Why does imaginary time behave like space?

[Ien Cleary]

I know what imaginary numbers are, but I'm struggling to understand why the Lorentz transformation makes a time-like dimension space-like. I suppose what I'm really asking is what is the difference between time-like and space-like. I've read that it has something to do with special relativity, but my math skills aren't up to it. Can anyone explain in terms of 'broad concepts'? Or do I just have to accept that, without the math, I'm never going to get a better grasp of the idea?

 

[Student100]

I'm not sure what you mean by "makes a time-like dimension space-like." There are lots of bits of subtly to this question.

In Minkowski space, the temporal component, ict, when compared to the spatial components can be thought of as a complex 90 degree phase relation between space and time. This arises into what we call flat space time(SR). When looking at flat space time from different inertial reference frames, they would be related to one another. In GR, when you curve space time, it no longer results in a 90 degree phase relation, it changes based on the local curvature of space from gravity, and GR formalism needs to be applied.

With the introduction of SR/GR, time could no longer be thought of as invariant and unchanging. It became something that could vary based on your frame of observation, a temporal dimension. There are also dimensional and geometric augments that could be made for why time is a dimension, but I'm not really sure this is hitting at what you're asking.

 

[bcrowell]

For simplicity, let's pick units where c=1, which is always a good idea in relativity.

If events A and B differ in position by x and in time by t, then the time interval s measured by a clock moving inertially from A to B is given by the relation $s^2=t^2-x^2$. Minkowski liked the idea of rewriting $s^2=t^2-x^2$ as $-s^2=x^2+(it)^2$, because then the equation has something more like the same form as the Pythagorean theorem, $A^2=B^2+C^2$. Minkowski's idea is neither right nor wrong, and it makes no new physical predictions. By about the middle of the 20th century, relativists had decided that it was more trouble than it was worth, so they stopped doing it. Textbooks tend to be several decades behind the times, and popularizations in turn tend to lag several decades behind the textbooks. If you came across a textbook or a popularization that still does the $it$ thing, then the book is stuck in the Eisenhower administration. Find a more modern book. There is no mind-blowing revelation about space and time there, just an outdated and clumsy notational convention.

 

[Nugatory]

then the book is stuck in the Eisenhower administration.

Maybe even a bit later. MTW came out during the Carter administration, and still needed a section titled "Farewell to $ict$".

 

[Dale]

I suppose what I'm really asking is what is the difference between time-like and space-like.

Timelike means that you can get from one event to the other by traveling slower than light. Spacelike means that you would have to travel faster than light. Lightlike or null means that you would have to travel at the speed of light.

 

[bcrowell]

Maybe even a bit later. MTW came out during the Carter administration, and still needed a section titled "Farewell to $ict$".

Along with "farewell to beatnik glasses" and "farewell to goldfish swallowing."

 

[PeterDonis]

MTW came out during the Carter administration

I thought it was published in 1973, which would be Nixon?

 

[Nugatory]

I thought it was published in 1973, which would be Nixon?

Hmm - you're right. I bought my copy in '77, must have been what I was thinking about.

 

[Stephanus]

Respon to Ien Cleary:

Timelike means that you can get from one event to the other by traveling slower than light. Spacelike means that you would have to travel faster than light. Lightlike or null means that you would have to travel at the speed of light.

Perhaps this ST diagram will help you understand it.

Or in English.
TIME LIKE: If you live in New Delphi, you have a friend in Jakarta 5000 km east from New Delhi, and your friend texts you "Happy New Year", Jakarta reads January 1st 00:00 AM, and it takes your friend 2 minute to type "Happy New Year" press "Send", processed by cell phone provider and it reaches your cell phone 3 minutes later, we call "New Year" event in Jakarta and reading your phone in New Delhi is time like. Actually it takes 16.7 miliseconds for light to reach New Delhi from Jakarta, while it takes you 3 minutes to realize that it's New Year in Jakarta, we call it time like.
SPACE LIKE: The distance between Jakarta and Bangkok is somewhat 2000 KM, but they are in the same time zone (don't let the term "time zone" confuses you), and New Year event in Jakarta happens simultaneously in Bangkok, altough there is a distance separated them, we call it space like.

It doesn't have to be simultaneusly. Event in Jakarta at 00:00 AM and event in Bangkok is 00:00 AM + 1 miliseconds, we still call it space like.
Jakarta at 00:00 AM and Bangkok at 00:00 AM + 6.6667 milisecond we call it light like.
Jakarta at 00:00 AM and Bangkok at 00:00 AM + 100 milisecond we call it time like,
see DaleSpam answer

 

[loislane]

...still needed a section titled "Farewell to $ict$".

That section gave several reasonable motives to abandon the imaginary time notation, but one of them was not mathematically correct. When talking about how the ict notation hides the Lorentzian metric structure it gives an example about null rays that lie on the light cone. In fact the topological requirements on Minkowski space as a manifold only admit the interior(timelike part) of the light cone, the rest of the causal structure determined by the Lorentzian metric doesn't correspond to either the manifold topology or finer like the path or Zeeman topologies.
https://en.wikipedia.org/wiki/Spacetime_topology and references therein.
Also: Malament; http://link.aip.org/link/?JMAPAQ/18/1399/1 ; J. Math. Phys. 18 7:1399-1404 (1977)