Einstein’s equation describing the equivalence of mass and energy is arguably the most famous equation in physics. But his beautifully elegant *derivation* of this formula (here is the English translation) from previously understood laws of physics is considerably less famous. (There is an amusing Far Side cartoon in this regard, with the punchline “squared away”, which you can find on-line by searching hard enough, though I will not link to it directly.)

This topic had come up in recent discussion on this blog, so I thought I would present Einstein’s derivation here. Actually, to be precise, in the paper mentioned above, Einstein uses the postulates of special relativity and other known laws of physics to show the following:

Proposition. (Mass-energy equivalence) If a body at rest emits a total energy of E while remaining at rest, then the mass of that body decreases by .

Assuming that bodies at rest with zero mass necessarily have zero energy, this implies the famous formula – but only for bodies which are at rest. For moving bodies, there is a similar formula, but one has to first decide what the correct definition of mass is for moving bodies; I will not discuss this issue here, but see for instance the Wikipedia entry on this topic.

Broadly speaking, the derivation of the above proposition proceeds via the following five steps:

- Using the postulates of special relativity, determine how space and time coordinates transform under changes of reference frame (i.e. derive the Lorentz transformations).
- Using 1., determine how the temporal frequency (and wave number k) of photons transform under changes of reference frame (i.e. derive the formulae for relativistic Doppler shift).
- Using Planck’s law (and
*de Broglie’s law*) and 2., determine how the energy E (and momentum p) of photons transform under changes of reference frame. - Using the law of conservation of energy (and momentum) and 3., determine how the energy (and momentum) of bodies transform under changes of reference frame.
- Comparing the results of 4. with the classical Newtonian approximations (and ), deduce the relativistic relationship between mass and energy for bodies at rest (and more generally between mass, velocity, energy, and momentum for moving bodies).

Actually, as it turns out, Einstein’s analysis for bodies at rest only needs to understand changes of reference frame at infinitesimally low velocity, . However, in order to see enough relativistic effects to deduce the mass-energy equivalence, one needs to obtain formulae which are accurate to second order in v (or more precisely, ), as opposed to those in Newtonian physics which are accurate to first order in v. Also, to understand the relationship between mass, velocity, energy, and momentum for moving bodies rather than bodies at rest, one needs to consider non-infinitesimal changes of reference frame.

*Important note*: Einstein’s argument is, of course, a physical argument rather than a mathematical one. While I will use the language and formalism of pure mathematics here, it should be emphasised that I am not exactly giving a formal proof of the above Proposition in the sense of modern mathematics; these arguments are instead more like the classical proofs of Euclid, in that numerous “self evident” assumptions about space, time, velocity, etc. will be made along the way. (Indeed, there is a very strong analogy between Euclidean geometry and the Minkowskian geometry of special relativity.) One can of course make these assumptions more explicit, and this has been done in many other places, but I will avoid doing so here in order not to overly obscure Einstein’s original argument.

— Lorentz transforms to first order —

To simplify the notation, we shall assume that the ambient spacetime S has only one spatial dimension rather than three, although the analysis here works perfectly well in three spatial dimensions (as was done in Einstein’s original paper). Thus, in any inertial reference frame F, the spacetime S is parameterised by two real numbers (t,x). Mathematically, we can describe each frame F as a bijection between S and . To normalise these coordinates, let us suppose that all reference frames agree to use a single event O in S as their origin (0,0); thus

(1)

for all frames F.

Given an inertial reference frame , one can generate new inertial reference frames in two different ways. One is by *reflection*: one takes the same frame, with the same time coordinate, but reverses the space coordinates to obtain a new frame , thus reversing the orientation of the frame. In equations, we have

if , then (2)

for any spacetime event E. Another way is by replacing the observer which is stationary in F with an observer which is moving at a constant velocity v in F, to create a new inertial reference frame with the same orientation as F. In our analysis, we will only need to understand infinitesimally small velocities v; there will be no need to consider observers traveling at speeds close to the speed of light.

The new frame and the original frame must be related by some transformation law

(3)

for some bijection . A priori, this bijection could depend on the original frame F as well as on the velocity v, but the principle of relativity implies that is in fact the same in all reference frames F, and so only depends on v.

It is thus of interest to determine what the bijections are. From our normalisation (1) we have

(4)

but this is of course not enough information to fully specify . To proceed further, we recall Newton’s first law, which states that an object with no external forces applied to it moves at constant velocity, and thus traverses a straight line in spacetime as measured in any inertial reference frame. (We are assuming here that the property of “having no external forces applied to it” is not affected by changes of inertial reference frame. For non-inertial reference frames, the situation is more complicated due to the appearance of fictitious forces.) This implies that transforms straight lines to straight lines. (To be pedantic, we have only shown this for straight lines corresponding to velocities that are physically attainable, but let us ignore this minor technicality here.) Combining this with (4), we conclude that is a linear transformation. (It is a cute exercise to verify this claim formally, under reasonable assumptions such as smoothness of . ) Thus we can view now as a matrix.

When v=0, it is clear that should be the identity matrix I. Making the plausible assumption that varies smoothly with v, we thus have the Taylor expansion

(5)

for some matrix and for infinitesimally small velocities v. (Mathematically, what we are doing here is analysing the Lie group of transformations via its Lie algebra.) Expanding everything out in coordinates, we obtain

(6)

for some absolute constants (not depending on t, x, or v).

The next step, of course, is to pin down what these four constants are. We can use the reflection symmetry (2) to eliminate two of these constants. Indeed, if an observer is moving at velocity v in frame F, it is moving in velocity -v in frame , and hence . Combining this with (2), (3), (6) one eventually obtains

and . (7)

Next, if a particle moves at velocity v in frame F, and more specifically moves along the worldline , then it will be at rest in frame , and (since it passes through the universally agreed upon origin O) must then lie on the worldline . From (3), we conclude

for all t. (8)

Inserting this into (6) (and using (7)) we conclude that . We have thus pinned down to first order almost completely:

(9)

Thus, rather remarkably, using nothing more than the principle of relativity and Newton’s first law, we have almost entirely determined the reference frame transformation laws, save for the question of determining the real number . [In mathematical terms, what we have done is classify the one-dimensional Lie subalgebras of which are invariant under spatial reflection, and coordinatised using (8).] If this number vanished, we would eventually recover classical Galilean relativity. If this number was positive, we would eventually end up with the (rather unphysical) situation of *Euclidean relativity*, in which spacetime had a geometry isomorphic to that of the Euclidean plane. As it turns out, though, in special relativity this number is negative. This follows from the second postulate of special relativity, which asserts that the speed of light c is the same in all inertial reference frames. In equations (and because has the same orientation as F), this is asserting that

for all t (10+)

and

for all t. (10-)

Inserting either of (10+) or (10-) into (9) we conclude that , and thus we have obtained a full description of to first order:

(11)

— Lorentz transforms to second order —

It turns out that to get the mass-energy equivalence, first-order expansion of the Lorentz transformations is not sufficient; we need to expand to second order. From Taylor expansion we know that

(12)

for some matrix . To compute this matrix, let us make the plausible assumption that if the frame is moving at velocity v with respect to F, then F is moving at velocity -v with respect to . (One can justify this by considering two frames receding at equal and opposite directions from a single reference frame, and using reflection symmetry to consider how these two frames move with respect to each other.) Applying (3) we conclude that . Inserting this into (12) and comparing coefficients we conclude that . Since is determined from (11), we can compute everything explicitly, eventually ending up at the second order expansion

(13)

One can continue in this fashion (exploiting the fact that the must form a Lie group (with the Lie algebra already determined), and using (8) to fix the parameterisation of that group) to eventually get the full expansion of , namely

,

but we will not need to do so here.

— Doppler shift —

The formula (13) is already enough to recover the relativistic Doppler shift formula (to second order in v) for radiation moving at speed c with some wave number k. Mathematically, such radiation moving to the right in an inertial reference frame F can be modeled by the function

for some amplitude A and phase shift . If we move to the coordinates provided by an inertial reference frame F’, a computation then shows that the function becomes

where . (actually, if the radiation is tensor-valued, the amplitude A might also transform in some manner, but this transformation will not be of relevance to us.) Similarly, radiation moving at speed c to the left will transform from

to

where . This describes how the wave number k transforms under changes of reference frame by small velocities v. The temporal frequency is linearly related to the wave number k by the formula

, (14)

(15+)

for right-ward moving radiation and by the (blue-shift) formula

(15-)

for left-ward moving radiation. (As before, one can give an exact formula here, but the above asymptotic will suffice for us.)

— Energy and momentum of photons —

From the work of Planck, and of Einstein himself on the photoelectric effect, it was known that light could be viewed both as a form of radiation (moving at speed c), and also made up of particles (photons). From Planck’s law, each photon has an energy of and (from de Broglie’s law) a momentum of , where h is Planck’s constant, and the sign depends on whether one is moving rightward or leftward. In particular, from (14) we have the pleasant relationship

(16)

for photons. [More generally, it turns out that for arbitrary bodies, momentum, velocity, and energy are related by the formula , though we will not derive this fact here.] Applying (15+), (15-), we see that if we view a photon in a new reference frame , then the observed energy E and momentum p now become

; (17+)

for right-ward moving photons, and

; (17-)

for left-ward moving photons.

These two formulae (17+), (17-) can be unified using (16) into a single formula

(18)

for any photon (moving either leftward or rightward) with energy E and momentum p as measured in frame F, and energy E’ and momentum p’ as measured in frame . Actually, the error term can be deleted entirely by working a little harder. From the linearity of and the conservation of energy and momentum, it is then natural to conclude that (18) should also be valid not only for photons, but for any object that can exchange energy and momentum with photons. This can be used to derive the formula fairly quickly, but let us instead give the original argument of Einstein, which is only slightly different.

— Einstein’s argument —

We are now ready to give Einstein’s argument. Consider a body at rest in a reference frame F with some mass and some rest energy . (We do not yet know that is equal to .) Now let us view this same mass in some new reference frame , where v is a small velocity. From Newtonian mechanics, we know that a body of mass moving at velocity v acquires a kinetic energy of . Thus, assuming that Newtonian physics is valid at low velocities to top order, the net energy E’ of this body as viewed in this frame should be

(19)

If assumes that the transformation law (18) applies for this body, one can already deduce the formula for this body at rest from (19) (and the assumption that bodies at rest have zero momentum), but let us instead give Einstein’s original argument.

We return to frame F, and assume that our body emits two photons of equal energy , one moving left-ward and one moving right-ward. By (16) and conservation of momentum, we see that the body remains at rest after this emission. By conservation of energy, the remaining energy in the body is . Let’s say that the new mass in the body is . Our task is to show that .

To do this, we return to frame . By (16+), the rightward moving photon has energy

; (20+)

in this frame; similarly, the leftward moving photon has energy

. (20-)

What about the body? By repeating the derivation of (18), it must have energy

(20)

By the principle of relativity, the law of conservation of energy has to hold in the frame as well as in the frame F. Thus, the energy (20-)+(20+)+(20) in frame after the emission must equal the energy E’=(19) in frame before emission. Adding everything together and comparing coefficients we obtain the desired relationship .

[One might quibble that Einstein’s argument only applies to emissions of energy that consist of equal and opposite pairs of photons. But one can easily generalise the argument to handle arbitrary photon emissions, especially if one takes advantage of (18); for instance, another well-known (and somewhat simpler) variant of the argument works by considering a photon emitted from one side of a box and absorbed on the other. More generally, any other energy emission which could potentially in the future decompose entirely into photons would also be handled by this argument, thanks to conservation of energy. Now, it is possible that other conservation laws prevent decomposition into photons; for instance, the law of conservation of charge prevents an electron (say) from decomposing entirely into photons, thus leaving open the possibility of having to add a linearly charge-dependent correction term to the formula . But then one can renormalise away this term by redefining the energy to subtract such a term; note that this does not affect conservation of energy, thanks to conservation of charge.]

## 90 comments

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3 June, 2011 at 8:09 pm

JonDear Terrence Tao,

Could you, pls, tell me what do I need to know to understand Einstein general relativity proof ?

I know that is necessary to understand Differential Geometry, right ?

What more ?

I want to set this as one of my lifetime Goals. :o)

Thanks

17 March, 2013 at 11:11 am

AnonymousStudy Tensors. Start with the metric tensor .

31 August, 2011 at 9:37 pm

Anonymouse=mc 2 is wrong …there must be a factor v that is subtracted from the velocity of light inorder to get the correct equation and that subtracted quantity is the limiting value for the reference frame …if we got that reference frame we can understand the success of reaction ….

8 December, 2014 at 6:49 pm

hoyhivE= mc q<- einstein forgot theory of relativity

24 October, 2011 at 5:17 am

amateurHow can someone explain changes in 4-d space-time (t,x(t),y(t),z(t)) mathematically since 1 is constant:

(1,dx/dt,dy/dt,dz/dt)

8 June, 2012 at 3:45 pm

ScottparkSo useful post.Thank you!!

25 July, 2012 at 1:39 am

Kuhanhttp://www.ams.org/bookstore/pspdf/mbk-59-prev.pdf

has similar article

2 October, 2012 at 7:51 pm

Einstein’s derivation of E=mc^2 revisited « What’s new[…] back in 2007, I wrote a blog post giving Einstein’s derivation of his famous equation . This derivation used a number of […]

3 October, 2012 at 6:37 pm

Wow Terrence Tao « The logic of images[…] back in 2007, I wrote a blog post giving Einstein’s derivation of his famous equation for the rest energy of a body with mass . […]

29 November, 2012 at 10:18 pm

MiladReblogged this on Milad Ebrahimpour 's Blog and commented:

You see how the universe is ruled by math?

22 December, 2012 at 2:40 pm

An introduction to special relativity for a high school math circle « What’s new[…] equivalence relation E=mc^2, largely following Einstein’s original derivation as discussed in this previous blog post); instead we will probably spend a fair chunk of time on related topics which do not actually […]

31 December, 2012 at 9:00 pm

Einstein’s derivation of E=mc^2 | What about being a physicist[…] Einstein’s derivation of E=mc^2. […]

8 January, 2013 at 3:13 am

bert latifgood justification about e=mcc,,,,,,

23 January, 2013 at 2:10 am

Everything is Relative. (Trippy) Consequences of Special Relativity. Prove E = mc2 | A Revolving Wheel[…] Terrytao: Deriving E=mc2 Ask a Mathematician: Q: Why does E=MC2? Wiki: Theory of Relativity Wiki: Special Relativity Wiki: Introduction to Special Relativity Wiki: Relativity of Simultaneity […]

17 March, 2013 at 11:07 am

jussilindgrenI’ve recently developed a possible model for gravitation that reduces to the Newtonian limit with low velocities/masses and yields also the mass-energy equivalence. The model is based on the idea to consider tensor products in flat Minkowski space. When one generalises then the concept of work-energy principle, one obtains the following nonlinear PDE system:

The respective energy equation is the following general wave equation

This is work in progress, I have a blog on the issue at jussilindgren.wordpress.com

2 April, 2013 at 3:46 am

OsumaWhat is the meaning of c in einstein equation

2 April, 2013 at 10:40 am

ReeceThe speed of light

23 April, 2013 at 5:54 am

johngokulI can’t believe it what a scientist was einstein. this website give chance to know about the universal equation.it will useful the student.

5 July, 2013 at 3:38 am

Aayush DhitalSir, I searched for the answer of the quesion “why mass causes curvature in space-time?” and I think I’ve found it. However, I’am not 100% sure on it. In my view every atoms contain a unit charge so any body with mass can be considered as a massive charge itself since it is the resultant of all the charges. In my view, this charge actually pushes the dark energy in the space and so does the dark energy. Hence, space-time is curved along the massive body….

16 August, 2013 at 6:15 pm

Anonymousthis cannot happen bcuz atom is electrically neutral

23 November, 2013 at 2:18 pm

socratusThe strange and magic E= Mc^2

1

In 1905 Einstein asked:

“ Does the inertia of a body depend upon its energy content?”

As he realized the answer was:

“ Yes, inertia depends on its energy content E= Mc^2 ”

Newton’s inertia doesn’t have force/energy,

Einstein’s inertia has energy. How to understand this difference?

2.

In 1928 Dirac said that E= Mc^2 can be as positive

as negative too. What is interaction between them ?

3 –

Sometimes E= Mc^2 can be ‘rest’ particle and sometimes

can be ‘active’ particle and can destroy cities like

Hiroshima and Nagasaki

Why E= Mc^2 is so strange ? Nobody gives answer

===.

All the best.

Israel Sadovnik Socratus.

==============..

25 January, 2014 at 7:51 pm

Ask a physicist anything. (8) - Page 61 - Christian Forums[…] Equation The original formula dealt with the rest mass of a body, forgotten in modern textbooks. https://terrytao.wordpress.com/2007/1…ation-of-emc2/ __________________ "If one closes their eyes they can imagine a universe of infinite […]

16 February, 2014 at 11:32 pm

James AllenWhere’s Leibniz? Tsk-tsk.

9 June, 2014 at 11:49 am

AnonymousStillni havevthe easiest way to prove it.

28 October, 2014 at 11:39 am

dominic nentawe.Please can someone deduce the einstein famous energy formula from the de broglie equation

17 December, 2014 at 12:27 am

yusrahow can we find the dimension of E=MC2

17 December, 2014 at 12:31 am

yusratell me quick.

10 January, 2015 at 8:13 am

On the Beauty Of Physical Theories | ENIGMA[…] https://terrytao.wordpress.com/2007/12/28/einsteins-derivation-of-emc2/ […]

15 January, 2015 at 7:41 pm

mjg0I urge everyone to read the paper by physicist Eugene Hecht entitled

“How Einstein confirmed E0=mc^2”, which you download from

http://www.loreto.unican.es/Carpeta2012/AJP(Hecht)Einstein2011.pdf.

Here is Hecht’s abstract:

The equivalence of mass m and rest-energy E0 is one of the great discoveries of all time. Despite the

current wisdom, Einstein did not derive this relation from first principles. Having conceived the idea

in the summer of 1905 he spent more than 40 years trying to prove it. We briefly examine all of

Einstein’s conceptual demonstrations of E0=mc2, focusing on their limitations and his awareness of

their shortcomings. Although he repeatedly confirmed the efficacy of E0=mc2, he never constructed

a general proof. Leaving aside that it continues to be affirmed experimentally, a rigorous proof of the

mass-energy equivalence is probably beyond the purview of the special theory [of relativity].

That last sentence is an important conjecture about the logic of SR.

A crucial point – which Terence emphasized he was avoiding – is whether mass changes with motion, i.e., whether mass is relativistic.

15 January, 2015 at 8:19 pm

mjg0Hecht’s article is critical of Einstein’s Sept. 1905 argument, which Terence presented above. Some of Hecht’s criticisms:

This derivation came very early in the development of

relativity, and the formal concept of “rest-energy” had not

yet evolved, nor had E0 been introduced to symbolize it.

Unless Einstein was willing to assume that whenever E0=0,

m=0—that all of the mass of a material entity was equivalent

to energy—he could not take the analysis any farther

than he had in September 1905. Unfortunately, he did not

overtly share his concerns about this issue with his readers.

it would not be until 1912 that he was writing statements

like: “According to this conception, we would have to view a

body with inertial mass m as an energy store of magnitude

mc^2 rest-energy of the body.” To attribute E0=mc^2 or

even worse, E=mc^2 to Einstein via the September

1905 paper is to do injustice to the gradual nature of his

development of the concepts.

Einstein made

two intuitive and seemingly innocent leaps that went beyond

what he could prove. “Since obviously here it is inessential

that the energy withdrawn from the body happens to turn into

energy of radiation rather than some other kind of energy, we

are led to the more general conclusion: The mass of a body is

a measure of its energy content [Energieinhalt]….”

The concept

of rest-energy was still unformulated, and he was not

even using the term. Without proof, Einstein guessed that the

loss of other forms of energy besides electromagnetic would

result in an equivalent loss of mass. That is no small point,

especially because many physicists at the time believed mass

was entirely electromagnetic.

An unspoken assumption was made early in the September

1905 paper that each emitted light pulse was not material,

in that it carried energy but had no intrinsic mass.

Throughout his work on the special theory, Einstein maintained

that light was other than matter. As he put it in 1911,

“the comparison of light with other ‘stuff’ is not

permissible.” Thus, Einstein avoided a more complicated

calculation by conjecturing, without discussion or proof,

that light was massless. But of course he could not prove

such a thing, one way or the other.

Finally, it should be noted that plane waves are a mathematical

contrivance. No real extended body can actually

radiate electromagnetic plane waves as required by this

thought experiment. Naturally enough Einstein said nothing

about the physical structure of the hypothetical emitting

body that could perform such a feat. This omission likely

troubled him because, as we will see, he returned to the issue

seven years later with a more elaborate emitter.

What Einstein proved in September 1905 was that if an

imaginary material body could somehow emit radiant energy

E0 which is itself massless in the form of plane waves,

that extraordinary object would diminish in mass by E0 /c^2.

He did not prove that every form of energy was equivalent to

mass; that, he simply proclaimed. Yet, he was quite aware

that his analysis had limitations, and in 1907 he discussed the

need for a more general approach.

23 November, 2015 at 9:07 am

milkias tesfalemit was amazing but you should write it in other African languages such as Amharic