Straight lines in space are the shortest possible distance; straight paths in space-time are the longest possible time. -Sean Carroll (Quanta Mag)
http://www.quantamagazine.org/how-to-think-about-relativitys-concept-of-space-time-20221114/
Consider two locations in space, such as your home and your favorite restaurant. What is the distance between them?
Well, that depends, you immediately think. There is the distance “as the crow flies,” if we could imagine taking a perfectly straight-line path between the two points. But there is also the distance you would travel on a real-world journey, where perhaps you are limited to taking public streets and sidewalks, avoiding buildings and other obstacles along the way. The route you take is always going to be longer than the distance as the crow flies, since a straight line is the shortest distance between two points.
Now consider two events in space-time. In the technical jargon of relativity theory, an “event” is just a single point in the universe, specified by locations in both space and time. One event, call it A, might be “at home at 6 p.m.,” and event B might be “at the restaurant at 7 p.m.” Keep these two events fixed in your mind, and think about a journey between A and B. You can’t hurry to get to B sooner; if you arrive at the restaurant at 6:45, you will have to sit around and wait until 7 p.m. to reach the event in space-time we have labeled B.
Now we can ask ourselves, just as we did for the spatial distance between home and restaurant, how much time elapses between these two events.
You might think this is a trick question. If one event is at 6 p.m. and the other is at 7 p.m., there is one hour between them, right?
Not so fast, says Einstein. In an antiquated, Newtonian conception of the world, sure. Time is absolute and universal, and if the time between two events is one hour, that’s all there is to be said.
An image of the cover of the book "The Biggest Ideas in the Universe: space, time, and motion' by Sean Carroll, author of something deeply hidden. There are different blue and grey circles on the cover and one notes that this is a NY Times Bestseller.
Relativity tells a different story. Now there are two distinct notions of what is meant by “time.” One notion of time is as a coordinate on space-time. Space-time is a four-dimensional continuum, and if we want to specify locations within it, it’s convenient to attach a number called “the time” to every point within it. That’s generally what we have in mind when we think of “6 p.m.” and “7 p.m.” Those are values of a coordinate on space-time, labels that help us locate events. Everyone is supposed to understand what we mean when we say “meet at the restaurant at 7 p.m.”
But, says relativity, just as the distance as the crow flies is generally different from the distance you actually travel between two points in space, the duration of time that you experience generally won’t be the same as the universal coordinate time. You experience an amount of time that can be measured by a clock that you carry with you on the journey. This is the proper time along the path. And the duration measured by a clock, just like the distance traveled as measured by the odometer on your car, will depend on the path you take.
That’s one aspect of what it means to say that “time is relative.” We can think both about a common time in terms of a coordinate on space-time and about a personal time that we individually experience along our path. And time is like space — those two notions need not coincide. (As the historian Peter Galison has pointed out, it’s not a coincidence that Einstein worked in a Swiss patent office at a time when rapid rail travel was forcing Europeans to think about what time it was in other cities across the continent, so that building better clocks became an important technological frontier.)
Still, there must be some way in which time is not like space, otherwise we’d just talk about four-dimensional space, rather than singling out time as deserving of its own label. And we’re not thinking of the arrow of time here — for the moment, we’re in a simple world with few moving parts, where entropy and irreversibility aren’t things we have to worry about.
Related:
Why Gravity Is Not Like the Other Forces
Gravitational Waves Should Permanently Distort Space-Time
Mass and Angular Momentum, Left Ambiguous by Einstein, Get Defined
The difference is this: In space, a straight line describes the shortest distance between two points. In space-time, by contrast, a straight path yields the longest elapsed time between two events. It’s that flip from shortest distance to longest time that distinguishes time from space.
By a “straight path” in space-time, we mean both a straight line in space and a constant velocity of travel. In other words, an inertial trajectory, one with no acceleration. Fix two events in space-time — two locations in space and corresponding moments in time. A traveler could make the journey between them in a straight line at constant velocity (whatever that velocity needs to be for them to arrive at the right time), or they could zip back and forth in a non-inertial path. The back-and-forth route will always involve more spatial distance, but less proper time elapsed, than the straight version.
Why is it like that? Because physics says so. Or, if you prefer, because that’s the way the universe is. Maybe we will eventually uncover some deeper reason why it had to be this way, but in our current state of knowledge it’s one of the bedrock assumptions upon which we build physics, not a conclusion we derive from deeper principles. Straight lines in space are the shortest possible distance; straight paths in space-time are the longest possible time.
It might seem counterintuitive that paths of greater distance take less proper time. That’s OK. If it were intuitive, you wouldn’t have needed to be Einstein to come up with the idea.
| 