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) 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.
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.
Speaking of special relativity, people learning basic special relativity might find this simple special relativity calculator I made useful [1].
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.
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
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".
This may be a bit pedantic, but I don't like the framing "its 'timeframe' slows down" which is similar to an often used phrase "when you go close to the speed of light, your clock runs slower" because it gets the situation backwards and actually misses Einstein's key insight about relativity.
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.
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.
This is a mostly incorrect approximation of the concept, but I think the visual representation is useful.
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.
This isn’t an incorrect approximation. It’s exactly how relativity was taught in my upper division physics class in college.
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.
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
Someone please correct me if I misunderstand, but it might be more useful to phrase it as, "the speed through time is a function of the speed through space"
The whole point of relativity is that space and time are part of the same fabric: spacetime. Movement through both is relative, and inseparable. The Lorentz factor is a factor for distance, time, and mass.
Even the word "movement" indicates this connection. You can not move if you have no time. And although it may be hard to imagine, the opposite is also true. Time doesn't really make any sense without space -- nothing can happen and therefore it can't take time. I am not a physicist, but I like to imagine that space actually created time. As soon as something separates from a singularity, you have time. It also makes me feel a lot more comfortable about quantum physics. Once you get smaller than a Lorenz wavelength, you don't really have space any more, just a probability that a particle may be somewhere and it magically teleports around. Well, you have to expect that, really, because if travel no distance, it can not (by definition) take any time. Probably not actually how things work, but it makes my brain happy anyway...
There are multiple references frames, and events can appear to be happening at different rates in different frames. So what you said actually does make sense because there are any number of references frames, each with their own view of how fast events are happening.
From the frame of reference of each proton, the opposing proton is going nearly the speed of light.
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.
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.
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.
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.
> 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.)
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.
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.
No, not unless you were making relativistic corrections for the vehicles, which would be silly[1].
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)
represents the velocities in fractions of speed of light. When u and v are much smaller than the speed of light, the denominator can be ignored, as it's close to 1, and we simply get (u+v).
[1] But still pedagogically sound when teaching relativity :)
Omitting conversion factors is a bit like omitting type information. If you already know what everything is, it doesn't add information, and can make the code look simpler. If you don't, not omitting them can help make sense of things when you're learning and lack context.
To get back a meaningful result for the car example, c=1 does not really help.
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.
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.
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.
Yes, if the cars are identical and collide perfectly enough that both crumple in exact symetry. Each absorbs and so the damage is split, as opposed to hiting a non-absorbing wall. But that is basically impossible to setup.
No. If the wall is stationary and infinitely massive, you’d bounce off at the same velocity; if it was moving at the same speed as you you’d bounce off at three times the original velocity.
Edit: if the other object is a car, then yes, there is no difference in either case from your perspective.
If the collision is completely inelastic, then there will be no bouncing. In a car, usually the crumple zone completely absorbs and dissipates all the energy kinetic.
Assuming similar vehicles that come to a stop, both cars experience a deceleration from 70 mph to zero. That is the same change of momentum per vehicle as 1 car hitting a stationary wall at 70.
Hmm, wouldn't a car of the same mass going the same speed joust counteract all the force exactly? If one of the cars had half as much mass, wouldn't that lessen the impact of the larger car and increase it for the smaller car, as the combined center of gravity afterwards would be moving inthe direction the larger car was traveling?
It's been far too long since I had a physics class...
Well, maybe we're using different terms- as the article notes, there's technically no such thing as a "such-and-such speed impact." I was speaking in terms of how much kinetic energy is liable to go towards damaging the cars involved (or into the fireball of a particle collision); in that case, if you have X energy in two cars, or 2X energy in one and 0 in the other, the sum is the same.
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.)
On a side note, if a massive 4wd heads-on collides with Corolla, the 4wd guy will have scratches and bruises, Corolla people would not suffer, instant death.
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.