I just read this fascinating article on Yahoo!, but originally on Popular Mechanics.
My questions:
OK, so let’s say you set up a three-dimensional coordinate system in your room. You’ve got one point at a position (x1, y1, z1), and another point at position (x2, y2, z2). If you want to know the distance between those two points, you can use the Pythagorean theorem to determine that the distance is D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}. Importantly, this distance is what’s called an invariant: If you set up some other, different coordinate system, both of those individual points would have different coordinates in that other system, but when you calculated the distance, it’d still be the same.
But that’s three dimensions. What if you have four dimensions, with time as the fourth? Then instead of two points, you have two events, where each one has three coordinates for its position, and one coordinate for when it happens. And you can still calculate an invariant distance between them, that doesn’t depend on what coordinate system you use (and here, that also includes moving coordinate systems, because with four dimensions, we can do that). But we learn from Special Relativity that the distance is not D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2+(t_2-t_1)^2}. Instead, it’s D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2-(t_2-t_1)^2} : Note that negative sign in front of the time part. This is a bit counterintuitive, but it’s definitely what we observe happening, as verified over more than a century through a great many experiments.
But maybe we can make it a little more intuitive. Remember imaginary numbers, from Algebra II? Where there’s a new number i, such that i=\sqrt{-1} ? Well, maybe there’s some sort of imaginary time variable \tau, where \tau = i\times t . In that case, then we could say that D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2+(\tau_2-\tau_1)^2}, and the distance formula looks like we expect it to (at the expense of introducing this new “imaginary time” variable).
Is this better than the formula with t in it? Well, mathematically, they both say the exact same thing, so there’s no basis to judge between them there. The only way to judge between them is which one is easier to think about. So, which one is easier to think about? That depends on who’s doing the thinking, and what context they’re thinking in. Sometimes it’s easier to use t, and sometimes it’s easier to use \tau.
Why did you change the - sign in front of (τ2-τ1 to +?
The Clockwork Rocket and its sequels by Greg Egan is set in a universe with the second distance metric Chronos mentions – the one where time enters with a + sign. There is an Appendix that discusses differences in the physics of that universe compared to ours.
That’s just how the math works out after the substitution. If:
\tau = i \cdot t
Then:
-i \cdot \tau = -i \cdot i \cdot t
So:
t = -i \cdot \tau
And considering:
D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2-(t_2-t_1)^2}
We can substitute the t_n with \tau_n:
D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2-(-i \cdot \tau_2 - -i \cdot \tau_1)^2}
A little algebra:
D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2-(-i)^2(\tau_2 - \tau_1)^2}
And:
D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2-(-1)(\tau_2 - \tau_1)^2}
Finally:
D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2+(\tau_2 - \tau_1)^2}
To add to the above about it being how the mathematics works, the deeper point is that this was the entire point of what was wanted.
This is just Pythagoras. But we have a slightly weird situation where if time intervals are measured as real numbers, we must subtract the squared value from the squared spatial intervals to obtain the final interval. If you really want to add the time intervals along with the length intervals, which might appear to make more intuitive sense, the answer is to use imaginary time. Neither is entirely free of intuitive difficulties. But that was the question, and this is the answer.
What distinguishes imaginary time from space, since they’re both orthogonal to ordinary time?
I don’t think anyone is talking about replacing t with it (Wick rotation) or a plus with a minus. Imaginary time would be a complex number, therefore has both real and imaginary components. In the link, the imaginary part corresponds to a frequency shift of the input pulse.
The research article discussed in the pop sci article is fairly niche and technical. It’s certainly a lot less interesting than the sci-fi-esque word choices imply. There is no actual imaginary time being measured. It also does not relate to spacetime distances in any way.
It’s also not about imaginary time as a coordinate measure but about an imaginary time delay in a scattering system. Much less foundational than it sounds without the word “delay”.
Imagine you watch a car enter and later exit a tunnel. Based on the car’s speed, you expect a certain amount of time to pass during the tunnel journey. If someone changed the road properties inside the tunnel – added a gravel patch, say – the car would come out later than you would normally expect. You could define a time delay t_d that gives the difference between the actual travel time and the “normal” travel time. In this car experiment, you could measure t_d experimentally and you could also predict it based on the road features inside the tunnel.
In the “gravel patch” scenario the time delay will be a positive real number. If the road were modified to be extra smooth or if there were some forward-facing wind turbines to help reduce air drag, the tunnel could lead to a negative time delay. This negative t_d could again be predicted and measured.
The next morning, a news article could come out saying, “Physicists have measured spooky negative time!” That’s literally what this pop sci article is doing, only it’s about going from real numbers to complex numbers instead of the car example’s positive (real) numbers to negative (real) numbers.
So, with excitement suitably quelled, I’ll give some additional technical details. To be sure, this is not a groundbreaking result, and there is no reason for hype, and certainly not for the reasons the pop sci article picked it up (which is entirely due to the words “imaginary” and “time” showing up in close proximity to one another, ignoring the fact that “delay” also shows up.)
A starting example related to imaginary frequency.
If you have an object attached to the end of a spring, and you displace the object from it’s happy equilibrium position and then let it go, it will oscillate back and forth at some frequency. If there’s also friction present, the oscillation amplitude will get smaller and smaller with time. The position of the object over time might look like the red curve here. The idealized version of this situation is relevant across lots of physics, and it is a canonical exercise in early physics studies.
One way to characterize the amplitude versus time is in two parts: (1) an oscillatory part which is a sinusoid with characteristic frequency f_0, and (2) a decaying “envelope” (depicted in blue in the above image) which is an exponential with characteristic decay time \tau.
Instead, you can bring complex numbers to bear (i.e., numbers that can have a real and an imaginary component). I’ll skip the details on the method unless asked, but in short, this system instead can be characterized as having a single complex frequency \tilde{f}_0 \equiv f_0 + i\frac{1}{\tau}. The “regular” oscillation part corresponds to the real part of the (complex) frequency and the decaying amplitude corresponds to the imaginary part of the frequency. And the math looks just like oscillations that actually have this complex-valued frequency \tilde{f}_0.
“Imaginary frequencies have been measured!”
Once you have the tools, you can extend further. For instance, the imaginary part of the frequency could be positive or negative, corresponding to amplitude decay or growth, with the latter relevant in other scenarios. Pretty soon you find that this complex-number approach is quite flexible in describing a wide array of systems.
To the article itself, with it’s complex time delay.
When waves enter and exit a system, it is sometimes called “scattering” of the waves. The properties of the waves are modified by the scattering, so the input and output waves differ. The scattering can be characterized by some mathematical engine S that depends on the input wave properties and the scattering system’s properties. One thing you can extract from S is how delayed the output wave will be (relative to some fundamental reference for the system).
If the scattering system is complicated – and especially if it has resonances and internal losses – it is helpful to invoke the power of complex numbers in characterizing the time delay. The real part of the complex time delay corresponds to, well, time delay, while the imaginary part corresponds to shifts in wave frequencies.
So, the research in the article corresponds to the sequence: (1) make a suitable scattering medium with non-trivial behaviors; (2) calculate S; (3) infer the complex time delay from S and pull out the imaginary part; (4) then measure frequency shifts in a real scattering experiment; (5) show that the measured shifts correspond to the imaginary time delay in the way that the math says it should.
