Hey! You Got Science in my Science Fiction (Essays)

Time is the Fourth Dimension

Many of you have probably heard the expression that time is the fourth dimension. You've probably noticed that it's different than its brethren, height, width, and depth. But how is it different and how is it the same?

Consider what we learned about relativity in the previous chapter, "at the same time" has no meaning. Or rather, every reference frame has its own idea of what "at the same time" means. Consequently, if Alice's starship has a clock on its bow and stern that are perfectly synchronized in her reference frame, they won't appear synchronized to Bob who's watching from Earth. The clock on the bow will appear to be slower than the clock on the stern.

Now imagine that we draw a line through spacetime, putting time on what we normally call the y-axis and space on what we normally call the x-axis. We call this a world line. The x-axis itself represents all of space at time zero, or "at the same time" in Bob's point of view.

Not surprisingly, would draw a different line to indicate "at the same time." If we draw the various notions of "at the same time" for a variety of reference frames and make a zig-zagging line through spacetime, always following some observer's notion of "at the same time" we can connect every point in the past and future.

Does that imply that, even though we experience time linearly, it's all still back there behind us and out there ahead of us at the same time? Mathematically, it's true. What this implies for reality and the ultimate nature of things, we can only guess and imagine.

However, there's another thing about time that's special. And for this, we need a few simple equations. Remember the Pythagorean Theorem from school?

That means that the change in the x-coordinate squared plus the change in the y-coordinate squared is equal to the distance between two points. If we rotate the coordinate system, tipping the X-axis on its side, we get a different change in x and y, but the same distance. We can add a change in z squared on if we want, and the distance will still work.

If time is a dimension, does that mean the Pythagorean Theorem will work for time as well? Almost. We get a slightly different equation.

Note in the above equation, the letter "c" stands for the speed of light. Why is it there? Because we need to measure things with the same units. The change in the x-coordinate is measured in meteres, but time is normally measured in seconds. You can't add meters and seconds, but if you multiply 1 second by 300 million meters per second (1 sec * 3e8 meter/sec), the "seconds" divide out giving us meters. Thus, "c t" represents time as measured in meters.

If that was confusing, don't worry about it. Just know that the "c" lets us measure time in meters. The really interesting thing about the equation above is the minus sign. That's a big difference compared to the normal Pythagorean Theorem. This notion of distance (usually called "the interval") is the same for all reference frames. Bob on Earth and Alice in her spaceship might have a different idea of what time and position each event in spacetime are, but they will agree on the distance.

 Pretty cool, eh?   

Part of me wonders, though, what would it be like if we had a + instead of a minus in front of the c t, if a reference frame was simply a rotation of the time and space axes? Basically, everything would work backward from what it does now... the clock on Alice's ship would run faster, not slower, and she'd get longer in the direction she was moving. She'd be able to travel infinitely fast according to Bob's reference frame, but she'd age one year for every light year she traveled.

Part of me thinks it would be cool to base a story in an alternate universe where the laws of physics are slightly different and that pesky speed of light thing doesn't get in the way.

Anyway, thanks for reading, and don't forget to click the star!