# Why does it require an infinite amount of energy to reach the speed of light?

• B
• mucker
In summary, the conversation discusses the concept of speed in space and the impossibility of reaching the speed of light. The speaker poses a hypothetical scenario of using rockets to increase speed and questions why it would require infinite energy. They also question the relationship between speed and mass, and the constant speed of light. The response suggests studying special relativity systematically and recommends textbooks for a better understanding. It also mentions that the impossibility of reaching the speed of light follows from the postulates of special relativity.
As you go faster, you get heavier (so you need more energy to go faster)
So, 0-10, "five figNewtons"
10-20, "five point, a lot of zeroes, one figNewtons"
20-30... so the faster you go the more figNewtons it requires to go faster again.

e=mc2, the more energy, the more mass. no idea how it works, if the inertia of the mass just increases or what.
You're referring to the concept of relativistic mass, which generally is no longer used in teaching SR. E.g. Helliwell has a good summary of its disadvantages. The reason you have no idea how it works is that there is no physical reason for mass to change!

We also have a Insight on it:

https://www.physicsforums.com/insights/what-is-relativistic-mass-and-why-it-is-not-used-much/

PeroK said:
You're referring to the concept of relativistic mass, which generally is no longer used in teaching SR. E.g. Helliwell has a good summary of its disadvantages. The reason you have no idea how it works is that there is no physical reason for mass to change!

We also have a Insight on it:

https://www.physicsforums.com/insights/what-is-relativistic-mass-and-why-it-is-not-used-much/
well I take it back, looks like i do have an idea how it works (it is inertial)

nice insight

Omega0, berkeman and PeroK
PeroK said:
You're referring to the concept of relativistic mass, which generally is no longer used in teaching SR. E.g. Helliwell has a good summary of its disadvantages. The reason you have no idea how it works is that there is no physical reason for mass to change!

We also have a Insight on it:

https://www.physicsforums.com/insights/what-is-relativistic-mass-and-why-it-is-not-used-much/
That’s a great Insight, but one thing I’ve wondered is why the concept of relativistic mass was even invented, other than the need to keep equations more familiar. Is there another more useful reason for it being created?

Grasshopper said:
That’s a great Insight, but one thing I’ve wondered is why the concept of relativistic mass was even invented, other than the need to keep equations more familiar. Is there another more useful reason for it being created?
It's not the worst idea, especially when you begin studying SR. The problem is that it can't be extended to GR and it's very clumsy when doing relativistic particle physics generally.

I think Don Lincoln in one of his videos quotes Corinthians in regard of relativistic mass as a childish thing:

When I was a child, I spoke and thought and reasoned as a child. But when I grew up, I put away childish things.

PeroK said:
It's not the worst idea
It is indeed not the worst, but there are a lot of bad ideas that are quite bad even though they aren’t the worst.

mucker said:
I don't need equations, I would just like to pose a question which contradicts the above statement (I know I am wrong btw, I want to see where I am going wrong).

This video explains it well.

Like I said:

for fast things, it's about 1 point a few zeroes 1 figNewtons, or you can convert that to graham-cracker force.

As you get faster - gamma goes higher, momentum = gamma * mass * velocity. this video shows this gamma value - and the effect is on the inertia.

Grasshopper said:
That’s a great Insight, but one thing I’ve wondered is why the concept of relativistic mass was even invented, other than the need to keep equations more familiar. Is there another more useful reason for it being created?
The concept of relativistic mass was invented in Einstein's very first paper on the subject in 1905. At this time he had not a complete understanding of the dynamics of a point mass. This was clarified pretty soon by Planck. The full understanding of the mathematical structure of special relativity has then be discovered by Minkowski in 1908, and since then all physicists should work in the most simple mathematical formulation, which is to write down covariant laws in Minkowski space. This also makes the physical concepts very clear.

All intrinsic properties of matter are defined in an appropriate rest frame of the matter. For a massive particle it's simply the rest frame of this particle. Then mass is defined in this frame of reference via ##E_0=m c^2##. Since energy and momentum together build a four-vector ##(p^{\mu})=(E/c,\vec{p})## you get the relation between energy and momentum in an arbitrary inertial frame by ##E=c \sqrt{m^2 c^2+\vec{p}^2}##. In covariant form this "on-shell condition" reads ##p_{\mu} p^{\mu}=m^2 c^2##. From this it's also clear that ##m##, the invariant mass of the particle, is the same quantity as in Newtonian physics.

In the (for relativity more appropriate) continuum-mechanical description it's the (local) restframe of the fluid/matter cell. That's why all thermodynamical quantities like temperature, chemical potentials, densities are defined as scalars.

Einstein himself also quite early abandoned the concept of "relativistic mass" (you'd even need more than one such "relativistic mass", not only depending on the magnitude of the non-covariant velocity but also on its direction, and that's too complicated to be useful):

https://doi.org/10.1063/1.881171

LBoy, Grasshopper, Dale and 3 others
vanhees71 said:
From the Physics Today website,
"During 2021, Physics Today is providing complimentary access to its entire 73-year archive to readers who register."From what appears to be (an archive of ) Okun’s website
http://www.itep.ru/science/doctors/okun/publishing_eng/em_3.pdf

From that article,
Okun quotes Einstein:
"It is not good to introduce the concept of the mass $M = m/(1 - v^2/c^2 )^{1/2}$ of a moving body for which no clear definition can be given. It is better to introduce no other mass concept than the 'rest mass' $m$ . Instead of introducing $M$ it is better to mention the expression for the momentum and energy of a body in motion." - Einstein (1948)

LBoy and vanhees71
Ibix said:
No one knows. It's a postulate of the theory that ##c## is the same in all inertial reference frames. That you can't travel at ##c## follows immediately - because you'd have to have an inertial reference frame where the speed of light is both zero (because you are traveling at the same speed as light) and 3×108m/s (because it is that in every inertial reference frame). But the only justification for the original postulate is post hoc: the predictions we make when we assume it match reality. Why reality is that way, though, no one knows.

As far as I remember from history of physics this postulate appeared after an attempt to explain the results of experiments of Michelson and Morley, without it the results didn't make sense.

For me it is a direct result of the diferent sign in space-time metrics, for example (-, +, +, +) - there exists "limiting velocity" for every moving body, but I can't say that I understand fully what it means, I can only calculate it without any further physical intuition.

This velocity (often denoted by c) is a part of geometric spacetime structure and it is not a "the velocity of light".

In certain materials light propagates with lesser speeds and other particles (electrons for example) can even exceed the velocity of light there. For reference anyone can find what is Cherenkow radiation. But in these materials there exists the "limiting velocity" as well, so moving electrons there cannot exceed it.

As ar as I remember all Lorentz transformations between reference frames also can be derived from the metric alone (I have it somewhere), so it looks like the half of STW (almost, without mass and energy) comes from the different sign assigned to time-like versor.

Ibix said:
To expand on this a bit, apply the equivalence principle and consider a rocket accelerating in flat spacetime. Inside you have a light strong frictionless horizontal rail suspended from a spring, and a small body of mass ##m## moving with (as measured in this frame) constant velocity along the rail. How much force would the spring exert in equilibrium? Since the spring's force is perpendicular to the direction of motion the answer is ##\gamma ma##, where ##a## is the "acceleration due to gravity".

So I think we can agree that the spring would extend the same for a mass ##m## doing ##v## and a stationary mass with a (rest) mass that happened to be ##\gamma m##. But we can (and, experience suggests we will) argue about whether that means that you are measuring a relativistic mass of ##\gamma m##, or that a spring balance is an inappropriate tool to measure the mass of a body in motion. I tend towards the latter view.

I think there may be a cosmic equivalent to this idea.

Take two stars (or two particles), of roughly equal mass and different velocities towards us (as measured by blueshift/redshift for example).

Let them move around some massive object that affects their trajectories. If relativistic mass makes sense (I mean ist is a real mass) then I guess the one with higher speed should be attracted more than the slower one.

I sense there is some mistake I'm making here, but it's the end of a long day and I'm not thinking clearly. I guess what I'm saying is that if velocity is relative and depends on the reference system then mass also depends, so a massive object can't attract two objects with different velocities to us with different force because that would mean that gravity depends on these two masses, massive object and us, measuring velocity of the objects - which doesn't make sense. I'll think it over tomorrow where the problem is here.

Delta2
LBoy said:
I sense there is some mistake I'm making here, but it's the end of a long day and I'm not thinking clearly. I guess what I'm saying is that if velocity is relative and depends on the reference system then mass also depends, so a massive object can't attract two objects with different velocities to us with different force because that would mean that gravity depends on these two masses, massive object and us, measuring velocity of the objects - which doesn't make sense. I'll think it over tomorrow where the problem is here.
Yes, you are mistaken here. "Mass" in the sense that the term is normally used is an invariant. It does not depend on the reference system.

If one chooses to define mass differently then it may not be invariant. But then it will not be the same mass the rest of us are talking about. In any case, gravity is not a force in the context of general relativity.

vanhees71 and LBoy
PeroK said:
It's not the worst idea, especially when you begin studying SR.
I am not sure about that. I would even say it is a bad idea. For a simple reason: It makes a lot of sense to introduce Lagrangian and Hamiltonian mechanics as soon as possible to "understand" the real laws of nature. It makes no sense to speak about the conservation of mass. It very much makes sense to speak about the conservation of momentum. I believe it is way better to prepare the student for a world which is dominated by energy and time as early as possible. This path allows a student to get into QM easier.
Just my opinion.

Delta2
Of course, one should introduce the action principle in Lagrange and Hamiltonian form as soon as possible in the university curriculum. It's impossible to do at the high-school level of course. The most important general subject to be taught early on in physics are symmetry principles, and indeed this should be done in the very first theory lecture on classical mechanics starting with Newton and then introduce also special relativity, of which one approach is to ask, whether Newtonian spacetime is the only possibility to realize the special principle of relativity, i.e., the principle of inertia, and the answer of course is that also Minkowski spacetime is a possibility, and then it's a matter of experiment to answer the question which spacetime concept is better to describe Nature with the obvious answer that it's Minkowski spacetime and thus relativistic physics.

As innocent as it might look, indeed the notion of mass is a problem in this approach, because to really understand it from the symmetry group-theoretical point of view, you need quantum theory. It's also important to keep in mind when teaching classical physics that this is an approximation and QT is the "real thing". So one should be careful not to introduce fundamental concepts that are wrong from the viewpoint of QT, and a relativistic mass is a paradigmatic example for a bad concept, because it doesn't match in any consistent way with the symmetry paradigm as the overarching conceptual framework of all modern physics.

From the point of view of classical Newtonian mechanics mass is pretty enigmatic. It's merely a property describing inertia (in Newton's Lex I) and active and passive gravitational mass (Newton's universal gravitational interaction law). Not to get things wrong from the group-theoretical perspective all you can do, what's anyway natural within Newtonian physics: Just treat it as a scalar to be introduced in finding the Lagrangian of a single particle fulfilling all 10 conservation laws from spacetime geometry, and you end up with ##L=m \dot{\vec{x}}^2/2##, where ##m## is a scalar proportionality constant, which turns out to have the meaning of inertial mass when considering also closed systems of 2 and more particles or open systems with particles moving under the influence of "external forces". Newtonian gravity is not a fundamental law but must be empirically introduced leading to the amazing conclusion that all three notions of mass (i.e., inertial, active and passive gravitational masses) are the same (Newtonian equivalence principle).

In this way you have as fundamental concepts of Newtonian mechanics the spacetime symmetries (Galilei symmetry) with the corresponding 10 conservation laws via Noether's theorem. There's no way to derive the conservation of mass, and that's already telling. Nevertheless you need it as another empirical assumption when considering continuum mechanics, where together with an equation of state it's needed to close the equations of motion in addition to the dynamical equations following from the 10 space-time conservation laws.

Further progress in the question of mass is then possible in two ways, depending on the order physics is taught as a whole. In the traditional way the next step is special relativity, and there you can start, as mentioned above, with the question, which spacetime models are possible given the Lex I only. Then you find that not only the Galilei group (and Newtonian spacetime) but also the Poincare group (and Minkowskian spacetime) are possible realizations of this symmetry principle. The very same symmetry analysis for a single particle than naturally leads to the notion of invariant and only invariant mass, i.e., mass stays a scalar quantity, but there's no empirical mass-conservation law anymore, i.e., you can use some example of relativistic reactions of elementary particles like ##\mathcal{e}^+ + \mathcal{e}^- \rightarrow \mu^+ + \mu^-## that clearly doesn't conserve mass but only energy, momentum, and angular momentum. For continuum mechanics instead of mass conservation you need to argue with some other conservation law like electric charge or baryon number together with some equation of state to close the equations.

From the symmetry point of view it's very clear that there's not even a useful idea behind the introduction of a non-covariant quantity called "relativistic mass". It also becomes clear that inertia is not (only) due to mass but due to energy and, as it turns out in the continuum mechanical context, also in some sense momentum and stress. This already hints at drastic changes necessary to also describe gravity. As we all know, it leads to General Relativity, which by construction makes all three kinds of Newtonian masses the same and just to a scalar parameter describing an intrinsic property of matter (as do the other fundamental properties like the different charges of the fundamental interactions described by the Standard Model).

With QT the role of mass becomes clearer, using the symmetry concepts. Here one needs the idea of unitary ray representations, leading to the conclusion that in general a classical symmetry can be realized by analyzing the Lie algebras of the classical spacetime symmetry groups leading then inevitably to the covering group of the corresponding symmetry group, which leads to the conclusion that instead of the rotation group of euclidean 3-space one can consider its covering group SU(2) leading to another intrinsic property of matter, the spin, which can be both integer and half-integer.

Analyzing then the Galilei group further it turns out that the Lie algebra admits a non-trivial central charge, which turns out to be the mass, and interestingly the case of 0 mass corresponding to a unitary representation of the covering group of the classical Galilei group doesn't lead to a useful dynamical quantum theory. So one has to extend the classical Galilei algebra by mass as central charge, leading to a superselection rule excluding superpositions of states belonging to different values of (total) mass, which explains the empirical mass-conservation law within relativistic mechanics.

The same analysis of the proper orthochronous Poincare group for relativistic QT leads to the conclusion that there are no (nontrivial) central charges of it's Lie group and mass is just a Casimir operator of the algebra with no need for a superselection rule nor an additional conservation law for mass in accordance with the empricial observation that in the relativistic realm mass is not conserved. Again there's no hint for why one should introduce an artificial concept of "relativistic masses" at all.

In summary, from a modern point of view thus there's no convincing theoretical or empirical argument for the introduction of a relativistic mass. To my knowledge there's not even a useful heuristic argument to introduce it!

LBoy, Omega0, Grasshopper and 1 other person
The invariant mass of an isolated system is conserved. It is not additive but it is conserved.

Delta2
It depends how you define "isolated system". E.g., some piece of matter gets a larger invariant mass when it gets warmer or a charged capacitor has a larger invariant mass than an uncharged one etc.

vanhees71 said:
It depends how you define "isolated system". E.g., some piece of matter gets a larger invariant mass when it gets warmer or a charged capacitor has a larger invariant mass than an uncharged one etc.
I've always understood "isolated" as the strongest form of a closed system there is. No mass transfer, no momentum transfer, no energy transfer, no charge transfer. Nothing gets in or out.

Motore
jbriggs444 said:
I've always understood "isolated" as the strongest form of a closed system there is. No mass transfer, no momentum transfer, no energy transfer, no charge transfer. Nothing gets in or out.
I myself have wondered if that's even physically possible, with all the quantum weirdness. I'm of course fully ignorant about QM, but the thought has crossed my mind. Not that it matters, since in physics simplifying models are used to isolate particular paths of inquiry.

Grasshopper said:
I myself have wondered if that's even physically possible, with all the quantum weirdness. I'm of course fully ignorant about QM, but the thought has crossed my mind. Not that it matters, since in physics simplifying models are used to isolate particular paths of inquiry.
Yes. I agree. Perfect isolation is likely unachievable. But it is a useful ideal which we can approximate in practice.

Grasshopper
Of course, in this idealized sense of "isolated system" the invariant mass is conserved, but it's not an independent conservation law but follows from energ-momentum conservation and the definition of invariant mass, ##p_{\mu} p^{\mu}=m^2 c^2##, where ##p## is the total four-momentum of the isolated system.

Grasshopper
vanhees71 said:
Of course, in this idealized sense of "isolated system" the invariant mass is conserved, but it's not an independent conservation law but follows from energ-momentum conservation and the definition of invariant mass, ##p_{\mu} p^{\mu}=m^2 c^2##, where ##p## is the total four-momentum of the isolated system.
Yes, which is precisely why I disagree with statements that mass is not conserved. If energy and momentum are conserved then conservation of mass follows. It is not independent so you cannot say that the former two hold and the latter does not.

Note further, that the idealized sense of isolated system is the same idealization for energy and momentum conservation. If you choose any non-ideal system where mass is not conserved then either energy or momentum (or both) will also not be conserved.

PeterDonis, Motore and vanhees71
As I tried to write in my long posting above the difference between Newtonian and relativistic physics concerning mass, from the point of view of the underlying symmetry principles, is that in Newtonian mechanics there is an additional mass-conservation law and that this can only be understood from a quantum theoretical argument. This additional conservation law is rather a super-selection rule then a usual conservation law from Noether's theorem. In relativistic physics mass is a Casimir operator and thus specifies the (irreducible) representations of the Poincare group as any Casimir operator of a symmetry group does. So there is no additional indepednent conservation law but it follows from energy-momentum conservation. The main point, however, is that via these symmetry arguments it is clear that conceptually the mass is to be defined as a Casimir operator of the Poincare group and thus as a scalar quantity, and quantities like various "relativistic masses" from the very early days of relativity are conceptually superfluous.

LBoy, Omega0, dextercioby and 1 other person
mucker said:
I don't need equations, I would just like to pose a question which contradicts the above statement (I know I am wrong btw, I want to see where I am going wrong).My understanding of space (not near any gravity and therefore no spacetime curvature) is that a body in motion will continue to move at said speed unless acted on by another force. So let's use an example. Let's say I am moving at 10mph through space but I want to reach the speed of light. (Now I'm going to use fictious numbers and equations to keep the math simple). Let's say in my spaceship I fire off my rockets to increase the speed, which requires 5 amount of Newtons (or whatever else the metric is for force – it doesn't really matter for the point I'll make). And that 5 amount of Newtons increases my speed by 10mph, so I'm now I'm going at 20mph. At this point I stop the rockets and i continue indefinitely at 20mph. So I now apply another 5 Newtons to reach 30mph, and so on until I reach the speed of light. This does not require an infinite amount of energy.Now I know the above is wrong (for many reasons) but bear with me for a minute. So from E=mc2 (I think) it states that the reason we can’t reach the speed of light is because as you get closer to the speed of light your mass increases, and therefore requires infinitely more energy – I get that. But the first issue I have here is that I thought speed was all relative. So in my example above I know that I am wrong when I keep saying my speed in increasing. But if my 30mph is the equivalent of being still in space, then how can my ship gain mass when increasing in speed? and therefore need infinitely more energy if speed is relative? To demonstrate more, imagine nothing is close by to judge our relative speed to it. To put it another it seems that speed is relative to everything but not light. I have come to the conclusion that when we say we can’t reach the speed of light that maybe it actually means the acceleration to the speed of light - because to reach the speed of light (relative to another body) we’d have to accelerate at an insane amount (which would require a lot of energy). So my question is, is it more accurate when we say “it requires an infinite amount of energy to reach the speed of light” to change that to “it requires an infinite amount of energy to reach the speed of light”
Or have I completely missed the point? Where I am coming from is that there is no speed in space unless it’s relative to another object. The other thing that is weird, is that, if light speed is always constant how can light still travel at said speed if I were to speed up just half the speed of light in the same direction as the light beam? I should perceive the light traveling at half the speed of light but I know (from reading up on GR) that light is always the speed no matter the reference frame. The explanation to this is that apparently that time is slowed down the faster you move – but again, I thought speed was relative.

In SR, if I see you are moving at half the speed of light (c), I will also see your clock run slower such that you will measure the light moving away from you at the speed of light (c). So don't matter at what speed you are traveling, light will always travel at c.

So at rest, light will travel at 670 000 000 mph away from you. So you should increase your speed with 6.7E8 mph to reach the speed of light. But when you reach 300 000 000 mph, you will still measure the speed of light at 6.7E8 mph and therefor still need to increase your speed with 6.7E8 mph. Even if you reach a speed of 600 000 000 mph, the speed of light will still be 6.7E8m mph and so on.

From an observer at rest, it will look like as if your acceleration is decreasing as you approach the speed of light due to the fact that the observer is also seeing your clock running slower. For example if you are acceleraing at 100 mph per hour, you will increase 100mph every hour. But because the observer see your hour going slower than his own - he will also see you accelerating slower.

Interesting fact: An person that is constantly accelerating, will follow a hyperbolic path and not an parabolic path. The asymptotes of the hyperbole is the speed of light

Last edited:
mucker said:
I don't need equations, I would just like to pose a question which contradicts the above statement (I know I am wrong btw, I want to see where I am going wrong).My understanding of space (not near any gravity and therefore no spacetime curvature) is that a body in motion will continue to move at said speed unless acted on by another force. So let's use an example. Let's say I am moving at 10mph through space but I want to reach the speed of light. (Now I'm going to use fictious numbers and equations to keep the math simple). Let's say in my spaceship I fire off my rockets to increase the speed, which requires 5 amount of Newtons (or whatever else the metric is for force – it doesn't really matter for the point I'll make). And that 5 amount of Newtons increases my speed by 10mph, so I'm now I'm going at 20mph. At this point I stop the rockets and i continue indefinitely at 20mph. So I now apply another 5 Newtons to reach 30mph, and so on until I reach the speed of light. This does not require an infinite amount of energy.Now I know the above is wrong (for many reasons) but bear with me for a minute. So from E=mc2 (I think) it states that the reason we can’t reach the speed of light is because as you get closer to the speed of light your mass increases, and therefore requires infinitely more energy – I get that. But the first issue I have here is that I thought speed was all relative. So in my example above I know that I am wrong when I keep saying my speed in increasing. But if my 30mph is the equivalent of being still in space, then how can my ship gain mass when increasing in speed? and therefore need infinitely more energy if speed is relative? To demonstrate more, imagine nothing is close by to judge our relative speed to it. To put it another it seems that speed is relative to everything but not light. I have come to the conclusion that when we say we can’t reach the speed of light that maybe it actually means the acceleration to the speed of light - because to reach the speed of light (relative to another body) we’d have to accelerate at an insane amount (which would require a lot of energy). So my question is, is it more accurate when we say “it requires an infinite amount of energy to reach the speed of light” to change that to “it requires an infinite amount of energy to reach the speed of light”
Or have I completely missed the point? Where I am coming from is that there is no speed in space unless it’s relative to another object. The other thing that is weird, is that, if light speed is always constant how can light still travel at said speed if I were to speed up just half the speed of light in the same direction as the light beam? I should perceive the light traveling at half the speed of light but I know (from reading up on GR) that light is always the speed no matter the reference frame. The explanation to this is that apparently that time is slowed down the faster you move – but again, I thought speed was relative.

With regard to increasing of mass, the notion that your mass increases as you approach the speed of light is a bit misleading (or leads to confusion)

From an observer at rest it is appearing that your mass is increasing as you approach the speed of light, because you appear to accelerate less (to the same force) as you approach the speed of light. From your own perspective it won't appear that your mass is increasing.

But relativistic mass becomes quite messy further on. So it is not a concept worth using: it is better to change the equation of momentum to momentum = gamma x mass x velocity

vanhees71 and Dale
In my opinion Einstein's 1905 paper remains the simplest and clearest explanation.

To make a long story very short the basic idea is that the laws of physics are the same in every frame of reference. Especially electricity and magnetism. Given this, special relativity was deduced. The math is quite simple, basically high school trigonometry. The difficulty lay in persuading people that the math matched reality.

vanhees71
@Dylan007 please do not quote an entire post when responding. Only quote the particular parts of the post that you are responding to.

Hornbein said:
In my opinion Einstein's 1905 paper remains the simplest and clearest explanation.

To make a long story very short the basic idea is that the laws of physics are the same in every frame of reference. Especially electricity and magnetism. Given this, special relativity was deduced. The math is quite simple, basically high school trigonometry. The difficulty lay in persuading people that the math matched reality.
Historically that's not entirely true, because almost immediately Einstein's view was accepted. The math was there even before (Lorentz, Poincare) but not the ingeniously simple physical interpretation of Einstein's 1905 paper. Immediately leading theorists of their time like Planck, von Laue, and Sommerfeld adapted these ideas and worked with them.

There were of course opponents among the physicists like the erratic ideologists of the "Deutsche Physik" movement like Lennard and Stark.

The greatest obstacle, however, have been philosophers. Particularly Bergson could never accept the notion of time according to the relativity theories. This had the bizarre consequence that Einstein did not get his Nobel prize for relativity, and that's even explicitly stated on the Nobel certificate.

PeterDonis, Dale and PeroK

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