Applications of Analogue Gravity Techniques

Application of analogue gravity techniques

Pratik TarafdarJunior 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

Motivation

Primordial black holes - Black holes formed under extreme conditionsduring the radiation dominated era - Survived Hawking evaporation byaccreting radiation.

Investigating primordial black hole accretion phenomenon from theperspective of analogue gravity.

Before moving on to primordial black holes, we need to understand thetechniques of analogue gravity by applying them to different models ofastrophysical 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 proceedto PBHs.


Analogue (acoustic) surface gravity (κ)

Following Unruh’s pioneering work, the surface gravity κ as well as theanalogue temperature TAH can be found to be proportional to the speedof propagation of the acoustic perturbation cs and the space gradient∂/∂η (taken along the normal to the acoustic horizon) of the bulk flowvelocity 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 onthe acoustic horizon rh. The acoustic horizon is the surface defined, forthe stationary flow configuration, by the equation

u2⊥ − c2s = 0, (2)

cs was assumed to be position independent. For position dependent soundspeed, the surface gravity as well as the analogue temperature can beobtained as,


Acoustic surface gravity (continued...)

[TAH , κ] ∝[cs∂

∂η(cs − u⊥)

]rh

. (3)

Expression for the surface gravity as well as the analogue temperature asdefined, corresponds to the flat background flow geometry of Minkowskiantype and its relativistic generalization may be obtained as,

κ =

[ √χµχµ

(1− cs2)∂

∂η(u⊥ − cs)

]rh

, (4)

where χµ is the Killing field which is null on the corresponding acoustichorizon.


Approach for finding space gradients of the dynamical andthe acoustic velocities at sonic points

Momentum conservation (Euler) and mass conservation (continuity)equations → First integrals of motion (Specific energy E and Massaccretion rate M which is equivalent to another constant known asentropy accretion rate).

Differentiating the derived constants, we obtain expressions for the spacegradients of dynamical velocity (u) and acoustic velocity (cs), which are ofthe form N

D .

Assuming that the flow is continuous at the critical points, such as in theabsence of shock, we equate both N and D to zero simultaneously toobtain the critical point conditions (expressions for u and cs and therelation between them at the critical points).


Approach for finding space gradients of the dynamical andthe acoustic velocities at sonic points (continued)...

Applying L’Hospital’s rule on the expressions for space gradients ofdynamical and acoustic velocities and then substituting the critical pointconditions, we get a quadratic equation in du

dr |c, which can be solvedanalytically for a given set of parameters (E , λ (specific angularmomentum) and γ (polytropic index)), to derive the space gradients of thevelocities at the critical points.

The critical points and the sonic points may not be equal for all the flowconfigurations (in both polytropic and isothermal case). In such scenarios,we have to integrate du

dr to generate the integral curve and find out thecorresponding sonic points.

Then by substituting for the sonic point in dudr we obtain the required

quantity and subsequently can calculate the acoustic surface gravity (κ).


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 : Verticalequilibrium model. Fig. IV : Comparision of acoustic surface gravity (κ) for allconfigurations in the adiabatic case.


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

I0.0000638

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 : Verticalequilibrium model. Fig. IV : Comparision of acoustic surface gravity (κ) for allconfigurations in the isothermal case.


Conclusion and future plan

Extension of the formalism to the case of radiation accretion ontoprimordial black holes.

Investigating effects of dispersion relation of the accreting fluid onanalogue Hawking temperature.

Towards an entirely analytical approach to study the parameter space.


References

1. Unruh, W. G. 1981, Phys. Rev. Lett. 46, 1351.2. Visser, M. 1998, Class. Quant. Grav. 15, 1767.3. Bilic, 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 ChicagoPress.10. Majumdar, A. S., 2003, Physical Review Letters, Vol. 90, Issue 3, id.031303.


Applications of Analogue Gravity Techniques

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