#
Ub-Ecm-Pf-04/33
When conceptual worlds collide:

The GUP and the BH entropy

###### Abstract

Recently, there has been much attention devoted to resolving the quantum corrections to the Bekenstein–Hawking (black hole) entropy. In particular, many researchers have expressed a vested interest in fixing the coefficient of the sub-leading logarithmic term. In the current paper, we are able to make some substantial progress in this direction by utilizing the generalized uncertainty principle (GUP). Notably, the GUP reduces to the conventional Heisenberg relation in situations of weak gravity but transcends it when gravitational effects can no longer be ignored. Ultimately, we formulate the quantum-corrected entropy in terms of an expansion that is consistent with all previous findings. Moreover, we demonstrate that the logarithmic prefactor (indeed, any coefficient of the expansion) can be expressed in terms of a single parameter that should be determinable via the fundamental theory.

## 1 Introduction

One of the most remarkable achievements in gravitational physics was the realization that black holes are thermodynamic objects with a well-defined entropy and temperature [1, 2, 3]. As our current interest is with the entropy, let us recall the famous Bekenstein–Hawking formula:

(1) |

To be perfectly clear, represents the cross-sectional area of the black hole horizon, is the Planck length, while the speed of light and Boltzmann’s constant are always set to unity. Also, we will always assume, for the sake of clarity, a macroscopically large Schwarzschild black hole in a four-dimensional spacetime.

It is worth noting that the main arguments in support of
equation (1)
are purely of a thermodynamic (rather than statistical) nature.
But let us suppose, as has become common in the literature,
that the Bekenstein--Hawking entropy can be attributed a definite statistical
meaning. ^{1}^{1}1However, for some commentary on
why this might not be the case, see [4].
Then how might one go about identifying these microstates and, even more
optimistically, counting them? The answer to this question presumably lies
within the framework of the elusive fundamental theory;
also known as quantum gravity. Indeed, the two leading candidate
theories — namely, string theory and loop quantum gravity
— have both
had success (albeit, with caveats attached) at statistically explaining
the entropy–area “law” (e.g., respectively, [5, 6]).

The proponents of either of these theories proclaim this success at
state counting to
be one of the major achievements of their favored program.
However, what about the unbias observer, who might actually prefer
if there was only one fundamental theory? In this regard, one might
be inclined to call upon the quantum corrections to (which, in spite
of its intrinsically quantum origins, is a tree-level quantity [7]).
On general grounds --- and has been verified in a multitude of
studies ^{2}^{2}2See [8] for an extensive list of references.
— one would expect that the quantum-corrected entropy takes on the
following expansive form:

(2) |

where the coefficients can be regarded as model-dependent
parameters.
Most of the recent focus has been on ; that is, the coefficient
of the leading-order
correction or the logarithmic “prefactor”. It has even been suggested that
this particular parameter might be useful as a discriminator of
prospective fundamental theories [9]. It is, therefore, appropriate
to reflect upon the loop-quantum-gravity prediction of
(according to the most up-to-date rigorous calculation
[10]); whereas string theory makes no
similar type of assertion that we are aware of. ^{3}^{3}3Note that the
loop-quantum-gravity result refers strictly to the microcanonical
correction. There will also be a canonical correction,
irrespective of the fundamental theory, that contributes
at least to [11, 12, 13].
However, without any further input, how can we say if
any particular value of the prefactor (such as )
is right or wrong? That is, unlike the tree-level calculations,
this type of discrimination is based on asking a question
for which we do not yet know the answer!

It becomes clear that, to proceed in this direction, one requires
a method of fixing (as well as the lower-order
coefficients) that does not depend on the specific elements
of any one particular model of quantum gravity. ^{4}^{4}4Let us
point out two recent studies that have endeavored
to at least put restrictions on the value of the logarithmic prefactor
[14, 15]. Nevertheless, the former was specific to
loop quantum gravity whereas the latter used a premise
that is believed to be contradictory with the same theory.
Hence, the general utility of their results are subject
to question. For instance, it might be hoped that the holographic principle
[16] could serve just such a purpose. In the current paper, we will
utilize a similarly general (i.e., model-independent) concept;
namely, the generalized uncertainty principle or the GUP
(see [17] for a general discussion).
The premise of the GUP (which will be presented in due course) is
that, as gravity is turned on,
the “conventional” Heisenberg relation is no longer
completely satisfactory (but still perfectly valid,
in an approximate sense, in low-gravity regimes). Although the GUP
has its historical origins in string theory [18, 19]
(as well as non-commutative quantum mechanics
[20, 21]),
it can also be argued for on the basis of simple Gedanken experiments
that make no reference to the specifics of the fundamental theory
[22, 23].
This means that the GUP is conceptually ideal for realizing the
discussed objective. ^{5}^{5}5Further
citations on the GUP can be found in
[24]. The reader might also find [25, 26, 27] as being helpful
precursors to the upcoming analysis.

The remainder of the paper goes as follows. Using the GUP
as our primary input, we present
a perturbative calculation of the quantum-corrected entropy,
which can readily be extended to any desired order.
Note that, here, we consider strictly the microcanonical
corrections, ^{6}^{6}6For an earlier discussion on interpreting black hole
thermodynamics in a mircocanonical framework, see [28].
For an example of calculating
quantum corrections to the entropy from a substantially different
perspective, see [29].
as the canonical corrections have been dealt with
elsewhere
(e.g., [11, 12, 13, 30]).
The paper concludes with
a discussion that emphasizes the relevance of our outcome.

## 2 Analysis

Let us begin the formal analysis by presenting the GUP as it typically appears in the literature (e.g., [17]); namely,

(3) |

Here, and are the position and momentum uncertainty for a quantum particle, and is a dimensionless (probably model-dependent) constant of the order unity. Let us point out that the combination can be replaced with the Newtonian coupling constant , thus implying that the “extra” (right-most) term is truly a consequence of gravity. Which is to say, the presumption of a gravitational modification to the quantum uncertainty principle — along with dimensional considerations — is sufficient to fix the form of equation (3); with the parameter reflecting our remaining ignorance. [It should be noted that, on general grounds, an infinite series of higher-order corrections can be anticipated on the right-hand side of equation (3). However, as explained in the footnote at the end of Section 2, this caveat does not affect our conclusions.]

By way of some simple manipulations, we can re-express the GUP in the following manner (subsequently setting ):

(4) |

where a sign choice has been made by imposing the correct classical () limit. Since is normally viewed as an ultraviolet cutoff on the spacetime geometry (e.g.,[31]), it should be safe to regard the dimensionless ratio as small relative to unity. [This is certainly true for our regime of interest — see equation (11) below.] Hence, it is natural to Taylor expand the square root and obtain

(5) |

Next, let us consider the following measurement process: a photon is used to ascertain the position of a quantum particle of energy . Starting with this setup, one can call upon a standard textbook argument [32] (also see [26]) to translate the “conventional” uncertainty principle (or ) into the lower bound . Now generalizing to the case of strong gravity and thus invoking the GUP, we have

(6) |

Let us now apply the above formalism to a specific case which is particularly relevant to our current interest. More to the point, we will consider a quantum particle that starts out in the vicinity of an event horizon and then is ultimately absorbed by the black hole. But let us first take note of the general-relativistic result that, for a black hole absorbing a classical particle of energy and size , the minimal increase in the horizon area can be expressed as [33]

(7) |

Given such a classical context, one is of course free to set , and so this is really no bound at all. Nonetheless, as originally observed by Bekenstein [2], there is no such freedom for a quantum particle since can never be taken as smaller than (i.e., the intrinsic uncertainty in the position of the particle). Hence, for the case of a quantum particle, one obtains the finite bound

(8) |

Substituting equation (6) into the above, we then have, as a consequence of the GUP,

(9) |

or

(10) |

In the latter form, is a yet-to-be-determined numerical factor that is greater than (but of the order of) . Note that, on statistical grounds, however, one might well argue for such that is a strictly positive integer [34].

The pressing concern is now what to take for . We can best address this issue by first reclarifying our objective: it is to deduce the entropy of a (macroscopically large) Schwarzschild black hole of fixed mass; that is, compute the microcanonical entropy. Such a framework necessitates a black hole that is (by some means) emersed in a bath of radiation at precisely its own temperature. (Otherwise, there would be a net gain or loss of mass with time, and a microcanonical framework would no longer be appropriate.) Hence, the particles that we are interested in have a Compton length on the order of the inverse of the Hawking temperature [3] (as measured by an asymptotic observer, which is implicit in calculations of this nature). Actually, the inverse surface gravity ( where is the Schwarzschild radius) is probably the most sensible choice of length scale in the context of near-horizon geometry. On this basis, let us choose

(11) |

(Note that an uncertainty of this order has been previously argued for on different but probably related grounds; e.g., [35, 36].) Granted, there is some degree of ambiguity here; nevertheless, as discussed below, this is not much of a concern.

Realizing that , we can rewrite equation (10) as follows:

(12) |

Admittedly, we have used an input [equation (11)] that is, quite possibly, off by an order-of-unity numerical factor. But let us take note of the form of the expansion in equation (10); in particular, we always have the ratio . Hence, it should be clear that any such numerical discrepancy in can be systematically “absorbed” (without loss of generality) into the already ambiguous parameter . Which is to say, we can still regard equation (12) as an accurate statement (up to the explicit order) modulo the intrinsic uncertainty in the parameters and .

In accordance with the ideas of information theory (e.g., [37]), one would anticipate that the minimal increase of entropy should be, irrespective of the value of the area, simply one “bit” of information; let us denote this fundamental unit of entropy as . (Typically, , but we need not be precise on this point.) Then, inasmuch as the black hole entropy should depend strictly on the horizon area [2], it follows that

(13) |

or

(14) |

Now integrating the above, we have (up to a constant term)

(15) |

where the Bekenstein–Hawking area law [1, 2, 3] has been used to calibrate . Note that one can also write

(16) |

where the expansions coefficients can always be
computed to any desired order
of accuracy. ^{7}^{7}7Let us now remind the reader that
the GUP should, itself, probably be regarded as a power-law expansion.
For instance, there is an alternative formulation of equation
(3) that goes as [20].
In this event, the calculation
of the numerical factors in the coefficients of equation (16)
would be technically more complicated than previously suggested.
Nonetheless, such a calculation could certainly be
carried out and our main observation, ,
would remain intact. Moreover, the logarithmic-order coefficient
is completely unaffected by such considerations.

## 3 Conclusion

In summary, we have utilized the generalized uncertainty principle (or the GUP)
to demonstrate an explicit form for the quantum-corrected
black hole entropy.
Let us now make some
pertinent comments about our result:

(i) We have obtained a leading-order correction to the classical
entropy–area law that goes as the logarithm of the area. This is consistent
with numerous other studies that have delved into this subject matter.
(Again, consult [8] for a list of references.)

(ii) Unlike many other treatments, we have achieved
a concise algorithm for calculating the sub-leading (or inverse power-law)
corrections to the entropy. Moreover, this calculation can be carried out to
any perturbative order; with the expansions coefficients depending on
only a single parameter — namely, .

(iii) Let us re-emphasize that our entropic
calculation was a microcanonical one.
This is consistent with
the logarithmic correction being negative; that is, the microcanonical
framework necessitates some type of additional boundary conditions
(or gauge fixing), which naturally implies a reduction in the entropy.
Moreover, a process of quantization (as was implicit in our use
of a quantum uncertainty relation) can be expected, on general grounds,
to remove entropy
from the system. On the other hand, any canonical corrections to
the entropy would most certainly be positive; as this class
can be attributed to thermal fluctuations in the horizon area.
(See [38, 39, 40]
for further elaboration on these points.)
If one were to (somehow) measure the entropy of a “real” black hole,
it is likely that both types of corrections would have to be accounted for.
Nonetheless, it is probably fair to say that the microcanonical class
is essentially the more fundamental one.

(iv) A controversial issue in the literature is the exact value of the logarithmic “prefactor” (i.e., the coefficient of the term). Here, we have found that it takes on the value (as had earlier been deduced in a related but distinct treatment [24, 41]); where is an order-of-unity parameter that reflects our ignorance about the exact form of the GUP — which is to say, our ignorance about the underlying theory of quantum gravity. In spite of the current lack of knowledge about , our calculation could still have merit as a discriminator of prospective theories of quantum gravity. To elaborate, the fundamental theory should (at least in principle) be able to make a precise statement about and, as a consequence of our calculation, a prediction about the logarithmic-order coefficient. Moreover, the theory of quantum gravity should also have something to say about this coefficient through more direct means; that is, through a process of state counting. The success (or failure) of these two calculations to match up could then be viewed as an important consistency check (or a revealing conceptual flaw) for any testable candidate theory.

## Acknowledgments

Research for AJMM is supported by the Marsden Fund (c/o the New Zealand Royal Society) and by the University Research Fund (c/o Victoria University). ECV would like to thank Professors Saurya Das, G. Amelino-Camelia, R. Adler, and P. Chen for useful correspondences and enlightening comments. Both authors thank Professor M. Cavaglia for making an important observation.

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