Question about event horizon and Rindler space

[off-diagonal]

I've got two questions here

1a What is the definition of an event horizon of black hole? In Carroll, he defines an event horizon as a boundary of causal past of future null infinitiy. What is the physical interpretation of that?

1b What is the difference between event horizon and Killing horizon? In static asymptotically flat, it seems to be the same thing right?

2 Is there any relevant between Rindler metric and surface gravity? I've seen that if the metric near the horizon can be approximated as a Rindler metric, then we get the entropy of the horizon. What does this mean?

 

[LAHLH]

1a) The event horizon can be thought of as the hypersurface that separates the set of points connected to timelike (null) infinity by timelike (null) trajectories, from those points that cannot reach timelike (null) infinity by timelike (null) lines. Take the example of Kruskal: http://tinyurl.com/63p9wzf . r=2GM defines the event horizon here, and separates the set of point that can reach timelike(null) infinity on timelike (null) trajectories: the right hand wedge, from those that can't: the interior . If you take any particle in the interior even if it were to travel at the speed of light (45 degree lines) it wouldn't make it to infinity, it would just run into the singularity at r=0.

Another way to think about it that I think Carroll mentions, is that beyond r=2GM, the r coordinate becomes timelike (because the coeff in front of dr^2 in the Schw metric flips sign when r<2GM) this means that a particle necessarily has to keep going in the decreasing r direction once then cross over the event horizon from outside. They can no more change direction in r than an exterior observer can change direction in time! The r=const lines go from being hyperbolic vertical type lines, to hyperbolic horizontal lines, that any future directed observer simply can't help but cross.

1b) A Killing Horizon (KH) is a surface (hypersurface) on which a particular KV say $$\xi$$ is null, i.e. $$\xi^{a}\xi_{a}=0$$ defines a KH for some KV $$\xi$$. KH are distinct things from EH, but in spacetimes with time translational invariance they are related. Carroll spells out exactly when the two definitions coincide in section 6.3, but essentially if a you have a stationary asymp flat spacetime, then every EH will be a KH for some KV field $$\chi$$, i.e. if your spacetime is stationary and asymp flat you can find some KV $$\chi$$ for which $$\chi^{a}\chi_{a} =0$$ holds on the EH.
If the spacetime is static this KV you find that has this property of making the KH coincide with the EH will be $$\partial_t$$ i.e. the KV representing time translations far away at infinity. (This is the case for Schwarzschild spacetime). If the spacetime is stationary not static, the KH for $$\partial_t$$ will no longer coincide with the EH (infact it forms a surface known as the ergosurface, see Carroll ch6 for more on this) that only coincides with the EH at the poles. But never the less in this case you can find a KV that does have a KH that coincides with EH, it will be a linear combo $$K^{\mu}+\Omega_H R^{\mu}$$ (again see Carroll), this is the situation for the Kerr spacetime of a rotating BH.

2) The surface gravity of the Rindler metric is $$\kappa=a$$ where a is the proper scalar acceleration, although there is no gravity, it characterizes the acceleration of the observer. There is an analogy between Rindler spacetime and Kruskal, the null line x=t plays the role of the r=2GM EH of Kruskal and is known as the rindler Horizon. The analogue of the Killing vector $$\partial_t$$ now is $$\partial_{\eta}=a(x\partial_t+t\partial_x )$$ i.e. the boost KV and once can check that x=t is a KH for this vector. The analogy is then that r=const observers in Kruskal are related observer of constant acceleration in Rindler.

Not sure if that answers your questions.

 

[off-diagonal]

Thank, LAHLH that's a great answer

but, I'm still doubt about the boundary of causal past of future null infinitiy which Carroll said in his book (ch6). What is its physical meaning? He said it is an equivalent definition of an event horizon.

One more question, if we begin with the metric. How can we define its horizon?
It's not always trivial as set $$g^{rr}=0$$
right?

 

[LAHLH]

Thank, LAHLH that's a great answer

but, I'm still doubt about the boundary of causal past of future null infinitiy which Carroll said in his book (ch6). What is its physical meaning? He said it is an equivalent definition of an event horizon.

hmm, I don't know how to say it any differently really, I'm not sure exactly what more you're after with 'physical meaning' of event horizon. You can't tell when you cross over the horizon so it has no physical presence in that sense. If you refer to fig 6.1 in Carroll all he is saying is the future event horizon is the boundary of the past of future null infinity, i.e. anything that is to the left of the future horizon in that picture is not in the past of scri (if you draw the lightcone of a paricle left of the horizon nothing within it (or even at its boundary) reaches scri plus. (remember in conformal diagrams light cones remain at 45 degrees). Another way: start at scri plus and trace back all the points in the spacetime that could reach it on lines at 45 degrees or less (from vertical), you should find that all such points are to the right of the horizon, this is the past of scri plus.
Hence the horizon is the boundary of the past.

The only physical meaning to take from this is that if you are to the left of the horizon, you are causally disconnected from future timelike/null infinity, you are bound for the singularity, you are trapped behind the horizon...

One more question, if we begin with the metric. How can we define its horizon?
It's not always trivial as set $$g^{rr}=0$$
right?

Yeah, Carroll discusses this I recall and he talks about the need to choose clever coordinates for this to apply. I'm not sure if there is a general prescription for finding the horizon other than this, or indeed how one chooses clever coordinates, so I would be interested in hearing the answer to this too. The only way I could think of would be just to draw the conformal diagrams and locate it that way.

 

[off-diagonal]

oh thank you very much. now I think I understand it more clearer.

Could you suggest me books or articles where the Penrose diagram is discussed in very neat detail?

 

[LAHLH]

Could you suggest me books or articles where the Penrose diagram is discussed in very neat detail?

The appendix in Carroll is actually quite good, I found that the most accessible so far. I think Wald also covers them but probably in a more complicated way...