The notes and calculations made by Albert Einstein sold for more than $13 million.
The manuscript was expected to sell for between $2.25m and $3.38m.
"This is the most valuable Einstein manuscript ever to come to auction," Christie's said in a statement before the sale.
The buyer's identity has not been revealed.
The 54-page document, which Einstein co-authored with his friend and colleague Michele Besso, is one of just two papers that show the theoretical physicist's early workings on his seminal theory of relativity.
The most important ideas in modern physics are still considered to be the theory published in 1915.
The fact that the manuscript survived and made its way to us already makes it extraordinary, said an expert at Christie's.
The pages were saved by Michele Besso.
The Einstein-Besso manuscript will be auctioned at Christie's in Paris.
There is a story about this.
Einstein and Besso wrote their notes in black ink.
According to Christie's, 26 of the pages had been written by Einstein, and 25 of them were written by Besso.
The manuscript isn't bound, and there are many different types of loose paper, so you get the impression of a working document that's full of energy, as if both men would grab the first page they could find to scribble their findings on.
The scientist making the initial calculations that would form the foundation for his theory made some mistakes along the way.
"Einstein makes errors in this manuscript, and that I think makes it even greater in a way, because we see the persistence, the thought that was in the process of being built, that is being corrected and redirected," said Belloy.
The experts said that the theory was the culmination of years of work.
The manuscript revealed the different styles of the two men.
The sense of personality that comes across in these pages is interesting.
You get the impression that Einstein was more confident in his calculations since his sheets are much lighter in terms of text and reserved for calculations. Besso added written notes in the margin.
The document is the most valuable of Einstein's notes. The scientist's handwritten letter containing his equation sold for more than a million dollars.
