Presentation Black Hole Scattering

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Scalar and Electromagnetic waves Scattering bySchwarzschild Black Hole

Dr. Muhammad Ayub Faridi

Centre for High Energy Physics, University of the Punjab, Lahore12/13/2015ICRA 2015 University of the Punjab, Lahore12/13/2015OutlineScattering Theory Schwarzschild Black HoleScalar Wave ScatteringElectromagnetic Wave ScatteringComparison b/w Scalar and electromagnetic waves ScatteringConclusion

22/13/2015ICRA 2015 University of the Punjab, LahorePhoton trajectories outside Schwarzschild black hole2/13/2015ICRA 2015 University of the Punjab, Lahore3

Scattering

A classical geometrical viewbImpact parameterScattering angleScatteringsolid angleIncidentcross-section

Differential cross-section

2/13/2015ICRA 2015 University of the Punjab, Lahore442/13/2015ScatteringA classical geometrical view

bImpact parameterScattering angleIncidentcross-sectionScatteringsolid angle

To be determined in specific situationsExample: Hard-sphere scattering

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Radiation zone

intermediate zoneScatteringPartial wave analysis

Develop the solution interms of spherical harmonics,solution to a spherically symmetrical potential

Scattering zone2/13/2015ICRA 2015 University of the Punjab, Lahore662/13/2015ScatteringPartial wave analysis

Intermediate zone

Scattering zoneSolve the Schrdingerequation with potential V Physical Solution

Hankel functionsGeneral Solution

Partial wave amplitudeGeometrical considerations

Radiation zone

2/13/2015ICRA 2015 University of the Punjab, Lahore772/13/2015ScatteringPartial wave analysisConnecting intermediate and radiation zone

Differential cross-section

Total cross-section

when

with

Orthogonality ofLegendre polynomials2/13/2015ICRA 2015 University of the Punjab, Lahore882/13/2015Schwarzschild MetricICRA 2015 University of the Punjab, Lahore92/13/2015Scattering by Schwarzschild Black Hole10

2/13/2015ICRA 2015 University of the Punjab, Lahore11In curved spacetime form of K-G equation changes into

Where Schwarzschild metric is given by

AlsoAs for the scalar field the equation is KleinGordon i.e

A scalar eld propagates in schwarzschild spacetime The equation which govern through the evolution of massless scalar eld is,ICRA 2015 University of the Punjab, Lahore2/13/2015

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substituting the Schwarzschild metric in the above equation.

The form along with the spherical symmetryof metric , suggest the decomposition

Using this in above equation it separates into the following radial and angular partICRA 2015 University of the Punjab, Lahore2/13/2015

With

Effective potential 13

The Schrodinger like equation 2/13/2015ICRA 2015 University of the Punjab, Lahore Regge-Wheeler and Zerilli PhysRev.108.1063

Amplitudes of Scattered and Transmitted Waves:

whereTransmission coecient for scalar wave for dierent values of l asa function of 2M14ICRA 2015 University of the Punjab, Lahore2/13/2015

Scattering coecient for scalar wavefor dierent values of l as a function of 2M.

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Scattering Amplitude or f( )can be written as fellow,


Absorption Cross Section:

Partial absorption cross section for dierent values of l as a function of M.Total absorption cross section asA function of M.17ICRA 2015 University of the Punjab, Lahore2/13/2015For a particular value of lScattering Cross Section:

Difference b/w Bardeen and Schwarzschild Space time:

Schwarzschild spacetimeBardeen spacetime18ICRA 2015 University of the Punjab, Lahore2/13/2015182/13/2015

In g. Partial scattering cross section for l = 2 at M = 1 for scalar wave. In this case we see maximum value of dierential cross section at angle = 0 and oscillatory pattern is not regular in this case. And in table a comparison is shown for the partial scattering cross section. The angle at which the maxima have been observed are almost same in both cases.

In g. below partial scattering cross section for l = 3 at M = 1 for scalar wave. In this case we see maximum value of dierential cross section at angle = 0 and more complicated oscillatory pattern has been observed as that of in pervious case. And in table a comparison is shown for the partial scattering cross section. The values have little agreement in this case.19ICRA 2015 University of the Punjab, Lahore2/13/2015

In g. below partial scattering cross section for l = 4 at M = 1 for scalar wave. In this case we see maximum value of dierential cross section at angle = 0 and more complicated oscillatory pattern has been observed as that of in pervious case. And in table a comparison is shown for the partial scattering cross section. The values have dierences because in case of schwarzschild is zero while in case of Bardeen has non-zero value.

Above four gures shows the behavior of partial scattering cross section as a function of angle () for four dierent partial waves from l = 1 up to l =4. On comparing these gures we can see that (i) when partial wave number l increase, more complicated oscillatory pattern arises. (ii) with the increase l the main scattering angle becomes narrower. (iii) scattering cross section increases with the increase of l (iv) at = 0 the oscillation of dierential cross section is symmetric and has maximum value.20ICRA 2015 University of the Punjab, Lahore2/13/2015Electromagnetic wave scatteringInclude spin part in the eective potential of the scalar wave eective potential. The Schrodinger eqn.

where

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Reection and Transmission Coecients:

where


as a function of l for the scattering of electromagnetic wave for dierent values of

23ICRA 2015 University of the Punjab, Lahore2/13/2015Absorption Cross Section:

Partial absorption cross section for l=1 up to l=5. From which we can see that corresponding partial absorption cross section starting from zero and then reaches a maximum values and then decreases asymptotically. The larger the value of (l ) the correspondence value of absorption cross section is smaller.Total absorption cross section as a function of M. In which summation in above eqn. is performed from l=0 to l=12. For larger the M the total absorption cross section falls. When M >> 1 the absorption cross section tends to the geometric- optic limit.24ICRA 2015 University of the Punjab, Lahore2/13/2015Scattering Cross Section:

In case of black hole for electromagnetic wave, the dierential electromagnetic scattering cross section


For Glory Scattering:

Luis et al. Scattering:

whereOur Work:



Fig. shows dierential electromagnetic scattering cross section at M = 2. In gure we made comparison of our numerical results(solid line) and glory approximation (dotted line). From this we can see that these are in not so good agreement for . This is more clearly seen in g. in which only comparison portion of is taken. . And in table we show numerical results for the electromagnetic cross section at M = 2 at dierent angles in (rad) . At some angles values match with each other but not completely.28ICRA 2015 University of the Punjab, Lahore2/13/2015

Fig. shows dierential electromagnetic scattering cross section at M = 3. In g. we made comparison of our numerical results(solid line) and glory approximation (dotted line). This is more clearly seen in which only comparison portion is taken. And in table we compare some numerical results for the electromagnetic cross section at M = 3 at dierent angles in (rad) .The agreement of values at some angles is good but near approaching to then there is huge dierence in ourvalue.29ICRA 2015 University of the Punjab, Lahore2/13/2015

Fig. shows dierential electromagnetic scattering cross section at M = 4. In g. we made comparison of our numerical results(solid line) and glory approximation (dotted line). This is more clearly seen in g.in which only comparison portion is taken. And in table we compare some numerical results for the electromagnetic cross sectionat M = 4 at dierent angles (in rad). The agreement is good as compared to all pervious cases.From all these gures and tables we can see that at higher values M of numerical results are in good agreement but when value is smaller for M the agreement between them is not good.302/13/2015ICRA 2015 University of the Punjab, LahoreComparison between scalar and electromagnetic wave scattering

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Fig. shows the behavior of scattering cross section at M = 4 for the scalar wave and electromagnetic wave.From this scattering ux is stronger and its width becomes narrower in forward direction. The glory phenomenon exists along the backward direction.At xed frequency the glory peak is higher and width becomes narrower. So along bothin forward and backward direction we can nd glory phenomenon.332/13/2015ICRA 2015 University of the Punjab, Lahore

In Fig. comparison of partial scattering cross section is shown for scalar and electromagnetic wave. At lower value of l up to 3 there is no dierence can be seen for the partial scattering cross section. But in case when l is 4 then we can see clear dierence in the peak. The dierence is more clear when l is 5. So from here we can infer that when l increases the dierence is greater and greater.Solid line in red colour for scalar wave while dotted line in black is for electromagnetic wave .342/13/2015ICRA 2015 University of the Punjab, LahoreConclusion:Partial scattering cross section for scalar wave is compared with Bardeen scattering cross section there is dierence between Bardeen spacetime in black hole and Schwarzschild spacetime. Which indicates actually dierence of alpha which have non zero value in case of Bardeen spacetime and is zero in case of Schwarzschild. For Electromagnetic wave partial scattering cross section are in good agreement at the lower value of l with the scalar wave partial scattering cross section but with the increase of l the dierence is appeared. Total scattering cross section is compared with the Glory scattering. we numerically carry out the absorption cross section and we fined that the larger the angular momentum quantum number l is, the smaller the corresponding maximum value of the partial absorption cross section


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Presentation Black Hole Scattering

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