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Comment by amluto | Hacker News Reader

> ask it for a formula for mass-energy equivalence

Way too easy. If you think that mass and energy might be equivalent, then dimensional analysis doesn’t give you too much choice in the formula. Really, the interesting thing about E=mc^2 isn’t the formula but the assertion that mass is a form of energy and all the surrounding observations about the universe.

Also, the actual insight in 1905 was more about asking the right questions and imagining that the equivalence principle could really hold, etc. A bunch of the math predates 1905 and would be there in an AI’s training set:

https://en.m.wikipedia.org/wiki/History_of_Lorentz_transform...

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It’s nontrivial why it’s mc^2 and not 1/2 mc^2, since kinetic energy generally is 1/2 mv^2

but e=mc^2 is just an approximation

e: nice, downvoted for knowing special relativity

Can you elaborate? How is E=mc^2 an approximation, in special relativity or otherwise? What is it an approximation of?

E^2 = m^2 + p^2 where p is momentum and i’ve dropped unit adjustment factors like c

this allows light to have energy even if its massless

e=mc^2 is only correct for objects at rest. The full equation takes into account velocity, but for "low" speeds where v<<c, the term is close enough to zero than E=mc^2 is still a good approximation.

E^2 = (mc^2)^2 + (pc)^2,

where p is momentum. When an object is traveling at relativistic speeds, the momentum forms a more significant portion of its energy

I didn't downvote it, but short comments are a very big risk. People may misinterpret it, or think it's crackpot theory or a joke and then downvote.

When in doubt, add more info, like:

But the complete equation is E=sqrt(m^2c^4+p^2) that is reduced to E=mc^2 when the momentum p is 0. More info in https://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalenc...

What I learnt is that there is a rest mass and a relativistic mass. The m in your formula is the rest mass. But when you use the relativistic mass E=mc² still holds. And for the rest mass I always used m_0 to make clear what it is.

sounds like you had a chemistry education. relativistic mass is IMO very much not a useful way of thinking about this and it is sort of tautologically true that E = m_relativistic because “relativistic mass” is just taking the concept of energy and renaming it “mass”

This is all sort of silly IMO. The equation, like basically all equations, needs context. What’s E? What’s m? If E is the total energy of the system and m is the mass (inertial or gravitational? how far past 1905 do you want to go?), then there isn’t a correction. If m is rest mass and E is total energy, then I would call it flat-out wrong, not merely approximate. After all, a decent theory really ought to reproduce Newtonian mechanics under some conditions beyond completely at rest.

IMO, when people get excited about E=mc^2, it’s in contexts like noticing that atoms have rest masses that are generally somewhat below the mass of a proton or neutron times the number of protons and neutrons in the atom, and that the mass difference is the binding energy of the nucleus, and you can do nuclear reactions and convert between mass and energy! And then E=mc^2 is apparently exactly true, or at least true to an excellent degree, even though the energies involved are extremely large and Newtonian mechanics can’t even come close to accounting for what’s going on.

inertial mass, rest mass, gravitational mass - these are essentially all the same thing. “relativistic mass” is an additional concept where we rewrite energy as mass and is considered archaic

I should have used m_0 to avoid confussion. Anyway, as he sibling comment says, most modern advanced books of special relativity try to avoid the relativistic mass. It's useful for some calculations, like synchrotron, but the problem is that for forward/backward acceleration you must use other number so the relativistic mass add confussion. https://en.wikipedia.org/wiki/Mass_in_special_relativity#His...

The next section of the wikipedia link discusses the low speed approximation, where sqrt(m^2c^4+(pc)^2) ≈ mc^2 + 1/2 mv^2.

Calling E=mc^2 an "approximation" is technically correct. It's the 0th order approximation. That's just pointlessly confusing. A better word choice would be "a special case".

i think we are venturing into pedantic territory - the point of my comment is that the full derivation is a little harder than just E=mc^2 dimensional analysis

Kind of agree. But pervasive downvoting by folks who don’t understand the subject is a form of forum rot. The risk is only that we expose the rot. Not such a terrible risk, because either the owners notice and fix the problem, or the forum continues to rot. In the latter case karma points wont be desirable in the long run.

This is why RLHF causes those overly verbose answers to simple questions, it's a fundamentally busted evaluation function so you wind up optimizing for the wrong thing

[Sorry for the dealy.]

In one extreme there are wall of text and in the other extreme very short answers that only the initiated understand (like inside jokes). Somewhere in between there is a sweet spot that helps everyone else to follow the discusion and gain a litle of knowdledge.

(I don't claim I get the best lenght in my comments, but I hope it's good enough.)

it's a special case, not an approximation

its not an either/or, it is both. regardless, my point is that you cannot simply dimensional analysis your way to special relativity or the energy-momentum relation

This thread has come up before(1), but I'll continue to argue that relativistic mass is a perfectly valid concept as much as any other, and if you disagree, you'll need arguments more substantial than it just being unpopular these days. Especially if you're trying to argue people out of using a concept that they personally find useful to aid their own understanding, just because it doesn't fit your own mathematical or aesthetic preferences.

1: https://news.ycombinator.com/item?id=38425252

sure but i think it

1. is not very intuitive/useful to have mass that varies on the direction (which is what this implies)

2. is somewhat tautological to define a new mass m_rel = E/c^2 and say that it satisfies the equation when this is not what most people understand mass to be. most people understand photons to be massless particles.

at minimum, relativistic mass should always be specified as m_rel to distinguish from what is typically referred to as mass.

but i don’t think relativistic mass is a wrong concept any more than any other mathematical convenience like virtual particles. the main question is how useful is it and should it be described using the word “mass” or is this confusing. there is value in having shared language, even if you can construct an alternate system of symbols and rules that can yield the same answer to every question. to the extent to which intent of the author matters at all (probably doesn’t), Einstein agreed that relativistic mass was not a useful concept.

i'll concede that the arguments in the thread you linked are not good

> most people understand photons to be massless particles.

I suspect that most people have no opinion at all and are generally unaware of the properties of light.

That being said, a (perfectly reflective or otherwise steady state, e.g. black body at equilibrium) box of photons has mass due to those photons. You can stick it on a scale or a pendulum and measure it. It attracts other nearby objects according to GR or, in the appropriate limit, Newtonian gravity, in accordance with the relativistic mass of those photons.

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