system
Given the following post, respond as if you're commenting on Reddit:
human
What does the term "imaginary time" mean Stephan Hawking uses in the context of cosmology (or if it's related what does it mean in quantum mechanics)?
gpt
It's a mathematical trick which you can use to simplify certain problems, or connect different areas of physics. The underlying idea is this: when we talk about space and time as this one thing called *spacetime*, what we mean is that we're defining a notion of distance that combines both *spatial distances* between places and *time intervals* between events. Remember the Pythagorean theorem? If you have two points separated on the x-axis by a distance x, and so on, then the distance s between them is given by

s^2 = x^2 + y^2 + z^(2).

Now let's say we have two *events*. They're separated spatially by distances x, y, and z along those axes, and they happen some time interval t apart. The *spacetime distance* between them is (ignoring gravity, which changes this)

s^2 = -(ct)^2 + x^2 + y^2 + z^(2),

where c is the speed of light. Notice the minus sign in front of time. That minus sign is what makes the time dimension "timelike." This means that if you replace time with an imaginary variable, i.e., write t = iτ where τ is some imaginary number, then we get rid of that minus sign, and the spacetime distance becomes just a spatial distance,

s^2 = (cτ)^2 + x^2 + y^2 + z^(2).

That's just the *spatial* Pythagorean theorem in four-dimensional space (not spacetime).

τ is imaginary time. If we write our problems in terms of τ, rather than t, then the time direction looks just like any spatial direction. It stops being special. This makes it easier to mathematically understand certain problems. But it's fundamentally just that, a mathematical tool.