Shadow of noncommutative geometry inspired black hole

0.42 MB·English

by  Shao-Wen Wei

Prepared for submission to JCAP
Shadow of noncommutative geometry
inspired black hole
5
1
0
2

g
u
A

7 Shao-Wen Wei, Peng Cheng, Yi Zhong, Xiang-Nan Zhou
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Institution,
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c
Theoretical Physics, Lanzhou University, Lanzhou 730000, People’s Republic of China
q
- E-mail:
r
g [email protected], [email protected], [email protected], [email protected]
[

2 Abstract. In this paper, the shadow casted by the rotating black hole inspired by noncom-
v
mutative geometry is investigated. In addition to the dimensionless spin parameter a/M
8 0
9 with M0 black hole mass and inclination angle i, the dimensionless noncommutative param-
2 eter √ϑ/M is also found to affect the shapeof the black hole shadow. The result shows that
0
6 the size of the shadow slightly decreases with the parameter √ϑ/M , while the distortion
0 0
. increases with it. Compared to the Kerr black hole, the parameter √ϑ/M0 increases the
1
deformation of the shadow. This may offer a way to distinguish noncommutative geometry
0
5 inspired black hole from Kerr one via astronomical instruments in the near future.
1
:
v Keywords: astrophysical black holes, gravity, GR black holes
i
X
r
a
Contents
1 Introduction 1
2 Null geodesics of noncommutative geometry inspired black hole 2
3 Effective potential and unstable circular orbits 6
4 Black hole shadow 8
4.1 Nonrotating case 8
4.2 Rotating case 8
5 Discussion 11
1 Introduction
Due to the quantum effects near the event horizon, a general relativity (GR) black hole was
foundtopossess Hawking radiation [1], whichis very differentfrom theclassical concept that
nothing can escape from a black hole. Subsequently, four laws of black hole thermodynamics
were established. They greatly improved our understanding of the black hole. However,
singularity of spacetime is one of the challenging problems in GR.
Many attempts were proposed to give a singularity free theory. For example, introduce
the higher derivative covariant generalizations of GR [2, 3], which is required to be well
behaved in the ultraviolet and reduced suitably to Einstein’s gravity in the infrared. Other
methods suggest introducing a minimal length to cure the singularity in a small distance,
which means that the singularity or ultraviolet divergence is a spurious effect resulting from
the inadequacy of the formalism at small distance scales [4, 5]. Noncommutative geometry
is shown to be a good candidate to implement a minimal length in physical theories.
From the fundamental view of noncommutative geometry, the picture of spacetime as
a manifold of points breaks down at Planck length. And a particle cannot be accurately
localized in the quantum phase space [6], since [xˆ ,pˆ ] = i~δ . Therefore, it is rational to
i j ij
replace a point in classical commutative manifold with a state in a noncommutative algebra.
Andapoint-likeobjectwillbereplacedwithasmearedone[7]. Ingeneral, therearetwoways
to construct the noncommutative quantum field theory: one is based on the Weyl-Wigner-
Moyal -product and the other on the coordinate coherent state formalism [7]. Following
∗
the latter way, one has [xˆµ,xˆν] = iϑdiag(ǫ , ,ǫ ). Then the Lorentz invariance can be
1 D/2
···
recovered by imposing a single noncommutative parameter ϑ, and unitarity was also checked
at the one-loop level with no violation [8].
Inspired by noncommutative geometry, the key idea to construct singularity-free black
holesisreplacingthepoint-likemasslocatedatthecentersingularitywithasmearedmass[7].
The scale of the distribution of the mass density is denoted with √ϑ, which has a dimension
of length. That parameter is a new fundamental length scale on the same ground as Planck
lengthl . Inordertokeepsomedegreeofgenerality, weconsideritasafree,model-dependent
P
parameter, and one can call it noncommutative parameter. Motivated by the result, various
blackholesolutionsinspiredbynoncommutativegeometry werefound[9–12],andtheseblack
hole solutions were extensively studied. At the terminal stage of the black hole evaporation,
– 1 –
the singularity problem can also be well cured [9]. Other properties such as thermodynamic
properties and black hole lensing were examined (see Refs. [13–15] and references therein).
All the results show noncommutative parameter dependent effects. And these black holes
becometherichestclass of quantumgravity black holes [16]. InRef. [14], theauthorsshowed
that the noncommutative parameter is measurable when √ϑ/M (0.24,0.52). Therefore,
0
∈
testing the existence of the noncommutative parameter through black hole lensing will shed
further light on such theory. However, the image of the source is a one-dimensional light ring
or arc making the test of √ϑ/M more difficult.
0
Compared with it, the black hole shadow is a two-dimensional dark zone seen from the
observer. So it can be easily observed from astronomical observation. By future interfer-
ometers [17–20], direct observation of a black hole will become possible in the near future.
Black hole shadow is the optical appearance casted by a black hole with the photon sources
located at infinity and distributed uniformly in all directions. For a nonrotating black hole,
the shadow appears as a perfect circle. While for a rotating black hole, it will be elongated
in the direction of the rotation axis due to the dragging effect [39]. The size and distortion
of the shadow are found to depend on the parameters of the black hole and the location of
the observer. So the test of the black hole shadow provides an efficient method to test the
nature of a black hole in the near future. And the topic has been examined by several groups
in the last few years [21–38].
So the main purpose of this paper is to investigate the effect of the dimensionless non-
commutative parameter√ϑ/M ontheblack holeshadow. Usingthenullgeodesic, weobtain
0
the observables proposed in Refs. [28, 36]. The shape of the shadow is found to closely de-
pend on √ϑ/M . And the result suggests that, compared with the Kerr black hole, the test
0
of √ϑ/M (0.4,0.52) through the observation of the black hole shadow is possible in the
0
∈
near future using the very-long baseline interferometry (VLBI).
Thepaper is structured as follows. In Sec. 2, we give a brief review of thenull geodesics
for the noncommutative geometry inspired black holes. With the effective potential, the
unstable circular photon orbits are studied in Sec. 3. Then in Sec. 4, we get the apparent
shape of the shadow casted by the black hole with or without the spin. The observables are
also obtained, and the comparison is made between the noncommutative geometry inspired
black hole and the Kerr black hole. The summary and remarks follow in Sec. 5. We adopt
the geometric units, i.e., ~= c = k = G= 1.
B
2 Null geodesics of noncommutative geometry inspired black hole
The key point to construct a noncommutative geometry inspired black hole is replacing the
point-like object with a smeared one. Therefore, in a spherically symmetric case, the mass
density of such an object is modified as a Gaussian density
ρ = M0 e r2/4ϑ, (2.1)
G −
(4πϑ)3/2
where M is the total mass of the object and ϑ measures the distribution of the mass. In the
0
spheroidal coordinates of the Minkowski spacetime, the mass density for a rotating object is
[40, 41]
r2
ρ = ρ . (2.2)
M r2+a2cos2θ G
– 2 –
Then the mass encoded in a radius r will be
r π
M(r) = 2π dr dθsinθ(r2+a2cos2θ)ρ
M
Z Z
0 0
2M
= 0γ(3/2,r2/4ϑ), (2.3)
√π
where γ(3/2,x) = xt1/2e tdt. Thus, the energy-momentum tensor for the generalized Kerr
0 −
black hole reads [1R2]
Tµ = (ρ+p )(uµu lµl ) p δµ, (2.4)
ν θ ν − ν − θ ν
where uµ = √ grr(δµ + a δµ) and lµ = 1 δµ. The invariant energy density
− t (a2+r2) φ −√ grr r
ρ= r4ρ /(r2+a2cos2θ)2. Imposing Tµν = 0, one−can obtain the pressure p .
G µ θ
∇
Finally, taking the choice of a “noncommutative geometry inspired” matter source as
the input of the Einstein equations, Smailagic and Spallucci obtained a metric of the non-
commutative geometry inspired Kerr black hole [12]:
∆ a2sin2θ 2a(r2+a2 ∆)sin2θ
ds2 = − dt2 − dtdφ
− ρ2 − ρ2
ρ2 (r2+a2) ∆a2sin2θ
+ dr2+ρ2dθ2+ − dφ2, (2.5)
∆ ρ2csc2θ
where the metric functions are given by
ρ2 = r2+a2cos2θ, ∆ = r2 2Mr+a2, (2.6)
−
a is the spin parameter of the spacetime, and M is given in (2.3) and denotes the mass
encoded in a sphere with radius r. In the limit √ϑ/M 0, this metric describes the
0
→
conventional Kerr spacetime. In Fig. 1(a), we show the mass distribution function M(r) as
a function of r. One can see that the mass dominates in small value of r for a small value
of √ϑ/M . It is clear that √ϑ/M measures the noncommutativity of spacetime and it is
0 0
reasonable to treat it as the noncommutative parameter of spacetime as suggested in Ref.
[14]. It is also worth to note that, for different values of √ϑ/M and a, the metric displays
0
different horizon structures, i.e., two horizons, one horizon, and no horizon. In this paper,
we only focus on the case that the black hole has, at least, one horizon, thus for fixed value
of the noncommutative parameter √ϑ/M , the spin parameter is required to 0 a a ,
0 max
≤ ≤
where a can be obtained by numerically solving ∆ = 0 and ∂ ∆ = 0. On the other hand,
max r
given a value of spin a/M for a black hole, there will be a maximum √ϑ/M . We show the
0 0
parameter space that corresponds to black holes in Fig. 1(b). Thus, for a black hole, the
parameters must be in the range
√ϑ/M (0,0.52), a/M (0,1). (2.7)
0 0
∈ ∈
We give a fitting form of the parameters for the extremal black hole (thick blue line in Fig.
1(b)),
√ϑ/M = 0.52+0.23a/M 6.91(a/M )2+84.52(a/M )3 560.97(a/M )4+2177.02(a/M )5
0 0 0 0 0 0
− −
5193.85(a/M )6+7711.70(a/M )7 6941.67(a/M )8+3465.86(a/M )9 736.26(a/M(2).180),
0 0 0 0 0
− − −
– 3 –
1.0
0.8
M0 0.6
(cid:144)
M
0.4 J(cid:144)M0=0.1
J(cid:144)M0=0.3
J(cid:144)M0=0.5
0.2
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
r(cid:144)M0
(a)
0.5
non-black hole
0.4
0
M
(cid:144)
J 0.3
black hole
0.2
0.1
0.0 0.2 0.4 0.6 0.8 1.0
a(cid:144)M0
(b)
Figure 1. (a) Mass distribution for different values of the noncommutative parameter √ϑ/M0.
(b) The parameter spaces of √ϑ/M0 and a/M0 describing a black hole and a non-black hole. The
boundary described by the blue line denotes the extremal black holes.
Then the maximum a/M or √ϑ/M can be obtained with this relation. In the rest of the
0 0
paper, we will focus on the case of the black hole with the parameters in the range Eq. (2.7).
In this spacetime, there are two Killing fields ξ = ∂ related to two conserved
t,φ t,φ
constants along the geodesics,
= g ξµpν, (2.9)
−E µν t
l = g ξµpν, (2.10)
µν φ
where and l represent the energy and angular momentum of the particle, respectively, and
E
pµ =g x˙ν is the four-momentum of a test particle.
µν
Next, we will employ the Lagrangian and Hamilton-Jacobi equation to obtain the equa-
tionsofmotionforaparticle. TheLagrangiandescribinganeutralparticleinthebackground
– 4 –
(2.5) reads
1
= g x˙µx˙ν
µν
L 2
1 ∆ a2sin2θ 2a(r2+a2 ∆)sin2θ
= − t˙2 − t˙φ˙
2(cid:18)− ρ2 − ρ2
ρ2 (r2+a2) ∆a2sin2θ
+ r˙2+ρ2θ˙2+ − sin2θφ˙2 ,
∆ ρ2 (cid:19)
where a dot over a symbol denotes the derivative with respect to an affine parameter λ. The
normalizing condition is g x˙µx˙ν = µ2 (µ2=-1, 0, and 1 for timelike, null, and spacelike
µν
−
geodesics, respectively). Regarding xµ as the generalized coordinates, we then obtain the
conjugate momenta
∂
p = L = g x˙ν. (2.11)
µ ∂x˙µ µν
And the Hamiltonian reads
1
= p x˙µ = gµνp p . (2.12)
µ µ ν
H −L 2
Next, we will consider the Hamilton-Jacobi equation for geodesic motion in the background
(2.5), which takes the following form:
∂ 1
S = = gµν(∂ )(∂ ), (2.13)
µ ν
∂λ −H −2 S S
where istheHamilton-Jacobifunction. WiththeformoftheHamilton-Jacobifunction,one
S
then can determine the generalized coordinates xµ as a function of time, which can uniquely
determine the motion of a particle. To solve the Hamilton-Jacobi equation, we first separate
the Hamilton-Jacobi function as
µ2
= λ t+lφ+S (r)+S (θ). (2.14)
r θ
S − 2 −E
Substituting it into (2.13) and solving (2.9) and (2.10), we get
dφ a
ρ2 = (lcsc2θ a )+ (r2+a2) al , (2.15)
dλ − E ∆ E −
(cid:16) (cid:17)
dt r2+a2
ρ2 = a(l a sin2θ)+ (r2+a2) al , (2.16)
dλ − E ∆ E −
(cid:16) (cid:17)
dr
ρ2 = σ √ , (2.17)
r
dλ ℜ
dθ
ρ2 = σ √Θ, (2.18)
θ
dλ
with and Θ given by
ℜ
= a2 al+ r2 2 ∆ +(l a )2+r2µ2 ,
ℜ E − E − Q − E
Θ = (cid:0) a2µ2cos2θ(cid:1) (lcsc(cid:0)θ a sinθ)2+(l a(cid:1))2.
Q− − − E − E
Thesignfunctionsσ = andσ = areindependentfromeach other. Sincethisspacetime
r θ
± ±
is of the Petrov type D, there are four conserved quantities along the geodesics of the test
particle, , l, µ2, and . Inthis paper,we adoptµ2 = 0for discussingtheblack hole shadow.
E Q
– 5 –
3 Effective potential and unstable circular orbits
In this section, we will give a brief study on the radial-motion of photons, which will be used
to determine the contour of the noncommutative geometry inspired black hole shadow.
The radial motion (2.17) can be written in the form
2
dr
ρ2 + (r) = 0, (3.1)
eff
(cid:18) dλ(cid:19) V
which is very similar to the equation of motion of a classical particle. The effective potential
is given by
eff
V
M r
(r)= η+(ξ a)2 r2 4 0 γ(3/2,r2/4ϑ)+a2
eff
V − (cid:18) − √π (cid:19)
(cid:0) (cid:1)
a2 aξ+r2 2, (3.2)
− −
(cid:0) (cid:1)
where we have introduced two parameters η = / 2 and ξ = l/ . The effective potential
Q E E
has the limit:
(0) = 0 and (r ) . (3.3)
eff eff
V V → ∞ → −∞
With a further analysis of (r), the motion of the test particle along the radial direction
eff
V
will be clear. We plot the effective potential against r in Fig. 2(a) with η, ϑ, and a/M
eff 0
V
fixed.
From thebehaviorof shownin Fig. 2(a), wegetthat therearethreedifferentkinds
eff
V
of orbit for a photon coming from infinity: (i) ξ > ξ . The photon starting from infinity will
c
meet a turning point, where (r)= 0, and then it turns back to infinity and be observed.
eff
V
(ii) ξ < ξ . In this case, there is no turning point, thus the photon will cross the horizon and
c
be absorbed by the black hole. Thus, the observer located at infinity will not observe such
photon. (iii) ξ = ξ . This is a critical case. Thephoton comes from infinity and approaches a
c
turningpointwithzeroradialvelocity. Theturningpointcorrespondstoanunstablecircular
orbit. The radius of the orbit is determined by
∂ ∂2
eff eff
(r )= V (r )= 0, V (r )< 0. (3.4)
Veff c ∂r c ∂r2 c
Here r is an important parameter to determine the shape of the black hole. For the
c
Schwarzschildblackhole,rc = 3M0;andfortheKerrblackhole,rc = 2M0 1+cos(32 arccos(±M|a0|))
for the equatorial circular orbit. For the noncommutative geometry ins(cid:0)pired black hole, it (cid:1)
cannot be written in an exact form, and we show it in Fig. 2(b) for the equatorial circu-
lar orbit, from which we see that r is almost the same for different values of ϑ with the
c
spin parameter a/M fixed. However, the deviation becomes notable when the black hole
0
approaches an extremal one. With r obtained, we can solve η and ξ form Eq. (3.4):
c
A B
η = , ξ = , (3.5)
C2 C
– 6 –
10
0
-10
eff
V
Ξ<Ξ
-20 c
Ξ=Ξ
c
Ξ>Ξ
c
-30
-40
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
r(cid:144)M0
(a)
4.0
3.01
3.5 3
2.99
3.0
-0.01 0 0.01
0
M
2.5
(cid:144)
c
r
J (cid:144)M0=0.1
2.0
J (cid:144)M0=0.45
J (cid:144)M0=0.48
1.5
1.0
-1.0 -0.5 0.0 0.5 1.0
a(cid:144)M0
(b)
Figure 2. (a) Different behaviors of the effective potential for ξ < ξ , ξ = ξ , and ξ > ξ ,
eff c c c
V
respectively. (b) The radius r of the equatorial circular orbit.
c
with the coefficients given by
r2 3 r2
A = −rc10−4rc6ϑ3/2e4ϑ (cid:20)√π 2a2+rc2 −6rcγ(cid:18)2,4ϑc (cid:19)(cid:21)
(cid:0) (cid:1)
4r3ϑ3er2cϑ2 4√π 2a2+3r2 γ 3, rc2 +πr3
− c (cid:20)− c (cid:18)2 4ϑ(cid:19) c
(cid:0) (cid:1)
3 r2 2
+36r γ , c ,
c
(cid:18)2 4ϑ(cid:19) (cid:21)
B = 2ϑ3/2e4rcϑ2 (cid:20)2 a2−3rc2 γ(cid:18)32, 4rϑc2(cid:19)+√πrc a2+rc2 (cid:21)
(cid:0) (cid:1) (cid:0) (cid:1)
+r3 a2+r2 ,
c c
C = a(cid:20)rc3(cid:0)−2ϑ3/2(cid:1)er4cϑ2 (cid:18)√πrc−2γ(cid:18)23, 4rϑc2(cid:19)(cid:19)(cid:21).
– 7 –
Using this expression, we will explore the black hole shadow in the next section by setting
M = 1 for simplicity.
0
4 Black hole shadow
The boundary of the black hole shadow is determined by the unstable circular orbit. In the
real observations, the shadow of a black hole is a dark region on the observer’s sky. So it is
natural to introduce the celestial coordinates
dφ
α = lim r2sinθ = ξcsci,
r (cid:18)− dr(cid:12) (cid:19) −
→∞ (cid:12)θ i
(cid:12) →
dθ
β = lim r2 =(cid:12) η+a2cos2i ξ2cot2i,
r (cid:18) dr(cid:12) (cid:19) ± −
→∞ (cid:12)θ i p
(cid:12) →
(cid:12)
where i is the inclination angle between the axis of rotation of the black hole and the line
of sight of the observer. The coordinates α and β are, respectively, the apparent perpendic-
ular distances of the image seen from the axis of symmetry and from its projection on the
equatorial plane. After substituting into Eq. (3.5), we get the explicit form of α and β:
B
α = csci, (4.1)
−C
1
β = A+a2C2cos2i+B2cot2i. (4.2)
±C
p
4.1 Nonrotating case
First, let us consider the nonrotating case, where the spin parameter a/M = 0. A simple
0
calculation shows
2r2 r6+4ϑ3e2rϑ2 πr2 12γ 3, r2 2 8r3ϑ3/2e4rϑ2γ 3, r2
(cid:18) (cid:18) − 2 4ϑ (cid:19)− 2 4ϑ (cid:19)
α2+β2 = (cid:16) (cid:17) (cid:16) (cid:17) . (4.3)
r3 2ϑ3/2e4rϑ2 √πr 2γ 3, r2 2
− − 2 4ϑ
(cid:16) (cid:16) (cid:16) (cid:17)(cid:17)(cid:17)
Since photons come from both sides of the nonrotating black hole have the same value of the
deflection angle, the shadow of such case is a standard circle with radius R /M = α2+β2
s 0
given in Eq. (4.3). It is easy to find that this radius depends of the inclination anglpe i, which
is owing to the spherically symmetric of the black hole with vanishing spin. The radius
of the shadow is also presented in Fig. 3. For small value of √ϑ/M , the radius almost
0
equals to R = 3√3M , which is exactly the resultof the Schwarzschild black hole. Although
s 0
the radius R has a noticeable decrease at √ϑ/M = 0.40 from the figure, the difference is
s 0
still indistinguishable from astronomical observations. For example, the difference between
√ϑ/M = 0.1 and 0.48 is 0.0013, between √ϑ/M = 0.1 and 0.52 is 0.0046.
0 0
4.2 Rotating case
For arotatingblack hole, thephotoncaptureradiiforco-rotating andcounter-rotating orbits
willbedifferent. From Fig. 2(b),weseethattheradiusofco-rotating orbitsisalways smaller
thanthat ofcounter-rotating orbits. Thisis auniversalresult. For thisreason, photonscome
from both sides of the black hole will have different values of the deflection angle, resulting
to a deformed circle. The shape of the black hole shadow is presented in Fig. 4. The shape
– 8 –
5.196
5.195
0
M 5.194
(cid:144)
s
R
5.193
5.192
0.0 0.1 0.2 0.3 0.4 0.5
J (cid:144)M0
Figure 3. Theradiusoftheshadowforanonrotatingnoncommutativegeometryinspiredblackhole.
depends on the parameter √ϑ/M , a/M and the inclination angle i. It is clear that for a
0 0
slow rotating black hole, the shape is an approximate circle. While for a fast rotating black
hole, the shape will deviate significantly from a circle. This deviation is found to increase
with the inclination angle i. It is also an interesting result that the shape of the black hole
shadow approaches a standard circle with the increase of √ϑ/M , even for a near extremal
0
black hole with small spin a/M .
0
In order to extract the information of an astronomical object from the shadow, the ob-
servables, which could bedirectly observed from the astronomical observation, are necessary.
Let first focus on the silhouette of the shadow. Similar to the Kerr case, for each silhouette
of the shadow, we clearly see that there are four characteristic points: the top point (α ,β ),
t t
bottom point (α ,β ), left point (α ,0), and right point (α ,0) of the shadow, and they cor-
b b l r
respond to the unstable retrograde circular orbit seen from an observer located at infinity.
Then a natural idea is that the observables can be constructed using these four points. In
Ref. [28], Hioki and Maeda proposed two observables, the radius R and distortion δ , to
s s
measure the shadow by a Kerr black hole. The observable R is defined as the radius of a
s
reference circle passing by the top, bottom, and right points. And δ gives its deformation
s
withrespecttothereferencecircle, measuredbytheleftpoint(α ,0) andpoint(α˜ ,0), which
l r
is the point where the reference circle cut the horizontal axis at the opposite side of (α ,0).
r
Hence R and δ measure the size and deformation of a shadow. With observables R and δ ,
s s s s
the Kerr black hole spin a/M and inclination angle i of the observer can be uniquely deter-
0
mined. However for the noncommutative geometry inspired black hole, there exists an extra
parameter √ϑ/M . The presence of the noncommutative parameter will bring the degener-
0
ate problem as presented in Ref. [36], which means that different pairs of (a/M , i, √ϑ/M )
0 0
may have the same shadow. In order to solve this problem, Tsukamoto-Li-Bambi studied the
fine structure of the shadow. They gave another distortion parameter ε [36], which measures
the distortion on the left side of the black hole shadow resulted by the unstable retrograde
circular orbit. And one more point is introduced, which we denote as (α ,β ). It is the
h h
point that the horizontal line of β = β /2 cuts the shadow at the opposite side of (α ,0).
t r
Combined with these three observables, the nature of black hole and inclination angle i of
the observer will be well determined for the noncommutative geometry inspired black hole.
After a simple algebra calculation and combining with the symmetry of the shadow, we
– 9 –