Black hole kinematics: The ``in''-vacuum energy density and flux for different observers

Black hole kinematics: The “in”-vacuum energy density and flux for different observers

Suprit Singh and Sumanta Chakraborty

Phys. Rev. D 90, 024011 – Published 3 July 2014

Abstract

We have investigated the local invariant scalar observables—energy density and flux—which explicitly depend on the kinematics of the concerned observers in the thin null shell gravitational collapse geometry. The use of globally defined null coordinates allows for the definition of a unique in-vacuum for the scalar field propagating in this background. Computing the stress-energy tensor for this scalar field, we work out the energy density and flux for the static observers outside the horizon and then consider the radially in-falling observers who fall in from some specified initial radius all the way through the horizon and inside to the eventual singularity. Our results confirm the thermal Tolman-shifted energy density and fluxes for the static observers which diverge at the horizon. For the in-falling observer starting from far off, both the quantities—energy density and flux—at the horizon crossing are regular and finite. For example, the flux at the horizon for the in-falling observer from infinity is approximately 24 times the flux for the observer at infinity. Compared with the static observers in the near-horizon region, this is quite small. Both the quantities grow as the in-fall progresses inside the horizon and diverge at the singularity.

Received 9 April 2014

DOI:https://doi.org/10.1103/PhysRevD.90.024011

Authors & Affiliations

Suprit Singh^* and Sumanta Chakraborty^†

IUCAA, Ganeshkhind, Pune 411007, India

^*suprit@iucaa.ernet.in
^†sumanta@iucaa.ernet.in

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Issue

Vol. 90, Iss. 2 — 15 July 2014

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Images

Figure 1
The Kruskal diagram of the gravitationally collapsing body showing two rays near the horizon—inside and outside. While the outside one goes off to asymptotic infinity and gives the standard Hawking effect, the inside one hits the singularity in a finite time. As the outside ray gives rise to the standard Hawking effect at infinity, one can ask about the effects of the inside ray.
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Figure 2
Penrose diagram for the Vaidya spacetime with two regions—Minkowski (interior) and Schwarzschild (exterior)—separated by null thin-shell (dashed) at $v = 0$ . The dotted line denotes the event horizon $H$ . Any event $x$ on the spacetime can be labeled by two globally defined null coordinates $(v^{+}, v^{-})$ , which correspond to a null ray coming directly from $J^{-}$ and a null ray traced back in the past to the vertical line ( $r = 0$ ) and then reflected off to $J^{-}$ , respectively.
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Figure 3
The trajectories of Static observer (cyan) and a radially in-falling observer who diverts from the static path at some time after $v^{+} > 0$ (orange).
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Figure 4
The figures show energy densities, $U$ and fluxes, $F$ perceived by a static observers specified by their radial positions along their trajectory as a function of $v^{+}$ which is proportional to the proper time. Both $U$ and $F$ are normalized to their value at infinity, $π T_{H}^{2} / 12$ and we have taken $M = 1$ so that $r_{s} = 2$ in our case. The energy density and flux show similar behavior of growing slowly early on and later making a transition to saturated thermal state for larger radii but as we near $r_{s}$ the differences come in. The energy density can be negative for region below certain radius (see text for details). Then we have the late-time energy density and flux for static observers as a function of $r$ showing that the energy density increases with decreasing $r$ to a maxima, then decreases thereon becoming negative at a point and diverging at $r = r_{s}$ . The flux on the other hand is positive throughout increasing with decreasing $r$ and diverging at $r = r_{s}$ .
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Figure 5
The plots show Energy density (U) and Flux (F) perceived by a radially in-falling observer normalized by $π T_{H}^{2} / 12$ and with $M = 1$ so that $r_{s} = 2$ . The first column gives both $U$ and $F$ as a function of parameter $η$ along the trajectory with the in-fall beginning at $r_{i} = 40 r_{s}$ . The second column compares $U$ and $F$ for the in-falling observer from $r_{i} = 20 r_{s}$ with those observed by the thermal static observers at different radii. The last column shows $U$ and $F$ for in-fall beginning close to the horizon at $r_{i} = 1.03 r_{s}$ . (See text for description).
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Figure 6
The first plot shows the energy density (U) and flux (F) perceived by a radially in-falling observer at horizon crossing as a function of the initial radius of the in-fall. Next, we plot the ratio of the flux seen by a static observer at different radii to the flux seen by a radial in-faller.
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