A letter penned by Albert Einstein in which he wrote down his famous ‘E = mc²’ mass–energy equivalence formula has sold at auction for the princely sum of $1.2 million after a furious bidding war.

Boston-based RR Auctions, which handled the sale, said it was more than three times what was expected for the rare letter, which went on sale earlier this month.

‘It’s an important letter from both a holographic and a physics point of view,’ Bobby Livingston, executive vice president at RR Auction said in an interview with the Associated Press.

Initially, there were five bidders for the rare item, but once the price of the item topped $700,000, just two bidders remained.

RR Auctions said the buyer of the Einstein letter has preferred to remain anonymous.

There are four known examples of the famous equation written in Einstein’s own handwriting, but this was the only one in a private collection.

It was held by the descendants of the man who received the Oct. 26, 1946 letter, Polish-American physicist Ludwik Silberstein.

A letter penned by Albert Einstein which is rare for containing his famous ‘E = mc²’ mass–energy equivalence formula (pictured) sold at auction for $1.2 million, more than three times what was expected

The German-born Einstein corresponded with Silberstein, telling him a question could ‘be answered from the E = mc² formula.’

‘Your question can be answered from the E = mc2 formula, without any erudition,’ Einstein wrote in the letter written on Princeton University letterhead.

‘If E is the energy of your system consisting of the two masses, E₀ the energy of the masses when they approach infinite distance, then the system’s mass defect is E₀ – E / c2,’ he continued.

Silberstein was a well-known critic and challenger to some of Einstein’s theories.

Following a more complex answer, Einstein went on to conclude that ‘one must first a theory that contains the correct unification of gravitation and electricity.’

The auction started on May 13 and finished on May 20.

The search for this ‘unified field theory’ would go on to consume the final third of the extraordinary physicist’s life.

Einstein demonstrated mass-energy equivalence in 1905 – his so-called ‘miracle year which also saw him publish ground-breaking papers introducing the theory of special relativity, explaining Brownian motion and outlining the photoelectric effect.

Special relativity – which involves the relationship between space and time – determined that the laws of physics are the same for all non-accelerating observers and that light’s speed in a vacuum is fixed, regardless of observer or source motion.

To this understanding Einstein later succeeded in factoring in acceleration and he published this in 1915 as his theory of general relativity, which explained that objects with mass distort the fabric of space and time, which we experience as gravity.

Dr. Silberstein cast doubt on general relativity in 1935–36 — even claiming in the press that the theory was ‘flawed’ — after developing a solution to Einstein’s field equations that he (erroneously) thought violated our understanding of gravity.

(The field equations are those that Einstein drew up to relate the geometry of spacetime to the distribution of matter within it — and thereby are what describe gravity as being a result the curvature of spacetime by mass and energy.)

However, by the time of Einstein’s correspondence in 1946, Dr Silberstein had reportedly come around to Einstein’s way of thinking.

Einstein sent a letter to Polish-American physicist Ludwik Silberstein in October 1946, telling him a question could ‘be answered from the E = mc² formula’

E = mc² — the mass–energy equivalence formula — details the relationship between mass and energy of a system at rest

‘It’s an important letter from both a holographic and a physics point of view, as it shows Einstein’s thinking on one of the most basic of all physical problems,’ said RR Auction’s executive vice president, Bobby Livingston

In fact, the Polish-American researcher is best known today for his work in introducing both Einstein’s general and special relativity into university courses.

The letter he received from Einstein has been put up for auction now by Dr Silberstein’s great-great-grandchildren.

‘It’s an important letter from both a holographic and a physics point of view, as it shows Einstein’s thinking on one of the most basic of all physical problems,’ said RR Auction’s executive vice president, Bobby Livingston.

Einstein, who passed away in 1955, would later go on to explain his general theory of relativity as ‘follow[ing] from the special theory of relativity that mass and energy are but different manifestations of the same thing.

‘Furthermore, the equation E is equal to m c-squared, in which energy is put equal to mass multiplied by the velocity of light squared, showed that a very small amount of mass may be converted into a very large amount of energy, and vice versa.’

In 2018, another piece of Einstein’s correspondence — the ‘God letter’ in which he called religion in general, and especially Judaism, a ‘childish superstition’ born of ‘human weakness’ — sold at auction in New York for $3 million (£2.1 million).