I’ve heard it said that no object with mass can travel at the speed of light, because accelerating such objects to light speed requires infinite energy. That is, more energy than is available in the entire universe.
I’ve also heard it said that electrons have mass, and that the Large Electron Positron collider was able to accelerate an electron to 99.9999999% the speed of light. The energy required to do this was 209 GeV (Giga electron Volts - a unit of measurement I’d never heard of). According to Chat GPT, 209GeV is enough to keep a standard 40 watt bulb lit for about 3 microseconds.
The speed of light is 186,282 miles per second. 99.9999999% of the speed of light is 186,281.99998137 or 2.998 centimetres per second slower than the speed of light.
So, why is it that moving an electron at 186,281.99998137 miles per second requires less energy than it takes to turn on a lightbulb for a fraction of a second, but accelerating it that extra 2.998 centimetres per second requires more energy than exists in the entire universe? It makes no sense.
P.S. - Please keep any answers as simple as humanly possible, the less math involved the better. I’ve no physics training whatsoever. Thanks.
A massive particle has a kinetic energy which depends on its mass and speed, you just need to estimate how much. This relationship is not linear; it is basically Einstein’s famous relation. If the particle is going really fast, the energy is approximately equal to its momentum times the speed of light. This will be proportional to \bigl(1-v^2/c^2\bigr)^{-1/2}, which grows without bound as the speed approaches the speed of light.
You can work through this step by step if you are open to a little physics; basically it is Newton’s Second Law, you just need to keep in mind the momentum of a relativistically fast particle (which, to reiterate, tends to infinity as the particle gets faster and faster). We are talking about a proton or an electron here; a particle like a photon will have a finite momentum even at the speed of light, because it has no mass.
Or, if you want a short answer, the velocities combine according to Lorentz’s transformations; you cannot just linearly add on an extra 2.998 cm/s or whatever seems insignificant. There is a reason why those particle accelerators cost real money (and it is more money than to turn on a 40-watt bulb).
And, just to add, the reason that a photon has any momentum at all despite having no mass is due to mass-energy equivalence. The normal equation for momentum, p = mv, is Newtonian and doesn’t apply at relativistic scales. The momentum of a photon is expressed as p = h/λ, where h is the Planck constant and λ is the photon’s wavelength. IOW, the photon’s momentum is a direct function of its frequency, because higher electromagnetic frequencies have higher energies.
@DPRK gave you the answer. But to put it simply, the energy required to accelerate something as tiny as an electron to very high speeds seems tiny at macro scales, but as it approaches the speed of light, the energy demands for higher speeds increase literally exponentially. You can approach c asymptotically, but you can never reach it. The interest in higher-energy accelerators is not to achieve higher particle speeds per se, but to achieve higher energies which may reveal new quantum phenomena.
On a slightly related note, there is something I have yet to see a clear explanation of.
We are told that neutrinos must have non-zero rest mass in order to be able to oscillate between the three ‘flavors’. Yet as far as I know nobody has measured a velocity for neutrino travel different from actual light speed. (Form supernova observations etc).
What gives? I imagine there is some handwaving loophole involving ‘quantum’…
This might be your biggest misconception. 209GeV was the energy of the electron stream. That’s not the energy required to get the electrons up to such a speed and fill them with so much energy. This is the difference between “how much energy does my car have when traveling at 100mph” and “how much energy does it take to get my car up to 100mph”.
The LEP probably consumed over 200 Mega Watts of electricity to get those tiny, nearly massless, electrons up to such a speed. That’s enough to power your 40W bulb for half a millenia.
We have measured the speed of some neutrinos but it is almost certainly a variable number depending on how much energy they got when created. Things that are massless must travel at the speed of light. Things with mass must travel below the speed of light. Here is one measurement that was made for neutrino speed:
The most precise agreement with the speed of light (as of 2012) was determined in 1987 by the observation of electron antineutrinos of energies between 7.5 and 35 MeV originated at the Supernova 1987A at a distance of 157000 ± 16000 light years. The upper limit for deviations from light speed was:
< snip >
thus more than 0.999999998 times the speed of light. This value was obtained by comparing the arrival times of light and neutrinos. The difference of approximately three hours was explained by the circumstance, that the almost noninteracting neutrinos could pass the supernova unhindered while light required a longer time.[7][8][9][10] - SOURCE
And neutrinos are bloody hard to detect.
Though if a lot of more sensitive future measurements fail to show them moving JUST a bit slower, we may be scratching our heads…?
What I find fascinating is the time dilation effect on really fast-moving objects. JWST has extended its reach to 13.something billion years, not long after the Big Bang, so a photon is generated and streaks literally across the universe for far longer than the sun has existed, never mind the earth nor the telescope whose detector it smacked into at long last.
No quantum handwaving. Neutrinos just go really, really close to the speed of light. Close enough that our instruments aren’t sensitive enough to tell the difference.
Of course, the further a neutrino travels, the easier it would be, so the neutrinos from Supernova 1987a would be where the distance in speed would have been most noticeable, as you referenced (that’s the only supernova we’ve had close enough to detect neutrinos from, in the time since we invented neutrino detectors). It still wasn’t good enough, though: Part of the difficulty is that supernovas are complicated, and the light and neutrinos weren’t emitted at exactly the same time to begin with, and not even all of the neutrinos at the same time. It’s like trying to time a race, where not all the runners started at the same time.
No, that’s the energy needed to get them up to that speed. It’s just that our processes for doing so are very inefficient, so you also need to waste a lot of energy in the process. But that energy wasn’t energy used for speeding up electrons; it was energy used for (inadvertently) heating up a bunch of liquid helium.
And powering the klystrons and RF fields in the acceleration cavities which directly contributed to particle acceleration. Even if you exclude all the energy (50+kW) used in the cooling systems, it still took a huge amount of power to accelerate the electrons and then to keep them at that speed. Saying it’s just due to inefficiencies is semantics. It takes a great deal of power to reach 209GeV. The OP’s statement claiming the “Large Electron Positron collider was able to accelerate an electron to 99.9999999% the speed of light[, and] [t]he energy required to do this was [equivalent to keeping] a standard 40 watt bulb lit for about 3 microseconds” represents a misconception worth addressing. It takes a massive amount of energy to do that.
And your claim that none of the energy was used to actually accelerate the electrons is completely false.
It’s too late to edit my statement that your claim is completely false, but I see what you mean by this now.
I just didn’t want to OP to think we can simply accelerate particles to such a high energy without consuming a significant amount of power in the process.
There have been cosmic rays detected with energies exceeding 10²⁰ eV. Which is, like, a lot, but not that much by ordinary macro standards. However, no one has a good idea of how to get that high; at least, you are not going to be doing it at home.
It’s not a problem, and nothing on tap will be able to measure the speed deviation from c. The expected difference is just too small.
The reason the neutrinos must have mass is that the neutrino flavor oscillations you mention require the three neutrino masses to be different from one another, and thus they can’t all be zero. Something involving “quantum” (keeping your scare quotes) does enter here. A core feature of quantum mechanics is that if you know something about a quantum system, there might be something else that you can’t know at the same time. For neutrinos, “mass” and “flavor” are a pair of things that you can’t know at the same time, and there are three possible values for each (three flavors and three masses). And if the possible masses are different, then a neutrino with a known flavor will soon not have a known flavor, since the different mass “states” that make it up are themselves evolving differently relative to one another as the neutrino travels. If the three masses were all the same – for instance if they were all zero – then the three underlying mass states would all behave equivalently for the present purposes, and thus a given quantum mechanical mixture of them would stay the same over time.
It is actually possible for the lightest of the three neutrino masses to be zero. It’s a bit counterintuitive, maybe, that two of them would have masses and one wouldn’t, but then, something similar occurs with the photon and the Z boson, which are very similar particles except that the Z is massive, so maybe.
Of course, this would mean that the expected value of the mass of any of the three flavors of neutrino (e, mu, or tau) would still be nonzero, since those are all mixes of the three different mass versions.
