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It is possible to figure out how time duration and space distance change when we are close to the speed of light - but I am not sure what happens to them if we travel in the speed of light.

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Hi time. Welcome to Phys.SE. This post asks hypothetical questions outside accepted standard physics theories, such as, e.g., asking about a massive object traveling at the speed of light. Fictional questions are off-topic, cf. faq. If you can modify your question, so it stays within the realm of standard physics, I would be happy to reopen it. –  Qmechanic Feb 27 '13 at 20:19
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closed as off topic by Qmechanic Feb 27 '13 at 20:21

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It is not possible for a massive object to move at the speed of light

see Accelerating particles to the speed of light for example

so the premise of your question is itself problematic. When we usually talk about time dilation or length contraction in a moving frame, we are referring to an observer measuring time or length in such a frame, and since it is not possible for an observer to move at the speed of light, it's unclear what the question means in the first place.

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