Super-Radiance homework

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In summary, Super-Radiance homework is a type of assignment given in physics or astronomy courses that involves researching and analyzing data related to the phenomenon of super-radiance. It is important because it allows students to gain a better understanding of this phenomenon and develop critical thinking and research skills. Typically, Super-Radiance homework is structured around conducting experiments or simulations and writing reports or essays. To successfully complete it, students should carefully read instructions, conduct thorough research, and seek assistance if needed. It can be challenging due to its complex nature, but with proper effort, students can successfully complete Super-Radiance homework and gain valuable knowledge and skills.
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latentcorpse
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I'm working through p90/91 of these notes:
http://arxiv.org/PS_cache/gr-qc/pdf/9707/9707012v1.pdf

In 4.58, he introduces this [itex]j^\mu[/itex] flux 4 vector.
(i) Where does this come from? Should I just accept it?

(ii) He says it's future directed since [itex]-k \cdot j>0[/itex]. I can't see why this inequality is true or why it implies future directed?

(iii)He then gets 4.59 using the divergence theorem for manifold integration
[itex]\int_M d^nx \sqrt{-g} \nabla_a X^a = \int_{\partial M} d^{n-1}x \sqrt{|h|} n_a X^a[/itex]
So I'm assuming that his measure [itex]dS_\mu = d^{n-1}x \sqrt{|h|} n_\mu[/itex], yes?

(iv) Then in 4.60, why do the second two terms have minus signs? I know it's to do with the way we integrate around that surface and the direction of the normals but I can't make sense of it.

(v)Why in 4.62 is the energy through the horizon [itex]E_1-E_2[/itex]? Is it basically saying that (looking at the diagram) we have some field below [itex]\Sigma_1[/itex] which enters the enclosed region, interacts and loses some energy such that when it comes out of [itex]\Sigma_2[/itex] it has energy [itex]E_1-E_2[/itex] that can be transferred through the horizon?

(vi)In 4.64, is he missing a [itex]dv[/itex] in the measure?

(vii) Also in 4.64, how does [itex]\xi_\mu j^\mu = ( \xi \cdot \partial \Phi)( k \cdot D \Phi)[/itex]

(viii) I also cannot show 4.67. I find:

[itex]P= \int dA \left( \frac{\Phi_0}{\omega} \sin{( \omega v - \nu \chi)} - \frac{\Omega_H \Phi_0}{\nu} \sin{(\omega v - \nu \chi)} \right) \frac{\phi_0}{\omega} \sin{(\omega v - \nu \chi)}[/itex]
But don't know how to proceed?

(ix) Also, does this whole super-radiance thing happen only in the ergoregion?

Thanks.
 
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(i) The j^μ flux 4-vector is derived from the conservation of energy-momentum, which states that the divergence of the stress-energy tensor is equal to zero. This means that there must be some flux of energy-momentum in and out of any volume element of space-time. j^μ is the 4-vector that represents this flux. (ii) -k⋅j>0 implies that the flux is future-directed because k is a future-directed vector. This follows from the fact that the dot product of two vectors is positive when they point in the same direction.(iii) Yes, his measure dSμ=d^n−1x√|h|nμ.(iv) The second two terms have minus signs because they represent outgoing flux through the boundary. The outward normal at a boundary is always pointing away from the enclosed region, so the sign of the flux term has to be negative for it to represent outgoing flux.(v) Yes, the energy through the horizon E1−E2 is the difference between the energy of the field entering the region through Σ1 and the energy of the field leaving the region through Σ2.(vi) No, he is not missing a dv in the measure.(vii) \xi_\mu j^\mu = ( \xi \cdot \partial \Phi)( k \cdot D \Phi) follows from the definition of the flux vector j^μ. (viii) To obtain 4.67, you need to use the identities cos(A+B)=cos(A)cos(B)−sin(A)sin(B) and sin(A+B)=sin(A)cos(B)+cos(A)sin(B).(ix) Super-radiance can occur in any region where there is an ergoregion, i.e. a region of spacetime where the Killing vector field associated with time translation is timelike.
 

1. What is Super-Radiance homework?

Super-Radiance homework refers to a type of homework assignment typically given in physics or astronomy courses. It involves researching and analyzing data related to super-radiance, which is a phenomenon where a large number of atoms or molecules collectively emit light at a higher intensity than they would individually.

2. Why is Super-Radiance homework important?

Super-Radiance homework is important because it allows students to gain a better understanding of this fascinating phenomenon and its applications in various fields such as optics, quantum mechanics, and astrophysics. It also helps students develop critical thinking and research skills that are valuable in any scientific career.

3. How is Super-Radiance homework typically structured?

Super-Radiance homework assignments may vary, but they usually involve researching and analyzing data from experiments or simulations related to super-radiance. Students may also be required to write reports or essays discussing their findings and the relevance of super-radiance in different contexts.

4. What are some tips for completing Super-Radiance homework successfully?

To successfully complete Super-Radiance homework, it is important to carefully read and understand the assignment instructions. Students should also conduct thorough research and use reliable sources to gather data. It is also helpful to work on the assignment in a timely manner and seek assistance from professors or classmates if needed.

5. Can Super-Radiance homework be challenging?

Yes, Super-Radiance homework can be challenging as it requires a solid understanding of physics concepts and the ability to apply them to real-world scenarios. It may also involve complex mathematical calculations and data analysis. However, with proper preparation and effort, students can successfully complete Super-Radiance homework and gain valuable knowledge and skills in the process.

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