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1. Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 15:25:17 UTC Robert McNees

   Happy 115th birthday, spacetime!

   Hermann Minkowski addressed the 80th Assembly of German Natural Scientists and Physicians #OTD in 1908, offering a radical four-dimensional reformulation of Einstein's theory of special relativity.

   His opening lines:

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 15:28:15 UTC Robert McNees

     in reply to

     As opening lines go, these are pretty good. Just comes right out and swings for the fences, boldly claiming that two concepts hardwired into your understanding of physical reality are "doomed to fade away into shadows."

     Today we refer to this 4-dimensional arena for special relativity, equipped with a peculiar notion of "distance," as Minkowski spacetime.

     Let me explain one of the ways in which it is different than the Newtonian picture of space and time that preceded it.

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 15:33:32 UTC Robert McNees

     in reply to

     Suppose I observe two events. From my point of view, each one happens at a particular time t and a specific place labeled by three coordinates x, y, z. Maybe they occur at the same time but different places, or vice-versa. Maybe they differ in both place and time.

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 15:34:04 UTC Robert McNees

     in reply to

     I'll record the info for these events so I have it all in one place, like so:

     Event 1: t1, x1, y1, z1
     Event 2: t2, x2, y2, z2

     Then I can unambiguously describe the where and when to anyone who asks. If they recorded the same info, we could even compare our descriptions.

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 15:34:36 UTC Robert McNees

     in reply to

     Let's refer to the time that passes between the two events as Δt.

     If the first event happens at 12:04pm and the second event happens at 12:05pm, Δt would be 60 seconds.

     We use Δ as shorthand for "the change in," so Δt just means "how much the clock changed" between the events.

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 15:35:31 UTC Robert McNees

     in reply to

     Likewise, I’ll use Δx, Δy, and Δz to refer to the differences in where the events take place.

     If the two events happen in the same place then all three quantities would be zero.

     If one of them happens 20m further away in the direction I’m calling “x,” then Δx=20m, and so on.

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 15:39:26 UTC Robert McNees

     in reply to

     Every one of us has our own personal way of describing where and when things happen. At the very least, we tend to put ourselves at the center of things. So when the minute hand on my watch (located on me!) hits 12:04pm, I'd probably label the “where” of that event as x=0, y=0, z=0.

     On the other hand, you'd probably describe that event differently. If I was 10m away from you in what you call the x direction, you would say the event takes place at x'=10m, y'=0, z'=0.

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 15:40:04 UTC Robert McNees

     in reply to

     (You record your "where" description as x', y', and z', so we don’t mix it up with my x, y, z.)

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 15:41:38 UTC Robert McNees

     in reply to

     A "Newtonian" would say that part makes perfect sense, but they would expect us to agree on *when* the watch ticked.

     After accounting for the light from my watch taking a split second to reach your eyes 10m away, you would also log that in your records as an event at 12:04pm.

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 15:42:49 UTC Robert McNees

     in reply to

     That’s true, the Newtonian would say, whether or not you and I see each other as being at rest or in motion.

     If we’re moving away from each other you’d keep revising the “where” of my watch — x=10m, then x=15m, then x=20m, and so on. But we’d still log the same times for events.

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 15:43:12 UTC Robert McNees

     in reply to

     (I'm using "Newtonian" to mean someone operating according to the ideas about space and time central to Newton's formulation of physics and generally accepted as true by other physicists until the early part of the 20th century. An equally good term here would be "Galilean.")

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 15:46:45 UTC Robert McNees

     in reply to

     So we always agree on when things happen. And if we see each other as not moving, we immediately agree on distances between events.

     I might say two events happen at x=0m and x=5m, so Δx=5m. You might be further away and record x'=10m and x'=15m, but you'd still say Δx’=5m.

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 15:48:52 UTC Robert McNees

     in reply to

     If we see each other as moving it’s a bit trickier, but we still know how to reconcile those descriptions.

     As you move away from me you might say x’=10m and x’=27m for those events.

     But once you account for 12m of relative motion you understand that from my point of view they were separated by only 5m.

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 15:50:59 UTC Robert McNees

     in reply to

     According to Einstein’s theory of special relativity, this isn’t right.

     Two people (“observers”) who are moving with respect to each other will in general give different accounts of both the where and when of an event, even if they gave the same description for a previous event.

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 15:54:14 UTC Robert McNees

     in reply to

     Take any two events. I say time Δt passes between them, and they are separated in space by Δx, Δy, and Δz.

     If you are moving relative to me (and I am moving relative to you) you would say time Δt’ passes between the events, with separation Δx’, Δy’, and Δz’.

     We would record different values for the time between those events, and for the part of the “where” involving the direction of our relative motion.

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 15:55:28 UTC Robert McNees

     in reply to

     How do we make sense of things if we disagree on these descriptions? How do we do physics?

     Especially the “when” part – can we disagree on really important things like which event happens first?

     Wouldn’t that screw up cause-and-effect? This all sounds very bad!

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 15:57:36 UTC Robert McNees

     in reply to

     Einstein explained how to reconcile these descriptions. I won’t give the full formulas here, but it's a set of rules analogous to (but not the same as) how you account for the extra 12m of relative motion in our earlier example.

     His explanation used rules first conceived by Dutch physicist Hendrik Lorentz. An important consequence of the rules is that different descriptions still agree on one thing. If c is the speed of light, then
     
     - c² Δt² + Δx² + Δy² + Δz² = - c² Δt'² + Δx'² + Δy'² + Δz’²

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 15:59:09 UTC Robert McNees

     in reply to

     This quantity is called the “spacetime interval.” We usually denote it by Δs².

     Δs² = - c² Δt² + Δx² + Δy² + Δz²

     Minkowski built the geometry of his four-dimensional spacetime around this invariant notion.

     "Invariant" in this context means that it stays the same if we switch from one observer's description to that of another observer moving relative to the first at constant speed along a straight line. I could be a little more precise here, but let's keep going.

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 15:59:50 UTC Robert McNees

     in reply to

     Earlier I referred to a “peculiar notion of distance.” Since Δs² has units of length², you might think its square root is probably some sort of distance.

     Except, Δs² can be positive, zero, or even *negative*. In the last case its square root wouldn't give a sensible distance.

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 16:01:36 UTC Robert McNees

     in reply to

     The meaning of Δs² doesn't exactly fit with our intuitive ideas about distance. (More on that later.)

     But the fact that it can be +,-, or 0 is how we sort out essential questions like “Can observers disagree on the order of events?"

     If two events have Δs² > 0 we say they are "spacelike separated.” There is always an observer who describes that pair of events as happening at the same time but at different places. Different observers may disagree on the order in which two spacelike events occur!

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 16:03:28 UTC Robert McNees

     in reply to

     If two events have Δs²<0 we say they are “timelike separated.”

     There is always an observer who describes them as happening at the same place but different times.

     Crucially, according to special relativity everyone agrees on the order of two events that are timelike separated.

     Finally, a pair of events with Δs²=0 are “null” or “lightlike separated.” Everyone agrees on their order, as well as the fact that a signal from the first would have to move at the speed of light to reach the second.

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 16:06:56 UTC Robert McNees

     in reply to

     So this is our notion of causality, which is essential for physics.

     Timelike and lightlike separated events can have a cause-and-effect relationship because there’s no question which one happened first.

     Events with spacelike separation cannot have a cause-and-effect relationship.

     Material objects must move slower than c, and everything happening to them at one instant is the effect of what was happening the previous instant. Therefore, material objects move along timelike trajectories.

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 16:08:52 UTC Robert McNees

     in reply to

     Minkowski had formulated this picture of spacetime, with its unusual geometry governed by Δs², by 1907.

     There is of course a lot more we can squeeze out of the spacetime interval, but I can't get into that right now because I have to go teach my students about the paths followed by light rays spiraling around a black hole!

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 21-Sep-2023 16:22:29 UTC Robert McNees

     in reply to

     • Stu

     @tehstu Thanks!

   • Stu (tehstu@hachyderm.io)'s status on Thursday, 21-Sep-2023 16:22:30 UTC Stu

     in reply to

     @mcnees These threads are such great scicomm, thanks for putting them together.

   • Richard Grant (richardgrant@mastodon.social)'s status on Friday, 22-Sep-2023 00:08:20 UTC Richard Grant

     in reply to

     @mcnees A terrific thread, and one of the most digestible discussions of these matters that I’ve seen. Also a fine candidate for the Mastodon bookmark feature

   • Robert McNees (mcnees@mastodon.social)'s status on Friday, 22-Sep-2023 00:42:41 UTC Robert McNees

     in reply to

     • Richard Grant

     @richardgrant Thanks!

   • Robert McNees (mcnees@mastodon.social)'s status on Friday, 22-Sep-2023 00:42:53 UTC Robert McNees

     in reply to

     • HomerHarlequin

     @HomerHarlequin Thanks!

   • HomerHarlequin (homerharlequin@mas.to)'s status on Friday, 22-Sep-2023 00:42:54 UTC HomerHarlequin

     in reply to

     @mcnees

     Interesting thread! Thanks for posting.

   • Robert McNees (mcnees@mastodon.social)'s status on Friday, 22-Sep-2023 22:06:26 UTC Robert McNees

     in reply to

     • Clifford Adams

     @wikicliff You've got it. The name Δs² is physicist notation, to remind us that what we're really doing is taking two copies of a “four vector” – an array containing the Δt, Δx, Δy, and Δz info for the two events – and combining them according to a specific rule.

   • Clifford Adams (wikicliff@fosstodon.org)'s status on Friday, 22-Sep-2023 22:06:27 UTC Clifford Adams

     in reply to

     @mcnees

     Maybe using some intermediate temporary variables could help me understand this? Let:

     P =Δx
     Q = Δy
     R = Δz
     U = Δt

     Then:

     D = Δs² = - (c² U²) + P² + Q² + R²

     D < 0: timelike
     D = 0: lightlike
     D > 0: spacelike

     In this, is "Δs²" just a name with the exponent stating the units? Or is the name "Δs"? Or am I even MORE confused? 🤪

   • Clifford Adams (wikicliff@fosstodon.org)'s status on Friday, 22-Sep-2023 22:06:28 UTC Clifford Adams

     in reply to

     @mcnees

     Thank you for another great thread. I have a quick math question about precedence of operations where you state:

     Δs² = - c² Δt² + Δx² + Δy² + Δz²

     I *think* (but am not sure) this means:

     (Δs)² = - (c² (Δt)²) + (Δx)² + (Δy)² + (Δz)²

     ...is this correct? (If not, spacetime becomes even MORE interesting. 😜 )

     EDIT: Now realizing that (Δs)² could never be negative, so I must be confused somewhere?