Since the big bang singularity has infinite density, gravity, and heat, am I correct to think that it has infinite energy? Since the black hole singularity also has infinite density, gravity, but has finite heat, does it have infinite energy as well? With regards to the black hole singularity, since it's gravity is infinite, and nothing can escape it, am I correct to think that even if my spaceship has infinite energy and thrust to escape the singularity, I cannot escape it no matter what? If this is the case, then, am I correct to think that the black hole has infinite energy?
General relativity does not have a definition of energy that applies to every spacetime. What it has is definitions of energy that apply to certain special cases, like stationary spacetimes and asymptotically flat spacetimes. Cosmological solutions are not stationary or asymptotically flat, and there is no known way to define the energy of a cosmological solution. We also don't know whether our universe is spatially infinite or spatially finite. Supposing that at some point in the future, somebody figures out a sensible way to define a conserved quantity of energy in GR in a way that applies to all spacetimes, then clearly it would have to be infinite in the case of a spatially infinite universe. No. The spacetime surrounding a black hole is asymptotically flat, and in that special case there is a definition of mass-energy that applies. According to that definition of mass-energy, the mass-energy of the black hole is finite. At large distances from the black hole, its gravitational effects are the same as those you would normally expect according to Newton's law of gravity from that amount of mass. Yes. No.
Thank you for your reply. In terms of heat energy, am I correct to think that since the big bang singularity has infinite heat, therefore, it's energy in terms of heat is infinite?
"Since the big bang singularity has infinite density, gravity, and heat, am I correct to think that it has infinite energy?" It doesn't necessarily have infinite anything..... Nobody knows what happened AT the big bang singularity.....nor what happens at the center of a black hole. Neither general relativity nor quantum mechanics covers those situations..they are unknowns. When people say "infinite" at the big bang, or at a black hole singularity, they typically mean 'unbounded' or 'divergent'...analogous to the description 1/x, as x approaches zero is "infinite".....nobody has a generally agree upon mathematical description of such singularities. But if the things you mention were infinite, it's reasonable to assume there was infinite energy as well....but that does not make it accurate, just a reasonable guess. Other models of the universe, less popular so far, posit a cyclic (repeating) universe, "ekpyrotic"...which does NOT require infinite anything. Neil Turok and Paul Steinhardt have such a model: http://en.wikipedia.org/wiki/Ekpyrotic_universe
Mathematics does not have a problem with functions that have singularities. The big problem with singularities in a classical field theory is a physical one, not a mathematical one. The problem is that initial-value problems don't have well-defined solutions (Earman's "lost socks and green slime"). Before you could discuss whether it was infinite, you'd have to be able to define it. There is no known definition of scalar mass-energy that applies to cosmological spacetimes -- even if all you want to do is to find the amount of mass-energy within a region of finite volume.
"Mathematics does not have a problem with functions that have singularities." Ben, can you provide a source that has either a GR or a quantum solution statement/description for the big bang? If so, what is the order of appearance of the four fundamental forces? Or am I misunderestimating (lol) your comment? My understanding is that neither theory has a mathematical solution that applies to either type singularity, that is, each theory breaks down at such singularities. "The spacetime surrounding a black hole is asymptotically flat, and in that special case there is a definition of mass-energy that applies." You don't mean at the singularity, right? How can it be flat with gravity present? I thought I understood your comment from above but now I'm not so sure: "General relativity does not have a definition of energy that applies to every spacetime. What it has is definitions of energy that apply to certain special cases, like stationary spacetimes and asymptotically flat spacetimes. Cosmological solutions are not stationary or asymptotically flat, and there is no known way to define the energy of a cosmological solution."
In rereading my original post, I could have been even more explicit about "infinite anything": Nothing has so far been encountered by scientific measurement (observation) that has infinite value. Now I guess it's possible the big bang will be an exception, but a good deal of skepticism is required about that at this point as it would be a REALLY unique exception.
I think I'm under-understanding your question :-) My statement was about math, but your question seems to be about physics...? When people say "the theory breaks down," etc., those are informal descriptions or descriptions aimed at a popular audience. Note that I am not making any claims about quantum-mechanical theories, just classical field theories. Here are a couple of books that discuss in detail how all this applies to GR: John Earman, Bangs, crunches, whimpers, and shrieks: singularities and acausalities in relativistic spacetimes, Oxford, 1995 Hawking and Ellis, The large scale structure of space-time, Cambridge University Press, 1975 http://en.wikipedia.org/wiki/Asymptotic_flatness
How the Universe, black hole can have a density higher than the Planck density? How was crossed the Planck wall? Who calculated?
The OP only makes sense as a question about general relativity, not about a hypothetical theory of quantum gravity (which we don't have).
If the highest density inside a black hole is the Planck density: When the core of the black hole attains the Planck density the core will ‘evaporate’ trough a burst of gamma rays (quasar)! What do you think about it?
Most likely it can't....as you suggest. If you are asking about the general creation of quasars, Planck density is not required. Quasars and radio galaxies are believed to be powered by the same black hole engine....as of 1994, Kip Thorne's excellent BLACK HOLES AND TIME WARPS, also says "It is still possible to explain,,,quasars using...a spinning,magnetized supermassive star...."
Ben posts: You can get a further insight here: http://en.wikipedia.org/wiki/ADM_energy and note that the vacuum energy density is CONSTANT.....not infinite....so in a finite universe the energy is finite....