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Matter cannot travel at velocity of light, certainly cannot travel faster of light. Inertial mass become infinite at velocity of light, at speed greater of light mass become imaginary, involve square root of minus one (i), cannot predict what happen. Cannot accelerate infinite mass, all energy of universe is finite, cannot accelerate it. Time passage slower at higher velocities, become zero at velocity of light. At speed greater of light time become imaginary have "i". Length at direction of travel become zero at velocity of light, become imaginary at velocity greater of light. You not comprehend Theory of Relativity.

The issue has nothing to do with mass increase, since it is "relativistic mass" (a rather unfortunate recently coined term that refers to momentum, but not mass as we normally describe it) that increases, not normal, every day rest mass. The problem lies in the inherent property of light - that it moves at a constant speed, regardless of the motion of the source or the observer. This means that if you set off toward a star - moving at .99 x the speed of light relative to that star - the light from the star would still pass you at the speed of light. If you are moving away from a star, the same thing happens. No matter how fast you are moving, the light will still pass you at the speed of light - relative to you. You cannot catch those photons. In a sense, you are no closer to actually reaching the speed of light in a high speed spacecraft than you are while sitting on your porch watching the sunset. In each case, the photons will pass you at the same speed, so it really makes little difference with respect to your motion or lack of motion.

This is not terribly intuitive or obvious, but it is nonetheless accurate, as demonstrated daily in laboratories and accelerators around the world. It is not a theory; it is just the way that light works. This is what Einstein started with, and it was fairly significant, when he released his work about 100 years ago..

No, it isn't possible, and you have given the answer yourself, indirectly. as you say, mass increases as speed increases, to infinity at v = c. Since mass is infinite, it takes an infinite amount of energy to produce any acceleration, and energy is not infinite. It is theoretically possible to acccelerate an object asymptotically towards c, but not to reach it. Photons travel at c because they are massless, and their mass can increase infinitely without becoming non-zero. (That isn't mathematically necessarily the case, but that's how it is in this case.)

There is another indication that time travel is nt just impossible now, but always will be (at least in the sense of science fiction stories). If it is some day going to be possible, where are the time tourists? Shouldn't the streets be full of them?

The problem is in your question:

How do you push an infinite mass any faster?

No matter how big an engine you have, or how much fuel you can burn... you don't have an infinite power to push an infinite mass any *closer* to the speed of light.

A photon can travel at the speed of light because a photon has no mass.However even a photon cant travel faster then the speed of light.Remember should you be able to travel at the speed of light you would be in all places at the same time.Now we get back to the same problem since you are not whitout mass,forget it.

with our current technology we dont even achieve 1% of the speed of light.

And time travel to the past is impossible, to the future, "technically" is possible, since if you travel fast enough time will slow down and lets say traveling 5 years would be like 50 years in earth.

You have included two complex subjects , time travel and light velocity. My book will expalin all of this in great detail.

1... it is possible to travel faster than light, and

2... it is possible to travel forward in time... not backward however.

I'm afraid that Einstein was an incomplete thinker. In fact he avoided quantum physics , saying it was too spooky. However quantum physics holds the key for much that we do not know !

You can't go back to the past. What happened HAPPENED, it's not alive in another dimension. It's gone.