A letter penned by Albert Einstein which is rare for containing his famous ‘E = mc²’ mass–energy equivalence formula has gone on sale for a staggering £282,000.

The German-born theoretical physicist corresponded with a fellow researcher in October 1946, telling him a question could ‘be answered from the E = mc² formula.’

The one-page letter, which is signed off ‘A. Einstein’, is being auctioned off by Boston-based RR Auction in a timed sale which will end next week, on May 20.

It is only one of four known examples of the mass–energy equivalence formula having been written down in Einstein’s own hand.

A letter penned by Albert Einstein which is rare for containing his famous ‘E = mc²’ mass–energy equivalence formula (pictured) has gone on sale for a staggering £282,000

Composed on Princeton University letterhead, the letter was written to the Polish-American physicist Ludwik Silberstein, who had previously contested part of Einstein’s theory of general relativity.

Translated into English, Einstein’s response read: ‘Your question can be answered from the E = mc² formula, without any erudition.’

‘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.

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 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.

The German-born theoretical physicist, pictured, corresponded with a fellow researcher 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.’