Black Holes Formalism: Overview of Misner Sharp Mass

[AHSAN MUJTABA]

TL;DR Summary

I want to know by just an introduction along with some example to visualize, that what is Misner Sharp mass?

I am currently studying the Vaidya metrics and I am a bit confused regarding the term Misner Sharp mass, and I am referring to Blau notes and in them the description is very deep which I surely don't need but just an overview. TIA

 

[PeterDonis]

I am referring to Blau notes

Can you give a specific reference?

 

[strangerep]

@PeterDonis : He's talking about Matthias Blau's GR Lecture Notes. Blau is head of the Gravity and String Theory Group within Bern University. I began using Blau notes some years ago and they have gradually become one of my top 3 go-to sources when I want to check something, or explore some aspect more deeply. I hope Blau eventually finds a publisher for these notes.

@AHSAN MUJTABA : I'm not sure why you're finding this so difficult. The Misner-Sharp mass is essentially just the "M" term which arises in the Schwarzschild solution, but becomes $M(t,r)$ in the more general case of a Vaidya metric if one has a spherically symmetric $T_{\mu\nu}$.

Can you give us more of a clue about what parts of Blau's treatment you find unclear? You said you were studying Vaidya metrics, but did you mean you're studying them via Blau's notes or are you also using another textbook? If so, which one(s)?

BTW, I think there might be a typo in Blau's eq(35.116) on p804, compared to a related earlier eq(24.82). Could someone else please check?

 

[PeterDonis]

He's talking about Matthias Blau's GR Lecture Notes.

Thanks for the reference! I have looked at these before, and I agree with you that they are an excellent source.

I think there might be a typo in Blau's eq(35.116) on p804, compared to a related earlier eq(24.82).

Do you mean you think there ought to be a definition symbol in between $M_{MS} (t, r)$ and $m(t, r)$ in eq. (35.116)?

 

[strangerep]

Do you mean you think there ought to be a definition symbol in between $M_{MS} (t, r)$ and $m(t, r)$ in eq. (35.116)?

Yes, or rather, an equivalence sign "$\equiv$" as in his eq(24.82).

 

[AHSAN MUJTABA]

Thanks dear for your reply I am actually reading section 40.2 "interpretations of mass functions in Vaidya metric" In the part of Vaidya Solutions

Thanks for the reference! I have looked at these before, and I agree with you that they are an excellent source.
Do you mean you think there ought to be a definition symbol in between $M_{MS} (t, r)$ and $m(t, r)$ in eq. (35.116)?

 

[AHSAN MUJTABA]

I am looking Blau and I need to present the topic with a fine background of general relativity(not too deep) as the audience will also be including undergrads and unfortunately I can't find much material for Vaidya solutions

 

[AHSAN MUJTABA]

There is a general question, when we say outgoing light rays define the evaporating black holes how can we reconcile it with the concept that no light can escape from the black hole?

 

[AHSAN MUJTABA]

I want other material as well to study the Vaidya solutions

 

[PeterDonis]

when we say outgoing light rays define the evaporating black holes how can we reconcile it with the concept that no light can escape from the black hole?

Outgoing light rays define the event horizon of a black hole. That is also true of an evaporating hole, but which particular outgoing light rays are the ones that define the event horizon is different for an evaporating hole than for a non-evaporating hole. The outgoing rays that would have been precisely on the horizon for a non-evaporating hole of a given mass are actually slightly outside the horizon for an evaporating hole of the same mass, so those light rays can escape. So the radiation that is "escaping" from an evaporating hole is not coming from on or inside the actual event horizon of the evaporating hole; it's coming from just outside that horizon.

 

[AHSAN MUJTABA]

outgoing means that oour event horizon is dynamical?

 

[PeterDonis]

outgoing means that oour event horizon is dynamical?

"Evaporating" means that the area of the horizon slowly decreases over time.

The area of a hole's horizon can also increase if something falls into it.

 

[AHSAN MUJTABA]

Outgoing light rays define the event horizon of a black hole. That is also true of an evaporating hole, but which particular outgoing light rays are the ones that define the event horizon is different for an evaporating hole than for a non-evaporating hole. The outgoing rays that would have been precisely on the horizon for a non-evaporating hole of a given mass are actually slightly outside the horizon for an evaporating hole of the same mass, so those light rays can escape. So the radiation that is "escaping" from an evaporating hole is not coming from on or inside the actual event horizon of the evaporating hole; it's coming from just outside that horizon.

you mean the light cones of an evaporating horizon at the horizon are tilted outside or some part of them is allowing the outgoing light rays to escape?

 

[PeterDonis]

the light cones of an evaporating horizon at the horizon are tilted outside

No, not at the horizon. Just outside the horizon.