Not Crunch Time

A familiar knock on my office door. “C’mon in, Jeremy, the door’s open.”

“Got a few minutes, Mr Moire?”

The second serious-sounding visitor today. I push my keyboard aside again. “Sure, what’s up?”

“I read your ‘Tops of Time‘ post and then I watched one of Katie Mack’s End of Everything‘ YouTube videos and now I’m confused. And worried.”

“I can understand that. Clearing up the confusion should be easy. Then I’ll do what I can about the worry part, okay?”

“That’d be great, sir.”

“So, imagine an enormous sheet of graph paper, and then imagine Puerto Rico laid down on top of that. You could use the graph paper to describe the latitude and longitude of any place on the island, right?”

“Sure, probably.”

“I happen to know that Playa Jobos is the northernmost point of the island. Does north stop there?”

“Nosir. The island stops there, but north keeps going.”

“Well, there you are.”

“Wait … oh, you’re saying that time by itself keeps going forever but what’s in the Universe might not and that’s what Dr Mack is talking about?”

“That’s the idea. More precisely, the ‘tops‘ I wrote about are different ways that spacetime’s time coordinate could play out in the future, or maybe not. Mack’s ‘end of everything‘ is about the future history of physical stuff laid on top of our mathematical spacetime constructs. Does that clarify things?”

“Mmm, yessir, but what about the ‘maybe not‘ you said?”

“This gets metaphysical, but cosmology often skates on that edge. Descartes and others maintained that space has meaning only when there are separate objects. If there was only one thing in the Universe you’d have nothing to compare sizes against and there’d be no point in measuring distances away from it. That’d be even more the case if there’s nothing. Same thing for time and events. From that perspective, if somehow the Universe emptied out then space and time sort of stop.”

“Just sort‑of stop, like Puerto Rico stops at that Playa place. Really they keep on going, I think, even if no‑one’s there to measure anything.”

“A perfectly reasonable position when there’s no evidence either way. Anyhow, a few of Mack’s scenarios wind up in that situation, right?”

“Umm… there’s the Big Crunch that reverses the Big Bang.”

“That one was popular before we got good data. The idea was that the Big Bang pushed everything apart but eventually gravity will slow outward momentum and pull everything back together again. The notion probably came from humanity’s experience with dirt falling back down after an explosion. The problems with that scheme are that the Big Bang wasn’t an explosion, outward momentum isn’t a thing and besides, we’ve got increasingly good data showing that between‑galaxy distances are getting wider, not shrinking. The last five billion years that’s sped up.”

“Wait, not an explosion? All the videos show it that way.”

“Chalk it up to artistic license. It’s hard to show everything moving away from everything else without making it look like the viewpoint’s simply diving into a static arrangement. No, an explosion comes out of a center and that’s not the Bang. Remember that huge piece of graph paper? Make it a balloon, tack Puerto Ricos all over it, then pump in some air. There’s no center, but every islander thinks their island is the center and every other island is running away from them. Really, all that’s happening is that the stretching rubber is creating new inter‑island space everywhere.”

“And that’s Universe expansion?”

“Mm-hm. Also known as Hubble Flow. We’ve looked very hard for a center of motion, haven’t found one.”

“If everything’s moving, why isn’t that momentum?”

“It is momentum, but only pairwise. For any two galaxies you can calculate mass times speed same as always. For really distant objects you’ve got to use a relativistic version. Anyway, in the cosmological context you’ve got to ask, momentum relative to what? Everyone has this picture that things came from a common center and will fall back there. The way Hubble expansion works, though, there’s no particular go‑back place.”

“Everything’s speeding up and going everywhere so no Big Crunch then.”

“Not on the original model, anyway.”

~~ Rich Olcott

Time Is Where You Find It

A familiar footstep in the hall outside my office, “C’mon in, Vinnie, the door’s open.”

“Got a few minutes, Sy?”

More than just “a minute.” This sounds serious so I push my keyboard aside. “Sure, what’s up?”

“I’ve been thinking about different things, putting ’em together different ways. I came up with something, sorta, that I wanted to run past you before I brought it to one of Cathleen’s ‘Crazy Theories‘ parties.”

“Why, Vinnie, you’re being downright diffident. Spill it.”

“Well, it’s all fuzzy. First part goes way back to years ago when you wrote that there’s zero time between when a photon gets created and when it gets used up. But that means that create and use-up are simultaneous and that goes against Einstein’s ‘No simultaneity‘ thing which I wonder if you couldn’t get around it using time tick signals to sync up two space clocks.”

“That’s quite a mix and I see why you say it’s fuzzy. Would you be surprised if I used the word ‘frame‘ while clarifying it?”

“I’ve known you long enough it wouldn’t surprise me. Go ahead.”

“Let’s start with the synchronization idea. You’re not the first to come up with that suggestion. It can work, but only if the two clocks are flying in formation, exactly parallel course and speed.”

“Hah, that goes back to our first talk with the frame thing. You’re saying the clocks have to share the same frame like me and that other pilot.”

“Exactly. If the ships are zooming along in different inertial frames, each will measure time dilation in the other. How much depends on their relative velocities.”

“Wait, that was another conversation. We were pretending we’re in two spaceships like we’re talking about here and your clock ran slower than mine and my clock ran slower than yours which is weird. You explained it with equations but I’ve never been good with equations. You got a diagram?”

“Better than that, I’ve got a video. It flips back and forth between inertial frames for Enterprise and Voyager. We’ll pretend that they sync their clocks at the point where their tracks cross. I drew the Enterprise timeline vertical because Enterprise doesn’t move in space relative to Enterprise. The white dots are the pings it sends out every second. Meanwhile, Voyager is on a different course with its own timeline so its inertial frame is rotated relative to Enterprise‘s. The gray dots on Voyager‘s track show when that ship receives the Enterprise pings. On the Voyager timeline the pings arrive farther apart than they are on the Enterprise timeline so Voyager perceives that Enterprise is falling farther and farther behind.”

“Gimme a sec … so Voyager says Enterprise‘s timer is going slow, huh?”

“That’s it exactly. Now look at the rotated frame. The pink dots show when Voyager sends out its pings. The gray dots on Enterprise‘s track show when the pings arrive.”

“And Enterprise thinks that Voyager‘s clock is slow, just backwards of the other crew. OK, I see you can’t use sync pulses to match up clocks, but it’s still weird.”

“Which is where Lorentz and Minkowski and Einstein come into the picture. Their basic position was that physical events are real and there should be a way to measure them that doesn’t depend on an observer’s frame of reference. Minkowski’s ‘interval‘ metric qualifies. After converting time and location measurements to intervals, both crews would measure identical spacetime separations. Unfortunately, that wouldn’t help with clock synchronization because spacetime mixes time with space.”

“How about the photons?”

“Ah, that’s a misquotation. I didn’t say the time is zero, I said ‘proper time‘ and that’s different. An object’s proper time is measured by its clock in its inertial frame while traveling time t and distance d between two events. Anyone could measure t and d in their inertial frame. Minkowski’s interval is defined as s=[(ct)²‑d²]. Proper time is s/c. Intuitively I think of s/c as light’s travel time after it’s done traversing distance d. In space, photons always travel at lightspeed so their interval and proper time are always zero.”

“Photon create and use-up aren’t simultaneous then.”

“Only to photons.”

~~ Rich Olcott

The Threshold of Stuffiness

<chirp chirp> “Moire here.”

“Hi, Sy, it’s Susan Kim. I read your humidifier piece and I’ve got your answer for you.”

“Answer? I didn’t know I’d asked a question.”

“Sure you did. You worked out that your humidifier mostly keeps your office at 45% relative humidity by moisturizing incoming air that’s a lot drier than that. As a chemist I like how you brought in moles to check your numbers. Anyway, you wondered how to figure the incoming airflow. I’ve got your answer. It’s a scaling problem.”

“Mineral scaling? No, I don’t think so. The unit’s mostly white plastic so I wouldn’t see any scaling, but it seems to be working fine. I’ve been using de-ionized water and following the instructions to rinse the tank with vinegar every week or so.”

“Nope, not that kind of scale, Sy. You’ve got a good estimate from a small sample and you wondered how to scale it up, is all.”

“Sample? How’d I take a sample?”

“You gave us the numbers. Your office is 1200 cubic feet, right, and it took 88 milliliters of water to raise the relative humidity to where you wanted it, right, and the humidifier used a 1000 milliliters of water to keep it there for a day, right? Well, then. If one roomful of air requires 88 milliliters, then a thousand milliliters would humidify (1000/88)=11.4 room changes per day.”

“Is that a good number?”

“I knew you’d ask. According to the ventilation guidelines I looked up, ‘Buildings occupied by people typically need between 5 and 10 cubic feet per minute per person of fresh air ventilation.‘ You’re getting 11.4 roomfuls per day, times your office volume of 1200 cubic feet, divided by 1440 minutes per day. That comes to 9.5 cubic feet per minute. On the button if you’re alone, a little bit shy if you’ve got a client or somebody in there. I’d say your building’s architect did a pretty good job.”

“I like the place, except for when the elevators act up. All that figuring must have you thirsty. Meet me at Al’s and I’ll buy you a mocha latte.”

“Sounds like a plan.”

“Hi, folks. Saw you coming so I drew your usuals, mocha latte for Susan, black mud for Sy. Did I guess right?”

“Al, you make mocha lattes better than anybody.”

“Thanks, Susan, I do my best. Go on, take a table.”

“Susan, I was thinking while I walked over here. My cousin Crystal doesn’t like to wear those N95 virus masks because she says they make her short of breath. Her theory is that they trap her exhaled CO2 and those molecules get in the way of the O2 molecules she wants to breathe in. What does chemistry say to that theory?”

“Hmm. Well, we can make some estimates. N95 filtration is designed to block 95% of all particles larger than 300 nanometers. A couple thousand CO2 molecules could march abreast through a mesh opening that size no problem. An O2 molecule is about the same size. Both kinds are so small they never contact the mesh material so there’s essentially zero likelihood of differential effect.”

“So exhaled CO2 isn’t preferentially concentrated. Good. How about the crowd‑out idea?”

“Give me a second. <tapping on phone> Not supported by the numbers, Sy. There’s one CO2 for every 525 O2‘s in fresh air. Exhaled air is poorer in O2, richer in CO2, but even there oxygen has a 4‑to‑1 dominance.”

“But if the mask traps exhaled air…”

“Right. The key number is the retention ratio, what fraction of an exhaled breath the mask holds back. A typical exhale runs about 500 milliliters, could be half that if you’ve got lung trouble, twice or more if you’re working hard. This mask looks about 300 milliliters just sitting on the table, but there’s probably only 100 milliliters of space when I’m wearing it. It’s just arithmetic to get the O2/CO2 ratio for each breathing mode, see?”

“Looks good.”

“Even a shallow breather still gets 79 times more O2 than CO2. Blocking just doesn’t happen.”

“I’ll tell Crys.”

~ Rich Olcott

It’s The Heat AND The Humidity

<from the casebook of Sy Moire, Consulting Physicist> Monday. Weather sunny, warm for this time of year. Dry, bad effects on nose and throat at wake‑up time. Bought a room humidifier Friday at Big Box — up‑scale, 4‑liter reservoir, ultrasonic but silent, WiFi‑enabled etc. Long way from the jug with boiler tube my folks used to use. WiFi’s a sneaky way to avoid building a remote control — just use the customer’s smart phone. Guess that keeps the price down.

Good news is, phone app does graphs of relative humidity against time. Had it in measure-only mode in office Monday night, baseline wobbled in the 32-36% range. Bad news is, when the device is running on Automatic it works toward a 45% target but had trouble getting near that high on its maiden effort on Saturday. Wondered, how much water would it have to send into the room to hit the target?

Start with the numbers. “45%” is forty‑five percent of what? Surely not some arbitrary maximum. Weather guy talks about relative humidity so, relative to what? Searched a little in the internet. All the sources say RH is a ratio of ratios, something over maximum something. “Something” is water vapor mass per unit volume or else water vapor mass per total gas mass in the same volume. Clouds, fog, raindrops and snowflakes don’t count. Then things get fuzzy. Some sources say the maximum is “saturation” which is just a tautology. The most precise definition says “the partial pressure of water in air over a large flat surface of pure water under laboratory conditions” <shudder>.

Doing engineer stuff here so keep it simple. Found a chart of water vapor content at 100% humidity at different temperatures. Will go with that. No surprise, the warmer the air, the more water mass it can hold before fogging up. I’m comfortable around 68°F which is 20°C. The chart says 100% saturated 20°C air holds 17.3 grams of water per cubic meter. 45% of that is 7.8 and 30% is 5.3. Need to know how many cubic meters in the office.

OK, measured the room as 10’×15′ with an 8′ ceiling. Ignore the space the furniture takes up. Total volume is 1200 cubic feet. Old Reliable says that’s 34 cubic meters. If the room’s at 100% humidity its air holds
  (34 m3)×(17.3 g/m3)
   = 588 grams of water.
At 1000 grams per liter that’s 0.588 liter or about a pint. Suppose humidifier starts when the room’s at 30% humidity. For a 15% bump to 45% the gadget has to vaporize
  (34 m3)×(17.3 g/m3)×(0.45-0.30)
   = 88 g = 88 milliliters.

Wow, that’s only about 6 tablespoons. Does that number even make sense? OK, air is about 80% N2, molecular weight 28. The other 20% is mostly O2, molecular weight 32. The average is near 29. Basic chem class stuff — at Earth‑typical room temp and pressure, a 22.4‑liter chunk of air has a mass near 29 grams, so office’s roomful of air would mass
  (29g/mole)×(1 mole/22.4 liters)
   ×(1000 liters/m3)×(34 m3/room)
    = 44 kilograms
The 588-gram number says that it’d get foggy in here if the moisture content ever got much above (588 g)/(44000 g) = 1.3% by mass, which sounds reasonable.

88 milliliters ain’t much, so how come the unit used up a liter of water in just one day?

Ah-hah. Air’s not sitting still. Ventilation system continually brings in low‑humidity outside air. Plus, clients complain in the wintertime about cold drafts leaking in around the door, transom and windows. If it weren’t for air shuttling in and out, we’d use up all the oxygen in here — that isn’t happening. Wonder how to calculate that flow. Bottom line is, humidifier doesn’t moisten what’s in the room so much as it loads up what comes in dry.

Problem — evaporating water cools air, Old Reliable says 2256 kilojoules per kilogram. 88 grams won’t have much effect, but a liter/day is a kilogram/day. An hour is 1/24 of a day. 1/24 liter means 94 kilojoules per hour of cooling. Air heat capacity is 1 kilojoule/(kilogram oC). (94 kJ/44 kg air)=2.1 degrees per hour. Suddenly I feel chilly.

~~ Rich Olcott

Shadow Plays

“A strawberry scone and my usual black, Al.”

“Sure thing, Sy, comin– Hiya, Cathleen, see my new poster? Event Horizon Telescope pictures of the two big‑guy black holes we’ve actually seen so far. Those white-hot blobs buried in those red rings. Ain’t it a beaut? What’ll you have?”

“They’re certainly wonderful graphics, Al. I’ll have a caramel latte, please, with a plain scone.” I’m waiting for it, because Cathleen never passes up a teachable moment. Sure enough — “Of course, neither one actually looks like that or represents what you think. Those images were created from radio waves, not visible light or even infrared. The yellows and whites don’t represent heat, and that darkness in the middle isn’t the black hole.”

“Whoa, don’t harsh Al’s happy, Cathleen. Maybe just go at it a step at a time?”

<sigh> “You’re right, Sy. Sorry, Al, I just get frustrated when press‑agent science gets in the way of the real stuff which is already interesting on its own. For instance, I haven’t seen anything in the pop‑sci press about the EHT people using the same 2017 data to produce both images, even though the two objects are almost 90° apart in the sky. I think about our optical telescopes and the huge high-tech motors it takes to point them in the right direction. These guys just re-work their data and they’re good for another round.”

“It’s a cute trick, alright, Cathleen, steering a distributed telescope with arithmetic.”

“OK, you guys are over my head — distributed telescope?”

“The EHT Collaboration works with eight radio telescopes scattered across the world. The signal from any point in the sky has a different time offset at each telescope depending on the angle to the point. If you know the baseline between each pair of scopes and you’ve got really good clocks keeping track of time at each location, when you combine the data from all eight locations it’s just arithmetic to pick out matching signals at the right set of offsets for any point of origin.”

“A lot of arithmetic, Cathleen.”

“I’ll give you that, Sy. Al, it took the researchers and some hefty compute facilities two years to boil down the data for the M87 monster. In principle, when they wanted to inspect the Milky Way’s beast all they had to do was run through the same data selecting for signal matches at the offsets pointing to Sgr A*. Awesome tech, huh?”

“Awesome, yeah, but if the colors aren’t heat, what are they?”

“Electron density, mostly. Your red‑and‑yellow Jupiter poster over there is like most heat maps. Researchers figure a pixel’s temperature by comparing data from multiple wavelengths with the Planck curve or some other calibrated standard. These images, though, came from a single wavelength, 1.3 millimeters. Light at shorter wavelengths can’t get past the dust, longer wavelengths can’t give us the image resolution. Millimeters waves are in the radio part of the spectrum — too low‑energy to detect moving charge inside atoms or between molecule components. The only thing that can give off those photons is free‑floating electrons. The brightest pixels have the most electrons.”

“So the hole isn’t the black hole?”

“Depends on your definition, I suppose. Everyone visualizes that black sphere, the event horizon, when they think ‘black hole.’ That’s not what the dark patches are. By my definition, though, a ‘black hole‘ is the whole package — central mass, event horizon, ergosphere if it’s spinning, a jet maybe and everything else that’s associated with the mass. It’s as much a collection of processes as a thing. Anyhow, the bright stuff in these images does come from accretion disks.”

“The dark patch is the disk’s inside edge?”

“Nope, it’s the shadow of the photon sphere. Before you ask, that’s a light‑trapping shell 1½ times the horizon’s diameter. Depending on its angle of approach, a photon that touches the sphere either spirals inward, orbits forever, or swerves outward. Going straight doesn’t happen. The shadow memorializes Earth‑bound photons that bounced away from us.”

“I guess my happy’s back, Cathleen, but it’s different.”

“You’re welcome, Al. Now how about the coffee and scones we asked for?”

~~ Rich Olcott

Credit: Event Horizon Telescope Collaboration
Image: Lia Medeiros, ISA, EHTC

The Tops of Time

Mr Feder doesn’t let go of a topic. He’s still stewing about Time. “Moire or somebody said the Big Bang is the Bottom of Time because there wasn’t any time before then. I guess I gotta buy that, but bottoms gotta have tops. What’s the Top of Time?”

“Whoa, Mr Feder, that’s a fuzzy question with a lot of answers, most of which are guesses.”

“No theories?”

“Not really, A few used to be called theories but people started muttering about testability so the theories got downgraded to hypotheses and now they’re guesses except for the ones that’ve been dropped altogether.”

“Like what?”

“Steady State, for one — the idea that Time has no end. That used to be popular, mostly because it was simple. Problem was, Edwin Hubble showed that other galaxies are separate from the Milky Way and in fact they’re receding from us. That clashed with the Cosmological Principle, the idea that on a large scale things are pretty much the same everywhere. Galaxies moving away from each other leave behind empty space that isn’t ‘pretty much the same.’ For the Steady State model to work, new matter would have to spring into existence between the departing galaxies.”

“Nature hates a vacuum, eh?”

“Apparently she doesn’t. Evidence has piled up against the Steady State model and in favor of the Big Bang. We still think the Cosmological Principle is a good assumption, but only on scales bigger than a few hundred million lightyears.”

“So Time has a Top, then.”

“Depends on how you define ‘Top.’ We’re now into Metaphysics territory, where theories come cheap and flimsy. It’s conceivable, for instance, that the Universe curves back onto itself along one or more of its dimensions. If it loops back along the time dimension then we’d be in an oscillating universe that cycles from Big Bang to Big Crunch and back out again. Time would have no Top or Bottom. Crosswise to time, some thinkers like the idea that the Universe circles back along a space dimension. If that’s true and we could see far enough we could inspect the back of our heads.”

“Wait, we’ve got lots of black holes. If their singularities are in the infinite future like you said, that’d stymie the circling.”

“Good point, Vinnie. As I understand the math, connectivity like that is possible if our 4D spacetime is embedded in a ‘bulk‘ with five or more dimensions. But that’s more complicated than I’m willing to accept without at least some evidence which no-one’s shown me yet. The endings of the 2001 and Interstellar movies don’t count.”

“What else you got?”

“What other theories, Mr Feder? How about block universes? Maybe the space dimensions are solid but only part of the time dimension is real. Some people opine that the only reality is ‘NOW,’ an infinitely thin slice of time evolving towards the future. A memory would only be a surviving imprint of things that stopped existing when Time was done with them.”

“I don’t like that one. For one thing, it doesn’t jibe with the ‘everyone’s got their own NOW‘ thing from relativity.”

“Einstein didn’t like it either. The easiest way to reconcile all those different versions of NOW is to assume that they all co‑exist permanently. I call that notion the closed block model. The idea is that all reality — past, present and future — is real and rigid. We perceive time as flowing only because consciousness floats upward along the time coordinate. The Top of Time is way up there, just waiting for us to arrive.”

“Why no sinking downward?”

“Good question, no good answer that I’ve seen. Besides, the closed block model doesn’t allow for free will. I like having free will.”

“Me, too. OK, if there’s closed block, what’s open block?”

“The future doesn’t exist yet. Picture the open block model as our 4D spacetime being a bowl with the Big Bang at the bottom. Time progressively fills the bowl like water. NOW is the Top of Time. Those relativity‑shifted NOWs only show up when we compare records of past observations.”

“Cheap and flimsy, but a pretty picture.”

Adapted from a public domain image,
Credit: NASA/WMAP Science Team

~~ Rich Olcott

Why I Never Know What Time It Is

It’s always fun watching Richard Feder (of Fort Lee, NJ) as he puts two and two together. He gets a gleam in his eye and one corner of his mouth twitches. On a good day with the wind behind him I’ve seen his total get as high as 6½. “I wanna get back to that ‘everybody has their own time‘ monkey‑business where if you’re moving fast your clock slows down. What about the stardates on Star Trek? Those guys go zooming through space at all different angles and speeds. How do they keep their calendars in synch?”

Trekkie and Astronomy fan Al takes the bait. “Artistic license, Mr Feder. The writers can make anything happen, subject to budgets and producer approval. The first Star Trek series, they just used random four‑digit numbers for stardates. That was OK because the network aired the episodes in random order anyway so no‑one cared about story arc continuity. Things were more formal on Captain Picard’s Enterprise, as you’d expect — five‑digit stardates, first digit always ‘4‘ for 24th Century, thousands digit was ‘1‘ for season one, ‘2‘ for season two and so on. Working up the other way, the digit right of the decimal point was tenths of a standard day, the units place counted days within an episode and the tens and hundreds they just picked random numbers.”

“I suppose that’s what they did, but how could they make it work? You guys yammer on about time dilation. Say a ship’s running at Warp Whoop‑de‑doo, relativity should slow its calendar to a crawl. You couldn’t get a whole fleet into battle position when some of the ships had to get started years ahead of time. And that’s just the dilation slow-down, travel time’s on top of that.”

“Travel time measured how, Mr Feder, and from where?”

“Well, there you go, Cathleen, that’s what I’m talking about!”

“You know that Arthur C Clarke quote, ‘Any sufficiently advanced technology is indistinguishable from magic‘? The Enterprise crew’s always communicating with ‘sub‑space radio’, which sure looks like magic to me. They could send sync pulses through there along with chatter. When you drop out of warp space, your clocks catch the pulses and sync up, I suppose.”

“There’s a deeper issue than that, guys.”

“What’s that, Sy?”

“You’re talking like universal time is a thing, which it isn’t. Hasn’t been since Einstein’s Special Relativity used Minkowski’s math to stir space and time together. General Relativity scrambles things even worse, especially close to a strong gravity center. You remember about gravity forcing spacetime to curve, right? The curvature inside a black hole’s event horizon gets so tight that time rotates toward the geometric center. No, I can’t imagine what that looks like, either. The net of it, though, is that a black hole is a funnel into its personal future. Nothing that happens inside one horizon can affect anything inside another one so different holes could even have different time rates. We’ve got something like 25000 or more stellar black holes scattered through the Milky Way, plus that big one in the center, and that’s just one galaxy out of billions. Lots of independent futures out there.”

“What about the past, Sy? I’d think the Big Bang would provide a firm zero for time going forward and it’s been one second per second since then.”

“Nup. Black holes are an extreme case. Any mass slows down time in its vicinity, the closer the slower. That multi‑galaxy gravitational lens that lets us see Earendel? It works because the parts of Earth‑bound light waves closest to the center of mass see more time dilation than the parts farther away and that bends the beam toward our line of sight.”

“Hey, that reminds me of prisms bending light waves.”

“Similar effect, Vinnie, but the geometry’s different. Prisms and conventional lenses change light paths abruptly at their surfaces. Gravitational lenses bend light incrementally along the entire path. Anyhow, time briefly hits light’s brakes wherever it’s near a galaxy cluster, galaxy or anything.”

“So a ship’s clock can fidget depending on what gravity it’s seen recently?”

“Mm-hm. Time does ripples on its ripples. ‘Universal Time‘ is an egregious example of terminology overreach.”

~~ Rich Olcott

Pushing It Too Far

It’s like he’s been taking notes. Mr Feder’s got a gleam in his eye and the corner of his mouth is atwitch. “You’re not getting off that easy, Cathleen. You said that Earendell star’s 66 trillion lightyears away. Can’t be, if the Universe’s only 14 billion years old. What’s going on?”

“Oops, did I say trillion? I meant billion, of course, 109 not 1012. A trillion lightyears would be twenty times further than the edge of our observable universe.”

“Hmph. Even with that fix it’s goofy. Sixty-six billion is still what, five times that 14 billion year age you guys keep touting. I thought light couldn’t travel that far in that time.”

“I thought the Universe is 93 billion light years across.”
  ”That’s diameter, and it’s just the observable universe.”
    ”Forty-seven billion radially outward from us.”
      ”None of that jibes with 14 billion years unless ya got stuff goin’ faster than light.”

“Guys, guys, one thing at a time. About that calculation, I literally did it on the back of an envelope, let’s see if it’s still in my purse … Nope, must be on my office desk. Anyhow, distance is the trickiest part of astronomy. The only distance‑related thing we can measure directly is z, that redshift stretch factor. Locate a familiar pattern in an object’s spectrum and see where its wavelength lies relative to the laboratory values. The go‑to pattern is hydrogen’s Lyman series whose longest wavelength is 121 nanometers. If you see the Lyman pattern start at 242 nanometers, you’ve got z=2. The report says that the lens is at z=2.8 and Earendel’s galaxy is at z=6.2. We’d love to tie those back to distance, but it’s not as easy as we’d like.”

“It’s like radar guns, right? The bigger the stretch, the faster away from us — you should make an equation outta that.”

“They have, Mr Feder, but Doppler’s simple linear relationship is only good for small z, near zero. If z‘s greater than 0.1 or so, relativity’s in play and things get complicated.”

“Wait, the Hubble constant ties distance to speed. That was Hubble’s other big discovery. Old Reliable here says it’s something like 70 kilometers per second for every megaparsec distance. What’s that in normal language? <tapping keys> Whoa, so for every lightyear additional distance, things fly away from us about an inch per second faster. That’s not much.”

“True, Sy, but remember we’re talking distant, barely observable galaxies that are billions of lightyears away. Billions of inches add up. Like with the Doppler calculation, you get startling numbers if you push a simple linear relation like this too far. As an extreme example, your Hubble rule says that light from a galaxy 15 billion lightyears away will never reach us because Hubble Flow moves them away faster than photons fly toward us. We don’t know if that’s true. We think Hubble’s number changes with time. Researchers have built a bucketful of different expansion models for how that can happen; each of them makes different predictions. I’m sure my 66 came from one of those. Anyhow, most people nowadays don’t call it the Hubble constant, it’s the Hubble parameter.”

“Sixty-six or forty-seven or whatever, those diameters still don’t jibe with how long the light’s had a chance to travel.”

“Sy, care to take this? It’s more in your field than mine.”

“Sure, Cathleen. The ‘edge of the observable universe‘ isn’t a shell with a fixed diameter, it simply marks the take-off points for the oldest photons to reach us so far. Suppose Earendel sent us a photon about 13 billion years ago. The JWST caught it last night, but in those 13 billion years the universe expanded enough to insert twenty or thirty billion lightyears of new space between between here and Earendel. The edge is now that much farther away than when the photon’s journey started. A year from now we’ll be seeing photons that are another year older, but the stars they came from will have flown even farther away. Make sense?”

“A two-way stretch.”

“You could say that.”

~~ Rich Olcott

  • Thanks to my brother Neil, who pointed out the error and asked the question.

A Thumbtack in A Needlestack

“What’re the odds?”

“Odds on what, Vinnie?”

“A gazillion galaxies out there, only 41 lensing galaxy clusters, but one of them shows us a singleton star. I mean, what’s special about that star? What are the odds?”

I can’t help it. “Astronomical, Vinnie.”

Cathleen punches my shoulder, hard. “Sy Moire, you be ashamed of yourself. That pun was ancient a century ago. Vinnie, the odds are better than they seem. We didn’t just stumble on Earendel and the Sunrise Arc, we found them in a highly targeted Big Data search for things just like that — objects whose light was extremely stretched and also gravitationally bent in our direction. The Arc’s lensing galaxy cluster has a spherical effect, more or less, so it also acts on light from other far-away objects and sends it in other directions. It even bends an image of our Milky Way towards Earendel’s galaxy.”

“I call weaseling — you used ‘more or less‘.”

“Guilty as charged, Vinnie. A nice, spherical black hole is the simplest case of gravitational lensing — just one mass at the center of its simple light‑bending gravity field. Same thing for a single star like our Sun. Clusters are messy. Tens or hundreds of billion‑star galaxies, scattered at random angles and random positions about their common center of mass. The combined gravity field is lumpy, to say the least. Half of that research paper is devoted to techniques for estimating the field and its effects on light in the region around the Arc.”

“I guess they had to get 3D positions for all the galaxies in the cluster. That’d be a lot of work.”

“It would, Al, but that’s beyond what current technology can do. Instead, they used computer models to do — get this, Sy — curve fitting.”

<chuckle> “Good one, Cathleen.”

“What’s so funny?”

“There’s a well-established scientific technique called ‘curve fitting.’ You graph some data and try to find an equation that does a respectable job of running through or at least near your data points. Newton started it, of course. Putting it in modern terms, he’d plot out some artillery data and say, ‘Hmm, that looks like a parabola H=h+v·t+a·t2. I wonder what values of h, v and a make the H-t curve fit those measurements. Hey, a is always 32 feet per second per second. Cool.’ Or something like that. Anyhow, Cathleen’s joke was that the researchers used curve fitting to fit the Sunrise Arc’s curve, right?”

“They did that, Sy. The underlying physical model, something called ‘caustic optics,’ says that—”

“Caustic like caustic soda? I got burnt by that stuff once.”

Image by Heiner Otterstedt,
under the Creative Commons Attribution-Share Alike 3.0 Unported license

“That’s the old name for sodium hydroxide, Vinnie. It’s a powerful chemical and yeah, it can give you trouble if you’re not careful. That name and caustic optics both come from the Greek word for burning. The optics term goes back to using a lens as a burning glass. See those focused patterns of light next to your water glass? Each pattern is a caustic. The Arc’s lensing cluster’s like any light‑bender, it’s enclosed in a caustic perimeter. Light passing near the perimeter gets split, the two parts going to either side of the perimeter. The Earendel team’s curve‑fitting project asked, ‘Where must the caustic perimeter be to produce these duplicate galaxy images neighboring the Arc?‘ The model even has that bulge from the gravity of a nearby foreground galaxy.”

“And the star?”

“Earendel seems to be smack on top of the perimeter. Any image touching that special line is intensified way beyond what it ought to be given the source’s distance from us. It’s a pretty bright star to begin with, though. Or maybe two stars.”

“Wait, you don’t know?”

“Not yet. This study pushed the boundaries of what Hubble can do for us. We’re going to need JWST‘s infrared instruments to nail things down.”

Al’s in awe. “Wow — that caustic’s sharp enough to pick one star out of a galaxy.”

“Beat the astronomical odds, huh?”

Adapted from a public-domain image.
Credit: Science: NASA / ESA / Brian Welch (JHU) / Dan Coe (STScI); Image processing: NASA / ESA / Alyssa Pagan (STScI)

~~ Rich Olcott