Does the LHC collide protons at twice the speed of light?
backreaction.blogspot.com
backreaction.blogspot.com
This amounts to measuring 'rapidity' w (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.
It's fairly primitive. The original intent was that it would be a base to play around with CSS on, and to maybe rewrite using various front end frameworks to learn them, and add features like named variables, showing Minkowski diagrams, cloning rows, and things like that. I never got any time for any of that, so it never got past the very simple stage, but is fine for doing things like figuring out the ladder paradox and things of that ilk when learning about special relativity.
[1] https://github.com/tzs/Physics-special-relativity-calculator
Edit: someone asked for a more detailed explanation, so here it is. Because of the equation in the article, it's impossible to see anything as moving faster than the speed of light in any reference frame; they just appear to asymptotically approach it. Both protons are moving at nearly the speed of light and perceive the other as moving at a speed that's slightly closer to 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...
Isn't 'speed' defined with the use of 'time'? Doesn't that make an expression such as 'speed of time' kind of meaningless?
You can in fact express this in terms of "seconds/second", as in how many seconds pass in some other reference frame for each second in yours. In that sense, the "speed of time" can in fact be meaningful.
Relativity provides a more rigorous way of modeling this, treating spacetime as a four-dimensional manifold. When you move through the spatial dimensions relative to some reference frame, the rate of your motion through time is reduced relative to that same reference frame. This is commonly modeled on a spacetime diagram: https://en.wikipedia.org/wiki/Minkowski_diagram
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.
Think of an object's "velocity" as a unit vector in a 2d circle. X axis is fraction of C(meter/second), Y axis is rate of time passage (seconds/second).
Every object no matter how slow moving has a velocity on the unit circle. An object that isn't moving is moving forward in time at the same rate as you, the observer. But an object that is moving very quickly moves much more slowly through time than you do as the observer.
So the key distinction is that time is not a fixed reference, it's subjective depending on who is watching.
General relativity basically starts with the assumption that everyone is moving through space-time at the speed of light. We sedentary material beings experience that mostly as moving through the dimension of time. Non material things like photons experience it as moving through space (because they lack inertia). When you accelerate to high speeds, you’re somewhere between the two extremes.
The issue is in your own frame (say if you're a muon or a person), unless you're accelerating, you're at rest with respect to your frame. Physics after Einstein's special relativity assigns to each reference frame a way to measure spatial differences and times (cute terms used in textbooks are rulers and a clock). Now, if another object is moving wrt you at a fast speed that you measure that is close to c, when you measure things that happen to it, it appears to happen slower as measured by your clock. So the reality is it appears slower to you in your frame. An good example of this is a particle like a muon will appear to have a longer lifetime if it were moving relativistically (near c). The converse is (sort of) true: for the moving object, in its frame, when accounting for the distances traveled, your clock also appears slower to it.
The reason framing it this way is important for me when I teach students is because it honors ones of the main points of special relativity that it did away with Newton's concept of a universal rest frame and made the important rest frame any inertial frame, and this helps demystify SR from being something esoteric but instead be based on simple logic: everyone is at rest in their own rest frame, so the physical laws in two frames moving respect to one another must be the same laws if you switch from one frame to the other. Since physics in any given rest frame seem to require light moves at c, certain other things must change in order to have both frames agree a light ray observed by both move at c.
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%
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.
> 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.)
http://blog.rongarret.info/2019/04/we-interrupt-this-blog-to...
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.
But if it has no mass, it shouldn't bring any energy, though.
Which leads to contradiction. Therefore, the initial assumption, that Earth is not flat, must be false. [0]
From the perspective of either vehicle, you do see an oncoming vehicle traveling at (approximately) 140 miles per hour.
When dealing with things traveling close to the speed of light, this doesn't work, because the relativistic correction term becomes significant. The equation they show on the blog
(u + v)/(1+uv)
[1] But still pedagogically sound when teaching relativity :)
Much easier to understand that the "simplified" version, which is usually the case.
It is much easier to plug in u and v directly into the simpler equation that contains c^2.
Otherwise (the indirect method), to get fractions, first you do a substitution a=u/c, b=v/c, arrive at:
c * (a * b) / (1 + a * b)
Then you convert the car speeds to fractions, only to multiply back by c=299792458.
Curiously enough, this sort of thing rarely occurs in the original publications, so they are usually easier to read than obfuscated textbooks or blog examples.
It's been far too long since I had a physics class...
If we consider debris or etc flying away after the collision, then you're right that they're different- the combined velocity of the two cars is 0mph in the both-moving case and 140mph (in some direction) in the one-moving case, and the velocity of all the debris will sum to that velocity. (And the sum of velocities would be weighted appropriately if one of the cars was more massive than the other.)
Choosing a car, choose tons, not stars :)
Mythbusters experiment: https://www.dailymotion.com/video/x2n9j62
Edit: if the other object is a car, then yes, there is no difference in either case from your perspective.
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 wall is complicated enough for me without thinking about multiple crumple zones, handbrakes, etc.
It does depend how much of that energy is dissipated in the collision though.