Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

A Near horizon limit and Hawking Temperature of the horizon

  1. Nov 21, 2016 #1

    ShayanJ

    User Avatar
    Gold Member

    One way that people introduce the Hawking temperature of an event horizon, is by taking the near-horizon limit of the BH metric and then do a Wick rotation of the time coordinate. Then, the regularity of the metric requires that the Euclidean time to be periodic. But how can this give us the temperature of the horizon? What's the relation between the periodicity of the Euclidean time and temperature?

    Thanks
     
  2. jcsd
  3. Nov 21, 2016 #2

    Haelfix

    User Avatar
    Science Advisor

    See note 3 and the path integral sections in the following course series:
    http://www.hartmanhep.net/topics2015/

    Alternatively, the full reasoning is given in this lecture (try to understand the Rindler path integral case first)
    https://arxiv.org/abs/1409.1231

    It is a peculiar but deep fact of gravity that an identification can be made between the thermal density matrix of finite temperature QFT and the reduced density matrix arrived from a computation of the gravitational path integral of gravity in Rindler space when you trace over the Rindler wedges.
     
  4. Nov 22, 2016 #3

    Demystifier

    User Avatar
    Science Advisor

  5. Nov 24, 2016 #4

    Ben Niehoff

    User Avatar
    Science Advisor
    Gold Member

    By the way, you don't have to take the near-horizon limit first! It's just often easier that way.
     
  6. Nov 24, 2016 #5

    ShayanJ

    User Avatar
    Gold Member

    Could you give a reference where it is done that way at some detail?
     
  7. Nov 24, 2016 #6

    Ben Niehoff

    User Avatar
    Science Advisor
    Gold Member

    Try it with Schwarzschild, it shouldn't be hard. Also try looking up "gravitational instantons".
     
  8. Nov 24, 2016 #7

    ShayanJ

    User Avatar
    Gold Member


    Looks like the transformation that takes ## ds^2=\frac{dr^2}{1-\frac{2m}r}+(1-\frac{2m}r)dt_E^2 ## to ## ds^2=d\rho^2+\rho^2 dT_E^2 ## is:
    ## \left\{ \begin{array}{c}\rho=r \sqrt{1-\frac{2m}r}-m\ln\left( \frac{\sqrt{1-\frac{2m}r}-1}{\sqrt{1-\frac{2m}r}+1} \right) \\ \rho T_E=\sqrt{1-\frac{2m}r}t_E\end{array}\right. ##
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Near horizon limit and Hawking Temperature of the horizon
  1. The horizon (Replies: 3)

  2. Cosmological horizon (Replies: 8)

Loading...