Quantum Mechanical Corrections to the Schwarzschild Black Hole Metric

Abstract

Motivated by quantum mechanical corrections to the Newtonian potential, which can be translated into an ℏ\hbar-correction to the g00g_{00} component of the Schwarzschild metric, we construct a quantum mechanically corrected metric assuming −g00=grr-g_{00}=g^{rr}. We show how the Bekenstein black hole entropy S receives its logarithmic contribution provided the quantum mechanical corrections to the metric are negative. In this case the standard horizon at the Schwarzschild radius rSr_S increases by small terms proportional to ℏ\hbar and a remnant of the order of Planck mass emerges. We contrast these results with a positive correction to the metric which, apart from a corrected Schwarzschild horizon, leads to a new purely quantum mechanical horizon. In such a case the quantum mechanical corrections to the entropy are logarithmic and polynomial.

Full text

arXiv:1605.06463v1 [gr-qc] 20 May 2016
Quantum Mechanical Corrections to the Schwarzschild Black Hole Metric
P. Bargue˜no,∗S. Bravo Medina,†and M. Nowakowski‡
Departamento de Fisica,
Universidad de los Andes,
Cra.1E No.18A-10, Bogota, Colombia
D. Batic§
Department of Mathematics, University of West Indies, Kingston 6, Jamaica
(Dated: May 23, 2016)
Motivated by quantum mechanical corrections to the Newtonian potential, which can be translated
into an ~-correction to the g00 component of the Schwarzschild metric, we construct a quantum
mechanically corrected metric assuming −g00 =grr. We show how the Bekenstein black hole
entropy Sreceives its logarithmic contribution provided the quantum mechanical corrections to the
metric are negative. In this case the standard horizon at the Schwarzschild radius rSincreases by
small terms proportional to ~and a remnant of the order of Planck mass emerges. We contrast
these results with a positive correction to the metric which, apart from a corrected Schwarzschild
horizon, leads to a new purely quantum mechanical horizon. In such a case the quantum mechanical
corrections to the entropy are logarithmic and polynomial.
PACS numbers: Valid PACS appear here
I. INTRODUCTION
The full theory of Quantum Gravity is one of the last unsolved challenges in fundamental science and is still eluding
us. Nevertheless, some effects of Quantum Theory do enter the gravitational interaction and can be handled in a
rigorous way without the knowledge of the full fledged theory. Such is the case of the Hawking radiation [1] or the
Unruh effect [2]. Apart from these paradigms there are some other interesting quantum effects related to gravity like
the absence of stable orbits of fermions around a black hole [3], the quantum correction to the Bekenstein entropy Sof
black holes which uses different approaches to Quantum Gravity [4–19] and the quantum correction to the Newtonian
potential or metrics [20–32, 36, 37] (for some applications of the new corrections see [33–35]). Indeed, the results
regarding the corrected Newtonian potential Φ spread over a period of the last forty five years starting with the early
seventies whereas the corrections to Sare a relative new undertaking. Whichever model one uses it turns out that S
receives corrections proportional to the logarithm of the black hole area and, in some models, also proportional to the
square root of this quantity. This is also the finding of our approach starting from a different context. We will make a
connection between the ~-corrected metric and the quantum mechanical corrections to the entropy. We will construct
our quantum mechanically corrected metric by demanding (i) that it reproduces the ~corrected Newtonian limit, (ii)
that it reproduces the standard result for the entropy of black hole including, in addition, the ~corrections which are
similar to results established elsewhere and (iii) that it passes some consistency checks regarding the geodesic motion
of a test particle moving in this metric. The point (i) which has to do with weak gravity, can be easily accommodated
by invoking the classical connection between the g00 metric component and the Newtonian potential. The second
point requires the determination of the horizons and probes into the strong regime of gravity. In principle, we cannot
infer the strong gravity effects from results zeroing around the weak regime as is the case of quantum corrections
to the Newtonian potential. However, we let ourselves be guided by the fact that in the most radially symmetric
metrics the time component is inverse of the radial one. We will take over this fact to the quantum mechanically
corrected metric and show that this step is sufficient to derive the quantum correction to the Bekenstein entropy.
Strictly speaking, this step is justified aposteriori as it enables us to obtain the right result. Finally, we check how
the equation of motion of a test particle gets affected by the quantum corrections. If overall the new metric is in
accordance with observational facts including the classical tests of General Relativity we can then consider such a
result as consistent.
∗Electronic address: p.bargueno@uniandes.edu.co
†Electronic address: s.bravo58@uniandes.edu.co
‡Electronic address: mnowakos@uniandes.edu.co
§Electronic address: davide.batic@uwimona.edu.jm

2
We note that we arrive at the standard results for the black hole entropy (logarithmic and polynomial corrections)
obtained in different ways elsewhere. This gives us some confidence about the ~-corrections to the metric and the way
we handle the calculation. Again, our results show that it is not necessary to invoke the full machinery of a particular
Quantum Gravity theory to derive a valid quantum mechanical result in gravity. Indeed, the quantum corrections to
Φ have been obtained by treating gravity as an effective field theory, which is a conventional approach.
The paper is organized as follows. In section II we will motivate the metric and spell out its full form. We give
here the first insight into the horizons connected with the metric. Section III is devoted to the thermodynamics of
the black hole governed by the quantum mechanically corrected metric. Here we calculate the corrections to the
Bekenstein black hole entropy. In the subsequent section we show by means of the heat capacity that a black hole
remnant emerges. To illuminate the role of the sign of the quantum corrections the next section treats a hypothetical
case. This is followed by a section in which we compare our results with results obtained in literature. The section VII
discusses the geodesic equation of motion resulting from the new metric. This serves as a consistency check to show
that no unwanted features will appear in the motion of a test particle. In the last section we draw our conclusions.
II. ~-CORRECTION TO THE METRIC
In the field theoretical language of an effective field theory of gravity the Fourier transform of an elastic scattering
amplitude gives the potential in r. The one loop correction being always proportional to ~represents then (after the
Fourier transform) the quantum mechanical correction to the potential under discussion. The result for gravity is
often written in the form
Φ(r) = −GM1M2
r1 + λG(M1+M2)
rc2−γG~
r2c3+...(1)
where the λand γare parameters which take different values depending on the author(s). Partly, we can attribute
the reason for these discrepancies to the precise coordinate definition used in the calculation [31]. The question about
the ambiguity of this potential due to the lack of clarity on the coordinates has also been risen in some related articles
[40], [31], [41]. It is argued that a redefinition r→r′=r(1 + aGM/r) would change the parameter λwithout affecting
the observables. The general consensus is that we can write the corrected potential as [31]
Φ(r) = −GM1M2
r1−γG~
r2c3+...(2)
taking rto be the distance between two objects, namely the static Schwarzschild r[24], [42]. The aforementioned
re-parametrization freedom still cannot account for all the discrepancies of the different γ’s found in the literature. A
number of errors have been identified [40], [20], but it is not clear if this accounts for all the different values available.
It is therefore fair to list some of the results (see Table I). In the table we have collected the different values for γ
which also vary in sign (we will see that the sign plays the most important role in the phenomenology derived from
these corrections).
Given the history of the subject, our approach here will be to take the latest value (see, however, [32]), as
the correct one, i.e, γ=−41/10. In any case, it is not so much the numerical value of γwhich affects the conclusions,
but the sign.
From the usual relation between g00 and the potential Φ, g00 =1 + 2φ
c2, the corrected g00 component of the
metric can be written as
g00 =1−2GM
rc2+2G2M~
c5
γ
r3.(3)
The procedure from here on will be to look for changes in the horizon of a black hole as a consequence of eq. (3).
This was suggested in [42] without giving the value of the new horizon and drawing the conclusion of the correction.
Let us mention that, without summing the entire perturbation series in ~, we should be careful in expanding our
results up to the first order in ~. Equally we can talk with confidence about the metric only at large values of rwhich
means quantitatively that we should respect r≫lp. Without the full quantum mechanical correction the issue of the
central singularity remains unknown.
We assume that the corrected metric is such that
−g00 =grr (4)

3
TABLE I: Different values of γfound in the literature.
(Year) Reference γ
(1994) [20] 127
30π2
(1995) [21] 122
15π
(1995) [22] −
17
20π
(1998) [23] 107
10π2
(2002) [24] −
121
10π
(2003) [40] −
41
10π
(2003) [39] −
167
30π
(2007) [25] −
41
10
(2007) [26] 107
30π
(2002) [27] 122
15π
(2012) [28] −
41
10π
(2015) [29] −
41
10
as in the classical Schwarzschild case. This assumption is not anymore of the form: classical result plus ~-corrections
as we have it in equation (3) (we will, however, expand the latter results consistently in ~as emphasized before). As
we mentioned in the Introduction this step is motivated by the classical Schwarzschild case where equation (4) holds
and justified aposteriori by the correct results concerning the corrections to the Bekenstein entropy of the black holes.
We look for the values of rthat will make grr = 0 and identify them as possible horizons for the black hole. The
relevant equation reads
1−2GM
rc2+2G2M~
c5
γ
r3= 0 (5)
It is convenient to work in dimensionless variables. To this end we introduce y≡r
rswith rs=2GM
c2and multiply on
both sides of eq. (5) by r3/r3
s. This way we arrive at
y3−y2+β= 0 (6)
with
β≡l2
p
r2
s
γ. (7)
where we introduced the Planck length lp=pG~/c3. To solve this polynomial equation we first reduce the equation
by the substitution ξ=y−1
3which results in
ξ3−1
3ξ−2
27 +β= 0.(8)
The discriminant Dof this cubic equation, namely
D=1
3−1
33
+1
2−2
27 +β2
=β2
4−β
27.(9)
gives us the necessary information on the number of the real roots [43]. We recall that:
•If D > 0 the polynomial in ξhas only one real solution.
•If D < 0 the polynomial in ξhas three real solutions.
•If D= 0 the polynomial in ξhas two real solutions.
In the following we will consider the two cases according to the sign of γ(and hence also of β) putting some emphasis
on case when γis positive.

4
III. THERMODYNAMICS
After taking β=−|β|, the reduced third order polynomial in ξcan be written as
ξ3−1
3ξ−2
27 − |β|= 0,(10)
and the discriminant
D=|β|2
4+|β|
27 .(11)
is always positive definite. Hence, only one real solution to the polynomial in ξ,ξ1exists. The solutions, expanded in
β, read
ξ1=2
3+|β|+O(β)3/2(12)
ξ2=−1
3+ip|β| − |β|
2+O(β)3/2(13)
ξ3=−1
3−ip|β| − |β|
2+O(β)3/2,(14)
The horizon receives a quantum mechanical correction of the form
rnh =rs+|β|rs=rs+l2
p
rs|γ|(15)
Within our previous assumption, −g00 =grr , the temperature of the black hole is given by T=~
2πc κ, where we
take kB= 1 and κis the surface gravity defined as
κ= lim
r→rnh
∂rg00
|g00grr |.(16)
After some algebraic manipulations we find the surface gravity to be
κ=GM
2GM
c2+~
2Mc |γ|2+3G2M~
c3|γ|
2GM
c2+~
2Mc |γ|4.(17)
The black hole temperature suitably expanded in ~takes the simple expression
T=~
2πc c4
4GM +c5~
16G2M3|γ|+O(~)2.(18)
Our main goal to calculate the black hole entropy is to use the relation dS =c2dM/T or, in integral form
ZdS =2πc3
~ZdM
c4
4GM +c5~
16G2M3|γ|+O(~)2.(19)
After expanding the expression inside the integral to first order in ~,
S=2πc3
~ZdM 4GM
c4−~
c3M|γ|+O(~2),(20)
and making the substitution
χ=M
mp
=MrG
~c,(21)
we obtain
S= 2πZdχ 4χ−|γ|
χ+O(~)2(22)
= 4πχ2−2π|γ|ln[χ] + O(~)2.

5
Going back to our usual variables we get the final result
S=SBH −π|γ|ln A
4l2
p+π|γ|ln [4π] + O(~)2,(23)
where we introduced the Schwarzschild black hole area, i.e. A= 4πr2
s=16πG2M2
c4and the Planck length, lp=qG~
c3.
The classical expression obtained by Bekenstein [48] is SBH =A/4l2
p.
We will comment on this result in section VI, but we note already here that there exists an overwhelming agreement
in the literature on logarithmic corrections to the black hole entropy. Whereas most of the results use a model for
Quantum Gravity, we have obtained corrections of the same form by analyzing ~corrections to the Newtonian potential
via an effective theory of gravity.
IV. HEAT CAPACITY AND THE BLACK HOLE REMNANT
Let us compute the heat capacity of the black hole using the standard expressions C=c2dM
dT . From eq. (18) we
can expand T−1up to order one in ~and deduce two solutions for Mas a function of T−1:
M±=
1
T±q1
T2+ 64 π2G
~c5|γ|
16πG
~c3
(24)
We note that taking γ= 0 forces us to consider the positive sign in order to recover the usual case for Schwarzschild
black hole. Therefore, the heat capacity turns out to be
C=−2π
m2
p"(−4M2+m2
p|γ|)2
4M2+m2
p|γ|#.(25)
At this point, let us define the remnant mass, Mr, by C(Mr) = 0. That is, when Mris reached, the black hole
evaporation stops. From eq. (25) we obtain
Mr=p|γ|
2mp.(26)
which is of the order of the Planck mass mp. We can relate this remnant mass to a maximum temperature, by taking
eq. (18) and replacing the value of the remnant mass. This yields:
T(Mr)≡Tmax =~1/2c5/2
4πG1/21
|γ|1/2+1
|γ|3/2(27)
=c2
4πmp1
|γ|1/2+1
|γ|3/2
which is of the order of T0=c2mp, a number suggested by Sakharov [45] for the the maximum temperature of thermal
radiation.
Both, the remnant black hole mass of the order of the Planck mass and the maximum temperature have analogies
in the literature. We point out that the existence of a remnant in the γ < 0 case is in complete agreement with
similar conclusions obtained within the quadratic GUP formalism [49, 50]. Hawking radiation formulated within the
formalism of a generalized uncertainty relation also indicates a black hole remnant as shown in [46]. Including the
cosmological constant Λ the generalized uncertainty relation not only gives the maximum temperature and minimum
mass, but in addition also a minimum temperature of the order of √Λ and a maximum mass proportional m2
p/√Λ
[47].
V. THE HYPOTHETICAL CASE γ > 0
We follow throughout the paper the latest state of art and consider the case of negative γ(to be precise, we should
take γ=−4/10) as the correct value. Nevertheless, it is illustrative to demonstrate how the physics changes when

6
going from a negative γto a positive result. We will discuss below some salient features of the sign of the parameter
γ. Indeed, as it will turn out it is the sign which changes the most important physical aspects.
For γ > 0 we simply have
D=β2
4−β
27 >0 if β > 4
27 (28)
D < 0 if β < 4
27 (29)
D= 0 if β=4
27.(30)
The value β1=4
27 ≃0.14848 is the benchmark which decides the number of horizons in this case. Before going
into the details of the relevant physical aspects let us complete the the solutions for the horizons. Applying standard
prescriptions one parametrizes the solutions by
u=1
27 −β
2+√D1/3
;v=1
91
27 −β
2+√D−1/3
.(31)
The three solutions to the polynomial equation (8) are then calculated to give
ξ1=u+v, (32)
ξ2=−u+v
2+u−v
2i√3,(33)
ξ3=−u+v
2−u−v
2i√3.(34)
The physics becomes interesting if we consider β << 1. Indeed, for the opposite case we cannot be sure if we are still
within range of the validity of our calculations. Three real solutions emerge now. Going back to the expressions in y
(y=ξ+1
3and recalling that y=r/rs) we obtain after expanding in β
y1= 1 −β+O(β)3/2,(35)
y2=−pβ+β
2+O(β)3/2,(36)
y3=pβ+β
2+O(β)3/2.(37)
We note here that y2can be discarded since it is negative for small values of β. Specifically,
y2=−pβ+β
2>0 if β > 4.(38)
Summarizing, we arrive at a quantum correction to the standard horizon, i.e.,
rnh+=rs−βrs=rs−l2
p
rs
γ(39)
and interestingly at a new purely quantum mechanical horizon which vanishes in the limit ~→0. It has the form
rnh−=pβrs+βrs
2=lp√γ+l2
p
2rs
γ(40)
It is remarkable that quantum mechanical corrections would reveal the existence of a new horizon (disregarding the
fact that we are discussing here a hypothetical case of γ). In theories inspired by non-commutative geometry [44] a
similar phenomenon occurs.
We saw that if γ > 0 we have two horizons provided β << 1. In this case the surface gravity is calculated taking
both of them into account as one does, for instance, in the Reissner–Nordstr¨om case. The relevant expression is,
however, quite simple
κ=c2
2
r+−r−
r2
+
(41)

7
where rnh+=r+and rnh−=r−. The full surface gravity reads therefore as
κ=c2
2


rs−lp√γ−3
2
l2
p
rsγ
rs−l2
p
rsγ2

.(42)
One can easily calculate the temperature which turns out to be
T=~c
4π

c2
2GM −c4qG~
c3γ
4G2M2+c3~
16G2M3+O(~)3/2
.(43)
Finally, with the full units the entropy can be computed along the same lines as in the case γ < 0. We quote the final
result
S=SBH +πγ
2ln A
4l2
p(44)
+ 2√γ√πsA
4l2
p−π
2ln [4π].
At this point several comments (especially concerning the new term proportional to √A) are in order. We postpone
a detailed discussion and a comparison with literature to the next section.
To make the analysis of the case γ > 0 complete, it remains to inspect the possibility of a black hole remnant. After
expanding 1/T in eq. (43) up to order ~, we find again two solutions for Mas a function of T−1. They read
M±=
1
T−4qG
~c3√γ
c
16πG
~c3
(45)
±
v
u
u
t 1
T−4qG
~c3√γ
c!2
−32 π2G
~c5γ
16πG
~c3
.
As before, to recover the standard case we choose the positive sign in this equation. The heat capacity is easily
calculable to be
C=−8M2π+ 4Mmp√γ+m2
pπγ2m2
pγ+ 8M2+m2
pγ−8M2
16M2m2
pπm2
pγ−8M2
.(46)
It is clear that no remnant mass shows up in this case, i.e. there does not exist a value for Mwhich makes C= 0.
Indeed, the γ > 0 case seems to be more subtle. In this situation, the logarithmic correction acquires the opposite sign
compared to that predicted by GUP and others, although the square root term shows up as in the linear GUP case,
as commented below. Albeit not inconsistent, it is evident that the case γ > 0 would reveal a completely different
physical scenario as compared to γ < 0.
VI. LOGARITHMIC CORRECTIONS TO THE BLACK HOLE ENTROPY IN DIFFERENT MODELS
AND DOMINANCE OF SQUARE ROOT
Different approaches to quantum gravity have predicted corrections to the Bekenstein–Hawking entropy in the form
[4–13]
S=A
4l2
p
+c0ln A
4l2
p+∞
X
n=1
cnA
4l2
p−n
,(47)
where the cncoefficients are parameters which depend on the specific model considered. Interestingly, loop quantum
gravity calculations are used to fix c0=−1/2 [14]. Moreover, the deformed commutation relations giving place to

8
a generalized uncertainty principle (GUP) have been also used to compute the effects of the GUP on the black hole
entropy from different perspectives (see, for example, [5, 15–18]). In this case the expression for the entropy reads
S=A
4l2
p
+√πα0
4sA
4l2
p−πα2
0
64 ln A
4l2
p+O(l3
p),(48)
where α0measures the deviation for the standard Heisenberg case, i. e.,
xi=x0i;pi=p0i1−αp0+ 2α2p2
0,(49)
where [x0i, p0j] = i~δij and p2
0=P3
j=1 p0jp0jand α=α0/mpc, with being α0a dimensionless constant. Even more,
polymerization (a non–standard representation of quantum mechanics that was inspired by loop quantum gravity,
LQG) also predicts logarithmic corrections to the black hole entropy [19] (it has been shown that polymerization
and quadratic GUP are equivalent provided α0and the polymerization parameter are proportional [38]). In fact, the
leading–order corrections to the entropy of any thermodynamic system due to small statistical fluctuations around
equilibrium, when applied to black holes, are shown to be of the form ln A[8].
Therefore, the corrections given by eq. (23), obtained from one–loop calculations, are consistent with different
approaches which incorporate, in some sense, some quantum gravitational considerations. Specifically, the correct
sign for the logarithmic term is obtained for the γ < 0 case. As we commented before, this is the case of LQG and of
quadratic GUP. Therefore, our approach is consistent with both of them provided |γ|= (2π)−1=α2
0/64. Interestingly,
the γ > 0 case predicts the existence of a second horizon, which gives place to the term proportional to √A. This
case corresponds to linear GUP provided γ=α2
0/64, as in the previous case. We would like to remark that contrary
to the logarithmic correction the interpretation of this square root term in the black hole entropy is awaiting a new
interpretation, as pointed out in Ref. [18]. In this sense, we conjecture that this leading order correction is related
to the appearance of the almost mass–independent term of the second horizon, rnh−=lp√γ+l2
p
2rsγ. In fact, for very
large black holes, rnh−→lp√γand rnh+→2Mand only a square root correction to the entropy appears.
In retrospect, the agreement with other findings on the corrections to the black hole entropy gives us some confidence
about the quantum mechanical corrections to the Newtonian potential and the conclusions drawn from it.
VII. THE GEODESIC EQUATION OF MOTION
One of the key observables in General Relativity is, of course, the particle trajectory once the metric in which the
particle moves has been given. From our point of view it is crucial to check whether the quantum mechanical corrections
proposed above will change the standard predictions in a drastic way. This would be the case if, for instance, new
circular (stable or unstable) orbits would appear leading to new phenomenological results. It is worthwhile to note
that even if the corrections to the geodesic equation of motion come out to be proportional to lp, there is no a priori
guarantee that all observables will receive small corrections and that no new features will emerge. Small quantum
effects on the three body Lagrangian points were recently found using the same corrections to the Newtonian potential
[51]. Mixing of scales can lead to new results as it happens, e.g., in the Schwarzschild-de Sitter metric where scales of
the cosmological constant combine with the Schwarzschild radius to reveal new aspects of the effective potential [52].
A. The quantum corrections to the effective potential
As far as the metric is concerned the results of the previous sections can be summarized by writing
ds2=B(r)dt2−A(r)dr2−r2C(r)[dθ2+ sin2θdφ2] (50)
where C(r) = 1, A(r) = 1
B(r)and
B(r) = 1−rs
r+rsl2
p
r3γ!.(51)
By using the standard methods one can cast the equation of motion in the form
˙r2
2+Veff (r) = const. (52)

9
in which the effective potential can be split into two sums indicating the classical and the quantum part
Veff (r) = V(~0)
eff (r) + V(~)
eff (r).(53)
The respective contributions read
V(~0)
eff (r) = (−GM
r+l2
2r2−GMl2
c2r3if m6= 0
l2
2r2−GMl2
c2r3if m= 0 (54)
and
V(~)
eff (r) = (G2M~
c3r3γ+G2Ml2~
c5r5γif m6= 0
G2Ml2~
c5r5γif m= 0 (55)
Evidently, the quantum correction vanishes as ~→0. In the following we will study the extrema and zeros of the
corrected effective potential. If no new zeros and local extrema emerge as a result of the quantum corrections and the
new zeros and extrema receive corrections of the order of ~, we can consider the theory based on (50) as consistent
and in accordance with observational facts. In the antipodal case additional extrema, even if as a small effect, would
imply new stable and unstable circular orbits.
B. The massless case
Based on
Veff (r) = l2
2 1
r2−rs
r3+l2
prs
r5γ!(56)
we look first for the zeros (r0) of this function. With r06= 0 we obtain:
r3
0−rsr2
0+l2
prsγ= 0 (57)
Dividing by r3
sand defining x≡r0
rs,β≡l2
p
r2
sγand setting y=x−1
3we arrive at a third order polynomial whose zeros
we wish to find, i.e.,
y3−1
3y−2
27 +β= 0 (58)
As usual it is the discriminant of this equation which is of importance. The latter is given by [43]:
D=1
4β2−1
27β. (59)
The case of relevance turns out to be D > 0 which implies one real solution of the cubic polynomial. The reason is
that
D > 0 implies β < 0 or 0 < β < 4
27,(60)
or equivalently
D > 0 if γ < 0 or γ > 0 and r2
s<27
4l2
pγ. (61)
On the other hand the case with three real solutions would lead to
D < 0 if γ > 0 and r2
s>27
4l2
pγ. (62)
Again it is the sign of γwhich is crucial here. Since we decided to focus on the latest (negative ) value of γit suffices
to handle the case D > 0. The only real zero is then calculated to be
r0=rs−l2
p
rs
γ+O(lp)3.(63)

10
The method to find the extreme is, in principle, very similar. Putting the derivative of the effective potential to
zero results in a third order equation in rmax. The latter can be transformed in a third order equation in the variable
ξ= (rmax/rs)−1
2:
ξ3−3
4ξ−1
4+5
2β= 0.(64)
The discriminant in this case can be calculated to be
Dmax =25
16β2−5
16β. (65)
The case distinctions for Dmax are similar to the discriminant of the zeros discussed above. In short, we can summarize
it as follows
Dmax >0 if β < 0 or β > 1
5⇒γ < 0 or r2
s<5l2
pγ(66)
Dmax <0 if 0 < β < 1
5⇒γ > 0 and r2
s>5l2
pγ(67)
Dmax = 0 if γ > 0 and r2
s= 5l2
pγ(68)
Continuing with γ < 0 we find again only one real zero. The radius of the unstable photon circular orbit receives a
small correction proportional to ~. In more detail, we obtain
rmax =3
2rs−10
9
l2
p
rs
γ+O(lp)3(69)
The first term in rmax is the standard result of the effective potential without quantum corrections.
C. The massive case
For the massive case we write the effective potential as follows
Veff (r) = l2
2 1
r2−rs
r3+rsl2
p
r5γ!−rs
2rc2+rsl2
p
2r3γc2.(70)
The search for the zeros as well as the extrema gives now a fourth order polynomial equation. Even though the latter
can be solved analytically the calculations are quite extensive and the steps not illuminating. We skip all the details
and quote the final result obtained with the help of MATHEMATICA. The physically relevant zeros, i.e., the zeros
of the effective potential which lie outside the horizon are
r1,2
0=l2
2c2rs∓1
2sl4
c4r2
s−4l2
c2−1
2
1±
l2
c2rs
ql4
c4r2
s−4l2
c2

l2
p
rs
γ+O(l4
p) (71)
As ~→0 we recover the usual roots. The extrema are located at
r1,2
max =l2
c2rs∓sl4
c4r2
s−3l2
c2−1
9
5±12rs+ 10 l2
c2rs
ql4
c4r2
s−3l2
c2

l2
pγ
rs
(72)
The restrictions to make both results real are actually restrictions on l2and are also present in the standard general
relativistic case without quantum corrections. For the r0values to exist we have to impose l2
c2<4r2
sand for the rmax
values to exist the inequality to be satisfied is l2
c2<3r2
s. This stronger case does not appear accidentally here. In the
classical case of the Schwarzschild metric, only respecting l < lcr it =√3rscleads to an effective potential without
local extrema.

11
VIII. CONCLUSIONS
We have explored the consequences of quantum mechanical corrections to the Newtonian potential. This correction
in tandem with −g00 =grr fixes the metric. We probe into the physics around the horizon of this metric. We find
a corrected Schwarzschild horizon where the correction is proportional ~. This was used to infer the corrections to
the black hole entropy. We derived logarithmic corrections in agreement with many other approaches. A black hole
remnant of the order of Planck mass emerges in this case. The (hypothetical) positive correction to the Newtonian
potential gives another picture. In addition to the quantum mechanically corrected Schwarzschild radius, a second
horizon of purely quantum mechanical nature (proportional to √~and ~) is possible. The Bekenstein entropy gets
corrected not only by a logarithmic term but a term with a square root of the area of the black hole also appears.
This term has been found also elsewhere in a completely different context. No remnant remains in this case. It is
obvious how much the sign of the correction affects the conclusions. Finally, we examine the consequences of the ~
correction in the geodesic equation of motion and find that that classical tests of General Relativity will be affected
only marginally.
In conclusion, the simple quantum mechanical correction to the Newtonian potential taken together with a rea-
sonable assumption on the grr component has remarkable consequences. Whether Hawking radiation or Bekenstein
entropy the quantum mechanics in the gravity of a black hole is centered at the horizon. We added to this list a
quantum mechanical correction of the horizon and connected it with the correction to the entropy.
Acknowledgment
P. B. and M. N. acknowledge the support from the Faculty of Science and Vicerector´ıa de Investigaciones of
Universidad de Los Andes, Bogot´a, Colombia. M. N. and S. B. M. would like to thank COLCIENCIAS for financial
support.
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