# B Hawking radiation / String Hagedorn temperature?

1. Apr 19, 2016

### TeethWhitener

I was playing around with numbers and found that the equivalent temperature for Hawking radiation from a Planck mass black hole is ~5×1030 K. Later, I saw that the Hagedorn temperature for strings (where the partition function is expected to diverge) is reported to be around ~1030 K. I thought "wow this is a really intriguing coincidence!" and then I started to wonder whether it's actually a coincidence. It could be that the string Hagedorn temperature demarcates a phase transition from "ordinary" matter to stringy black hole matter at sufficient energy density. If so, that (in my opinion) would mark a plausible and fairly impressive quantum-ish explanation of how black holes are formed.

Since I don't really know how the Hagedorn temperature was calculated, my question is this: Is it a coincidence? Or does this aspect of string theory actually predict a phase transition at the same order of magnitude that you would expect quantum effects to dominate a gravitational system (Hawking radiation from a Planck mass black hole)? Or is it a sleight of hand: maybe string theorists used what they know about Hawking radiation to come up with a plausible value for the Hagedorn temperature (which might give an estimate for some other parameter--such as string tension--that they'd like to know)?

A note: I have an advanced degree (in chemical physics), but I know nothing about string theory other than the pop-sci stuff, which is why I marked the thread "Basic." Be gentle :)

2. Apr 19, 2016

### MacRudi

A very good question. I don't know it too. Never calculated anything like this with numbers. And the question would be how? And with what kind of model?
D9 Brane/Antibrane? Axion dilaton field with axion hair? AdS Atick-Witten theory? Instanton D field? ...
I think there will be a difference between Superstringtheory and M theory. But which and why, I don't know too.
Would be really interesting to get to know this. But we have some String Cracks here, who might help

3. Apr 21, 2016

### Demystifier

That's not coincidence. They are both of the order of Planck energy (divided by Boltzmann constant).

Since you are a chemical physicist, it may be illuminating for you to know that Planck distance (namely, inverse Planck energy times $\hbar c$) is for gravity and string theory what Bohr radius is for chemistry and atomic theory.

Last edited: Apr 21, 2016
4. Apr 21, 2016

### TeethWhitener

At least from what I've seen, they're both 2 orders of magnitude less than the Planck energy. To me, this isn't really "of the order of Planck energy." That's why I'm wondering how the string Hagedorn temperature was estimated.

5. Apr 22, 2016

### Demystifier

When one performs actual calculations, one gets additional factors such as
$$\frac{E_{Planck}}{(4\pi)^2}$$
In this way one easily gets a result which is two orders magnitude smaller, despite the fact that the relevant energy is still the Planck energy $E_{Planck}$.

6. Apr 25, 2016

### TeethWhitener

So basically, since the Hagedorn temperature is related to the self-dual radius, and since we choose the self-dual radius to be on the order of the Planck length, the Hagedorn temperature will be on the order of the Planck temperature? Is that somewhat right?

7. Apr 26, 2016

### Demystifier

Yes.

8. Apr 26, 2016