# Question about Einstein's Original E=mc^2 derivation

1. Dec 27, 2009

### jwdink

Hi, I'm a non-physics student trying to use primarily Einstein's original 1905 essay to write a paper on the philosophical questions that E=mc^2 brings up. However, I've hit a snag which I can't seem to get past.

Einstein essentially uses what he's proven about the Doppler shift to show that a moving body emits a more intense light flash than does a stationary one. Since the internal energy of the body remains the same, he asserts that the increased energy of light must have come from the kinetic energy of the body-- and since KE=1/2mv^2, and v didn't change, therefore m changed.

My question: how can we assert the internal energy of the body does not change when the body moves? I suppose we could say that, "by definition, 'internal energy' means 'that energy which does not change in motion.'" But then we'd be questioning the equation for KE energy above-- we'd be saying that kinetic energy is 1/2mv^2 PLUS some other energy which doesn't necessarily relate to the mass, but increases with increased velocity. We could, I suppose, assert that the equation above is an approximation for very low velocities, but this runs us into trouble, because the Doppler shift only occurs at very high velocities! That is, if we want to say that KE only deviates from the classical equation by a negligible amount, we have to say that the light's energy only deviates a negligible amount as well.

My paper, unfortunately, depends on the validity of the original Einstein argument--NOT as a generalized or perfectly logical proof, but as an instance of science's empiricism telling us we're wrong about something that we didn't realize we could be wrong about ("mass" not being a constant which is synonymous with "stuff"). Is the Einstein argument an appropriate approximation which points us to the more general conclusion, which we can grant has not yet been proven? Or is it not even that?

Thanks!!

EDIT: Does it have something to do with the Taylor theorem expansion?

Last edited: Dec 27, 2009
2. Dec 28, 2009

### Cleonis

The most extensive discussion I know of the Einstein 1905 mass-energy treatment is the one by Kevin Brown:
http://www.mathpages.com/home/kmath600/kmath600.htm" [Broken]

A quote from that discussion:
"None of this is to suggest that Einstein was fully satisfied with his 1905 derivation, which was after all just a heuristic argument for a tremendously profound principle. In 1906 he published another argument in support of the proposition that a body’s inertia depends on its energy content."

E=mc^2 is a very general assertion. There are many kinds of energy, so asserting that all kinds of energy have inertial mass is an assertion with a very wide scope.

While E=mc^2 is a theorem of relativistic physics it seems it cannot be derived in one big sweep. It seems it can only be demonstrated for specific thought experiments, with the details coming out differently for different cases.

As I understand it, the 1905 assertion of E=mc^2 has a strong element of physics intuition. It's the kind of physics property that is either not valid or universally valid, but not something in between. Once the property is demonstrated for several thought experiment cases it is safe to assume it is a universally valid property.
And indeed to this day no counterexamples of E=mc^2 have been encountered

Cleonis

Last edited by a moderator: May 4, 2017
3. Dec 28, 2009

### jwdink

Yeah, Kevin Brown's discussion was helpful, but it was also the article that led me into this crisis.

"One might also challenge the premise that the internal energy of a body is independent of its state of motion. If, instead, we were to postulate that mass (for example) is independent of the state of motion, then the same argument would force us to conclude that the internal energy of a body varies with motion. However, the invariance of internal energy with motion is not really a postulate, it is a definition. We are certainly free to say the total energy of an object consists of two parts, one of which varies with motion and the other of which does not. This then returns us to consideration of the part that does vary with motion, and the assumption that it has the form mv^2/2 in the limit as v goes to zero. This form follows directly from the definition of inertial mass as the resistance to acceleration, and the idea that work equals force times distance."

This, I think, is not a sufficient reply. If we call "kinetic energy" "that energy which a body has due to its movement", then its not fair to say that it will have very little internal energy augmentation due to its small velocity-- because it ALSO has very little doppler intensity shift due to its small velocity. Either they're both negligible, or neither is.

This is all fair, and indeed, the best I could hope for. Unfortunately, merely stating it doesn't quite resolve the problem I'm having, which is that it's unclear that Einstein's initial paper even got as far as demonstrating mass is equivalent to this one type of energy.

4. Dec 28, 2009

### Cleonis

I suspected that.
When I reread Kevin Brown (after posting my reply) I noticed his formulation "However, the invariance of internal energy with motion is not really a postulate, it is a definition." Your turn of phrase was similar to that, so I wondered.

I'm struck by a similarity between dictionaries and scientific theories.

A dictionary is highly self-referential. All the words in the dictionary are described using other words that are in the dictionary. At worst you'll have two words, synonyms of each other, referring to each other for their meaning. The reason that a dictionary is not purely circular is that it reflects something that exists in the larger world: a living language.

Scientific theories are highly self-referential too. The concept of force is defined by Newton's third law; that is the only way to define the concept of force at all. Physics theories use a range of abstract concepts (such as force) as elements, and at the same time the only way to define those concepts is by invoking the theory.

Being highly self-referential, any theoretical demonstration borders on being circular.
This doesn't bother me. Our theories reflect the existing world, if we design a machine, such as a particle accelerator, we find that once it's build it performs pretty much as designed. Obviously if our theories would be detached from physical reality that wouldn't happen.

Cleonis

5. Dec 28, 2009

### jwdink

The question of whether definitions of fundamental concepts like "force," "energy," and "mass" will always be referring back to each other in a circular manner is central to my paper. I think the above problem is different, however. If we could really say that a body's energy content changed with its motion, then his proof wouldn't work.

I'm beginning to think we can't say this, however. I haven't quite finished the thought process yet.

6. Dec 28, 2009

### jwdink

Okay, I think I figured it out.

The body's internal energy increasing with motion can't explain the doppler shift. Imagine the inverse situation: two flashlights shine on opposite ends of an object. They're moving in the same direction, with the object in the middle-- so one's moving away, the other's moving towards. One will have more intense light, which I suppose we could give an ad-hoc explanation of as some "internal energy" which increased with the flashlight's motion--but the other one will have LESS intensity, even though the flashlight's moving. It turns out bare "motion" just ain't gonna cut it-- we have to talk about vectors, or, in other words, we have to talk about how the flashlight's are imparting momentum to the light. But if light has momentum, we've willingly put ourselves on the slippery slope towards giving all energy inertia, and therefore inferring that mass's only property--inertia--should just mean we call mass a type of energy.

Anyways, it's 5 am. I'm very tired, sorry if that's incoherent. Gnight!

7. Dec 28, 2009

### edpell

Why is this a problem. Yes, there is a deep symmetry between inertia and energy.

8. Dec 28, 2009

### jwdink

I never said it was a problem. Did you read my original post? It's a good slippery slope-- I just wanted to make sure it was valid.

Last edited: Dec 28, 2009
9. Dec 28, 2009

### edpell

Your phrase "slippery slope" led me to think you were unhappy.

I find it interesting that he uses a Taylor expansion and then drops all but the first non-canceled term. To me it seems the result is valid only for v=0. That is the energy of the rest mass is $$mc^2$$. If an object is moving it gets more complex.