Negative Temperature, Entropy and Imaginary time| Wednesday Q & A

In this post, I will answer some popular physics questions. If you have questions you would like to see me answer next week, please leave them in the comments.

What does the negative temperature mean?

Negative temperature is not cold. In fact, in some sense its hotter than infinite temperature!! So what the hell do I mean by that?

Like all things in life, it comes down to how you define things. In statistical physics temperature is a measure of the number of possibilities. More precisely, it is a measure in the increase in possibilities as the energy goes up. There are a lot of state that the world can be in that require a lot of energy. For example, in a gas, as the energy goes up the number of possibilities goes up. That’s because these particles can move faster, some of them can still move slowly but there are more speed values available to them now, and that’s why the temperature goes up!!

This is clearly not what we think temperature is when we say something is hot. We mean that when we touch it a lot of energy flows to us and we feel that “burn”. But the reason that happens is that the energy flowing to us after we touch it makes the particles of our body move faster. That’s actually why heat flows from higher temperature to a lower temperature, it increases the number of possibilities, and that’s why it is more likely to happen.

What is imaginary time?

This is another example that sounds cooler than it actually is. It’s again a matter of definition and math.

To understand what an imaginary time is, you first have to understand what an imaginary number is. An imaginary number is a number where if you square it gives you a negative number. You probably remember that if you square any real number the answer is positive. So how can there be a number that you square and gives you something negative? Well, it’s not a real number it’s an “imaginary number”. You just define an object abstractly that when you square would give you -1, and then you give that a name usually the square root of -1 is called i. And that’s that.

This might sound weird to you. Where did this number come from? what does it mean? how can you picture it? remember that is also weird to define the square root of 2, there’s no integer that satisfies x² = 2. So how did we get to square root of 2? We made it up! we defined it.

Another ingredient to understand imaginary time is Pythagorean theorem or Euclidean distances: R² = T² + x² + y². The distance squared to a point is the squared sum of it’s coordinates. When you rotate something this distance is unchanged. In Einstein's theory of relativity the thing that is unchanged is actually R² = -t² + x² + y². Notice that the symmetry which actually defines relativity, this equation results in the strange behavior of relativity is similar to rotational symmetry of lengths. All we have to do to make them the same is go to imaginary time: t = i T.

I hope that made sense and thank you for watching!!