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What is a White Hole?
There are a variety of ways to explain a white hole. Generally speaking, they could be described as a theoretical exit for all the matter sucked into a black hole; or even more generally as an event where matter is being spewed outward at one end. This exit would either be shooting the matter into another part of our universe or possibly shooting matter into another universe. --Whereas a black hole looks somewhat like a reverse or inverted water fountain, with the water falling into some hole, the white hole would resemble the typical type of water fountain we are used to seeing, with the water being shot out into the air.--So there is matter entering something similar in structure to a worm hole (or an Einstein-Rosen bridge) via a black hole mouth, and then what comes in is projected out through a white hole on the other end of the worm hole. However, there are a few issues relating to the passage of matter through a black hole to be delved into in the second section.
Another way of looking at a white hole is as part of a geodesically-complete system. What this means generally is that the equations which allow for black holes to exist, also require the existence of white holes elsewhere in existence to complete the geodesic map. And thus, we get the idea of white holes being described as "time-reverse" black holes: White holes are what black holes become when you reverse the direction of time in the same equations; black holes absorbing matter become white holes emitting matter. This is all part of geodesic completeness: In order for a model to be "complete" it must also include the negative part of the scale.
It is also worth mentioning here that there is some speculation out there that our universe is the product of a white hole. Some suppose that the Big Bang was actually the explosion of a white hole. There is some thought out there that white holes are a possible point of origin for universes in a multiverse set up for the universe.
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How Plausible Is the Existence of White Holes?
Although, when talking about white holes and their plausibility, some are quick to harp on the fact that black holes were once thought to be outlandish, and they then apply this example to the white hole case to keep our minds open to the possibility of white holes being existent--and it is usually good to keep our minds open--there is currently not much evidence to support the existence of white holes nor to provide much hope for their actuality. It's definitely not impossible, of course. As for them being the other side of a blackhole, one major issue arises: The singularity that caused the black hole still remains within the black hole, blocking the passage way; so anything that enters the black hole is "stacked" upon that singularity. So unless over some grandiose time scales there occurs some reaction within the black hole which allows for the exit of the matter, possibly some quantum processes, it seems as if black holes don't allow for an exit. However, to keep some balance here, I would say that it's fair to say our understanding of the universe is almost laughably and helplessly jejune in terms of our ability to formulate both the physical dynamics and areas of the universe. So, it's not fair to rule much of any theory out at this point.
However, just to mention here while we are discussing plausibility, there is the presupposition of a multiverse in much of white hole theory. Which of course isn't ridiculous, but it definitely should be mentioned. Also, it appears that as far as we have calculated thus far, white holes appear to violate the second law of thermodynamics. And, their very existence as the time reverse of a black hole in under scrutiny, due to the argument that black holes achieve thermal equilibrium by emitting Hawking radiation and that in a state of thermal equilibrium the time reverse of a black is again a black hole, because the time reverse is invariant.
So for now, the white hole remains a diverse theory and an equation.