Why did we need relativity to derive $E=mc^2$? Okay, so the way I understand one of the "derivations" of $E=mc^2$ is roughly as follows:


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*We observe a light bulb floating in space. It appears motionless. It gives off a brief flash of light. We measure the frequency of this light, and conclude that the light bulb lost some amount of energy $E_1$.

*Next, we imagine flying by this light bulb at 90% the speed of light (or something similar), and again observe the flash. Due to the Relativistic Doppler Effect, the frequency of the light is decreased, and so we observe that the bulb only lost a smaller amount of energy $E_2$.

*Since the total amount of energy lost must be the same in both reference frames, we conclude that the bulb must have lost kinetic energy in the moving reference frame. Kinetic energy is $\frac{1}{2}mv^2$. Since $v$ did not decrease, $m$ must have decreased. Thus, when bodies lose energy, they also lose mass. QED.


So obviously it's a little more complicated than that, but my question is: why do we even need special relativity? Can't we replace "relativistic doppler effect" with "classical doppler effect" and still get a similar result? The exact relationship between $E$ and $m$ will change, but shouldn't the insight that a relationship exists be accessible even with only classical physics?
 A: The argument is Einstein is a little annoying to read, here's a simpler version (I put it on Wikipedia under mass-energy equivalence a long time ago. It also appears somewhere on Terry Tao's blog, and it's the right way to make the argument).
Consider a mass M which is stationary. It emits two identical pulses of light in opposite directions, each of equal energy E/2. After the emmission it still isn't moving.
Now consider the frame where the mass is moving with velocity v to the right (so the frame is moving to the left compared to the original frame). It emits two pulses of light in opposite direction, but now the light moving to the right is blueshifted and the light moving to the left is redshifted. The frequency shift is by the doppler shift formula, the shift factor is v/c, and the momentum is E/2c. So the right moving light has more momentum by Ev/2c^2 and left moving light has less momentum by Ev/2c^2, so the total momentum of the light is
Ev/2c^2
to the right. This means that the body has it's momentum reduced by this amount, but the velocity didn't change, so the mass went down. The amount of  the mass decrease, to conserve momentum, must be so that the new momentum of the body is:
P = (m - E/c^2) v
so the mass is decreased by the outgoing energy. Einstein uses the energy of the body, not the momentum, but Einstein's argument is equivalent to the one above by 4-vector nature of energy momentum. So Einstein's argument is fine, and there should be no controversy.
Let's consider the same experiment using sound in a nonrelativistic theory of a mass in air. A body emits two pulses of sound to the left and to the right with total energy E/2. Now we shift frames to where the body and the air are moving with velocity v. But now we can't conclude anything--- the air is moving, there is no relativity, the background the body is moving in isn't invariant to boosts.
But consider the momentum balance in the moving frame anyway: the sound moving to the right is blueshifted, the sound moving to the left is redshifted by v/c, but there is no imbalance in momentum, because in Galilean transformations, if the momentum of two things is balanced in one frame, it is balanced in every frame.
Einstein's argument requires the principle of relativity, but he used a small v limit where all the equations are nonrelativistic. The only relativistic equation you use in the whole thing is the relation between energy and momentum of light E=pc. That doesn't mean it's a nonrelativistic argument, because the principle of relativity is invoked multiple times, and the only reason the momentum doesn't balance in the moving frame is because the light is relativistic.
I should point out that Einstein already knew about photons, and for sure was considering two photon emission to come up with this. From the photon principle, he knew that frequency and energy have the same transformation. Otherwise it would be harder to know what the change in momentum energy under blueshifting/redshifting will be.
A: These thought experiments should always be made as simple as possible. Emitting in all directions is a needless complication. Consider Einstein's original derivation of E=mc^2 with only two light rays pointing in opposite directions (as Ron said above, this is a necessary assumption- it means the momentum of the light rays won't kick the bulb in one direction). You will find that, if you only put in the first term of the Relativistic Doopler Effect (the v/c term that -alone- is the classical doopler effect), it cancels out. Only the v^2/c^2 and higher terms remain. Using only the classical doopler effect produces no result. EDIT: The +/-v/c terms produces no net energy from the point of view of the moving frame because adding the energies results in the cancellation of +/-v/c terms (see Einstein's original paper), but these terms do produce a net change in the momentum of light in the right direction from the point of view of the moving system (See * below for how this net momentum is used in Rohrlich's derivation of E=mc^2)
On the cancellation of the v/c term: (http://www.science20.com/curious_inquirer/blog/was_einstein_wrong-90405)
On Einstein's original paper:(http://www.science20.com/curious_inquirer/most_famous_equation_physics_and_its_derivation-89948)
The Original Paper on E=mc^2:( http://www.fourmilab.ch/etexts/einstein/E_mc2/www/)
*BUT! There is a derivation of E=mc^2 using the classical doopler effect:(http://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence#Alternative_version) 
[The last comment was EDITED due to Alfred Centauri's comment below.].
A: If I understand your question correctly, you're asking if there is an argument using light for some kind of mass - energy equivalence in the context of non-relativistic mechanics.
I think you might be forgetting that the classical theory of light, Maxwell's equations, are relativistically covariant and it was this tension between Maxwell's theory and non-relativistic mechanics that led Einstein to his relativistic mechanics.
One way to see the problem is to consider that the Galilean transformation is recovered in the limit as the invariant speed $c \rightarrow \infty$.  So, from this "Galilean" angle, non-zero mass represents infinite energy.
What's going on here?  The speed of light is observationally finite so we have that in a non-relativistic context, the speed of light is not invariant.  If you go with that and, as Ron suggested, you actually use two oppositely directed pulses of light, the sum of the energy of the two light pulses is invariant.  So, there's no change in kinetic energy required to balance the books, no need for a mass - energy relation.
But, observationally, the speed of light is invariant so we have a conflict and a need for a new mechanics to do this problem.
A: Direct answer without arguing about your points (thanks for showing YouTube video which is origin of problem):


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*Classical physics can't answer such problems if you forcefully apply your point 2 (classical doppler effect can't cause lesser energy loss). With classical physics, we start like this: m is constant, so there must be change in v. If not, we need to in-invent physics. Mind it, particle mass is constant in classical physics and has higher priority than v.

*Special Theory of Relativity wasn't introduced to create E=mc^2. It was introduced because classical physics failed to explain relativistic phenomena due to wrong understanding of space and time scales. New "Spacetime" scale was introduced to measure everything with new way. And, with new scale, we found E=mc^2.
Remember: Relativistic Doppler effect needs time dilation for understanding (which requires Special Theory of Relativity). In that YouTube video, mass is motion dependent which is also from Special Theory of Relativity.
