I am always confused which kind of article I should put in the following case:
This paper shows an/the inverse relationship between gravity and air density.
At first, "the inverse relationship" sounded correct to me but now I have no idea.
Any principle or rule of thumb behind this usage?
This is fairly basic (famous last words) but, as a scientist, I would differentiate between the two as follows:
This paper shows the inverse relationship between gravity and air density
Here what is implied is that it is already well known that an inverse relationship of this kind exists. Perhaps the formula is given in text books or a specific relationship has just been referred to. Hence one is writing about one particular relationship, so this requires the definite article when refering to its appearance in the paper. (It is perhaps difficult to see the context in which this would be used — more likely in a sentence such as “This paper shows how to derive the inverse relationship…”.)
This paper shows an inverse relationship between gravity and air density
Here the implication is that there is no common understanding of or consensus on whether there is a relationship between gravity and air density. If there is a relationship its nature is unknown. So there may be many different types of relationship possible (direct, inverse, logarithmic), or none. Hence the use of the indefinite article in the sentence.
So, the answer to the poster’s question depends on the context.