matrices are symmetric, since for example A(1,2) = A(2,1) is just the angle between beacons 1 and 2. So actually each matrix contains only 10 numbers rather than 16. So altogether, you need to transmit 20 numbers to the rescuer, together with instructions about what they mean and what the rescuer needs to measure (i.e. get .

QUOTE
Today (BBC):
Dead stars 'to guide spacecraft' ( http://www.bbc.co.uk/news/science-environment-17557581 )
!
C2.

Jackpot! It looks like they're working on the same idea, although they're focusing on locating yourself and I'm focusing on communication across frames. Yes, pulsars are the way to go.

And now, in my next post, my version of the solution.....

--Stuart Anderson
Confused1
There might be a new player in the game. It would be interesting to find out what he (or she) comes up with.
Meanwhile..
I am reminded of a story about the time when barbed wire fences were replaced by electric fences in Australia (?). On the first day many buffalos (?) simply walked or ran (whatever buffalos do) through the electric fences and learned that V_electric_fence/R_buffalo=I_never_want_to_do_that_again. The point of the story is that word about the hostile nature of electric fences spread through the buffalo population faster than .. with hindsight I note that buffalos aren't generally known for their ability to talk .. so I am left with .. the information spread through the buffalos faster than would have been possible even if buffalos could talk. The point (of the point) is that once an idea exists (its time has come) it can (sometimes) spring up 'everywhere'.
-C2.
Confused1
We have a 'pass' from the new player. I (and the new player) look forward to seeing the solution.
-C2.
QUOTE (mr_homm+Mar 26 2012, 06:13 PM)
The puzzle is this:  you are in a disabled interstellar spacecraft, and you wish to broadcast a message to a rescue craft (let's ignore the fact that rescue will take decades to reach you and you will probably be dead -- this is just a puzzle).  You do not know the velocities or locations of potential rescuers.

Part 1:  Can you (i.e. is it theoretically possible to) send a general distress message that will contain enough information for a rescue craft to find your present location? (Remember that, since their velocity relative to you is unknown, you cannot simply send your coordinates, as they will be unable to Lorentz transform them into their own frames.  So the question is really, can you express your position in a way that DOES NOT require the use of a Lorentz transformation in order for the receiver to find it?)

Easy.

Quantum entangled particle transceiver that exist in both ships.

Edit;- end of story;- thanks for that 0.0001µs of puzzlement.

Robittybob1
QUOTE (Lady Elizabeth+Mar 31 2012, 03:13 AM)
Easy.

Quantum entangled particle transceiver that exist in both ships.

Edit;- end of story;- thanks for that 0.0001µs of puzzlement.

But what happens if that hadn't been invent at the time the first craft left on it's journey?
Had you spent 0.0007sec thinking of that eventuality?
QUOTE (Robittybob1+Mar 31 2012, 05:36 AM)
But what happens if that hadn't been invent at the time the first craft left on it's journey?
Had you spent 0.0007sec thinking of that eventuality?

Nah, realized I'd already invented it (0.000000000000000000000000001ns).

Hyperspatial Communication
Robittybob1
QUOTE (Lady Elizabeth+Mar 31 2012, 10:25 AM)
Nah, realized I'd already invented it (0.000000000000000000000000001ns).

Hyperspatial Communication

I see you describe yourself as "Another stupendously average" way back then. So how come you have become such a genius now?

QUOTE (Robittybob1+Mar 31 2012, 10:51 AM)
I see you describe yourself as "Another stupendously average" way back then. So how come you have become such a genius now?

Genius? no, merely a potty-mouthed crank, but @ least I'm honest.

Robittybob1
QUOTE (Lady Elizabeth+Mar 31 2012, 11:32 AM)
Genius? no, merely a potty-mouthed crank, but @ least I'm honest.

I didn't want to read that. Far too humble.
mr_homm
QUOTE (Investigator+Mar 30 2012, 05:27 PM)
Thankyou sir mr homm. Welcome. Another try? If all intra-galactic travel use constant update communications broadcast for any trip then easily extrapolate from trip history broadcasts of ongoing position, speed and etcetera from all ships? So any ship can calculate any other ship details and stranded ship only need say when and what grid reference they were last in data set to date of stranding or present position when making last and only one broadcast? This too practical and not need mathematics for solution, so not what you want for relativity principles mathematics solution? Sorry if too trivial after then. Thankyou for good game anyway sir. I wait for your relativity solution to enjoy it.

This would work quite well, although again, it depends on maintaining an elaborate set of data. Actually doing this would require that everyone transform their coordinates into a single standard coordinate frame which everyone agrees upon in advance. Once you do this, everyone can communicate effectively, but this doesn't really address the original puzzle question, because you need to use Lorentz transformations to compute things in the agreed frame, but the puzzle requires you not to use the Lorentz transformation.

So, your solution would work, but doesn't quite answer the original puzzle.

--Stuart Anderson
mr_homm
QUOTE (Lady Elizabeth+Mar 30 2012, 07:13 PM)
Easy.

Quantum entangled particle transceiver that exist in both ships.

Edit;- end of story;- thanks for that 0.0001µs of puzzlement.

Why don't we just teleport to safety while we're at it? Actually, your response doesn't answer the question, because the puzzle was about the content of the message, and you've described the transmission medium. It's analogous to my asking "what do you want me to say in my letter?" and your answer is "airmail."

Anyway, +1 funny. Thanks!

--Stuart Anderson

Next: my solution
mr_homm
OK, here is my version of the solution to Part 2:
('m running this puzzle on more than one board, and this is a cross-post of the solution I've given elsewhere. Just a disclaimer.)

First, use pulsars, which are plentiful, fairly evenly distributed across the galaxy, and (by the time we have interstellar spaceships) well charted. Each pulsar has both a known rotation period and a known spectrum, so it can be identified by giving either of these numbers. I'll use the spectrum, since I have other uses for the rotation period.

Second (and this is the core of the answer), you must send the values of the spacetime intervals between your ship and events at FIVE pulsars, together with identifying spectra for each.

Why 5? it's fairly simple really: On a plane, distances from 2 points wil pin your location, on a plane, distances from 3 points will do the job, and so on. It takes one more point than the dimension of the space you are in. And yes, things are different in space-time, because you can't use the Euclidean metric, but must use space-time interval instead. However, I've checked that it works, which is basically the same as having the answerr to Part 3.

Why space-time intervals? Because they are invriant! They have the same numerical value in ANY frame, so if I say my STI from pulsar #1 is 10^25 meters, then in the rescuer's frame, it will also be 10^25 meters. Of course, how the rescuer parses this into space and time components will be different from how you do, but the numbers themselves are unambiguous.

Why events? Because in space-time a steady beacon is a line (specifically a world-line) rather than a point, and as such it does not pin down your location in space-time. You must make a 3-d slice trhough space-time (determined by your velocity) to find what loks like space to you. Then the world-lines become points, and you can locate yourself as expected, but this only locates you on a specific slice through space-time, and the rescuer doesn't know what slice you're on! Dopplet shifts to express your velocity would take care of this, but it's messy. So I just decided to use events rather than whole world-lines as the external reference.

So that's the answer to Part 2... except for one thing:

(For those who want some practical details: ) How do you measure your spacetime intervals to the pulsars? This is actually incredibly easy. Pulsars not only have a very precise rotation rate, each pulsar also slows down at a fixed rate, due to the energy loss from gravitational radiation. These rates of slowdown are absolutely uniform over decades at least, according to our measurements. Therefore, you can do this: For each pulsar, look up what the star's spin rate was at that date and time your reference tables were compiled. Put that in your message, together with the known spectrum. Then the rescuer can identify the star, and also the particular moment in its history which you are taking as your beacon event.

Next, measure its spectrum and spin rate directly. The spectrum gives you the time dilation factor, which you apply to the observed spin rate. After all, if you see the frequencies in its spectrum as shifted somewhat, it means that everything you observe is shifted by the same amount. It's as if you were watching a movie of a car driving by a clock, and someone is playing the movie too fast or too slow. By noting how much time passes on the movie clock, and the distance traveled by the movie car, you can correctly determine the car's speed in the movies original reference frame, even if the movie is being played at the wrong speed (because of time dilation). So now you know the true rate of spin of the pulsar in its own frame.

Now use the known rate of spin and rate of slow-down to compute the date and time at which the pulsar had this) rotation speed. This tells you how long before or after the reference event that the light you are seeing now was emitted. So if the reference event was at (t0,x0,y0,z0) and the light you are seeing was emitted at (t0+T,x0, y0, z0) and your current location and time are (t1,x1,y1,z1), then the space-time interval from the beacon event to you is (ct1-ct0)^2 - (x1-x0)^2 - (y1-y0)^2 - (z1-z0)^2. But it gets easier than this! The light that you're seeing now traveled from its point of emission to you across a space-time interval (ct1-c(t0+T))^2 - (x1-x0)^2 - (y1-y0)^2 - (z1-z0)^2, which you know is ZERO, because it is a light ray! So subtracting the two space-time interval formulas will not change the value of your space-time interval, but will simplify the formula:
((ct1-ct0)^2 - (x1-x0)^2 - (y1-y0)^2 - (z1-z0)^2) - ((ct1-c(t0+T))^2 - (x1-x0)^2 - (y1-y0)^2 - (z1-z0)^2) = ((ct1-ct0)^2 - ((ct1-c(t0+T))^2 = c(t1^2 - 2*t1*t0 + t0^2) - c(t1^2 - 2*t1*t0 - 2*t1*T + t0^2 +2*t0*T + T^2) = cT(t0-t1+T).

So that's it: transmit the identifying spectra and rotation rates of 5 pulsars, together with cT(t0-t1+T) for each.

Now, on to Part 3, which is about the rescuer decoding this!
--Stuart Anderson
QUOTE (mr_homm+Mar 31 2012, 10:13 PM)
Why don't we just teleport to safety while we're at it? Actually, your response doesn't answer the question, because the puzzle was about the content of the message, and you've described the transmission medium. It's analogous to my asking "what do you want me to say in my letter?" and your answer is "airmail."

Anyway, +1 funny. Thanks!

--Stuart Anderson

Next: my solution

To Homm it may concern,

Anyone could've suggested pulsars .... it's insanely common knowledge, and I was far too horrendously embarrassed to mention the obvious

EXHIBIT A;- BBC Report (JUST YESTERDAY)

ps;- I wanted you to say "airmail" in your letter.

mr_homm
QUOTE (Lady Elizabeth+Mar 31 2012, 02:59 PM)
To Homm it may concern,

Anyone could've suggested pulsars ....  it's insanely common knowledge, and I was far too horrendously embarrassed to mention the obvious

EXHIBIT A;- BBC Report (JUST YESTERDAY)

ps;- I wanted you to say "airmail" in your letter.

I love "To Homm it may concern." In all my posts here, no one has ever come up with this one before.

As to "pulsars" -- yes, they are common knowledge, but "pulsars" isn't the answer to the puzzle, just one practical consideration relevant to its implementation. The solution I gave is that you must transmit the spacetime intervals between you and 5 known reference events, together with identifying information for the events.

About the URL you gave -- yes, C1 brought that to my attention earlier in the thread. Of course, I started the thread before that news item came out. BTW, it is not really relevant to the puzzle, because their method depends on calculating offsets from a basepoint on earth, rather than being fully independent of a coordinate system. This is how they achieve the great accuracy they mention, since they only have to compute offsets across the solar system, rather than across the full interstellar distances to the pulsars.

"Airmail." There, I said it.

Now, since Part 2 was so easy, how about giving the mathematical algorithm for decoding the 5 space-time intervals back into coordinates in the rescuer's frame, which is Part 3 of the puzzle. I'll expect your answer in about 0.001µs, since this is about 10 times harder than Part 2.

--Stuart Anderson
Confused1
1st buffalo> What do you think about this electric fence thing that's going up all round us?
2nd buffalo> Sorry, I can't talk right now.
1st buffalo> You can't talk because you're too busy?
2nd buffalo> No, I can't talk because I'm a buffalo.

Edit: Sorry.. among the LE stuff (a fertile brain if ever there was) I missed Mr H's solution to part 2.
QUOTE (mr_homm+Apr 1 2012, 12:10 AM)
Now, since Part 2 was so easy, how about giving the mathematical algorithm for decoding the 5 space-time intervals back into coordinates in the rescuer's frame, which is Part 3 of the puzzle. I'll expect your answer in about 0.001µs, since this is about 10 times harder than Part 2.

--Stuart Anderson

Au contraire,

It took me a mere 0.001ys to realise I've not a blooming clue how to calculate that algorithm.

ps;- great thanks for writing "airmail", you've simply made my day.

Will now, both crawl back in my illustrious filth pit and entirely cease to disrupt your excellent thread.

mr_homm
Solution time! Part 3:

You have received a transmission containing the spacetime interval values between a disabled ship and 5 pulsar events. You know the coordinates of these 5 events in your own frame, and you want to find the coordinates of the disabled ship, also in your own frame. How do you do it?

1: Notation: Let the events be E0, ... E4, and let the coordinates of each event be (t0,x0,y0,z0), ... (t4,x4,y4,z4) in your frame. Let the intervals transmitted by the disabled ship to each event be T0, ... T4. Let the unknown coordinates of the disabled ship be (t,x,y,z), which is what you want to solve for.

2: Pick E0 as your new origin of coordinates. All calculations will be in this new coordinate system. You then can get back to coordinates in your original system by adding the coordinates of E0 to your answer. (This all takes place inside your frame, no Lorentz transformation is used. All we're doing is shifting the location of the origin for convenience.)

3: Calculate the coordinates of E1, ... E4 in the coordinate system based at E0. (Just subtract (t0,x0,y0,z0) from each of (t1,x1,y1,z1), ... (t4,x4,y4,z4). Let's still call these coordinates (t1,x1,y1,z1), ... (t4,x4,y4,z4), and just remember that we have to add (t0,x0,y0,z0) to whatever we calculate, in order to get back to our original system.

4: Write down the formulas for T0, ... T4 in terms of coordinates:
T0 = (ct-ct0)^2 - (x-x0)^2 - (y-y0)^2 - (z-z0)^2 (actually, t0,x0,y0,z0 are all zero, since E0 is our origin)
T1 = (ct-ct1)^2 - (x-x1)^2 - (y-y1)^2 - (z-z1)^2
T2 = (ct-ct2)^2 - (x-x2)^2 - (y-y2)^2 - (z-z2)^2
T3 = (ct-ct3)^2 - (x-x3)^2 - (y-y3)^2 - (z-z3)^2
T4 = (ct-ct4)^2 - (x-x4)^2 - (y-y4)^2 - (z-z4)^2
These look like 5 simultaneous quadratic equations, which would be messy but could be done. But....

5: Subtract the T0 formula from each of the others to get 4 formulas that turn out to be linear, not quadratic! This is because the quadratic part of each term cancels; for instance (x-x1)^2 - (x-x0)^2 = (x^2 - 2xx1 +x1^2) - (x^2 - 2xx0 +x0^2) = -2x(x1-x0) + x1^2 - x0^2. Since x0=0 this is x1^2 - 2xx1. The same thing happens to the y, z, and ct terms in each equation, so you get the results:
T1-T0 = ((ct1)^2 - 2ctct1) - (x1^2 - 2xx1) - (y1^2 - 2yy1) - (z1^2 - 2zz1)
T2-T0 = ((ct2)^2 - 2ctct2) - (x2^2 - 2xx2) - (y2^2 - 2yy2) - (z2^2 - 2zz2)
T3-T0 = ((ct3)^2 - 2ctct3) - (x3^2 - 2xx3) - (y3^2 - 2yy3) - (z3^2 - 2zz3)
T4-T0 = ((ct4)^2 - 2ctct4) - (x4^2 - 2xx4) - (y4^2 - 2yy4) - (z4^2 - 2zz4)
Notice that thes all look very much alike; it's just the event number that changes in each equation.

6: Notice all the square terms in each equation collect together into the spacetime intervals between E0 and E1, ... E4. For instance, (ct1)^2 - x1^2 - y1^2 = z1^2 = S01, the spacetime interval between E0 and E1. Let's do this for each equation and then move the terms to the left, to get
T1-T0 - S01 = -2(ctct1 - xx1 - yy1 - zz1)
T2-T0 - S02 = -2(ctct2 - xx2 - yy2 - zz2)
T3-T0 - S03 = -2(ctct3 - xx3 - yy3 - zz3)
T4-T0 - S04 = -2(ctct4 - xx4 - yy4 - zz4)

7: After you divide both sides by -2, the right hand side is just what you'd get if you made a matrix A whose rows were the coordinates of E1, ... E4, multiplied by a column vector V of the unknown coordinates. The left hand side is a column vector B whose entries are (S01 +T0 - T1)/2, ... (S04 + T0 - T4)/2. The whole thing is just a matrix equation B = AV, and it can be solved by just inverting the matrix A and computing V = B^-1 A.

Summary: Calculate the spacetime intervals between E0 and the othe events and make a column of these numbers. Subtract T0 from each entry in the column. Make a column of T1 through T4, and take the difference at each place between this and the previous column and divide all entries by 2. This is B. Now take the coordinates of each event relative to E0 and lay them out in rows, so that the rows are ct1, -x1, -y1, -z1, and similarly for the rest, to fill up a 4 by 4 matrix. This is A. Now compute the inverse of A, and multiply this by B. The answer is a column ct,x,y,z containing the coordinates of the disabled ship. Finally, add the coordinates of the E0 in your frame to this to get the coordinates of the ship in your original coordinate system.

All done!
Confused1
QUOTE (Mr_Homm+)
So that's it: transmit the identifying spectra and rotation rates of 5 pulsars, together with cT(t0-t1+T) for each

I think we need to decode these cT(t0-t1+T) into something like:

CODE

-cT1^2 + x1^2 +y1^2 +z1^2
-cT2^2 + x2^2 +y2^2 +z2^2
-cT3^2 + x3^2 +y3^2 +z3^2    x Something = 0
-cT4^2 + x4^2 +y4^2 +z4^2
-cT5^2 + x5^2 +y5^2 +z5^2

the last bit looks possible (given time) apart from the extra player at position number 5. Maybe attempt the decode (anyone else?) and worry about No 5 later.

-C2.
mr_homm
@C1:

Definitely on the right track, and within 10 minutes of my posting my solution. Please see the post just above yours for my details. I'm sorry if I cut you off, since it looks like you were onto something, but my post came just before yours, probably while you were writing, so I didn't know you were preparing something.

Now that a day has passed, and Part 4 has had no takers, I'll post my (quite simple) solution to it:

First, get the space-time intervals to your 5 events, then wait a week or month or whatever time period seems feasible (you'll be waiting years for rescue anyway), and get the space-time intervals again. Transmit both the old and new intervals in your message, clearly labeled as two sets of data. It is not necessary to tell how much time passed on your clock, nor to tell the receiver which data was taken earlier. Just send two sets of 5 space-time intervals measured at different times.

The receiver can decode the first set into the space-time coordinates of an event, and similarly decode the second set. Now the rescuer has his coordinates for two space-time events, and these two points determine your trajectory. (I'm assuming that "disabled ship" means that you have no thrust, so you are coasting. Coasting is motion at constant velocity, and it plots out as a straight line in space-time. The rescuer now has two points on this line, and these determine the whole line.) Furthermore, it's extremely easy to write a formula for the coordinates of an arbitrary event on the line, simply by taking (ct, x, y, z) = (ct0, x0, y0, z0) + k*(c(t1-t0), x1-x0, y1-y0, z1-z0), where k is an arbitrary number and ct0, etc and ct1 etc are the coordinates of the two events. Want to know the ship's coordinates in your frame at time t? just solve t = t0 + k(t1-t0) for k, which gives k = (t - t0)/(t1 - t0). Now compute (x, y, z) = (x0, y0, z0) + k*(x1-x0, y1-y0, z1-z0), and you have the ship's location in your coordinates at any time t on your own clock.

All done!

PS: after doing this puzzle, I thought of an even easier solution, in the same vein as what I posted for Parts 2 and 3, but with simpler algebra. Anyone want to try to figure out what the improved scheme is? (Hint: look for messy algebra, and add an assumption which deletes a bunch of the variables....)

--Stuart Anderson
Confused1
@Mr homm,

My cache didn't have your solution in it until now - I'm quite happy to be cut off - I don't think I'd get there in finite time.

Excellent problem - many thanks for posting it.

-C2.
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