I am a science fiction writer.
I know how to time travel to the future (using black hole or speed). But I need to argue in my novel that time travel to the past might not be possible. Is there a way to do this without using a formula? And how basic would argument look like ?
Time travel is difficult to prove or reject, even by using a formula (at this time there is no decisive proof for or against it). There are solutions of Einstein's equations which allow closed timelike curves, like Gödel's universe, Kerr black holes, wormholes, etc.
As mentioned in other answers, chronology protection conjecture, grandfather paradox and other causality paradoxes, and the absence of visitors from the future are the most used arguments against time travel. Each of them can be avoided, by creating alternative universes in which you don't exist because you killed your grandfather, or by protecting the past using Novikov's self-consistency principle, or a time patrol or similar solutions, etc.
Also, there are arguments against particular time travel methods. For instance, that you can't use wormholes, because they will become singular or/and self-destroy. There are theorems suggesting that in reasonable conditions closed timelike curves are accompanied by singularities, so if there are no singularities, there is no time travel.
Alternatively, you can make the characters discover fictional laws forbidding time travel.
I am very curious why a science fiction writer would want to reject time travel.
Update
If you are interested in various aspects of time travel (involving formulae), check this conference, The Time Machine Factory, 2012. The proceedings just appeared and are available online.
Update 2
The introduction of this paper by Seth Lloyd et. al. contains a brief review of many results concerning time travel, and references.
Well, the most trivial one is "Where the hell are all the time travellers from the future today?"
One additional nice feature to consider in rejecting time travel is global entropy of the universe. As far as it is known today, in the past universe had lower entropy and it is impossible to reduce the entropy of the system without dumping the "excess entropy" (plus, whatever additional entropy created in the process) elsewhere.
The same consideration applies to why we can not precisely reconstruct the past (that is, perfect historical knowledge is not possible, no matter how hard we try). Entropy will always prevail and eat some of the precious information.
The most basic argument I can think of is the Chronology protection conjecture. In essence, any time machine allows a quantum feedback cycle that will result in the immediate destruction of the machine.
One theory states that time travel requires cooperation from both sides, which explains why no time travelers have been seen yet. You can only travel as far back as the invention of the first time machine. And even then, it would be restricted to places and times where a "receiver" was set up.
That's of course very convenient for SF writers; it gives them a degree of freedom in deciding precisely where and when time travel fits the plot.
Have a look at http://en.wikipedia.org/wiki/Grandfather_paradox. I think you can find a nice argument there. You can use the argument that past is unchangeable so it is pointless to go to the past if it can't be changed. Or you can use the parallel universe argument, meaning that somebody who goes in the past and alter something there ends up in a parallel universe where the effect of his alteration creates a completely different present than the present he left from. The parallel universe is different than ours where he can't return to ours.
For clarity, those are NOT scientific arguments. Those are arguments which were made to solve the Grandfather paradox.
