If we could extract all the energy from the mass of our baseball - anti-baseball pairs in a usable form, we'd need about 20,000 of them to cover what we, as a species, use in a year.

This isn't possible, either. In nuclear fission only a fraction of the energy released is used to produce electricity. But let's not get hung up on details.

Well, a baseball has a mass of around 0.15 kg, give or take. According to special relativity, this is equivalent to an amount of energy

0.15 kg * (3 x 10⁸ m/s)² = 1.35 x 10¹⁶ J.

One Joule, "J", is the amount energy you'd expend lifting a 1kg mass about 10cm here on Earth.

So you bring your baseball and your anti-baseball together and get twice that - a bit less than 3 x 10¹⁶ J.

How much is that? A few years ago, in 2019, global energy consumption was about 600 exajoules, or 6 x 10²⁰ J.

There aren't really any practical methods for converting macroscopic amounts of mass entirely into energy. A particle and its anti-particle will annihilate and convert entirely into energy, but this makes it pretty hard to get your hands on a macroscopic amount of anti-particles.

Imagine that, somehow, we could produce an anti-baseball. Just a regulation baseball, where every particle has been replaced by its associated anti-particle. What could we do with that?

Some of the mass of unstable nuclei comes from binding energy, and fission in a reactor releases this as energy. Some of that energy can then be used to produce electricity.

It's just a small fraction of the mass that gets converted, but that factor of c² is awfully big (c is 3 x 10⁸ m/s) so a little bit of mass goes a long way.

Einstein was the first to propose that the relationship goes beyond energy lost to particular types of radiation, and the first to arrive at the correct form of the equivalence.

Most importantly, his arguments were based on a physical principle – relativity.

So what can we get out of this equivalence?

Einstein wasn’t the first person to propose a relationship between changes in an object’s energy and inertia. A number of 19th century scientists suggested various relationships between a particle's mass and its energy content.

But those proposals were poorly motivated, or imprecise, or both. And in most cases they applied only to certain sorts of mass or radiation.

Einstein jumps right in and picks up a thread from the first paper. He considers the energy of a massive object in two reference frames – one with the object is at rest, one where it moves at speed v – and considers a process where it emits energy.

The image in the first post shows his conclusion: “If a body gives off the energy L in the form of radiation, its mass diminishes by L/V².” (In modern notation, E/c²). Einstein concludes that “The mass of a body is a measure of its energy-content.”

As you may recall, yesterday was the 118th anniversary of the publication of Einstein's first special relativity paper in the journal.

That paper lays out many of the results we associate with special relativity, except for the most famous one!

The mass-energy equivalence result was submitted immediately after the first manuscript appeared in print.

https://mastodon.social/@mcnees/111132114549253218

If there is one physics formula everyone knows, it's E = mc².

Einstein proposed the famous equivalence between energy and mass in his second special relativity manuscript, which he submitted to Annalen der Physik #OTD in 1905.

https://www.fourmilab.ch/etexts/einstein/E_mc2/www/

@drewolbrich If I was made of sound waves? It would depend on the direction of the other sound wave.

As a sound wave my speed is fixed by the properties of the medium. So I'm moving at basically 340 m/s through air in some direction. Any other sound wave is, as well. But this is with respect to the medium; it provides a fixed reference frame to refer to. If the other wave is moving in the same direction as me, I'd say its speed relative to me is zero. If the opposite direction? ~680 m/s.

@Edelruth You're welcome!

@jamesjm Submission versus final publication

Happy 118th birthday, special relativity!

Notice that Einstein's result for the kinetic energy looks like something minus mc²! That makes sense, right? Everyone has heard of “E=mc²” so this must be where he introduces the famous result.

But E=mc² doesn't show up until his next paper, titled "Does the Inertia of a Body Depend Upon Its Energy-Content?" That paper was submitted to Annalen der Physik the day after this one appeared.

But as v approaches the speed of light, the kinetic energy grows without bound.

Apparently, no amount of work can ever get a massive object up to c. In Einstein's words, "Velocities greater than that of light have – as in our previous results – no possibility of existence."

In 2018, total global energy consumption was closing in on 6x10²⁰ Joules. That's about the amount of work you'd need to get a 1000 kg space probe up to a speed of 0.99c, according to Einstein's result.

Here μ is the particle's mass, v is its speed, and V is c.

Once again, he obtains a result that agrees with our expectations when applied to everyday situations.

If the object's velocity is much smaller than c, he recovers the usual non-relativistic kinetic energy: (1/2) μ v².

@rmathematicus Is that even the right spelling?

Finally, Einstein considers a “slowly accelerated” electron.

Here, “slowly” means he will ignore the energy it radiates as it accelerates. Instead, he wants to focus on the work required to bring the object up to some speed v. This is just the object's kinetic energy.

Einstein notes that his calculation applies to any "ponderable material point" whether or not it is charged. So it doesn't have to be an electron; the result would be the same for any massive particle.

He obtains:

Having fixed kinematics so that it reflects Nature rather than our intuition, Einstein moves on to electrodynamics.

He works out the transformation of Maxwell’s laws, and derives (among other results) the special relativistic form of the Doppler shift.

Unlike sound waves, where the source or listener might be moving relative to the medium, the special relativistic Doppler shift depends only on the relative motion.

Because c is very big (about 3x10⁸ m/s), Einstein's new rule is still compatible with our everyday experience.

Suppose I'm standing on the side of the road and see you drive by at 60mph. If you say someone passes you at 20mph, I would see them moving at 79.99999999999989 mph.

You could forgive me for describing that as the 80mph given by my common-sense rule for adding velocities.