[davidmanescu]

Equalities that involve units aren't straightforward. Not a physicist of any sort but don't all the 1's still fundamentally change the equality since they change the units? I would guess they therefore implicitly change the intuition behind these equations?

[psyc]

I would say it's the opposite. A unit conversion is not fundamental at all, but a matter of arbitrary convention. 0C and 32F are exactly the same temperature. Mass and energy are exactly the same property. c^2 is a constant, so E doesn't even vary as the square of anything - it's just a linear unit conversion. If you follow it further, you find c is only even in there because some other units are defined in terms of it. That is, the ME equivalence doesn't really have anything to do with the speed of information propagation per se - but our chosen unit conventions do.

It's because of people's fondness for legacy and backwards compatibility that we don't use natural units all the time, and still write "e=mc^2". I find this insane, personally, but that's just the way it goes.

I guess I should add that apparently this is still controversial? But in my mind, the only reason for this is that people find meaning in units. In this case, ME being the same property implies duration being exactly the same as length, which implies spacetime is a unified, regular 4D manifold. ME not being the same property implies time is special. I've been fully persuaded that duration is fundamentally exactly the same thing as length, so I've chosen my side, in case there's any serious controversy.

[Xcelerate]

> Mass and energy are exactly the same property

Definitely not. Everything couples to the stress-energy tensor, but not every type of particle has rest/invariant mass. No one really speaks in terms of relativistic mass anymore. (So a photon has energy but no rest mass.)

[psyc]

Relativistic mass is the one that is equivalent to energy, so if no one speaks of it, they'll perpetuate the legacy sloppiness I referred to.

[jessriedel]

Mass and energy are not the same property at all. E=mc^2 is only applicable at the limit of zero speed.

[jschwartzi]

...In the same frame of reference as the observer.

[beojan]

That's what makes natural units natural. Certain quantities (like mass and energy, or distance and time) are physically related such that the distinction is, in a sense, artificial. The conversion factors between the units used for these pairs of quantities are the fundamental physical constants that get set to unity in a natural system of units.

[jessriedel]

The distinction between distance and time is not artificial at all. It is encoded in the metric, at the most fundamental level. The existence of symmetries is not the same as equivalence.

[coldtea]

Parent is not talking about the "distinction between distance and time" but about the multipliers that come into play into their equations (which are artificial and based on the base units selected).

[jessriedel]

The distinction between feet and seconds is just as real as the distinction between distance and time, and is much more real than the distinction between feet and meters. Look at the parent's literal words and pretend you did not already know what's going on.

I'm well aware of the practical value and conceptual clarity of natural units. They are just not being explained well in this thread. "What the teacher really meant was..." is not a good defense when the student doesn't understand.

[firethief]

> The distinction between feet and seconds is just as real as the distinction between distance and time

There is a "real" distinction between [light]seconds of distance and seconds of time, but it's a difference in what is being measured, not in how it is quantifiable. Measuring them with the same units doesn't imply that they are perfect substitutes, any more than one could arbitrarily replace an ounce of gold with the same weight of feathers; in practice, they are non-interchangeable enough that gold is even measured with a different type of "ounce". Likewise with spacetime: for most purposes human non-physicists think about, they are completely non-substitutable. So practically, it works well to use a "Troy" sort of system for one. But it's a purely human distinction.

[jessriedel]

> but it's a difference in what is being measured, not in how it is quantifiable. Measuring them with the same units doesn't imply that they are perfect substitutes

I agree! Please read my comments carefully. I am not arguing against the usefulness or deep conceptual importance of natural units! I am critiquing the terrible explanations being given in this thread.

> Likewise with spacetime: for most purposes human non-physicists think about, they are completely non-substitutable. So practically, it works well to use a "Troy" sort of system for one. But it's a purely human distinction.

No! Space and time are not interchangeable in any universal sense even if they are universally linked by a symmetry. A space-like interval and a time-like interval cannot be interchanged with each other by a Lorentz transformation. The distinction between space-like and time-like intervals is observer independent.

[ChuckMcM]

Would you agree that if there was a unit for 299,792,458 meters called 'q' then the "speed of light" could be expressed as 1 q/s? Then e = m*c^2 is e = m x 1 or e = m (but the units of q^2/s^2 are still there, just the arithmetic is simplified)

[jessriedel]

I am not arguing against natural units. I use them daily, and they are more than just practically useful. I am critiquing bad explanations for their interpretation and justification.

[beojan]

"are physically related such that, in a sense..." I didn't say they were equivalent, I said they were connected. You can't transform a time-like interval into a space-like one, but you can transform one time-like interval into another such that the separation on time increase and the separation in space decreases.

There's only so much clarity you can fit into a HN comment.

[marcosdumay]

> don't all the 1's still fundamentally change the equality since they change the units?

Not more than changing unities from meters to feet, or kg to pounds.

When two unities are related by a constant, they are measuring the same thing.