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Hi,
I'm trying to understand the argument between eqns 4.117 and 4.119 in this paper http://arxiv.org/abs/1501.06570 as to why the penrose process is not possible for a boosted black string.
I understand the basic idea is to show using an argument similar to 4.33 that [tex]|E| < |p_z|[/tex] (see 4.118) and then from the relation [tex]E^2=p^2+m^2[/tex] that [tex]|E| \geq |p_z|[/tex] thus obtaining a contradiction and showing that the negative energy particle (required for Penrose process) could never have existed in the first place.
My problem is with the derivation of 4.117 using an argument similar to 4.33. My problems are the following:
1, Why is there a minus sign in front of the integral in 4.32?
2, On the right hand side of 4.32, he has [tex]\delta E - \Omega_H \delta L[/tex]. I don't understand the signs here either - why is the first term positive and the second term negative?
3, This is then integrated to give [tex]E_H=E-\omega_H L[/tex] which is apparently positive since the locally measured energy must be positive. Why is this the case?
4, To me it seems that in the Kerr case (4.33) he arrives at [tex]E_H=E-\omega_H L[/tex] whilst in the linear case we should get [tex]E_H=E-v_H p_z[/tex]. Then what he does is note above 4.117 that [tex]v_H=-v[/tex] for an observer on the horizon with no linear momentum i.e. comoving. If this observer is comoving then surely they should have the same momentum as the horizon i.e. [tex]v_H=v[/tex]?
Can anyone explain this? It doesn't seem like particularly difficult calculation - just some missing intuition...
Thanks
I'm trying to understand the argument between eqns 4.117 and 4.119 in this paper http://arxiv.org/abs/1501.06570 as to why the penrose process is not possible for a boosted black string.
I understand the basic idea is to show using an argument similar to 4.33 that [tex]|E| < |p_z|[/tex] (see 4.118) and then from the relation [tex]E^2=p^2+m^2[/tex] that [tex]|E| \geq |p_z|[/tex] thus obtaining a contradiction and showing that the negative energy particle (required for Penrose process) could never have existed in the first place.
My problem is with the derivation of 4.117 using an argument similar to 4.33. My problems are the following:
1, Why is there a minus sign in front of the integral in 4.32?
2, On the right hand side of 4.32, he has [tex]\delta E - \Omega_H \delta L[/tex]. I don't understand the signs here either - why is the first term positive and the second term negative?
3, This is then integrated to give [tex]E_H=E-\omega_H L[/tex] which is apparently positive since the locally measured energy must be positive. Why is this the case?
4, To me it seems that in the Kerr case (4.33) he arrives at [tex]E_H=E-\omega_H L[/tex] whilst in the linear case we should get [tex]E_H=E-v_H p_z[/tex]. Then what he does is note above 4.117 that [tex]v_H=-v[/tex] for an observer on the horizon with no linear momentum i.e. comoving. If this observer is comoving then surely they should have the same momentum as the horizon i.e. [tex]v_H=v[/tex]?
Can anyone explain this? It doesn't seem like particularly difficult calculation - just some missing intuition...
Thanks