In the old children’s TV show Lambchop’s Singalong, the adorable Lambchop and her woodland friends famously sang “The Song that Doesn’t End”:
🎶 This is the song that doesn’t end; (Yes, it goes on and on my friend.) Some people started singing without knowing what it was; And they’ll continue singing it forever, just because…” 🎵
The song then repeats. 44 notes, lasting roughly 12 seconds per cycle. It was cute. At first.
Just watch for one minute… you’ll see
But the singalong was unsustainable, you see: every 12 seconds, Lambchop returned to the beginning of the song but raised the key by a half step. Starting in the key of G, the highest note (the syllable “star” in “some people started singing”) was E4, which has a frequency of 323.63 Hertz. In the second pass, 12 seconds later, that E4 turned to an F4, with a frequency of 342.88 Hertz, an increase by a factor of 2^(1/12) or approximately 1.0595. After 12 cycles, 144 seconds later, the frequency of the high note had risen by a full factor of 2 (now E5).
But this is The Song that Doesn’t End. (Yes, it goes on and on my friend.) So the singing continued.
After 468 seconds (7.8 minutes), the high note was a G7, matching Maria Carey’s famous 2003 rendition of the Star Spangled Banner. One minute later, Lambchop matched the world record C8. On and on (my friend), beyond what was thought to be possible, the woodland creatures reached D10# (a frequency of about 20000 Hertz = 20 kHz), achieving the highest note a typical human can hear a mere 14 minutes from the start of the singalong. For a few minutes, still a handful of animals could hear the high note, but before long the song became silent to all.
But this is The Song that Doesn’t End. (Yes, it goes on and on my friend.) So the singing continued.
Lambchop and her friends continued to sing inaudibly (at frequencies known as “ultrasound”) for several minutes, but even this was not sustainable. The maximum frequency that air can support can be determined roughly as the speed of sound in air (about 300 meters/second) divided by the mean free path of air molecules (about 30 nanometers at sea level). The Singalong reached this limit of 10^10 Hz = 10 GHz roughly 1 hour after it began. They moved the Singalong underwater to buy some more time: there they could achieve a frequency up to 1000 GHz = 1 THz, but they hit this mark only 15 minutes later. By singing into periodic crystal lattices, their song could continue for another 10 minutes, but soon no substance on Earth would carry their pressure waves.
But this is The Song that Doesn’t End. (Yes, it goes on and on my friend.) So the singing continued.
The woodland creatures had anticipated this inevitability. They developed technology that allowed them to transfer their song from sound waves into light waves. They sent laser pulses with frequencies matching each note out into the vast expanse of space. The high note, now at 10 THz, was not yet visible, but it entered the visible band a mere 5 minutes after transitioning to light; 2 minutes later, it entered the ultraviolet range. It didn’t matter; they had started singing it not knowing what it was, and they’ll continue singing it forever (just because). 13 more minutes of singing, and they were emitting X-rays; 20 more minutes, gamma rays. To conserve energy they reduced the intensity of the emission, releasing one photon at a time whose energy was now about 10 mega electron volts (MeV), greater than the binding energy of nuclei.
But this is The Song that Doesn’t End. (Yes, it goes on and on my friend.) So the singing continued.
At roughly the 200 minute mark, Lambchop et al. were emitting gamma rays of the highest energy every recorded, above 100 TeV. At 4 hours, each photon had energy of order 1 Joule, enough to heat 1 gram of water by 0.25℃. Soon they needed to harness significant power reserves to source their ever-growing need for energetic photon emission. Civilizations on other planets would have soon been rained upon by mighty photons carrying The Song… if there had been any within a few light minutes of Earth. Thankfully, none were.
On and on and on they sung until 5.3 hours into the singalong, when they reached another fundamental limit: the high note frequency was given by a photon of energy 12 billion Joules, or 12 GJ. The wavelength of the photons was nearing the Planck length, the theorised tiny distance scale at which our classical notion of spacetime breaks down. Lambchop and her friends worked to unite quantum mechanics with Einstein’s general relativity, only to learn that even higher energy photons could not be emitted, even in principle.
But this is The Song that Doesn’t End. (Yes, it goes on and on my friend.) So the singing continued.
No higher frequencies were possible for individual particles, but energy is a close enough proxy to frequency that the Singers found this an acceptable substitute for continuing the Eternal Song; nonetheless, they had to harness new forms of energy emission to continue. They began launching rocks (asteroids really), whose energy was equal to the now-necessary minimum energy of 12 GJ, while modulating the launch speed so that the total kinetic energy of each rock was equal to the energy-frequency of each note in the song. (The extra kinetic energy difference between notes was now essentially negligible, though with high-precision instruments, one could in principle distinguish them.) In the first such cycle, the rock energy of 12 GJ corresponded to 10^10 kg; they needed 44 rocks for the 44 notes in the song. So in that first cycle, they emitted about 10^12 kg of mass from the Earth. In the next cycle, more, and more, and more.
The Earth has a total mass of roughly 6*10^24 kg. So it was only about 90 minutes after the emission of the first rock-song (6.75 hours from the start) that the Lambchop Singers were emitting 1% of the mass of the Earth per note. 12 song-cycles later, 2% per note. Near the 7 hour mark, the mass of the Earth was depleted and the Singers found themselves without a planet to sing on.
But this is The Song that Doesn’t End. (Yes, it goes on and on my friend.) So the singing continued.
Next they would harness the mass of the sun for their Eternal Song, tearing it apart note by violent note, but this bought them less than an hour. Before reaching the 8th hour of singing, they already needed 1 full solar mass per note, so they traveled to other star systems. Our Milky Way galaxy is believed to be home to 100 billion stars (10^11); at 9 hours, Lambchop needed 1 billion (10^9) of these per note. Soon the mass of all the stars in the galaxy wasn’t enough, and Lambchop needed to harness the mass of the dark matter (about 10 times more mass than all the stars). 20 more minutes and the Milky Way was no more, shattered into trillions of melodious parts.
There were more galaxies, but they did not last long. 100 billion galaxies housing 100 billion stars (a total of 10^22 stars, an overestimate of the mass of the universe) each, would only allow the Singers to reach 10.5 hours of total singing. The dark matter bought them a few extra minutes, but before long there was not enough mass in the universe for even a single note. The universe, its matter content now united in glorious Song, had been ripped to shreds.
But this is The Song that Doesn’t End. (Yes, it goes on and on my friend.) So the singing continued.
The largest contribution to the energy density of the universe is Dark Energy. Even better, as the universe expands, ever more Dark Energy is created in the vacuum of empty space. The Singers realized that this was the ultimate source of power for the Eternal Song. They learned its secrets, developed tools to extract this energy from the very vacuum of spacetime, and convert it into impossibly powerful energy sources, alive with the sound of music. With each cycle, they extracted ever more energy, whilst the universe expanded and fed them more power, then more, and more, and more.
It’s hard to know precisely how long this went on. The energy density in Dark Energy is, as far as we know, only 10^(-29) grams/cm^3, but if it permeates the entire universe of radius 10^29 cm, then the total amount of Dark Energy could be only as large as 10^56 kg, or 10^26 stars (again, likely an overestimate). More could be created as spacetime expands, but the rate was insufficient. No later than 11 hours from the start of The Song, the Singers needed the full present-day energy of the universe to sing a single note.
There was no energy left in our universe. It could not be replenished fast enough. The Song could not be sustained…
…
Some ‘where’, in a ‘nearby’ corner of the Multiverse, an advanced race of intelligent beings watches as our universe crunches into nothingness, with no energy remaining to sustain it.
“Shame,” one Being telepathically communicates to the Hive Mind. “They had promise.”
“Wonder what happened,” intuits another in reply.
“The only signal We detect appears to be an encoded transmission, some last message sent from the final survivors.”
“It has a… musical pattern.”
“Well, even though We don’t know what it is, perhaps We should see what it sounds like… just because.”
🎶 “This is The Song that Doesn’t End.” 🎶
🎶 (Yes, it goes on and on my friend.) 🎶