The other day I had to give a semi-popular talk about "Special Relativity". I aimed

to explain E=mc^2 in simple terms. To my horror I could not find a simple

enough explanation. Of course I can derive it with 4 - momentum or photon momenta.

But this is not what I was looking for. So for a change I post a question here:

Who can provide me with the most simple explanation: Why is E=mc^2 ?

Find below the slides I used for my talk. The crucial point of special relativity

is really to rethink the meaning of time and how to define the simulaneity of two events.

Well at the end I give a derivation for E=mc^2, which goes back to a 1906 paper by Einstein, but it uses the momentum of a photon. I would like to see it purely from mechanics.

My textbook in highschool had an elementary derivation but I can't remember the details. I think it first derived time dilatation and the studied a cannon ball that hits a wall and destroys it (I am sure there are more pacifist examples along the same lines). The fact, that the wall gets destroyed is supposed to be a measure of the cannon ball's kinetic energy. Then you look at the same process from the perspective of a moving observer (IIRC you should move transversally to the ball). However the fact that the wall gets knocked down in both reference frames tells you something how the kinetic energy transforms (using the transformation of the velocity that you worked out earlier).

Somehow, you could compute E=mc^2 from that (be sure you know what you want to call E and m, E is something like restmass plus kinetic energy and m is the 'moving mass', mass decorated with a gamma factor).

Maybe there is also a way to get this relation by doing a gedankenexperiment that is based on the fact the energy and time are conjugate variables and of course the symplectic structure is invariant. Of course you should not say it that way in a public lecture...

Posted by: Robert | October 27, 2005 at 05:34 AM

probably the simplest explanation is GUAM.

Posted by: m. visaya | October 27, 2005 at 08:07 AM

B=((3/4)(sqrt(3)-3i)(mgv)), then

(B^T)^nT (time-T map dual)

Posted by: m.visaya | October 27, 2005 at 10:43 AM

I like m.visaya's answer. I don't know what it means, but I like the look of it (lol).

p.s. Thanks for your continuous effort.

Posted by: Fred | October 27, 2005 at 11:16 AM

The best thing to say to an audience about E = mc^2 is this:

OK people, I know why you care about this, it's because you think it has something to do with bombs and nuclear reactors. Well, that's wrong. A nucleus of uranium is held together by something called the strong force. Those bonds store energy, just as chemical bonds store energy. In a nuclear weapon that energy gets released and turned into another kind of energy. Absolutely nothing to do with E = mc^2 you see. No mass is being turned into energy, it's just the same old boring story: energy stored in one form released in another form. So you are all here for the wrong reason, please go home now.

Posted by: jack | October 27, 2005 at 09:57 PM

The PBS.org site has a page where 10 different physicists give simple explanations of the equation. If you haven't seen it, it's worth reading. Don't know if it helps your question.

http://www.pbs.org/wgbh/nova/einstein/expe-text.html

Posted by: Brad | October 28, 2005 at 11:35 PM

Hi Jochen,

I was having the exact same problem a few weeks ago: I was preparing lecture notes for "Physics for Poets" here in Princeton and got frustrated that I couldn't find a simple derivation of E=mc^2. This is interesting since everything else in special relativity really can be shown in quite simple terms (with little math and little prior knowledge) to non-scientists.

Finally, I discussed this with Richard Gott and we came up with a simple thought experiment. Unfortunately, it also involves photon momentum, so it doesn't qualify as something you are looking for. However, I prefer it to the center of mass argument you use. I will send you our notes in a separate email, but basically the idea is the following:

Imagine a particle travelling at low speed v and spontaneously emitting two photons. In the particle rest frame the photons have equal and opposite momenta, so the particle remains at rest. Now analyze this in the lab frame. The two photons are Doppler shifted and therefore impart unequal momenta on the two sides of the lab. But momentum has to be conserved. Where is the change in momentum of the walls of the lab coming from? The speed of the particle can't change (by symmetry of the situation in its rest frame; no kick in any direction), so its mass must change. Include the details of Doppler formulae and you get

E=mc^2

Hope you enjoy being back in London.

Best, Daniel

Posted by: Daniel | October 30, 2005 at 06:45 AM

In a book The Einstein Paradox

by Colin Bruce

page 163 the equations for deriving e=mc^2.

is it correct.

How does he get momentum to = e/c/m ?

Posted by: Edward Sawka | November 01, 2005 at 05:29 PM

In a book The Einstein Paradox

by Colin Bruce

page 163 the equations for deriving e=mc^2.

is it correct.

How does he get momentum to = e/c/m ?

Posted by: Edward Sawka | November 01, 2005 at 05:30 PM

Hi Edward

this is exactly the problem I was running up against, that most books assumed the expression

for momentum.

For a derivation of this you can look in

Feynman lectures.

Posted by: Jochen Weller | November 03, 2005 at 05:34 AM

I think time dilation is wrong due to inertia.

Posted by: vishal | November 04, 2005 at 01:00 AM

Hi,

I'm a Grade 12 student who is just beginning to learn about Quantum Physics and E= mc^2. My teacher wanted the class to find an explanation of the equation over the christmas holidays: is there an explanation that a class of Grade 12 Physics students could understand?

From my research, I found out that Einstein was trying to make a connection between Mass and Energy, and that he basically tried to explain Mass in terms of Energy. I understand how the speed of light is the constant in the equation: it doesn't change in any frame of reference, while the mass and energy can change: they are the variables. What I don't fully understand is how and why Mass can be turned into Energy and back again.

Thanks a lot!

Nandini

p.s. also, what is this GUAM that m. visaya mentioned in one of the past comments??

Posted by: Nandini | January 05, 2006 at 03:30 PM

By the way, I've written out everything I understand on this topic so far on my blog space, if you'd like to see it:

http://spaces.msn.com/members/nanzspace

Thanks again!

Posted by: Nandini | January 05, 2006 at 03:35 PM