1. Application of analogue gravity techniques
Pratik Tarafdar
Junior Research Fellow
Dept. of Astrophysics and Cosmology.
Supervisor : Dr. Archan S. Majumdar
S. N. Bose National Centre for Basic Sciences.
February 5, 2013
Pratik Tarafdar (SNBNCBS) Analogue Gravity February 5, 2013 1 / 10

2. Motivation
Primordial black holes - Black holes formed under extreme conditions
during the radiation dominated era - Survived Hawking evaporation by
accreting radiation.
Investigating primordial black hole accretion phenomenon from the
perspective of analogue gravity.
Before moving on to primordial black holes, we need to understand the
techniques of analogue gravity by applying them to different models of
astrophysical black hole accretion, which are well understood.
We have particularly focussed on the aspect of analogue surface gravity,
since it shall be the characteristic property of our interest when we proceed
to PBHs.
Pratik Tarafdar (SNBNCBS) Analogue Gravity February 5, 2013 2 / 10

3. Analogue (acoustic) surface gravity (κ)
Following Unruh’s pioneering work, the surface gravity κ as well as the
analogue temperature TAH can be found to be proportional to the speed
of propagation of the acoustic perturbation cs and the space gradient
∂/∂η (taken along the normal to the acoustic horizon) of the bulk flow
velocity u ⊥ measured along the direction normal to the acoustic horizon
[TAH, κ] ∝
1
cs
∂u⊥
∂η rh
, (1)
where the subscript rh indicates that the quantity has been evaluated on
the acoustic horizon rh. The acoustic horizon is the surface defined, for
the stationary flow configuration, by the equation
u2
⊥ − c2
s = 0, (2)
cs was assumed to be position independent. For position dependent sound
speed, the surface gravity as well as the analogue temperature can be
obtained as,
Pratik Tarafdar (SNBNCBS) Analogue Gravity February 5, 2013 3 / 10

4. Acoustic surface gravity (continued...)
[TAH, κ] ∝ cs
∂
∂η
(cs − u⊥)
rh
. (3)
Expression for the surface gravity as well as the analogue temperature as
defined, corresponds to the flat background flow geometry of Minkowskian
type and its relativistic generalization may be obtained as,
κ =
√
χµχµ
(1 − cs
2)
∂
∂η
(u⊥ − cs)
rh
, (4)
where χµ is the Killing field which is null on the corresponding acoustic
horizon.
Pratik Tarafdar (SNBNCBS) Analogue Gravity February 5, 2013 4 / 10

5. Approach for finding space gradients of the dynamical and
the acoustic velocities at sonic points
Momentum conservation (Euler) and mass conservation (continuity)
equations → First integrals of motion (Specific energy E and Mass
accretion rate ˙M which is equivalent to another constant known as
entropy accretion rate).
Differentiating the derived constants, we obtain expressions for the space
gradients of dynamical velocity (u) and acoustic velocity (cs), which are of
the form N
D .
Assuming that the flow is continuous at the critical points, such as in the
absence of shock, we equate both N and D to zero simultaneously to
obtain the critical point conditions (expressions for u and cs and the
relation between them at the critical points).
Pratik Tarafdar (SNBNCBS) Analogue Gravity February 5, 2013 5 / 10

6. Approach for finding space gradients of the dynamical and
the acoustic velocities at sonic points (continued)...
Applying L’Hospital’s rule on the expressions for space gradients of
dynamical and acoustic velocities and then substituting the critical point
conditions, we get a quadratic equation in du
dr |c, which can be solved
analytically for a given set of parameters (E, λ (specific angular
momentum) and γ (polytropic index)), to derive the space gradients of the
velocities at the critical points.
The critical points and the sonic points may not be equal for all the flow
configurations (in both polytropic and isothermal case). In such scenarios,
we have to integrate du
dr to generate the integral curve and find out the
corresponding sonic points.
Then by substituting for the sonic point in du
dr we obtain the required
quantity and subsequently can calculate the acoustic surface gravity (κ).
Pratik Tarafdar (SNBNCBS) Analogue Gravity February 5, 2013 6 / 10

7. Results
0.015
0.02
0.025
0.03
0.035
0.04
3.52 3.54 3.56 3.58 3.6 3.62 3.64 3.66
κ
λ
I
0.049
0.05
0.051
0.052
0.053
0.054
0.055
0.056
3.52 3.54 3.56 3.58 3.6 3.62 3.64 3.66
κ
λ
II
0.193
0.1935
0.194
0.1945
0.195
0.1955
0.196
0.1965
0.197
0.1975
3.52 3.54 3.56 3.58 3.6 3.62 3.64 3.66
κ
λ
III
0
0.05
0.1
0.15
0.2
3.52 3.54 3.56 3.58 3.6 3.62 3.64 3.66
κ
λ
IV
Figure : Fig. I : Constant height disc. Fig. II : Conical disc. Fig. III : Vertical
equilibrium model. Fig. IV : Comparision of acoustic surface gravity (κ) for all
configurations in the adiabatic case.
Pratik Tarafdar (SNBNCBS) Analogue Gravity February 5, 2013 7 / 10

8. Results (continued...)
0.00002248
0.00002248
0.00002249
0.00002249
0.00002250
0.00002250
0.00002251
3.65 3.7 3.75 3.8 3.85 3.9 3.95 4
κ
λ
I
0.0000638
0.0000638
0.0000639
0.0000639
0.0000639
0.0000639
0.0000639
0.0000640
3.65 3.7 3.75 3.8 3.85 3.9 3.95
κ
λ
II
0.0001816
0.0001817
0.0001818
0.0001819
0.0001820
0.0001821
0.0001822
0.0001823
0.0001824
0.0001825
3.65 3.7 3.75 3.8 3.85 3.9
κ
λ
III
2e-005
4e-005
6e-005
8e-005
0.0001
0.00012
0.00014
0.00016
0.00018
0.0002
3.6 3.65 3.7 3.75 3.8 3.85 3.9 3.95 4
κ
λ
IV
Figure : Fig. I : Constant height disc. Fig. II : Conical disc. Fig. III : Vertical
equilibrium model. Fig. IV : Comparision of acoustic surface gravity (κ) for all
configurations in the isothermal case.
Pratik Tarafdar (SNBNCBS) Analogue Gravity February 5, 2013 8 / 10

9. Conclusion and future plan
Extension of the formalism to the case of radiation accretion onto
primordial black holes.
Investigating effects of dispersion relation of the accreting fluid on
analogue Hawking temperature.
Towards an entirely analytical approach to study the parameter space.
Pratik Tarafdar (SNBNCBS) Analogue Gravity February 5, 2013 9 / 10

10. References
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2. Visser, M. 1998, Class. Quant. Grav. 15, 1767.
3. Bili´c, N. 1999, Class. Quant. Grav. 16, 3953.
4. Barcelo, C., Liberati, S., and Visser, M., 2005, ‘Analogue Gravity’,
Living Reviews in Relativity, Vol. 8, No. 12.
5. Novikov, I.D., & Thorne, K.S., 1973, Black Holes, ed. C. de Witt &
B.S. de Witt (New York: Gordon & Breach) 343.
6. Abramowicz, M.A., Lanza, A., & Percival, M.J., 1997, ApJ, 479, 179.
7. Bondi, H., 1952, MNRAS, 112, 195.
8. Landau, L. D., & Lifshitz, E. D., 1959, Fluid Mechanics, New York:
Pergamon.
9. Wald, R. M., 1984, General Relativity, Chicago, University of Chicago
Press.
10. Majumdar, A. S., 2003, Physical Review Letters, Vol. 90, Issue 3, id.
031303.
Pratik Tarafdar (SNBNCBS) Analogue Gravity February 5, 2013 10 / 10