Relativity (4 of n): The Storrow Drive Theorem

[wrog]

My explanation of Special Relativity, trying to keep it geometric, continued from part 3 wherein I describe the Interval between two events, a kind of "spacetime measurement" that everyone agrees on and that, depending on the trajectory between the events, can represent either a (proper) distance or a (proper) time lapse.

It is time to expand our universe a bit.

A New Dimension

Let's try adding a dimension. We move from the John Hancock Tower to the Flatiron Building (which, if you haven't seen it, is this mostly flat thing standing on end, hence the name). Each floor is a single south-to-north corridor of offices.

(…or, at least, so it was when I saw it back in the 1980s when Stuart worked there for St. Martins, which had two or three floors to themselves. Tom Doherty had a killer "corner office" with a 250+° view. Evidently, it's all being converted into condos now. Sigh.)

Coordinates continue to be nice to have, so we assign office numbers (y) increasing northward. For sanity's sake, we use the same distance units north-south as for the up-down direction and the future-past direction.

Space being 2-dimensional makes the spacetime diagram 3-dimensional (surprise). The new direction, north, comes directly out of "the sheet" on which we drew the previous 2D spacetime diagram, said sheet representing the entire column history of the stack of offices at $y=0$, this being a 1D sub-universe where our previous understanding of how things work carries over. And the same goes for all of the other sheets stacked on top or underneath, i.e., the column histories for all other values of y.

There will be lots of questions. But, first things first:

The Stationary People Expand Their Empire

Nothing about what we were doing with our clocks, screens, and projectors to establish Other People being Stationary or to get our clocks in synch depended on the universe being 1-dimensional.

Or, at least, it won't once we clarify that screens need to be more like white surfaces rather than mirrors, i.e., the lit-up parts re-send light out in all directions, which matters since whoever sent the light could now be moving some direction other than towards or away, and we'll also have to assume observers can cope with whatever distortions occur from viewing a screen at a bad angle. They will also have a wider variety of directions — not just "up" or "down" anymore — from which they can receive light, directions which need to be recorded in order to nail down where/when the bounces happened.

The World Is Flat

At some point we probably need to talk about curvature. E.g., if I and two of my Stationary friends discover we're at the vertices of a triangle whose angles aren't adding up to 180°, or on a circle whose circumference isn't 2π times the radius, what then?

This sort of thing can actually happen in real life: If I'm in Greenwich, England, and my two friends are at Puerto Ayora, Ecuador (near 0°,-90°) and at São Tomé e Príncipe (off the coast of Africa near 0°,0°), our triangle is adding up to about 230°. Earth's surface is curved; film at 11.

Best we can do here is say that if I stick with strictly local stuff and only try to measure my backyard or my town and don't try to cover distances more than 10 km or so, Earth-surface measurements will be Euclidean to within maybe 0.1% and I'll be mostly fine.

Really, that's all that General Relativity tries to do, and this turns out to be good enough. As far as we know (experiments!) things are pretty damn flat locally, skipping for now how we define "locally", but I'll just note that, out in interstellar or intergalactic space, "locally" can go for very long distances (many light years).

But the real point is we have to understand How Flat Stuff Works before we can understand How Curved Stuff Works. Special Relativity is about Flat Stuff, so we assume flatness (Pythagorean Theorem, triangles adding to 180°, circumferences being 2π times radius, and so on) and see what that looks like (it's plenty weird, as you've already seen).

Eventually, I get an entire circle of directions in which I can be looking out at all of my Stationary friends in all of the places I care about, know all of the distances, having all of our clocks synchronized, thus establishing a full Stationary cubic spacetime grid over everywhere/everywhen we care about.

Knowing how 2D spacetime for the up-and-down Moving People maps into the Stationary scheme, we now have to figure out how to extend this into the new direction. Fortunately, most of our answers come out of this one scenario:

Storrow Drive

Imagine that we have an epidemic of college students using rental trucks to measure the height of an overpass (Yes, the article points out that, contrary to popular belief, it's mostly not college students doing this, but I like ragging on college students anyway.) Interesting and fun facts emerge.

TIMTOWTDI

As it happens, Einstein and many physics texts begin with the Storrow Drive scenario (even if they don't call it that), immediately dismissing all possibilities of weirdness in the transverse direction with a symmetry argument. Which everyone quickly accepts because they haven't seen the weird stuff yet. And then once you can rely on transverse distances, you use a tubes oriented that way with a light ray in it bouncing back and forth to make a clock, then use that to derive the rate of time dilation, and only then derive what happens in the 1D universe.

Which you can do; it is perhaps more economical. But it does seem to make what happens in a 1D universe dependent on what happens in a 2D universe, which, to me, is weird and backwards. And you also don't get to fully appreciate just how weird it is that transverse direction/time measurements don't actually get messed up.

For Total Economy, there are books just start off with a general definition of the interval, completely unmotivated, and then just show that it (mysteriously) works.

As the Perl People would say, there is more than one way to do things.

Do Moving People, who we already know will disagree on the length of the truck, see different heights for it as well? Or see the front of the truck as being slanted differently? How do these effects depend on velocity?

The answers, perhaps surprisingly, turn out to be, "Nope," "Nope," and, "Velocity has no effect as long as it's STL."

(This is as much of a statement of the actual theorem as you're going to get from me. Sorry.)

For the sake of being able to more easily use our prior results (since our seemingly clever previous choice to orient our 1D universe "up-down" now bites us in the ass), we will now flip this scenario (i.e., exchange the up and north directions)…

Vertical Storrow Drive

Imagine that we have an epidemic of college students piloting rental elevators of nonzero width up or down an elevator shaft at $y=0$, attempting to ram them past a horizontal obstacle in the shaft on some particular floor at some particular distance y to the north.

We get a collision event, i.e., the obstacle contacts the elevator — call this the 'Overpass' event — happening at distance y north out of the sheet, according to The Stationary People. Where and when does this happen for various up-down Moving People?

Within the $y=0$ sheet, The Stationary People will be able to identify a corresponding 'Ground Zero' event that is simultaneous (same t) and has the same altitude (z) as 'Overpass'. Or, if you prefer not thinking in terms of coordinates, then imagine all of the lines through 'Overpass'; one of them is perpendicular to the sheet and there'll be an intersection point/event.

How does somebody stuck in the sheet bounce a light ray off of 'Overpass'? It gets 'Sent' from wherever and whenever. The path in space will generally be diagonal, going vertically some distance ${\mathrm{\Delta z}}_s$ (subscript indicating these are all Stationary People numbers) to 'Ground Zero', and also y northward out of the sheet. This takes some amount of time ${\mathrm{\Delta t}}_s$, such that ${{\mathrm{\Delta t}}_s}^2=y^2+{{\mathrm{\Delta z}}_s}^2$ (yay, constant speed of light = 1 and Pythagorean Theorem). Rearranging gets us

$${{\mathrm{\Delta z}}_s}^2-{{\mathrm{\Delta t}}_s}^2=-y^2$$

The left side of this should look familiar. It is the interval between 'Sent' and 'Ground Zero', two events in the sheet where we already know how intervals work, and therefore any up-down Moving People in the sheet get the same number using their own coordinates. So, really it's

$${\mathrm{\Delta z}}^2-{\mathrm{\Delta t}}^2=-y^2$$

and the same goes for a light ray from 'Overpass' to 'Received', wherever and whenever that might be in the sheet.

We now pick an up-down Moving Person trajectory in the sheet. Their snapshot through 'Ground Zero' picks out a simultaneous 'Halfway Tick' event on the trajectory, at some proper distance Δz away, which, for these Moving People, will be the spatial separation from their 'Ground Zero' Person for all events along the trajectory (recall: the Moving People don't think they're moving).

Proceeding backwards in time from 'Halfway Tick' increases the elapsed proper time Δt to get to 'Halfway Tick' until it eventually surpasses Δz by enough that we get an interval of $-y^2$. Going any farther back would make the interval more negative, so this identifies 'Sent', the only event on this trajectory where a light ray can be sent that will reach 'Overpass'.

Proceeding forwards in time from 'Halfway Tick' similarly gets us to the only 'Received' event where a light ray can arrive from 'Overpass'. Because this interval with 'Ground Zero' also has to be $-y^2$ and the Moving People Δz is the same, the elapsed (proper) time Δt from 'Halfway Tick' also has to be the same. So, these Moving People will infer:

• 'Halfway Tick' and 'Overpass' are simultaneous, and
• the 'Halfway Tick'–'Overpass' distance is Δt.

Restricting this to the case of the Moving Person who passes through 'Ground Zero' (and is presumably driving the doomed elevator), for whom $\mathrm{\Delta z}=0$, means

• 'Ground Zero' and 'Overpass' are simultaneous, and
• the 'Ground Zero'–'Overpass' distance is y.

Recalling that ${\mathrm{\Delta t}}^2={\mathrm{\Delta z}}^2+y^2$, the angle between 'Halfway Tick' and 'Overpass' at 'Ground Zero' now has to be 90° (yay, Backwards Pythagorean Theorem). Therefore, these Moving People also infer that

• 'Overpass' and 'Ground Zero' are the same altitude.

There being nothing special about the 'Overpass' event we originally chose, or the y value that identifies that sheet, we can now group together corresponding snapshot lines and floor history lines in all of the sheets into planes, and thus

All of the up-down Moving People snapshot and floor history planes are perpendicular to the (constant y) column history planes.

Meaning when you're looking at this

you should now be seeing planes coming perpendicularly out of the page. Also

The unit spacing in the north-south direction is the same for The Stationary People and all up-down Moving People

and, therefore,

For any pair of events, The Stationary People and all up-down Moving People will agree on their north-south separation (Δy)

which, because they also have to agree on the interval, as previously defined, that you get when you project one of the pair into the column history sheet of the other, that means they also all have to agree on this:

$${\mathrm{\Delta y}}^2+({\mathrm{\Delta z}}^2-{\mathrm{\Delta t}}^2)=({\mathrm{\Delta y}}^2+{\mathrm{\Delta z}}^2)-{\mathrm{\Delta t}}^2={\mathrm{\Delta s}}^2-{\mathrm{\Delta t}}^2$$

which can now serve as our new definition of interval that applies to all pairs of events. Here, Δs is just the spatial separation generally, now that we have two dimensions of space (again, yay, Pythagorean Theorem), and we no longer have to care which direction is "up" or "north".

Which means we can now compare notes with The Rotated People, the political dissident subset of The Stationary People who wanted "up" to be in some different direction, which then enables them to relate to all of the corresponding Moving Peoples with velocities in that direction, giving them results analogous to those above re what those column histories, floor histories, and snapshots look like, and similarly arrive at their own definition for the interval that happens to be the same as ours, no matter which "up" direction was chosen, so

For any pair of events, The Stationary People and all (STL) Moving People (in any direction) will agree on their interval

at which point we can re-apply the previous logic that agreeing on intervals means agreeing on who is and is not in The STL Club and from there getting that the initial choice of Stationary People, once again, does not matter.

One more to go…

Adding back the x direction is straightforward, you can use the same logic as above to show that Stationary and up-down Moving People similarly agreeing on distances and simultaneity of events in the x direction for those pairs of floors and times that corresponded to each other in the 1D universe. Agreeing on distances and simultaneity of events in both the x and y directions (and there being zero curvature) means you've got agreement over the whole x-y plane, i.e., any direction perpendicular to the Moving People's direction of motion and that's everything we need for mapping the Moving People spacetime onto the Stationary People spacetime.

I won't draw anything because my brain explodes when I try to do visualizations in 4D.

(Next up, hyperbolas and velocity angles, I think…)