Do black holes really exist?

You're talking about material that exists within time and space. A black hole could be a tear in the fabric of time and space from collapsed stars. Tearing that fabric opens up a passage between dimensions, you need to be able to conceive a plane of existence that is not framed by time and space.

OK, let's not confuse theoretical mathematical models with reality.

There is a static solution to the Einstein field equations that has a gravitational singularity in the center, and vacuum everywhere else. This is usually called the "black hole" solution. But, can any arrangement of matter actually get into that configuration? The answer is: no, and for the reasons you describe in your question. Does this make any difference? The answer is again: no (mostly), because a collapsing star, outside the event horizon, behaves so similarly to a theoretical black hole that the practical differences are negligable. Admittedly, there are always some tiny differences between the gravitational field of a collapsing star and a theoretical black hole, but those differences decay rapidly (in a matter of a few light crossing times of the event horizon, which is to say a few seconds). A collapsing star, after a very short time, looks and behaves exactly like a theoretical black hole. This is true even if you enter the black hole to check it out inside, and get stuck to the singularity. So the theoretical solution for a black hole is actually quite useful for calculating almost anything you might want to know about a collapsing star.

Black holes absolutely positively without question do exist as proven by science. The theory in which you state by Einstein may have been misunderstood. The event horizon is the black holes point of no return and once an object moves into that boundary it vanishes to never return. We don't know all the facts about black holes and they may even defy laws of physics so its hard to state any of the relativity theory as more then theory and to be fact. The facts are that black holes exist and are still in the first stages of research and there is much more to learn.

http://www.space.com/scienceastronomy/060109_event...

Since time is relative and not absolute, it is admittedly difficult to think of things existing within time frames that are different from ours, or perhaps even non existent relative to us. Distant galaxies, however, are experiencing time differently than we do; that is a result of math similar to what you have described, but based on relative motion rather than mass. The fact that they are clearly on a different time scale than we experience does not make them any less real within their own reference frame, despite the fact that we cannot observe that reality due to the limitations of distance and the speed of light.

In the case of black holes, it is true that we cannot observe the reality beyond the event horizon, any more than we can observe the "current" reality within distant galaxies. But it is equally difficult to argue with the evidence that they do appear to exist at the center of most galaxies. It is not clear to me what else could be occurring, when giant stars in the region are suddenly wrenched from their trajectory by unseen objects that in some cases emit synchrotron radiation in the process. The evidence is clear if incomplete.

Wiki says:

"Astronomers have identified numerous stellar black hole candidates, and have also found evidence of supermassive black holes at the center of galaxies. After observing the motion of nearby stars for 16 years, in 2008 astronomers found compelling evidence that a supermassive black hole of more than 4 million solar masses is located near the Sagittarius A* region in the center of our own Milky Way galaxy."

So yes, they do excist.

We know that black holes exist. We know how they are born, where they occur, and why they exist in different sizes. We even know what would happen if you fell into one. Our discoveries have revealed one of the strangest objects in the Universe, and there's still much we don't know.

http://hubblesite.org/explore_astronomy/black_hole...

there are 10 million of them in our own galaxy

but the chance of earth to die due to that is close to 0

Black holes are mathematical and elegant theoretical entities,but practically they cannot exist.