Hawking radiation / String Hagedorn temperature?

[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 :)

 

[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

 

[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.

 

[TeethWhitener]

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

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.

 

[Demystifier]

At least from what I've seen, they're both 2 orders of magnitude less than the Planck energy.

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 $E_{Planck}$.

 

[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?

 

[Demystifier]

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?

Yes.

 

[TeethWhitener]

Yes.

Excellent, thanks for your help!