So non-relativistic matter has p more or less 0 compared to ρ, and the equation above tells us that the energy density for non-relativistic matter dilutes like ρ ~ 1 / a³ as the universe expands.

That makes sense! The energy content of non-relativistic matter is dominated by mc², the energy density is ρ = mc² / V, and physical volumes V in our expanding universe grow like a³.

Most sources of energy that affect cosmological expansion exert a pressure p that is proportional to their energy density ρ. This is called a "linear equation of state."

For non-relativistic matter the pressure is much, much smaller than the energy density. Atoms might knock around and collide and produce a pressure. But since they move much slower than the speed of light, this is negligible compared to their mc² energy content.

A basic principle of thermodynamics says that changes in volume or entropy change a system's internal energy.

Imagine a system, like our little chunk of universe containing some energy, expanding a tiny bit while its entropy doesn't change.

The change in the energy is

dE = -p dV.

Now the energy in our volume of space V is E = ρV, and the
volume V is growing like a³, so this basic statement of thermodynamics becomes

a³ dρ + 3 (ρ + p) a² da = 0

We can learn quite a bit from this!

Physical distances are proportional to one factor of a(t); spatial volumes are proportional to a(t)³

The scale factor has grown over the history of the Universe – there's an equation that relates this growth to the density of matter and energy on cosmological scales.

Now imagine a volume of space containing a source of energy whose density is ρ. This source of energy may also exert a pressure p. As you might guess, the expansion of space grows the volume and might dilute the energy density.

As we've known since the 1920s, the universe is expanding. We describe this by relating cosmological lengths, areas, and volumes at different times with a time-dependent "scale factor."

Let's denote this scale factor by a(t). The "a" is the symbol that represents the scale factor, and the "(t)" means that it depends on time ("is a function of time"). Sometimes I will just write it as "a" with the "(t)" part implicit.

What does combining relativity and thermodynamics do for you?

I’m glad you asked.

We can motivate some oft-repeated facts about cosmology this way, using just a few basic results from classical physics. Let me explain.

On cosmological scales, the universe is more or less spatially homogeneous and isotropic. At any given time big regions of space contain similar distributions of matter and energy, and wherever you are things look similar in any direction. This is the "Cosmological Principle."

“Relativity, Thermodynamics, and Cosmology” was well-received. Einstein (who may have been a little biased) praised it as "reliable, detailed, and clear" in a review for Science.

Among the results established in Tolman's book, he shows that blackbody radiation cools but remains thermal as the universe expands. This anticipates the Cosmic Microwave Background, the most precise blackbody spectrum ever measured!

Figure: Ned Wright (https://www.astro.ucla.edu/~wright/cosmo_01.htm)

But the work for which Tolman is best known involves the application of thermodynamics to relativistic systems.

In his monograph “Relativity, Thermodynamics, and Cosmology,” Tolman reviews relativity, develops an extension of thermodynamics to special and general relativity, and then applies it to cosmological models.

A Dover edition of the book lists for $19.95 but can often be found for less. I learned about Tolman when I spotted this book at the UT-Austin bookstore, back in grad school.

Most of Tolman's work is what you would call “theory.” But he produced a very important experimental result in 1916!

Working with T. Dale Stewart, Tolman confirmed the popular view that electrical current in a metal is due to the flow of electrons.

https://journals.aps.org/pr/abstract/10.1103/PhysRev.8.97

Physicist and chemist Richard Chace Tolman was born #OTD in 1881. He showed that electricity in metal is the flow of electrons, developed relativistic applications of thermodynamics, and served as science advisor to General Leslie Groves during WWII.

Image: Caltech / Oregon State

@johncarlosbaez Oh yes, I really enjoyed this book. I think I gave it a go not long after it first came out, when I really wasn't equipped to make much sense of it. But I came back to it at different points in my career, and got something new from it each time.

Stephen Hawking's "A Brief History of Time" was published #OTD in 1988.

I read it a few months later, excitedly underlining and starring passages that wowed me, and spent the summer daydreaming about big bangs and black holes.

https://www.hawking.org.uk/in-print/books/a-brief-history-of-time?position=13

@nigenet @knud It really is bonkers.

@knud Not so fast! I think it's probably a case of simultaneous invention, but the parallels are almost preposterous:

https://mastodon.social/@mcnees/109768169079981704

Chicago area friends!

I'm hosting an "in conversation" event with Fermilab physicist @ChrisQuigg, this evening at Bookends & Beginnings in Evanston.

We'll be talking about his new book with Robert Cahn, "Grace in All Simplicity." Hope to see you there.

https://www.bookendsandbeginnings.com/event/store-book-talk-grace-all-simplicity-chris-quigg-robert-mcnees

It’s a leap year!

[SHOWTURTLE]
🐢

That means we get an OTD for February 29th.

[FORWARD 4]
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[DRAW BIRTHDAY CAKE]
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Seymour Papert, co-inventor of LOGO, was born OTD in 1928.

[BACK 4]
🐢 🎂

This one’s for you, nerds of a certain age.

Image: Cynthia Solomon

The sky above the port was the color of Apple TV+, tuned to a long awaited adaptation of Neuromancer.
https://variety.com/2024/tv/news/apple-neurmancer-series-william-gibson-1235925640/

Currently snowing here on the north side of Chicago. It got up to 75 F yesterday.

Good afternoon.

Channeling this version of Lake Michigan in February, after spending two hours in a lab with a broken thermostat stuck on "hot, aggressive" when it is already 75 F outside.