
Does the LHC collide protons at twice the speed of light? - jacobedawson
http://backreaction.blogspot.com/2019/04/dear-dr-b-does-lhc-collide-protons-at.html
======
ajkjk
Intuition for relativistic velocity works a lot better if you imagine changing
units so that velocities between (0,c) become velocities in (0, infinity).
Then two infinities added is just infinity, and adding two numbers which are
close to infinity is a number even closer to infinity.

This amounts to measuring 'rapidity' w
([https://en.wikipedia.org/wiki/Rapidity](https://en.wikipedia.org/wiki/Rapidity))
instead of velocity. Rapidity captures what velocity 'actually is': a
hyperbolic angle between the spatial coordinates and the time axis, given by w
= arctanh(v/c). For low velocities, w = v/c, so you can write v = wc and just
treat it like a regular velocity and it basically adds as you'd expect. But as
v/c -> 1, w -> infinity.

(Unfortunately, non-trivial rapidities along different spatial directions
don't exactly add, and it's complicated. Boosting in x and y can result in a
rotation in the xy plane also. Or something like that; I haven't looked at the
math in a while.)

I sometimes wonder if it would make more sense to define cw = v as the
'correct' extension of velocity to high energies, and just say "oh, meters per
second -- that was wrong. That's not what velocity is". Thinking of the speed
of light as 'infinite' seems very appealing.

~~~
noobermin
The problem of course is when position is actually important in a problem, you
need to integrate velocities instead of rapidity. The reason velocity addition
trips people up is because adding velocities inherently means "seeing velocity
as measured by another frame," which then requires Lorentz transform. For many
problems, you confine yourself to a single inertial frame, you can still do
physics comfortably and you never have to worry about adding velocities.

------
kmm
Well in a sense yes. The distance between the two protons decreases at almost
exactly twice the speed of light, as seen from the rest-frame of the detector.
That's not a violation of relativity as there is no physical object or
information moving that fast. And of course either proton sees the other
approaching at just a little less than c

~~~
iamgopal
Explain your last sentence ?

~~~
AceJohnny2
As an entity approaches the speed of light, its "timeframe" slows down. So the
timeframe of one proton is slowed down so much that, to it, the speed of the
incoming proton is only the speed of light.

A key insight of Einstein's relativity is that the speed of time isn't fixed
and absolute. Hence, "relativity".

I highly recommend Chris Lee's _excellent_ 2012 article on the history of "the
speed of light" at Ars Technica: [https://arstechnica.com/science/2012/09/the-
complicated-trut...](https://arstechnica.com/science/2012/09/the-complicated-
truth-behind-scientific-findings/)

~~~
jujulet
> speed of time isn't fixed and absolute

Isn't 'speed' defined with the use of 'time'? Doesn't that make an expression
such as 'speed of time' kind of meaningless?

~~~
garmaine
No, the insight of Einstein is that speed (of light) is fixed and space and
time are defined with respect to it. Is that weird? Yes. But that says more
about our evolutionary instincts being off than anything else.

Btw our atomic standard of length is defined with respect to the speed of
light, making speed (he first derivative of space with respect to time) more
fundamental than space itself. This isn’t just abstract theory.

~~~
jujulet
Ahh, I see. Thanks!

------
Gibbon1
I think it would worth pointing out that the two protons smash into each other
with a lot more energy than you would think if you thought kinetic energy was
proportional to speed squared.

~~~
Gibbon1
Coming back to link to this. Indicative of the amount of energy being pumped
into the beam. And this is 'just' SLAC.

[https://today.slac.stanford.edu/feature/beam-
dump.asp](https://today.slac.stanford.edu/feature/beam-dump.asp)

------
JshWright
If you find these sorts of questions interesting, I can't recommend the PBS
Space Time YouTube channel highly enough.

[https://www.youtube.com/channel/UC7_gcs09iThXybpVgjHZ_7g](https://www.youtube.com/channel/UC7_gcs09iThXybpVgjHZ_7g)

------
p1mrx
> you will see that adding twice 99% of the speed of light brings you to
> something like 99,9999% of the speed of light

That seems like too many nines... why write the formula, and then not bother
running the numbers through it?

(.99+.99)/(1+.99*.99) is roughly 99.995%

------
krick
That's a really bad explanation IMO. The author provides some weird formula
which you have to use when u and v are "near the spead of light", w/o actually
explaining how near the speed of light you have to be to use (u + v) / (1 +
uv) instead of u + v or even explaining why do you have to use another formula
at all. She basically says "wow, look at all this, what a nonsense! but I
promise it's true". Meanwhile the problem is really really simple.

No, Einstein hasn't "taught us nothing travels faster than the speed of
light". Instead, he taught us that neutonian velocity-addition formula is
imprecise, instead of u + v it's _always_ (u + v)/(1 + uv/c^2). It just
doesn't usually matter, since "wrong" u + v formula works good enough for our
everyday needs, because "c" is so huge compared to our usual "u" and "v" (so
uv/c^2 is usually close to zero). And that's it, that's why velocity-addition
of 0.99c and 0.99c gives us 0.9999c. Not all that "sometimes 1 + 1 equals 1"
nonsense.

------
sytelus
Here's another way to think about this: Your velocity is time dependent
quantity. When you are moving faster time for you is simultaneously slowing
down. When you almost approach speed of light relative to anything, time for
you slows down to almost zero relative to that thing. So everything runs in
super slow motion. Another particle coming at you at the speed of light
towards you will appear almost still. So you total relative velocity wrt to
other particle is still just speed of light. The key is again, its because
time has slowed down for you.

~~~
gus_massa
This comment is confusing/wrong.

> _time for you slows down to almost zero relative to that thing_

This is at least confusing. You would not fell anything special. If all the
windows are covered, you will not note that you are moving fast or slow.

> _Another particle coming at you at [almost] the speed of light towards you
> will appear almost still._

If you are moving at a speed of 99% of c in the "laboratory" reference frame,
and the other particle is still in the in the "laboratory" reference frame,
then (if the front windshield is not covered) you will see that the other
particle comes to you at a speed of 99% of c.

If you are moving at a speed of 99% of c in the "laboratory" reference frame,
and the other particle is moving in the oposite direction in the in the
"laboratory" reference frame, then (if the front windshield is not covered)
you will see that the other particle comes to you at a speed of 99.995% of c.
(See the calculation details in the article.)

------
lisper
Heh, I just now published a blog post of my own about this same topic:

[http://blog.rongarret.info/2019/04/we-interrupt-this-blog-
to...](http://blog.rongarret.info/2019/04/we-interrupt-this-blog-to-bring-
you.html)

------
pontifier
One strange thought that struck me when studying relativistic time dilation
and length contraction is about light from distant stars.

For an observer traveling with a photon from a distant star that reaches your
eye, they would see a collision between the atom that emited the photon, and
the back of your eye because there is both no space, and no time separating
them.

------
ryanthedev
Terrible analogy. 1+1=1. More like Ax + Bx = 1.

------
revskill
If photon travels at speed of light, then it has no mass ?

But if it has no mass, it shouldn't bring any energy, though.

~~~
gus_massa
The protons in this device travel _almost_ at the speed of light.

------
auslander
> .. 1 plus 1 is not equal to 2.

Which leads to contradiction. Therefore, the initial assumption, that Earth is
not flat, must be false. [0]

[0]
[https://en.m.wikipedia.org/wiki/Proof_by_contradiction](https://en.m.wikipedia.org/wiki/Proof_by_contradiction)

~~~
dylan604
Of course 1 + 1 != 2. 1 + 1 = 10. Somebody forgot to carry the 1

------
cs02rm0
This is the same physics I was taught in school as for two cars crashing into
each other at 70mph I presume? Neither has a 140mph impact, although a 140mph
impact would involve more than double the energy anyway AIUI.

~~~
maxander
No, I don't think so- two cars impacting head-on at 70mph _would_ have a
"140mph impact," insofar as that it would be essentially equivalent to one car
traveling at 140mph hitting a stationary car.

~~~
thatswrong0
I could have sworn that a perfect head on collision was basically the same
thing as a collision at the same speed into a perfect stationary wall?

Mythbusters experiment:
[https://www.dailymotion.com/video/x2n9j62](https://www.dailymotion.com/video/x2n9j62)

~~~
Dylan16807
Hitting a car is not the same as hitting a wall.

All three of these do similar damage, if the cars have equal mass:

    
    
      * 70mph car vs. 70mph car
      * 140mph car vs. parked car
      * 70mph car vs. wall.
    

The only real question is whether you consider "140mph impact" to mean "vs. a
similar-mass car" or "vs. a wall". These are wildly different impact
intensities that should not be confused.

~~~
cs02rm0
FWIW I was thinking immovable wall rather than parked car for the 140mph
impact.

The wall is complicated enough for me without thinking about multiple crumple
zones, handbrakes, etc.

