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The entropy of a black hole is given as $$S \propto k_B \frac{A}{l_P^2}$$ where $k_B$ is Boltzmann's constant, $l_P = \sqrt{G \hbar/c^3}$ is the Planck length and $A$ is the area of the event horizon of a black hole and it must be of this form from dimensional analysis. Hawking showed the constant was $1/4$.

However, is it possible to have a "black line", where the entropy would presumably be of the form $$S \propto k_B \frac{L}{l_P}$$ where $L$ would be the length of the singularity?

(I am aware of the following question and admit my knowledge in these areas are negligible and I am not sure if it is the same question.)

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2 Answers 2

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OP is apparently looking for 1D structures:

  1. On one hand, a 3+1D Kerr black hole has a 1D ring singularity.

  2. On the other hand, an event horizon has always codimension 1 in spacetime. In 3+1D we speak of the event-horizon as a 2D area, i.e. we are implicitly suppressing 1 dimension. According to that terminology the event-horizon becomes a 1D length in 2+1D.

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  • $\begingroup$ An event horizon has codimension one. Space-like slices of the horizon have codimension two. $\endgroup$
    – MBN
    May 20, 2018 at 12:17
  • $\begingroup$ @MBN: Ups. Thx. $\endgroup$
    – Qmechanic
    May 20, 2018 at 12:34
  • $\begingroup$ Out of curiosity are disk singularities not possible? $\endgroup$
    – Jus12
    Jun 10, 2018 at 20:26
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How long would this line be? How would it end? The simplest possibility is that it turns and closes up on itself. Since objects of infinite thinness are physically not possible what we have is a ring. A black ring.

These aren't possibilities in classical GR. However they are possibilities in string theory. But there's a catch, this would be in a higher dimensional space. If I recall rightly at least 5d. So maybe still not physically appropriate.

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