This question was posed in an Astronomy Cast episode a while back. It offers an interesting thought experiment, although a reasonably definitive answer to the question can be arrived at.
Imagine a scenario where a spacecraft gains relativistic mass as it approaches the speed of light, while at the same time its volume is reduced via relativistic length contraction. If these changes can continue towards infinite values (which they can) – it seems you have the perfect recipe for a black hole.
Of course, the key word here is relativistic. Back on Earth, it can appear that a spacecraft which is approaching the speed of light, is indeed both gaining mass and shrinking in volume. Also, light from the spacecraft will become increasingly red-shifted – potentially into almost-blackness. This can be partly Doppler effect for a receding spacecraft, but is also partly a time dilation effect where the sub-atomic particles of the spacecraft seem to oscillate slower and hence emit light at lower frequencies.
So, back on Earth, ongoing measurements may indicate the spacecraft is becoming more massive, more dense and much darker as its velocity increases.
But of course, that’s just back on Earth. If we sent out two such spacecraft flying in formation – they could look across at each other and see that everything was quite normal. The captain might call a red alert when they look back towards Earth and see that it is starting to turn into a black hole – but hopefully the future captains of our starships will have enough knowledge of relativistic physics not to be too concerned.
So, one answer to the Astronomy Cast question is that yes, a very fast spacecraft can appear to be almost indistinguishable from a black hole – from a particular frame (or frames) of reference.
But it’s never really a black hole.
Special relativity allows you to calculate transformations from your proper mass (as well as proper length, proper volume, proper density etc) as your relative velocity changes. So, it is certainly possible to find a point of reference from which your relativistic mass (length, volume, density etc) will seem to mimic the parameters of a black hole.
But a real black hole is a different story. Its proper mass and other parameters are already those of a black hole – indeed you won’t be able to find a point of reference where they aren’t.
A real black hole is a real black hole – from any frame of reference.
(I must acknowledge my Dad – Professor Graham Nerlich, Emeritus Professor of Philosophy, University of Adelaide and author of The Shape of Space, for assistance in putting this together).