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I'm currently reading Carlo Rovelli's Seven Brief Lessons on Physics, and I find it quite fascinating. Of course, the book gives mostly an overview of modern physics without going too much into details or equations and the like, stressing more the journey through physics that we have undertaken in the last century.

However, in the book Rovelli explains that the studies of thermodynamics and statistical mechanics "were extended to electromagnetic and quantum phenomena. Extension to include the gravitational field, however, has proved problematic. How the gravitational field behaves when it heats up is still an unsolved problem." (pgs 57-58) He then describes how heat causes the electromagnetic waves in an electromagnetic field to vibrate, but adds that it is unknown as to how heat affects a gravitational field. Since gravity by Einstein's theory of general relativity is in essence the curvature of space-time, heating a gravitational field would also result in the heating of time. However, as Rovelli states, "what is a vibrating time?" (pg 58)

Later on in the book, however, Rovelli mentions that Stephen Hawking calculated that "black holes are always 'hot.' They emit heat like a stove." (pg 63) From my perspective, couldn't the heat emitted from the black hole affect its gravitational field? From my knowledge, black holes have an enormous gravitational field as to not allow light to escape its grasp, and time in a black hole essentially stops. It seems to me, then, that two things are at work here: the gravitational field of the black hole, and the heat of the black hole. Since heat can, apparently, affect a gravitational field (in a way that is currently unknown), is it possible that time as witnessed near a black hole (where it essentially stops) is a result of the black hole's gravity, of its heat, or both? Could it also add to our understanding of the way heat affects a gravitational field?

Note: my knowledge of black holes is fairly limited, as it is of modern physics, so criticism on my logic is requested as is a refined understanding pertaining to any misconception I may have.

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    $\begingroup$ I haven't read this book, but it seems that the concept of "heat" is not well defined in hour question. Can you clarify what physical observable you mean by "heat", especially at the quantum level? For example, due to the Pauli exclusion principle, the Fermi temperature of a dense fermionic matter can be astronomically high, but the heat temperature of the same object can be near absolute zero. $\endgroup$ – safesphere Feb 6 '18 at 7:07
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    $\begingroup$ What precisely is meant by "heat affecting a gravitational field" needs to be more precisely defined - I don't see any obvious interpretation. As we lack a quantum theory of gravity, I'm not sure we can make definitive statements about thermodynamics and gravity. Hawking radiation temperature would be minute for normal black holes - not hot, but colder than CMB. I don't see BH's as being "like a stove". $\endgroup$ – StephenG Feb 6 '18 at 8:20
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1. A cosmological black hole is a (charged), rotating black hole with an accretion disk where matter in the state of plasma emits thermal radiation, essentially electromagnetic radiation generated by the thermal motion of charged particles. Then the Hawking radiation, explained by quantum effects near the event horizon, adds further energy emission from the black hole. However this latter is negligible for massive or super massive black holes.

2. SR (special relativity) shows that energy and inertial mass are equivalent. The Einstein equivalence principles state that the inertial mass and gravitational mass of any object are equal (Weak equivalence principle (WEP)) and that all forms of nongravitational energy and momentum gravitate (Einstein equivalence principle (EEP)). I overlook the Strong equivalence principle (SEP) as not necessary in this argumentation.

As per 1. and 2., black hole mass, angular momentum (and charge), thermal radiation from the accretion disk and Hawking radiation from near the horizon, all of them gravitate, meaning that they shape the geometry of the surrounding spacetime. If the conditions are stationary the metric tensor is time independent, otherwise it may be characterized by vibrations. Consequently time, as measured by a far away observer, is affected. However the proper time, e.g. the time measured by a clock on board of an audacious spaceship approaching or even crossing the horizon does not stop. As for the spaceship, apart from the tidal force, the horizon is not a special region of spacetime.

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One thing you (or may be Rovelli) get wrong in this situation is the "perceived" part. Time, as perceived near the black hole never stops. You can move as close to a horizon as you like and never notice anything strange going on with time (i'm not taking into account your height and width and consider you to be point-like). Time will only slow from the point of view of some faraway observer. Looking at you, he will see that your clocks (and your heartbeat and your breath yadda yadda) becoming gradually slower as you get closer to the horizon.

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    $\begingroup$ Good point but this doesn't completely answer the question. This question requires a much more detailed answer I think - just because of the effort s/he put in coming up with this interesting question. $\endgroup$ – LostCause Feb 6 '18 at 7:04

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