# 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.”

“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

# Now And Then And There

Still at our table in Al’s otherwise empty coffee shop. We’re leading up to how Physics scrambled Now when a bell dings behind the counter. Al dashes over there. Meanwhile, Cathleen scribbles on a paper napkin with her colored pencils. She adds two red lines just as Al comes back with a plate of scones. “Here, Sy, if you’re going to talk Minkowski space this might be useful.”

“Hah, you’re right, Cathleen, this is perfect. Thanks, Al, I’ll have a strawberry one. Mmm, I love ’em fresh like this. OK, guys, take a look at Cathleen’s graphy artwork.”

“So? It’s the tile floor here.”

“Not even close, Mr Feder. Check the labels. The up‑and‑down label is ‘Time’ with later as higher. The diagram covers the period we’ve been sitting here. ‘Now‘ moves up, ‘Here’ goes side‑to‑side. ‘Table‘ and ‘Oven‘, different points in space, are two parallel lines. They’re lines because they both exist during this time period. They’re vertical because neither one moves from its relative spatial position. Okay?”

“Go on, Moire.”
”Makes sense to me, Sy.”

“Good. ‘Bell‘ marks an event, a specific point in spacetime. In this case it’s the moment when we here at the table heard the bell. I said ‘spacetime‘ because we’re treating space and time as a combined thing. Okay?”

“Go on, Moire.”
”Makes sense to me, Sy.”

“So then Al went to the oven and came back to the table. He traveled a distance, took some time to do that. Distance divided by time equals velocity. ‘Table‘ has zero velocity and its line is vertical. Al’s line would tilt down more if he went faster, okay?”

“Mmmm, got it, Sy.”
”Cute how you draw the come-back label backwards, lady. Go on, Moire.”

“I do my best, Mr Feder.”

“Fine, you’ve got the basic ideas. Now imagine all around us there’s graph paper like this — except there’s no paper and it’s a 4‑dimensional grid to account for motion in three spatial dimensions while time proceeds. Al left and returned to the same space point so his spacetime interval is just the time difference. If two events differ in time AND place there’s special arithmetic for calculating the interval.”

“So where’s that get us, Moire?”

“It got 18th and 19th Century Physics very far, indeed. Newton and everyone after him made great progress using math based on a nice stable rectangular space grid crossed with an orderly time line. Then Lorentz and Poincaré and Einstein came along.”

“Who’s Poincaré?”

“The foremost mathematician of nineteenth Century France. A mine safety engineer most days and a wide‑ranging thinker the rest of the time — did bleeding‑edge work in many branches of physics and math, even invented a few branches of his own. He put Lorentz’s relativity work on a firm mathematical footing, set the spacetime and gravity stage for Minkowsky and Einstein. All that and a long list of academic and governmental appointments but somehow he found the time to have four kids.”

“A ball of fire, huh? So what’d he do to Newton’s jungle gym?”

“Turned its steel rod framework into jello. Remember how Cathleen’s Minkowski diagram connected slope with velocity? Einstein showed how Lorentz’s relativity factor sets a speed limit for our Universe. On the diagram, that’d be a minimum slope. Going vertical is okay, that’s standing still in space. Going horizontal isn’t, because that’d be instantaneous travel. This animation tells the ‘Now‘ story better than words can.”

“Whah?”
”Whah?”

“We’re looking down on three space travelers and three events. Speeds below lightspeed are within the gray hourglass shape. The white line perpendicular to each traveler’s time line is their personal ‘Now‘. The travelers go at different velocities relative to us so their slopes and ‘Now‘ lines are different. From our point of view, time goes straight up. One traveler is sitting still relative to us so its timeline is marked ‘v=0‘ and parallels ours. We and the v=0 traveler see events A, B and C happening simultaneously. The other travelers don’t agree. ‘Simultaneous‘ is an illusion.”

~~ Rich Olcott

# Now And Then

“Alright, I suppose there’s no going down below the Universe’s Year Zero, but what about the other direction? Do you physics guys have a handle on Time’s Top?”

“That’d be Cosmology, Mr Feder. We physicists avoid theorizing about stuff we can’t check against data. Well, except for string theory. The far past leaves clues that astronomers like Cathleen can gather. Sad to say, though, we barely have a handle on Now.”

Cathleen grins. Al and Mr Feder go, “Whaaat?”

“No, really. One of Einstein’s insights was that two observers randomly and independently flying through space won’t be able to agree on whether two external events occurred simultaneously. They can’t even agree on what time it is now.”

“Oh, yeah, I know about that. I’ve read about how the GPS system needs to make corrections to account for what relativity does to the satellite timings.”

“You’re right, Al, but that’s a different issue. Some of that relativistic correction has to do with space compression because of Earth’s mass. The simultaneity problem is strictly about rapid motion and geometry.”

“Wait — geometry?”

“Relativistic geometry, which is a bit different from the kind that Descartes built.”

“Whoa, Sy, slow down there. Descartes was the ‘I think therefore I am‘ guy, right? What’s that got to do with geometry?”

“I guess I got a little ahead of myself there, didn’t I? OK. Yeah, Al, same Descartes. Grew up Catholic in France, was a professional mercenary soldier in the Thirty Years War, wound up fighting first on the Catholic French side and later on fought on the Protestant Dutch side but cross‑over was common, both directions. He realized he was in an ostensibly religious war that was really about who ruled over whom. That may have had something to do with him becoming a professional philosopher who rejected all religious dogmas in favor of what he could learn solely from logic and his own senses. That’s where his famous mantra came from — he started by proving to himself that he existed.”

“Logic led to geometry, I suppose.”

“Indeed, but a new kind, one that required a few innovations that Descartes developed. On the one hand, mathematicians traditionally expressed algebraic problems in words and some of them were doozies, like saying ‘the zenzizenzizenzic‘ where we’d just say x8. We got that simple but <ahem> powerful notation from Descartes. On the geometry side, he’d ditch all the confusing line-ending markers in a diagram like this one. Instead, he’d label the whole line representing a known quantity with a front-of-the-alphabet letter like a or b or c. A line representing an unknown quantity would get its label from the alphabet-trailers like x, y and z. Then he used the same character conventions and his new power notation to write and manipulate algebraic expressions. Those notational inventions were foundational for his bridge between algebraic and geometrical problems. Draw your problem with lines and curves, transform it to algebraic equations, solve that problem exactly, transform it back to geometry and you’re done. Or vice-versa.”

“That goes back to Descartes, huh?”

“Mm-hm. His big innovation, though, arose from a borrow from an early Greek gadget called a mesolabe. He proposed an idealized version that would let someone break a line into exact fractions or compare a length against a unit length. That broke the rules of classical Geometry but setting his mesolabe’s Y‑angle to 90° prompted him to name points by their distance along the x– and y‑axes. That’s the nub of the Cartesian coordinate system — a rectangular grid of numbered straight lines that go on forever. Graph paper, right? Wrap the grid around the Earth and you’ve got latitudes and longitudes. Add more numbered grid lines perpendicular to either grid and you’ve got z‑axis coordinates. Three coordinates let you name any point in space. Newton and all the physicists who came after him until the dawn of the 20th Century assumed Descartes’ nice, stable coordinate system.”

“20th Century — that’s when Einstein came on the scene. He broke that system?”

“Sure did. You’ve heard about bent space?”

“Who hasn’t?”

“Well, fasten your seat belts, it’s going to be a fun ride.”

~~ Rich Olcott