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A bunch of LHC relativity questions 
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#1
Dec1709, 10:11 PM

P: 19

Hello all. First time posting here.
I have a basic understanding of special relativity and I have few questions about what the Large Hadron Collider (LHC) looks like to a proton going around inside it. As I thought more and more about it the few questions turned into a bunch of questions. I know that special relativity might not apply everywhere since the proton is accelerated by traveling in a circle. I am not sure where special relativity can be used and where general relativity would be used. What shape does the LHC have to a proton traveling the 27km circumference with a Lorentz factor of 7500 (in about 89 ns)? I am guessing it would look like a very flat oval with a diameter contracted by 7500 in the direction the proton is moving and the diameter perpendicular to that (about 8.6km) would not be contracted. Is that correct? At least, that is what it would look like to a proton traveling straight and tangent to the LHC with the same speed. Does traveling in an arc make a difference? By the way, assuming it's an oval, is the oval an ellipse or some other shape? (When I say "looks like" I mean according only to whatever relativistic transformations would apply, not what it looks like to a camera, but that would be another interesting question.) Also, would a proton see clocks on the circumference of the LHC running slower or faster? The moment a proton passes by a clock on the circumference it seems that special relativity would say it would be running slower. Is that correct? But it also seems that when a proton makes its 12ns (89us/7500 I'm assuming) cycle according to the proton's clock, an LHC clock on the circumference would have to show 89us having passed when a proton passes by a second time making the clock appear to run faster. How is this resolved? How would a clock at the center of the LHC or on the opposite side of the LHC appear to run to a proton? What about a proton's clock on the opposite side? How long is the 27km LHC circumference as seen by the proton? It seems it could be 12 light nanoseconds (3.6m) considering the proton sees itself traveling in the LHC at near the speed of light for 12ns to make a cycle, or it could be about 2 x 8.6km = 17.2km looking at the very flat oval with the circumference being about two noncontracted diameters, or it could be 27km since it seems the proton is always at the most outer point of the oval from its point of view. I'm not sure if this is a meaningful question but, if a proton were to slowly feed out a string, with 1 meter markings, behind itself as it went around, how much string would be fed out, by the markings, when the string gets all the way around? I am guessing that it could be 27km x 7500 = 20,2500km, calculating from the LHC point of view if the string is contracted. If the proton looks across the LHC to the other side, what marking does it see on the string. It seems it would be the halfway mark, or could it be something else? Are the 1 meter string markings on the opposite side contracted even more than 7500 since the string is traveling in the "oncoming" direction? If so, how much? From the four different numbers the circumference, are some more correct or meaningful than the others? Also, what radius of curvature and centrifugal force (how many g's) does the proton experiences as it goes around? And finally, what would everything look like to a proton's video camera? 


#2
Dec1809, 12:11 AM

P: 3,967

Hi, welcome to PF :)
Now if the circulating proton considers itself at rest, and measures how long it takes for a mark on the perimeter of the LHC ring to complete a circuit it will calculate the length of the LHC ring to be about 3.6 metrs and it will put this down to length contraction of the ring due to the motion of the ring relative to the proton. It could also calculate that the proper length of the ring is 27kms when length contraction is allowed for. 


#3
Dec1809, 02:14 AM

P: 1,235

Teve,
General Relatvity is irrelevant to this question. Special relativity just does the job from A to Z. It is a widespread misconception that SR cannot deal with acceleration. Actually, this is even not a misconception, this is plain wrong. Michel 


#4
Dec1809, 03:39 AM

P: 19

A bunch of LHC relativity questions
If I did my math right, I think the instantaneous length contraction and time dilation for two protons at near light speed, each with Lorantz factor 7500 is approximately 15000. I think that would apply for instantaneous clock rates for nearby oncoming protons. Is that correct? I think it would also apply to the marked meters on the string. But two oncoming protons should encounter each other twice a cycle and both should see their clocks showing 6ns having elapse since their last meeting. It seems that observed clock rates have to run faster for clocks while they are separated before they later meet up again.
Also, does a counter circulating proton looking across the LHC see noncontracted meter markings on the string since they are moving in the same direction? Back to clocks. What clock rate does a proton see (not by camera but by relativistic transformation) for protons on the other side going in either direction, and what clock rate does the proton see for an LHC clock on the other side and at the center of the LHC? Counter circulating protons on opposite sides would have the same instantaneous velocity, but then they are separated in their "gavitational" fields. For the center LHC clock it seems the clock rate the proton sees is 7500 times faster than its own. How much elapsed time is seen by a proton on a proton clock or LHC clock when it is on the other side? (Again, not by camera but by transformation.) It seems that counter circulating protons would each have to see 3ns (1/4 cycle) of elapsed time on each others (and their own) clock when they are opposite each other just after they last met. If two counter circulating protons synchronize their clocks to 0 when they meet, as a function of the proton's time, what is the time on the other proton's clock? I don't think they read the same (from one proton's view) shortly after the meet, but they will be synchronized when they meet again and when they are opposite each other it seems. It seems it has involve a periodic function of some sort. Protons circulating in the same direction and on opposite sides are "oncoming" but never meet. It seems they can synchronize using the center LHC clock. How do they see each others clock rates? It seems they would both see the same increased clock rate for the center clock. This seems to suggest that proton clocks always on opposite sides, and all proton's circulating in the same direction for that matter, see each others clocks running at the same rate. Is this right? But is seems the same can be applied to the counter rotating protons so that all proton clocks run at the same rate relative to each other. This contradicts the idea that oncoming protons near each other see different clock rates between them. What am I missing? I have heard that special relativity can be used for "no gravity/mass" situations but I am not sure how it would be applied to all the questions above. Specifically the LHC circumference clocks and maybe the counter circulating clocks which are time dilated when near by but have to be the reverse of time dilated (expanded?) at other times. Also, one more question. If the LHC were on a very dense planet with protons accelerated to a Lorentz factor of 7500, and I were hovering above it far away so that the gravitational time dilation was, say, a factor of 10, is the overall time dilation I see for the protons just the product of the two, or 75000, or is the general formula more complicated? 


#5
Dec1809, 04:21 AM

Sci Advisor
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#6
Dec1809, 06:05 AM

Mentor
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#7
Dec1809, 06:20 AM

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#8
Dec1809, 06:25 AM

Mentor
P: 16,985

Even just that is already complicated because the definition of simultaneity is continuously changing.



#9
Dec1809, 07:25 AM

Sci Advisor
P: 1,892

Hmm..
I'd say that kev is talking about a snapshot, where the proton's motion can be approximated by a straight, inertial line: 


#10
Dec1809, 05:05 PM

P: 3,967

There are quite a few tricky issues here (and a lot of questions!) and I may have got some of my answers wrong. Hopefully the other members here will pick up on my errors. 


#11
Dec1809, 08:09 PM

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#12
Dec1809, 08:37 PM

P: 3,967

As you say, a sphere looks like sphere (from a camera point of view) at relativistic speeds just as sphere looks the same from any angle so the visual rotation that occurs has no effect on the visual appearance of a sphere if we ignore surface markings. On the other hand a circle looks like a line from the point of view of a camera moving in the plane of the circle and continues to look like a line (of the same length) at relativistic speeds. Square and rectangular boxes do appear distorted to a camera at relativistic speeds because of the lack of rotational symmetry. I think it is important to make clear that length contraction can be measured by multiple observers in the same frame piecing there local measurements together, while the view of an relativistic object from a single viewpoint such as a camera is distorted by light travel times from distant parts of the object and when those light travel times are taken into account, the length contracted shape of the object becomes apparent. 


#13
Dec1909, 12:34 PM

P: 19

Thanks kev.
I stand corrected on the combined Lorentz factor. I did the math (algebra) wrong. If I did it right this time, the Lorentz factor for two oncoming protons each with Lorentz factor k=7500 is 2*k*k1 = 112,499,999 which agrees with what kev came up with. I did not completely follow the Doppler discussion. I will have to give some more thought to how protons observe each others clocks and perimeter clocks. I am fairly convinced (I think) that all protons observe the center clock running 7500 time faster, circumference clocks running 7500 times slower as they pass by, and that oncoming proton's see each others clocks slowed by 112.5 million. But at some time later before the next meeting, an oncoming proton's clock and circumference clock must then be observed to be running fast to make up for running slow in order to show 6ns at the next proton meeting, and likewise 12ns for the next circumference clock meeting. I would like to know where/how/why the time is made up. I still can't find much on what centrifugal force/acceleration the protons experience. Maybe their are experts here who could comment. What things look like to a camera would be interesting but I consider that another layer of complexity. For now I be happy to understand how things are observed or determined relativistically only. Another question that came to my mind is how long does a light pulse emitted forward and backward from a proton (say, inside a mirrored beam pipe) take to get back to the proton. I don't know if this would lead to any help answering my remaining questions. (By the way, from http://public.web.cern.ch/Public/en/LHC/Factsen.html the exact LHC circumference is 26659m.) From the LHC frame of reference, the time for a forward light pulse to go around and catch up with the proton it started from should be 26659m/(cv) = 26659m/(c*(1sqrt(11/(7500*7500)))) = 10004s. From the proton's point of view the forward light pulse travels along the string (described in my first post) which is 7500 times the actual circumference. I would think the pulse travels locally along side the string at the speed of light. So the time is 7500 x 26659m / (299,792,458m/s) = 666.9ms on the proton's clock, a factor of 15000 from the LHC calculation. I was expecting 7500. What did I miss? 


#14
Dec1909, 02:14 PM

P: 4,663

This might help. A while back experimenters at Brookhaven National Laboratory stored radioactive muons with a gamma (γ) of about 29.4 in a magnetic storage ring. The natural restframe lifetime of the muon is τ=2.2 microseconds. This is an important invariant "clock" for the muon. The circulating velocity of the muons in the ring was about βc. The observed laboratoryframe lifetime by the experimenters was γτ = 64.7 microseconds, and the average distance the muons traveled before decaying was βcγτ.
Bob S 


#15
Dec1909, 03:07 PM

P: 19

Bob S, that seems to be similar to the calculation I made from the LHC frame of reference which I think is correct. I am a little more suspicious of my second calculation.



#16
Dec1909, 03:30 PM

P: 3,967

[EDIT] I have checked your calculations and they seem to be OK. The calculations assume the speed of light is measured as being the same in both frames which addresses the issue I mentioned above. That means we are left with explaining why the difference in the time interval between the frames is twice the amount that can be accounted for by simple velocity time dilation by straight forward Special Relativity calculations. Didn't Einstein find he was out by a factor of 2, in his first draft of General Relativity when working out how much star light should be deflected by the Sun during an eclipse? I guess you would have to take into account whether the mirrors are attached to the tunnel in which case they would be moving relative to the proton or if they are attached to the string. I think it might be worth having a look at the Sagnac effect in this context too. 


#17
Dec2009, 05:01 AM

P: 3,967




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