Do white holes exist and if they do can we see them?
Thanks for the question.
For clarity, I'm assuming that you're taking the definition of a white hole to be the one given here: http://en.wikipedia.org/wiki/White_hole
It was long thought that the laws of physics should be the same whether time is running forward or backward. That is, say you make a movie of yourself gently tossing a ball up in the air and then catching it. Now play the movie backwards. Does one look different from the other? Not really—the laws of physics in this case are the same whether time is running forward or backward, so you can't easily tell which movie is which.
Now, there's good evidence that black holes exist, so people thought that there should be a “time reversal” of a black hole. This object would spit out matter into our universe. That's the theoretical justification for putting forward the idea of white holes.
However, white holes have never been observed. Furthermore, there's now good reason to think that the fundamental laws of physics are not the same whether time is running forward or backward. Therefore the idea of white holes loses some of its theoretical justification.
For more about the whether the laws of physics are the same whether time is running forward or backward, check out the second law of thermodynamics and something called CP violation. CP violation is very interesting but unfortunately a very technical subject. I encourage you to do some reading about it but I can't go into it now.