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In this article discussing this and related papers, it is explained among other things, how the neighborhood of an observer's worldline can be approximated by a region of Minkowsky spacetime.

If I understand this right (corrections of confused fluff and misunderstandings are highly welcome), a coordinate transformation which depends on the observer's current location $p_0$ in the classical backround spacetime, to a free falling local Lorentz frame is applied. In this reference frame, local coordinates ($\tau$, $\theta$, $\phi$) together with a parameter $\lambda$ (which describes the location on the observer's worldline?) can be used. As $\lambda$ deviates too mach from $\lambda(p_0)$, the local proper acceleration $\sqrt{a_{\mu}a^{\mu}}$ becames large and approaches the string scale (is this because flat Minkowsky space is only locally valid?) and stringy effects kick in.

The authors postulate that at these points (called the gravitational observer horizon) some microscopic degrees of freedom have to exist that give rise to the Beckenstein-Hawking entropy describing the entropy contained in spacetime beyond the gravitational observer horizon (?).

This is quite a long text to introduce my question, which simply is: Can these microstates be described by the fuzzball conjecture or what are they assumed to "look" like?

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Can these microstates be described by the fuzzball conjecture or what are they assumed to "look" like?

We don't know. The gravitational observer horizon is supposed to be a place where low-energy physics becomes invalid (i.e. one shouldn't trust GR and quantum field theory of a spacetime background). For an observer far from a black hole, this horizon roughly agrees with the usual black hole horizon, and something like the fuzzball scenario may be appropriate. However, the paper remains agnostic about the details of the high-energy physics (it can hopefully be described well in string theory). For now, the only thing we can say with (a reasonable level of) certainty is the number of degrees of freedom in an observer horizon.

together with a parameter λ (which describes the location on the observer's worldline?)

I think you've misunderstood the meaning of $\lambda$. Take a look at the figure in the paper. It is an affine parameter the goes down the past light cone of the observer. (The observer is at $\lambda=0$.) The observer horizon occurs when a trajectory of constant $\lambda$ but changing $\tau$ accelerates too much to be described safely by GR.

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