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TABLE IV: The first lowest QN frequencies for gravita- tional vector (j = 2) perturbations of the five dimensional Schwarzschild black hole. The frequencies are normalized in units of black hole temperature, so the Table really shows =, where Ty is the Hawking temperature of the black hole.

Table 4 IV: The first lowest QN frequencies for gravita- tional vector (j = 2) perturbations of the five dimensional Schwarzschild black hole. The frequencies are normalized in units of black hole temperature, so the Table really shows =, where Ty is the Hawking temperature of the black hole.

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FIG. 1: Scalar (s = 0), electromagnetic (s = 1) and gravitational (s = 2) QN frequencies of a four-dimensional Schwarzschild black hole. Note that the dots indicate the QN frequencies there, and the lines connecting the dots only help to figure out to which multipole (1, s) they belong to. We show the lowest 500 modes for the lowest radiatable multipoles of each field, i.e., ! = 0 for the scalar field, | = 1 for the electro- magnetic and / = 2 for gravitational. However, the asymp- totic behaviour is l-independent. It is quite clear from this plot that highly damped electromagnetic QN frequencies have a completely different behaviour from that of scalar or gravita- tional highly damped QN frequencies. In fact, the electromag- netic ones asymptote to tt — i(2n)m ,n — ov, as predicted by Motl and Motl and Neitzke 3. The scalar and grav- itational ones asymptote to F — In3+i(2n+1)m ,n > ow.
FIG. 1: Scalar (s = 0), electromagnetic (s = 1) and gravitational (s = 2) QN frequencies of a four-dimensional Schwarzschild black hole. Note that the dots indicate the QN frequencies there, and the lines connecting the dots only help to figure out to which multipole (1, s) they belong to. We show the lowest 500 modes for the lowest radiatable multipoles of each field, i.e., ! = 0 for the scalar field, | = 1 for the electro- magnetic and / = 2 for gravitational. However, the asymp- totic behaviour is l-independent. It is quite clear from this plot that highly damped electromagnetic QN frequencies have a completely different behaviour from that of scalar or gravita- tional highly damped QN frequencies. In fact, the electromag- netic ones asymptote to tt — i(2n)m ,n — ov, as predicted by Motl and Motl and Neitzke 3. The scalar and grav- itational ones asymptote to F — In3+i(2n+1)m ,n > ow.
TABLEI: The correction coefficients for the four-dimensional Schwarzschild black hole, both numerical, here labeled as “Corr)’” and analytical, labeled as “Corr#”. These results refer to scalar perturbations. The analytical results are ex- tracted from (see also formula (29) below). Notice the very good agreement between the numerically extracted re- sults and the analytical prediction. These values were obtained using the first 5000 modes. toting to In3 x Ty. Electromagnetic QN frequencies, on the other hand, have a real part asymptoting to 0. This is clearly seen in Fig. [J] For gravitational perturbations Nollert found, and we confirm, that the correction term ain @ is
TABLEI: The correction coefficients for the four-dimensional Schwarzschild black hole, both numerical, here labeled as “Corr)’” and analytical, labeled as “Corr#”. These results refer to scalar perturbations. The analytical results are ex- tracted from (see also formula (29) below). Notice the very good agreement between the numerically extracted re- sults and the analytical prediction. These values were obtained using the first 5000 modes. toting to In3 x Ty. Electromagnetic QN frequencies, on the other hand, have a real part asymptoting to 0. This is clearly seen in Fig. [J] For gravitational perturbations Nollert found, and we confirm, that the correction term ain @ is
FIG. 2: Electromagnetic QN frequencies of a four-dimensional Schwarzschild black hole.
FIG. 2: Electromagnetic QN frequencies of a four-dimensional Schwarzschild black hole.
TABLE II: The correction coefficients for the four- dimensional Schwarzschild black hole, both numerical, here labeled as “Corr}’” and analytical, labeled as “Corr”. These results refer to gravitational perturbations. The analytical results are extracted from (see also formula (29] below). Notice the very good agreement between the numerically ex- tracted results and the analytical prediction. These values were obtained using the first 5000 modes.
TABLE II: The correction coefficients for the four- dimensional Schwarzschild black hole, both numerical, here labeled as “Corr}’” and analytical, labeled as “Corr”. These results refer to gravitational perturbations. The analytical results are extracted from (see also formula (29] below). Notice the very good agreement between the numerically ex- tracted results and the analytical prediction. These values were obtained using the first 5000 modes.
TABLE III: The first lowest QN frequencies for scalar and gravitational tensor (j = 0) perturbations of the five dimen- sional Schwarzschild black hole. The frequencies are normal- ized in mans of black hole temperature, so the Table really shows =, where Ty is the Hawking temperature of the black hole.
TABLE III: The first lowest QN frequencies for scalar and gravitational tensor (j = 0) perturbations of the five dimen- sional Schwarzschild black hole. The frequencies are normal- ized in mans of black hole temperature, so the Table really shows =, where Ty is the Hawking temperature of the black hole.
FIG. 3: The QN frequencies of scalar (j = 0), gravitational vector (j = 2) and electromagnetic vector (j = 2/3) pertur- bations of a five-dimensional Schwarzschild black hole. We only show the lowest radiatable multipoles, but the asymtp- toic behaviour is [-independent.
FIG. 3: The QN frequencies of scalar (j = 0), gravitational vector (j = 2) and electromagnetic vector (j = 2/3) pertur- bations of a five-dimensional Schwarzschild black hole. We only show the lowest radiatable multipoles, but the asymtp- toic behaviour is [-independent.
TABLE V: The first lowest QN frequencies for electromag- netic vector (7 = 2/3) perturbations of the five dimensional Schwarzschild black hole. The frequencies are normalized in units of black hole temperature, so the Table really shows =, where Ty is the Hawking temperature of the black hole.
TABLE V: The first lowest QN frequencies for electromag- netic vector (7 = 2/3) perturbations of the five dimensional Schwarzschild black hole. The frequencies are normalized in units of black hole temperature, so the Table really shows =, where Ty is the Hawking temperature of the black hole.
FIG. 4: The QN frequencies of scalar and gravitational ten- sor perturbations (j = 0) of a five-dimensional Schwarzschild black hole. Notice that for frequencies with a large imaginary part the slope is J-independent and equals 3/2.
FIG. 4: The QN frequencies of scalar and gravitational ten- sor perturbations (j = 0) of a five-dimensional Schwarzschild black hole. Notice that for frequencies with a large imaginary part the slope is J-independent and equals 3/2.
FIG. 5: The QN frequencies of gravitational vector perturba- tions (j = 2) of a five-dimensional Schwarzschild black hole.
FIG. 5: The QN frequencies of gravitational vector perturba- tions (j = 2) of a five-dimensional Schwarzschild black hole.
TABLE VII: The correction coefficients for the five- dimensional Schwarzschild black hole, both numerical, here labeled as “CorrX” and analytical, labeled as “Corr?”, for gravitational vector perturbations (j7 = 2). The analytical re- sults are extracted from the analytical formula (2). Notice the very good agreement, to within 0.5% or less, between the numerically extracted results and the analytical prediction.
TABLE VII: The correction coefficients for the five- dimensional Schwarzschild black hole, both numerical, here labeled as “CorrX” and analytical, labeled as “Corr?”, for gravitational vector perturbations (j7 = 2). The analytical re- sults are extracted from the analytical formula (2). Notice the very good agreement, to within 0.5% or less, between the numerically extracted results and the analytical prediction.
TABLE VI: The correction coefficients for the five- dimensional Schwarzschild black hole, both numerical, here labeled as “Corr” and analytical, labeled as “Corr#”. These results refer to scalar or gravitational tensor perturbations (j = 0). The analytical results are extracted from the analyt- ical formula (BI). Notice the very good agreement, to within 0.5% or less, between the numerically extracted results and the analytical prediction.
TABLE VI: The correction coefficients for the five- dimensional Schwarzschild black hole, both numerical, here labeled as “Corr” and analytical, labeled as “Corr#”. These results refer to scalar or gravitational tensor perturbations (j = 0). The analytical results are extracted from the analyt- ical formula (BI). Notice the very good agreement, to within 0.5% or less, between the numerically extracted results and the analytical prediction.
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