So all this talk of the fourth dimension and gateways to 600BC has got me thinking we should discuss a few of the standard objections to time travel that have less to do with physics and more to do with logic. Here are a few I've come across as well as a couple I've thought of myself.
1. Grandfather Paradox. Pretty much everyone has heard of this or some variation of it. You go back in time to kill your baby grandpa which results in your not being born to kill him which results in your being born to go back in time to kill him. Alternatively, you go back in time to "murder" yourself.
2. The universe will allow only some kinds of time travel; for example, the kind that doesn't allow you to create a grandfather paradox.
3. By time traveling, you would be putting matter into the universe that wasn't there before and by doing so, break the law of conservation of energy. This is not possible therefore neither is time travel.
4. The future does not exist since it hasn't happened yet. Therefore there is no place to travel to so time travel into the future isn't possible.
5. Einstein theorized and it has been proven that gravity and speed cause time dilation, but can we say that it doesn't affect time with a capital T, but rather just processes that occur within time?
6. Applying the classic Zeno paradox of halving distances in space to time. That which is in locomotion must arrive at the half-way stage before it arrives at the goal.
7. If time travel was ever invented, then, in an infinite universe, there would be an infinite number of time travel trips that would fill up every instant in the universe. Since this has not happened yet, it will not happen, regardless of the possibility of time travel.
So respond to any and all of these and throw out some more!