### An Indian scientist’s disputes with Einstein’s mass-energy equivalence betray a misreading of how one of history’s most famous equations came to be

To quote John Cutter (Michael Caine) from *The Prestige*:

Every magic trick consists of three parts, or acts. The first part is called the pledge, the magician shows you something ordinary. The second act is called the turn, the magician takes the ordinary something and makes it into something extraordinary. But you wouldn’t clap yet, because making something disappear isn’t enough. You have to bring it back. Now you’re looking for the secret. But you won’t find it because of course, you’re not really looking. You don’t really want to work it out. You want to be fooled.

*The Pledge*

Ajay Sharma is an assistant director of education with the Himachal Pradesh government. On January 10, the Indo-Asian News Service (IANS) published an article in which Sharma claims Albert Einstein’s famous equation E = mc^{2} is “illogical” (republished by *The Hindu*, *Yahoo!* News, *Gizmodo India*, among others). The precise articulation of Sharma’s issue with it is unclear because the IANS article contains multiple unqualified statements:

Albert Einstein’s mass energy equation (E=mc

^{2}) is inadequate as it has not been completely studied and is only valid under special conditions.Einstein considered just two light waves of equal energy emitted in opposite directions with uniform relative velocity.

“It’s only valid under special conditions of the parameters involved, e.g. number of light waves, magnitude of light energy, angles at which waves are emitted and relative velocity.”

Einstein considered just two light waves of equal energy, emitted in opposite directions and the relative velocity uniform. There are numerous possibilities for the parameters which were not considered in Einstein’s 1905 derivation.

It said E=mc

^{2}is obtained from L=mc^{2}by simply replacing L by E (all energy) without derivation by Einstein. “It’s illogical,” he said.Although Einstein’s theory is well established, it has to be critically analysed and the new results would definitely emerge.

Sharma also claims Einstein’s work wasn’t original and only ripped off Galileo, Henri Poincaré, Hendrik Lorentz, Joseph Larmor and George FitzGerald.

*The Turn*

Let’s get some things straight.

**Mass-energy equivalence** – E = mc^{2} isn’t wrong but it’s often overlooked that it’s an approximation. This is the full equation:

E^{2} = m_{0}^{2}c^{4} + p^{2}c^{4}

(Notice the similarity to the Pythagoras theorem?)

Here, m_{0} is the mass of the object (say, a particle) when it’s not moving, *p* is its momentum (calculated as mass times its velocity – *m***v*) and *c*, the speed of light. When the particle is not moving, *v* is zero, so *p* is zero, and so the right-most term in the equation can be removed. This yields:

E^{2} = m_{0}^{2}c^{4} ⇒ E = m_{0}c^{2}

If a particle was moving close to the speed of light, applying just E = m_{0}c^{2} would be wrong without the rapidly enlarging p^{2}c^{4} component. In fact, the equivalence remains applicable in its most famous form only in cases where an observer is co-moving along with the particle. So, there is no mass-energy equivalence as much as a mass-energy-momentum equivalence.

And at the time of publishing this equation, Einstein was aware that it was held up by multiple approximations. As Terence Tao sets out, these would include (but not be limited to) *p* being equal to *mv* at low velocities, the laws of physics being the same in two frames of reference moving at uniform velocities, Planck’s and de Broglie’s laws holding, etc.

These approximations are actually inherited from Einstein’s special theory of relativity, which describes the connection between space and time. In a paper dated September 27, 1905, Einstein concluded that if “a body gives off the energy*L* in the form of radiation, its mass diminishes by *L*/c^{2}“. ‘L’ was simply the notation for energy that Einstein used until 1912, when he switched to the more-common ‘E’.

The basis of his conclusion was a thought experiment he detailed in the paper, where a point-particle emits “plane waves of light” in opposite directions while at rest and then while in motion. He then calculates the difference in kinetic energy of the body before and after it starts to move and accounting for the energy carried away by the radiated light:

K_{0} – K_{1} = 1/2 * L/c^{2} * v^{2}

This is what Sharma is referring to when he says, “Einstein considered just two light waves of equal energy, emitted in opposite directions and the relative velocity uniform. There are numerous possibilities for the parameters which were not considered in Einstein’s 1905 derivation.” Well… sure. Einstein’s was a *gedanken* (thought) experiment to illustrate a direct consequence of the special theory. How he chose to frame the problem depended on what connection he wanted to illustrate between the various attributes at play.

And the more attributes are included in the experiment, the more connections will arise. Whether or not they’d be meaningful (i.e. being able to represent a physical reality – such as with being able to say “if a body gives off the energy *L* in the form of radiation, its mass diminishes by *L*/c^{2}“) is a separate question.

As for another of Sharma’s claims – that the equivalence is “only valid under special conditions of the parameters involved, e.g. number of light waves, magnitude of light energy, angles at which waves are emitted and relative velocity”: Einstein’s theory of relativity is the best framework of mathematical rules we have to describe all these parameters together. So any *gedanken* experiment involving just these parameters can be properly analysed, to the best of our knowledge, with Einstein’s theory, and within that theory – and as a consequence of that theory – the mass-energy-momentum equivalence will persist. This implication was demonstrated by the famous Cockcroft-Walton experiment in 1932.

**General theory of relativity** – Einstein’s road to publishing his general theory (which turned 100 last year) was littered with multiple challenges to its primacy. This is not surprising because Einstein’s principal accomplishment was not in having invented something but in having recombined and interpreted a trail of disjointed theoretical and experimental discoveries into a coherent, meaningful and testable theory of gravitation.

As mentioned earlier, Sharma claims Einstein ripped off Galileo, Poincaré, Lorentz, Larmor and FitzGerald. For what it’s worth, he could also have mentioned William Kingdon Clifford, Georg Bernhard Riemann, Tullio Levi-Civita, Gregorio Ricci-Curbastro, János Bolyai, Nikolai Lobachevsky, David Hilbert, Hermann Minkowski and Fritz Hasenhörl. Here are their achievements in the context of Einstein’s (in a list that’s by no means exhaustive).

**1632, Galileo Galilei**– Published a book, one of whose chapters features a dialogue about the relative motion of planetary bodies and the role of gravity in regulating their motion**1824-1832, Bolyai and Lobachevsky**– Conceived of hyperbolic geometry (which didn’t follow Euclidean laws like the sum of a triangle’s angles is 180º) over 1824-1832, which inspired Riemann and his mentor to consider if there was a kind of geometry to explain the behaviour of shapes in four dimensions (as opposed to three)**1854, G. Bernhard Riemann**– Conceived of elliptic geometry and a way to compare vectors in four dimensions, ideas that would benefit Einstein immensely because they helped him discover that gravity wasn’t a force in space-time but actually the curvature of space-time**1876, William K. Clifford**– Suggested that the forces that shape matter’s motion in space could be guided by the geometry of space, foreshadowing Einstein’s idea that matter influences gravity influences matter**1887-1902, FitzGerald and Lorentz**– Showed that observers in different frames of reference that are moving at different velocities can measure the length of a common body to differing values, an idea then called the FitzGerald-Lorentz contraction hypothesis. Lorentz’s mathematical description of this gave rise to a set of formulae called Lorentz transformations, which Einstein later derived through his special theory.**1897-1900, Joseph Larmor**– Realised that observers in different frames of reference that are moving at different velocities can also measure different times for the same event, leading to the time dilation hypothesis that Einstein later explained**1898, Henri Poincaré**– Interpreted Lorentz’s abstract idea of a “local time” to have physical meaning – giving rise to the idea of relative time in physics – and was among the first physicists to speculate on the need for a consistent theory to explain the consequences of light having a constant speed**1900, Levi-Civita and Ricci-Curbastro**– Built on Riemann’s ideas of a non-Euclidean geometry to develop tensor calculus (a tensor is a vector in higher dimensions). Einstein’s field-equations for gravity, which capped his formulation of the celebrated general theory of relativity, would feature the Ricci tensor to account for the geometric differences between Euclidean and non-Euclidean geometry.**1904-1905, Fritz Hasenöhrl**– Built on the work of Oliver Heaviside, Wilhelm Wien, Max Abraham and John H. Poynting to devise a thought experiment from which he was able to conclude that heat has mass, a primitive synonym of the mass-energy-momentum equivalence**1907, Hermann Minkowski**– Conceived a unified mathematical description of space and time in 1907 that Einstein could use to better express his special theory. Said of his work: “From this hour on, space by itself, and time by itself, shall be doomed to fade away in the shadows, and only a kind of union of the two shall preserve an independent reality.”**1915, David Hilbert**– Derived the general theory’s field equations a few days before Einstein did but managed to have his paper published only after Einstein’s was, leading to an unresolved dispute about who should take credit. However, the argument was made moot by only Einstein being able to explain how Isaac Newton’s laws of classical mechanics fit into the theory – Hilbert couldn’t.

FitzGerald, Lorentz, Larmor and Poincaré all laboured assuming that space was filled with a ‘luminiferous ether’. The ether was a pervasive, hypothetical yet undetectable substance that physicists of the time believed had to exist so electromagnetic radiation had a medium to travel in. Einstein’s theories provided a basis for their ideas to exist *without* the ether, and as a consequence of the geometry of space.

So, Sharma’s allegation that Einstein republished the work of other people in his own name is misguided. Einstein didn’t plagiarise. And while there are many accounts of his competitive nature, to the point of asserting that a mathematician who helped him formulate the general theory wouldn’t later lay partial claim to it, there’s no doubt that he did come up with something distinctively original in the end.

*The Prestige*

To recap:

Albert Einstein’s mass energy equation (E=mc

^{2}) is inadequate as it has not been completely studied and is only valid under special conditions.

Claims that Einstein’s equations are inadequate are difficult to back up because we’re yet to find circumstances in which they seem to fail. Theoretically, they can be made to *appear* to fail by forcing them to account for, say, higher dimensions, but that’s like wearing suede shoes in the rain and then complaining when they’re ruined. There’s a time and a place to use them. Moreover, the failure of general relativity or quantum physics to meet each other halfway (in a quantum theory of gravity) can’t be pinned on a supposed inadequacy of the mass-energy equivalence alone.

Einstein considered just two light waves of equal energy emitted in opposite directions with uniform relative velocity.

“It’s only valid under special conditions of the parameters involved, e.g. number of light waves, magnitude of light energy, angles at which waves are emitted and relative velocity.”

Einstein considered just two light waves of equal energy, emitted in opposite directions and the relative velocity uniform. There are numerous possibilities for the parameters which were not considered in Einstein’s 1905 derivation.

That a *gedanken* experiment was limited in scope is a pointless accusation. Einstein was simply showing that A implied B, and was never interested in proving that A’ (a different version of A) did not imply B. And tying all of this to the adequacy (or not) of *E* = *mc*^{2} leads equally nowhere.

It said E=mc

^{2}is obtained from L=mc^{2}by simply replacing L by E (all energy) without derivation by Einstein. “It’s illogical,” he said.

From the literature, the change appears to be one of notation. If not that, then Sharma could be challenging the notion that the energy of a moving body is equal to the sum of the energy of the body at rest and its kinetic energy – letting Einstein say that the kinetic energy on the LHS of the equation can be substituted by L (or E) if the RHS is added to E_{0} (energy of the body at rest): E = E_{0} + K. In which case Sharma’s challenge is even more ludicrous for calling one of the basic tenets of thermodynamics “illogical” without indicating why.

Although Einstein’s theory is well established, it has to be critically analysed and the new results would definitely emerge.

The “the” before “new results” is the worrying bit: it points to claims of his that have already been made, and suggests they’re contrary to what Einstein has claimed. It’s not that the German is immune to refutation – no one is – but that whatever claim this is seems to be at the heart of what’s at best an awkwardly worded outburst, and which IANS has unquestioningly reproduced.

A persistent search for Sharma’s paper on the web didn’t turn up any results – the closest I got was in unearthing its title (#237) in a list of titles published at a conference hosted by a ‘Russian Gravitational Society’ in May 2015. Sharma’s affiliation is mentioned as a ‘Fundamental Physics Society’ – which in turn shows up as a Facebook page run by Sharma. But an ibnlive.com article from around the same time provides some insight into Sharma’s ‘research’ (translated from the Hindi by Siddharth Varadarajan):

In this way, Ajay is also challenging the great scientist of the 21st century (sic) Albert Einstein. After deep research into his formula, E=mc

^{2}, he says that “when a candle burns, its mass reduces and light and energy are released”. According to Ajay, Einstein obtained this equation under special circumstances.This means that from any matter/thing, only two rays of light emerge.The intensity of light of both rays is the same and they emerge from opposite directions. Ajay says Einstein’s research paper was published in 1905 in the German research journal [Annalen der Physik] without the opinion of experts. Ajay claims that if this equation is interpreted under all circumstances, then you will get wrong results.Ajay says that if a candle is burning, its mass should increase.Ajay says his research paper has been published after peer review. [Emphasis added.]

A pattern underlying some of Sharma’s claims have to do with confusing conjecturing and speculating (even perfectly reasonably) with formulating and defining and proving. The most telling example in this context is alleging that Einstein ripped off Galileo: even if they both touched on relative motion in their research, what Galileo did for relativity was vastly different from what Einstein did. In fact, following the Indian Science Congress in 2015, V. Vinay, an adjunct faculty at the Chennai Mathematical Institute and teacher in Bengaluru, had pointed out that these differences in fact encapsulated the epistemological attitudes of the Indian and Greek civilisations: the TL;DR version is that we weren’t a proof-seeking people.

Swinging back to the mass-energy equivalence itself – it’s a notable piece but a piece nonetheless of an expansive theory that’s demonstrably incomplete. And there are other theories like it, like flotsam on a dark ocean whose waters we haven’t been able to see, theories we’re struggling to piece together. It’s a time when Popper’s philosophies haven’t been able to qualify or disqualify ‘discoveries’, a time when the subjective evaluation of an idea’s usefulness seems just as important as objectively testing it. But despite the grand philosophical challenges these times face us with, extraordinary claims still do require extraordinary evidence. And at that Ajay Sharma quickly fails.

*Hat-tip to @AstroBwouy, @ainvvy and @hosnimk.*