Robert McNees
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
Robert McNees
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.
Robert McNees
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.
Robert McNees
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)
Robert McNees
“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.
Robert McNees
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."
Robert McNees
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.
Robert McNees
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.
Robert McNees
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!
Robert McNees
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.
seachanged
It's over a century old, so I doubt that its a big moneymaker for a paywall, so I poked around and found an open copy of it:
Robert McNees
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.
Robert McNees
On the other hand, the energy density associated with radiation (which includes photons, as well as particles like neutrinos that move so fast their energy is much larger than their mc²) does something different.
A standard result from electrodynamics is that EM radiation exerts a pressure equal to 1/3 its energy density.
If we use p = (1/3) ρ in the our dE = -p dV equation, we find that the energy density for radiation dilutes like ρ ~ 1 / a⁴.
Robert McNees
The reason is easy to understand! Space expands, so we expect the density in a volume V to dilute like 1/a³.
But cosmological expansion also stretches out the wavelength of radiation. Like other physical lengths it grows with a single power of the scale factor. The energy of a photon is inversely proportional to wavelength, so E decreases ~ 1/a as the universe expands.
Together, these effects cause the energy density of radiation to dilute like (1/a³)x(1/a) = 1/a⁴ as the universe expands.
Robert McNees
Right away, you see that radiation was more important for cosmological dynamics when the Universe was smaller, but eventually became irrelevant compared to matter.
This is the first example of how combining basic thermodynamics with some rudimentary cosmology and relativity can explain a few things you've heard about the role of matter and radiation in the evolution of the Universe.
What else can it tell us?
Robert McNees
The "cosmological constant" was first proposed by Einstein in 1917 as a mechanism for balancing instabilities in his early attempts at relativistic cosmological models.
It can be thought of as an energy density that is "built in" to space.
That is, its energy density ρ does not change as space expands. If we revisit our dE equation, dρ = 0 and we have
(ρ + p) dV = 0
In other words, a cosmological constant is associated with a pressure equal to minus the energy density: p = - ρ.
Robert McNees
If the cosmological constant exists and is positive, which seems to be the best explanation for current cosmological observations*, then it produces a negative pressure which contributes to the expansion of space.
Sources of energy like matter and radiation have the opposite effect. But as the Universe expands their energy density dilutes as described above. The energy density associated with the cosmological constant, on the other hand, stays the same.
* You should google "Hubble Tension."
Robert McNees
Matter and radiation are trying to slow down the cosmological expansion, but lose gravitational potency as expansion dilutes their density.
The positive cosmological constant has been drinking green tea all goddamn day and just picked up a parking cone.
In an expanding universe it is difficult to overcome the eventual dominance of a positive cosmological constant. Given the apparent rate of expansion of our Universe, this would require more matter and radiation than seem to be present.
Robert McNees
There are lots of caveats and details that I glossed over in the arguments above.
But this is quite a bit of info to extract from the combination of thermodynamics, relativity, and the Cosmological Principle!
Marius Fortuna_3
@mcnees Thanks for that update, Bob.
Robert McNees
In the late 1930s, Tolman (remember him?) worked on astrophysical and cosmological solutions of Einstein's equations.
A 1939 paper, published alongside an article by Oppenheimer and Volkoff, considered static, spherical distributions of "perfect fluid" like radiation or dust.
An immediate application was modeling stellar structure. Oppenheimer and Volkoff's paper applied Tolman's solutions to a degenerate Fermi gas of neutrons that might exist at the core of a collapsed star.
Robert McNees
Equation (10) in the O&V paper is now known as the Tolman-Oppenheimer-Volkoff equation.
It relates the pressure and energy density of a self-gravitating, spherically symmetric distribution of perfect fluid in hydrostatic equilibrium.
Robert McNees
A neat application of their work has to do with the entropy of self-gravitating systems.
When we ignore gravity, we expect the entropy of a thermodynamic system in a box to be proportional to the volume of the box. For a sphere of radius R that would be S ~ R³.
So people often express surprise when they learn that the entropy of a black hole is instead proportional to the area of its event horizon. For a spherically symmetric Schwarzschild black hole of radius R, this is S ~ R².
Robert McNees
(There's no obvious meaningful notion of the volume inside a black hole, so one shouldn't expect an entropy that is proportional to a volume, but that's another discussion.)
Robert McNees
This is usually framed as "Wow, weird, S ~ R² rather than the naive expectation S ~ R³."
But you can use Tolman, Oppenheimer, and Volkoff's work to show that S ~ Volume should not be the expectation when gravity is involved!
Self-gravitating systems typically have an entropy that grows with size much more slowly than the volume. A little thermodynamics and the TOV equation shows that a self-gravitating sphere of radiation with radius R has an entropy that grows like R to the 3/2 power.
Robert McNees
The surprising thing about a black hole's entropy isn't that it grows much more slowly than a naive volume scaling. Rather, it's that it grows so much *faster* than the entropy of any other self-gravitating system!
Robert McNees
Anyway, as I mentioned way back at the beginning of this thread, Tolman was science advisor to General Leslie Groves during WWII. He often had to help military and intelligence personnel get up to speed on scientific issues related to the Manhattan Project.
Tolman would take these folks out on his sailboat, far from shore. Only then, when he could be sure that no one was around to overhear, would be begin to teach them the physics behind the atomic bomb.
https://www.aip.org/history-programs/niels-bohr-library/oral-histories/25565
Robert McNees
Richard Tolman's former home used to be near the Caltech campus, but now finds itself *on* campus because of another sort of expansion.
It has been converted to a little museum and meeting facility for the Keck Institute for Space Studies. A lovely place to sit and think about the Universe.
Here's a page where you can learn more about the Tolman / Bacher House:
franco_vazza
@mcnees Nice thread about the equation of states of the different constituents of the Universe!
quite some time ago, I did something vaguely similar here:
https://mastodon.social/@franco_vazza/109647449680240271
Robert McNees
👆A little thermodynamics can tell you a lot about the evolution of the Universe! A thread on Richard Chase Tolman.
Tolman's ideas still lead to surprising perspectives on problems at the forefront of the theoretical physics. One of his papers, written 85 years ago, turns the naive framing of black hole entropy on its head.
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