To make it as simple as possible: In the diagram, the wiggly line going bottom to top is inside the Black hole, and could be a singularity, but that's not important. The two curved lines represent the horizon envelopping the BH: crossing that line is a point of no return. The curved line with the letters a and b is on a Rindler wedge ( I've used Rindler coordinates in my blog You are not allowed to view links.
Register or Login, equation 6). And R is Hawking radiation at an earlier time.

First, I would like to congratulate you for your having mastered so much math and science. I would like to have had the capacity to achieve such mastery, but I don't have that capacity. Never-the-less, I do have a fair amount of curiosity about how things work. I have read through Brian Greene's books that are popularizations of current microphysics and macrophysics. I understand that such popularizations suffer from things like analogies that imply ontological status claims that are not supported by the math (or at least the math implies ambiguity about ontological status), so I always am cautious of taking such popularizations too literally. I try to take them as giving me a rough approximation of the state of ideas in physics as seen by current physicists.

One thing that I've tried to grasp in at least a rudimentary way, is what is meant by time in General Relativity and what is meant by time in quantum mechanics. I kind of get what time is taken to be in General Relativity - time is what is measured by a clock in an inertial frame. That is probably not completely right and may be a total misconception on my part. If that is basically what time is in GR, though, then I must admit that it doesn't actually help me understand what time actually is - especially with respect to what is probably suspectly analogously referred to as the "fabric" of space-time. With respect to quantum mechanics, I feel like I understand that even less (if that is possible). I have gotten the impression from some readings that I have done, that time is taken to be a kind of "absolute" - that is, that there is a fixed simultaneity throughout the universe. But that doesn't seem like it can be right because that seems to contradict the notion of time in GR.

I have done multiple Google searches with many different search parameters, but I haven't found any web site that answers my question about how the conceptions of time in GR and quantum mechanics compare.* And I'm curious that if they do have different conceptualizations of time, what implications follow from those differences - say with respect to trying to "unify" the two.

Anyway, when I read the passage of yours that I quoted above, it occurred to me that if there's one place where the nature of time in quantum mechanics and GR would have to be manifest the same way, it seems like it would be a singularity. If you are so inclined, I would appreciate any help you could give me in assuaging my curiosity about conceptions of time. A link to a site that would be likely to have the answers to my questions would be very helpful if you happen to know of any.

*I did run into this web page that I found pretty baffling but understood enough of it to be really intrigued:

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You get can get really bogged down with the notion of time. Just recently Lee Smolin published his You are not allowed to view links.

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Login. I must confess that I didn't read it, but this is just to show that physicists are still grappling with the notion of time.

Now leaving all philosophical musings aside, in Relativity, time is simply what you would measure on a clock, which is made of matter, with something inside ticking with a regular beat. So when we look at time dilation, it means for a moving observer, his clock would slow down. Ditto for an observer moving into a greater gravitational field. You can debate ad infinitum whether this is real or apparent, but we do know that for communication satellites involved in GPS, their clocks have to take into account these two effects, otherwise they will get out of synchronization withing minutes.

Now in QFT, the notion of time only rears its ugly head when we look at commutation relationship. For instance, the position q and its conjugate momentum p, follows this simple rule:

[x,p] = i ?, where the square brackets means xp - px, the reason being is that these are operators in QM and don't commute.

These commutator relationship must be taken at equal time interval in order not to violate causality. This is taken care by defining 4-vectors x = (t,x,y,y,z) and p = (p[sub:2yo9xepr]t[/sub:2yo9xepr],p[sub:2yo9xepr]x[/sub:2yo9xepr],p[sub:2yo9xepr]y[/sub:2yo9xepr],p[sub:2yo9xepr]z[/sub:2yo9xepr]). By making everything 4-d tensors, QFT becomes Lorentz invariant, and SR is immediately incorporated into QFT. So a good deal of time in QFT is to make sure that such things as the Lagrangian and the Hamiltonian functions are Lorentz invariant. That way, time is effectively treated on equal footing with space, as Relativity demands.