Henry Poincaré wrote exactly this in 1902 in "the science and hypothesis" Nous sommes acculés à la définition suivante, qui n'est qu'un aveu d'impuissance : les masses sont des coefficients qu'il est commode d'introduire dans les calculs.[0]

I am also happy for the reference to the excellent blog of Matt Strassler

[0] https://en.wikipedia.org/wiki/Science_and_Hypothesis

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.

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.)

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.

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.

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.

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

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.

Yes but that is still true (E=m) if c<>1, since c is a constant. It's just a question of scale, isn't it?

James Joule discovered that heat and work can be related by:

  h = kw

But really, this isn't a relationship between heat and work. This is a statement of equivalence. The constant of 4.184cal/J just tells you how the two units differ. They both describe the same property. Really, h = w. The version with k is only needed if you insist on using the traditional, different units on the left and right sides.

The same is true of e = mc^2. This does not tell you how two different fundamental properties are related, it merely tells you how two units, traditionally considered separate, actually measure the same underlying property. Really, e = m. The c^2 is only needed if you insist on using J on one side and kg on the other.

That is to say, you absolutely can have a kilogram of energy. If you have an object with a mass of one kilogram, you do have a kilogram of energy. Physicists and engineers don't typically use that unit for energy, but there's no reason they couldn't, and likewise no reason they couldn't use joules for mass.

So any time two physical quantities are related by a constant factor -- even if said constant factor has what we currently think of as "units" (and even particularly weird units like squared speed) -- they are fundamentally equivalent. I'm still mulling this over, but it's a pretty fascinating mental revelation for me.

Edit: I also want to say that I didn't find the comment I replied to rude in any way. It's direct and straightforward, but that's pretty common around here.

In natural units, c is "unitless" (AKA dimensionless), by definition.

Maybe someone has a link that goes into more depth on why all of this is okay?

When using a pseudo-Riemannian manifold one uses a metric signature where (keeping it simple, cf. "metric signature" on wikipedia) coordinates on orthogonal dimensions of can take one sign, or the opposite sign and a constant multiplier. In the Lorentzian case there will be one dimension taking one sign, and one or more taking the other; the choice of whether the solitary dimension takes a + or - sign is a matter of convention or preference. Conventionally the solitary dimension also takes the constant multiplier and is called the timelike dimension. A Lorentzian signature (minuses, 1) or (1, plusses) guarantees that one can describe paths through the manifold as null, timelike, or spacelike, and this gives one a causal structure.

Our universe can be well represented by a Lorentzian manifold (3, 1) or (1, 3), and this has been tested to exquisite precision. It does not tell us the value of the constant c, but we can determine that from tests of causal relations, the boundaries of which will be null. Alternatively, a massless pointlike object will always travel on null geodesics.

One runs into c being set to unity in systems geometrized units in relativity often; it's very handy to have mark off coordinates as e.g. "-seconds" vs "c seconds" (-,+,+,+ aka (1,3)) as long as one doesn't mess up one's dimensional analyses (which is unfortunately easy).

As a concrete example, the line element for (1,3) flat spacetime using Cartesian coordinates is dS^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2. Compare the formula for Euclidean distance in the (Euclidean flat) plane between points p = (p1,p2) and q = (q1,q2) for (x,y) coordinates: d(p,q) = sqrt((q1-p1)^2 + (q2-p2)^2), which we could rewrite as dx = q1-p1, dy = q2-p2, ds = sqrt(dx^2 + dy^2) or ds^2 = dx^2 + dy^2. In Euclidean 3-space, we add another axis: ds^2 = dx^2 + dy^2 + dz^2. In Lorentzian 4-spacetime, we have to change the sign, so ds^2 = dx^2 + dy^2 + dz^2 - c^2dt^2. Note that we are not restricted to use any particular unit of distance or system of units; dx could be in metres, miles, astronomical units, light-years, gigaparsecs or practically anything else, while dt could be in seconds or fortnights or any other handy unit of time. When setting c to unity, we do need to choose appropriate units for dx (and dy and dz ...) vs dt. In SI units, that's seconds and light-seconds, but we could use another system if we wanted.

Notably we aren't restricted to Cartesian coordinates, however if we were to change to some other system of coordinates (e.g. polar ones) the line element would need to reflect that. For example, in the Lorentzian 4-spacetime case, we would write ds^2 = dr^2 + r^2 * dtheta^2 + r^2sin^2(theta)dphi^2 -c^2dt^2.

Finally, when using SI units (for example), one can see very clearly that a path taken by an object moving much slower than the speed of light is totally dominated by the amount of time between starting point and finishing point, because the value of c is large. Using the (+,-,-,-) metric signature [ds^2 = c^2dt^2 - dx^2 - dy^2 - dz^2], a large c helps make it clear that lightlike paths through spacetime are shorter, and purely timelike paths (where dx=0, dy=0, dz=0, dt != 0) have extremized length, since we don't subtract anything from cdt. This is the root of the explanation of the twin paradox in flat spacetime: the twin moving quickly compared to the speed of light takes a shorter path between together1 = (x1,y1,z1,t1) and together2 = (x2,y2,z2,t1) than the twin moving slowly compared to the speed of light. In the extreme, twin A holds x=const,y=const,z=const, whereas twin B's x coordinate is only equal to twin A's at the start and end of the journey. This holds up under any system of coordinates; we could consider r=const vs changing r in spherical coordinates, for instance.

That is clever, but you've already adjusted the mass to take into account the fact that you are now setting c=1. This reminds me of computer programming where, for the sake of an easy-to-read clarity, I decide to take one line of code and make it two. Basically, instead of doing this:

e=mc^2

You have decided to make it two lines of code:

m=mc^2

e=m

I often do that, especially if I'm working with junior devs and I want the code to be as easy as possible for them to read (90% of the time).

It's clever, but everyone should remember the assumption that allows this: the mass has already been adjusted to reflect the reality that of c=1.

Inertia is not merely something that depends on mass. It is mass.

(though from a brief web search, it is not typically presented as such)

For example, I said E=M. But that's for objects at rest. For an object in motion it's E^2=m^2c^4+p^2c^2. In natural units that turns into E=m+p. The energy of a photon is hf, where h is Planck's constant and f is the frequency. That energy is entirely the photon's momentum. And that's without getting into the issues of invariant mass or relativistic mass increases...

E² = m²c⁴ + p²c²

https://en.wikipedia.org/wiki/Unicode_subscripts_and_supersc...

A number of early computer systems used teletypes for I/O, and had programming languages in which exponentiation was expressed by moving a half line up, and array subscripts by moving a half line down.

But yeah, having a "next-is-superscript"-token would probably be cleaner than individual code points for every little glyph that might be a superscript...

https://en.m.wikipedia.org/wiki/Blackletter#Unicode

Since mass also determines the speed of time to the observer, isn't a time machine basically just a thing that converts energy into mass?