##E= -T^uP_u ##,(adsbygoogle = window.adsbygoogle || []).push({});

where ##T^u## is the time-like killing vector associated with the Kerr Metric and ##P_u## is the 4-momentum of the particle. ##E## is the energy.

Outside the ergosphere ##T^u## is time-like and inside the ergosphere it is space-like. Therefore it can be arranged within the ergosphere that ##E= -T^uP_u <0. ## *

The condition for which this can be arranged is described so far as :particle's motion must be against the angular momentum of the black hole..

My Question:

I looked at a quick proof justifying that assuming negative energy of a particular particle within the ergosphere, ##E'##, its angular momentum ##L'## is negative (against the black hole's rotation) using the fact that the particle crosses the event horizon and by defining a new killing vector - linear combination of the time-like and angular killing vector, with the angular velocity of the event horizon -##\omega## - being a multiplicative constant:

##L'< \frac {E'}{\omega} ##

Question:

But, if ##L'## is negative, we can make no conclusion on the sign of ##E'##.So my interpretation so far is that the particles motion to be opposing the angular momentum of the black hole is a necessary but not sufficient condition for the particle to have negative energy.

My question then is,must we impose so that * is space-like - and what do they physically correspond to ? I.e- I see that the scalar product of a TL vector with a SL can be null, SL or TL- how do you formulate the relationship / relationships between the components of the vectors/vectors forming the scalar product so that we know what the resulting nature will be andwhat further conditions, if this is formulated for * does this give the correspondingphysical conditions that must additionally be imposed so that the particle's energy is negative?

(I've had a google around but can't find anything).

If anyone could help me ut or lead me to some helpful resources that would be extremely appreciated ! Thank you !!!

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# Black hole, Penrose Process, SL, TL-Vector quick question

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