Einstein's Rationale

[aggle-rithm]

I was recently re-reading David Bodanis' "E=mc squared". He explained somewhat vaguely how Einstein had discovered the link between matter and energy, and said that if you wanted to know more, you could go to Bodanis' web site and the full rationale for Einstein's conclusions would be explained.

Well, that information may have been there once, but it's not now. If I want to learn more about Einstein's equation, all that web site now tells us is to buy the book that led me to the site in the first place. I've already done that.

So my question is, how exactly did Einstein go from "the speed of light is always the same relative to the observer" to "matter and energy are interchangeable"?

Here is how I understand it, but I recognize that my understanding could be way off:

The speed of light is always the same relative to the observer. Therefore, the observer, and anything else that has mass, cannot reach the speed of light. Since objects cannot reach the speed of light, then there must be some physical mechanism that prevents massive things from reaching that speed. That mechanism is that, as an object approaches the speed of light, the energy that would normally go into accelerating it further starts to be converted into matter instead, adding to the mass of the object and making it more difficult to accelerate. As the object approaches the speed of light, its mass approaches infinity, making it impossible to achieve light speed.

Is this anywhere close to what Einstein was thinking? There seems to be a step missing somewhere; the mechanism just seems to have been arbitrarily constructed. I know that observations have now proved it correct, but how did Einstein get to this point just by using mathematics?

 

[Ziggurat]

Here is how I understand it, but I recognize that my understanding could be way off:

The speed of light is always the same relative to the observer. Therefore, the observer, and anything else that has mass, cannot reach the speed of light. Since objects cannot reach the speed of light, then there must be some physical mechanism that prevents massive things from reaching that speed. That mechanism is that, as an object approaches the speed of light, the energy that would normally go into accelerating it further starts to be converted into matter instead, adding to the mass of the object and making it more difficult to accelerate. As the object approaches the speed of light, its mass approaches infinity, making it impossible to achieve light speed.

What you mention here is what's commonly refered to as "relativistic mass". Relativistic mass is something that Einstein talked about, but nobody does any more. And the reason is because it's simply redundant with energy and completely unnecessary. You can do every calculation you could ever think of doing with an invariant (velocity-independent) mass, a.k.a. "rest mass". In fact, if a physicist just says "mass" without any qualifier, he's almost certainly talking about invariant mass. In that formulation, E=mc² is only true at zero velocity. For non-zero velocities, the more general equation E² = m²c⁴ + p²c², where p is the (3D) momentum. You can see that when p=0, this reduces to the famous equation, but now we don't need to vary mass when p != 0, or when dealing with photons (which have energy and momentum, but have no invariant mass).

Is this anywhere close to what Einstein was thinking? There seems to be a step missing somewhere; the mechanism just seems to have been arbitrarily constructed. I know that observations have now proved it correct, but how did Einstein get to this point just by using mathematics?

Yes, there are some missing steps. In fact, the actual derivation is rather mathematical. It has to do with the fact that in relativity, pretty much all your vectors of importance are actually 4-vectors (they have 3 spatial components along with an additional time component). In particular, momentum (which is still a conserved quantity in relativity) actually has a temporal component as well. And that temporal component is the energy (multiplied by a scale factor).

 

[Dr. Trintignant]

Is there "really" such a thing as rest mass, or is it just a useful approximation for composite objects?

For instance, the thermal energy in an object will show up as additional mass. You could fold that into rest mass, but really it's relativistic mass from the motion of the particles.

Likewise, I'd think, for the motion of subatomic particles: the electrons whirling around the nuclei, the nucleons jostling, the quarks jiggling, and so on.

I suppose that even once you subtract this motion out, there's still mass left. Any idea how much that might be?

- Dr. Trintignant

 

[elgarak]

I was recently re-reading David Bodanis' "E=mc squared". He explained somewhat vaguely how Einstein had discovered the link between matter and energy, and said that if you wanted to know more, you could go to Bodanis' web site and the full rationale for Einstein's conclusions would be explained.

Well, that information may have been there once, but it's not now. If I want to learn more about Einstein's equation, all that web site now tells us is to buy the book that led me to the site in the first place. I've already done that.

So my question is, how exactly did Einstein go from "the speed of light is always the same relative to the observer" to "matter and energy are interchangeable"?

Here is how I understand it, but I recognize that my understanding could be way off:

The speed of light is always the same relative to the observer. Therefore, the observer, and anything else that has mass, cannot reach the speed of light. Since objects cannot reach the speed of light, then there must be some physical mechanism that prevents massive things from reaching that speed. That mechanism is that, as an object approaches the speed of light, the energy that would normally go into accelerating it further starts to be converted into matter instead, adding to the mass of the object and making it more difficult to accelerate. As the object approaches the speed of light, its mass approaches infinity, making it impossible to achieve light speed.

Is this anywhere close to what Einstein was thinking? There seems to be a step missing somewhere; the mechanism just seems to have been arbitrarily constructed. I know that observations have now proved it correct, but how did Einstein get to this point just by using mathematics?

I guess your problem (and that of most other non-physicists and non-mathematicians) is the "just math" part: You do not understand math, and think it is an artificial construct. It's not.

Because Einstein was 'just' doing math. Nothing else. That's it. He was not thinking anything.

If you formulate "c = const in all reference frames" mathematically, and plug them into physical equations that have been shown to be right repeatedly (such as the Maxwell Equations), the "E=mc^2" just pops out.

Most other physicists had stopped along the way, and thought about the implications that they couldn't believe because they went against everyday experience. They thought they had made an error somewhere in the math.

Einstein trusted his math skills, and followed the math to its logical end, and simply ignored the implications until there was nothing else to calculate. Just then he set down, and discussed the implications with peers, and not while doing the math as most others.

 

[Ziggurat]

Is there "really" such a thing as rest mass, or is it just a useful approximation for composite objects?

It's not an approximation at all. It's exact, even for composite objects.

For instance, the thermal energy in an object will show up as additional mass. You could fold that into rest mass, but really it's relativistic mass from the motion of the particles.

Part of the thermal energy will indeed be increased kinetic energy for the particles, but part of it is also increased potential energy (as particles spend more time away from their minimum energy configuration). But figuring out the division between those categories is actually irrelevant. The important thing is that this increase in energy, UNLIKE the center-of-mass kinetic energy of a moving system, is reference frame invariant. If you heat up an object, you add energy to it regardless of the reference frame you look at it from. If you speed up an object, it will only gain energy in some frames, in other frames it will lose energy. That's why the term "invariant mass" is used, and why I find it preferable to the term "rest mass".

 

[Dr. Trintignant]

If you heat up an object, you add energy to it regardless of the reference frame you look at it from.

Hmmm. I will have to think about it more to convince myself that it's true, but it seems more reasonable now.

I'm thinking in terms of a system of two atoms--say, an H2 molecule. You heat it up. From the reference frame of the center of mass, each atom will now be jiggling with a certain average displacement, depending on the temperature. From the reference frame of one atom, that atom will have less energy and thus a lower mass, but the other atom will be jiggling with a larger average displacement. I think I can buy that the total energy comes out the same no matter how you do the math.

- Dr. Trintignant

 

[Dr. Trintignant]

There seems to be a step missing somewhere; the mechanism just seems to have been arbitrarily constructed.

Been thinking about this a bit more...

Ziggurat is certainly correct in that the actual derivation is somewhat more calculated, but just the same I think there is a thought experiment that you can use to convince yourself that it's correct.

First, you must prove to yourself that time slows down for moving observers. There is a simple way to derive Lorentz's equation with a geometrical argument (assuming a constant SoL for all observers, of course). I can show you this argument if you like. So now, why does mass seem to increase as time slows?

Imagine two pool balls (A and B), each moving toward each other on the Y axis at 1 m/s. Due to conservation of momentum, when they collide, they will bounce apart, again going at 1 m/s. Basic stuff.

Now imagine that ball A also has an X component to the velocity--again, 1 m/s. And it is positioned such that the collision happens in the same place. What happens?

From the perspective of ball B, nothing has changed. There was a momentum transfer in the Y axis, and again it finds itself moving at 1 m/s in the opposite Y direction after the collision. Ball A is still going at 1 m/s in X.

Ok, one step further. Ball A starts off with no X component, but again is traveling at 1 m/s in Y. It's suddenly accelerated to 86.6% of the SoL. Still, as far as its concerned, it's still traveling at 1 m/s in Y. It was never pushed in a Y direction, and due to conservation of momentum, it must be going at the same speed.

But things look different to an outside observer. Ball A's clocks have slowed down by a factor of 2, and instead of traveling 1 m/s in Y, it's actually only going 0.5 m/s. Where did that momentum go?

The answer is relativistic mass. Since momentum is just p=mv, if v goes down by a certain factor, mass must go up by the same factor if we expect to conserve it. Remember that ball B doesn't care how fast ball A is going in the X direction. It only cares about the Y component of ball A's momentum, and that number better stay the same if the behavior is to remain the same.

So relativistic mass (which Ziggurat correctly notes is really just energy) must go up by the same Lorentz factor that time goes down by, if conservation of momentum is to hold.

- Dr. Trintignant

 

[aggle-rithm]

I guess your problem (and that of most other non-physicists and non-mathematicians) is the "just math" part: You do not understand math, and think it is an artificial construct. It's not.

Because Einstein was 'just' doing math. Nothing else. That's it. He was not thinking anything.

Just to clarify, by using the phrase "just using mathematics", I wasn't disparaging mathematics, I simply meant that Einstein was a theoretical rather than an experimental physicist. There were no direct observations on which he based his conclusions.

 

[aggle-rithm]

Been thinking about this a bit more...

Ziggurat is certainly correct in that the actual derivation is somewhat more calculated, but just the same I think there is a thought experiment that you can use to convince yourself that it's correct.

First, you must prove to yourself that time slows down for moving observers. There is a simple way to derive Lorentz's equation with a geometrical argument (assuming a constant SoL for all observers, of course). I can show you this argument if you like. So now, why does mass seem to increase as time slows?

Imagine two pool balls (A and B), each moving toward each other on the Y axis at 1 m/s. Due to conservation of momentum, when they collide, they will bounce apart, again going at 1 m/s. Basic stuff.

Now imagine that ball A also has an X component to the velocity--again, 1 m/s. And it is positioned such that the collision happens in the same place. What happens?

From the perspective of ball B, nothing has changed. There was a momentum transfer in the Y axis, and again it finds itself moving at 1 m/s in the opposite Y direction after the collision. Ball A is still going at 1 m/s in X.

Ok, one step further. Ball A starts off with no X component, but again is traveling at 1 m/s in Y. It's suddenly accelerated to 86.6% of the SoL. Still, as far as its concerned, it's still traveling at 1 m/s in Y. It was never pushed in a Y direction, and due to conservation of momentum, it must be going at the same speed.

But things look different to an outside observer. Ball A's clocks have slowed down by a factor of 2, and instead of traveling 1 m/s in Y, it's actually only going 0.5 m/s. Where did that momentum go?

The answer is relativistic mass. Since momentum is just p=mv, if v goes down by a certain factor, mass must go up by the same factor if we expect to conserve it. Remember that ball B doesn't care how fast ball A is going in the X direction. It only cares about the Y component of ball A's momentum, and that number better stay the same if the behavior is to remain the same.

So relativistic mass (which Ziggurat correctly notes is really just energy) must go up by the same Lorentz factor that time goes down by, if conservation of momentum is to hold.

- Dr. Trintignant

This helps a great deal. Thanks!

 

[Dr. Trintignant]

This helps a great deal. Thanks!

Glad to help out!

I had actually started trying to explain it in terms of an oscillating mass on a string. My thought was that there needed to be some explanation for why the oscillator slows, and that increased mass would be it (since the frequency of a mass-spring system is proportional to √(1/m)).

But the math didn't turn out right, and as it turns out, the mass is only partially responsible for the slowing oscillation. The other factor was the spring constant, which goes down with the Lorentz factor. That was a bit harder to explain, so I went for the momentum method.

I understand your desire for a "bottom up" explanation, since I'm the same way. I've been slowly reading through Feynman's Lectures on Physics, and think he does a great job in this regard, deriving basic principles from simple examples of physical systems.

Still, at some point, bottom-up explanations become untenable, and you have to just accept that the equations are right. It's a bit like explaining why perpetual motion machines don't work. It can be a lot of work finding the specific flaws in them. Although it might be fun to do that as an exercise, it's easier and just as correct to say they violate the laws of thermodynamics.

- Dr. Trintignant

 

[Cuddles]

Is there "really" such a thing as rest mass, or is it just a useful approximation for composite objects?

Rest mass is somewhat similar to absolute zero. We know what it is, but there's no way that anything can ever reach it. Rest mass does really exist, but only in the sense that we can say "If this particle were at absolute zero, its total mass would be its rest mass".

 

[Ziggurat]

Rest mass is somewhat similar to absolute zero. We know what it is, but there's no way that anything can ever reach it. Rest mass does really exist, but only in the sense that we can say "If this particle were at absolute zero, its total mass would be its rest mass".

This is incorrect. Rest mass DOES increase with temperature. I've already detailed why above. It is a reference-frame invariant quantity (which is what "rest" refers to, and why I prefer the term "invariant mass"), but there's no reason it needs to be temperature-independent (and it isn't). It is, therefore, not at all like absolute zero temperature. Every object always has a rest mass, regardless of what temperature it's at, so there's never a question about whether or not an object has "reached" it.

 

[Cuddles]

This is incorrect. Rest mass DOES increase with temperature. I've already detailed why above. It is a reference-frame invariant quantity (which is what "rest" refers to, and why I prefer the term "invariant mass"), but there's no reason it needs to be temperature-independent (and it isn't). It is, therefore, not at all like absolute zero temperature. Every object always has a rest mass, regardless of what temperature it's at, so there's never a question about whether or not an object has "reached" it.

No. It is impossible for an object to ever have zero energy. Therefore it always has greater mass than its rest mass. Whether the rest mass can also change is irrelevant.

 

[Ziggurat]

No. It is impossible for an object to ever have zero energy. Therefore it always has greater mass than its rest mass.

Your second sentence simply does not logically follow from the first sentence, and it's false. I suspect that you're caught up in the notion of "rest", and the idea that because of thermal energy nothing is ever at rest. Well, that's irrelevant. "Rest mass" is not a particularly good term. The better, and more precise, term is "invariant mass". And whether or not components of an object ar jiggling around or not, that object has an invariant mass which is well defined. And if we pick the appropriate reference frame, then the relativistic mass will exactly match the invariant mass. That's how these things are defined.

It's a little clearer if you can think of the 4-vector picture: the relativistic mass is the time component of the momentum 4-vector, and the invariant mass is the length of the momentum 4-vector (using the Minkowski metric). When we choose a frame in which the momentum 4-vector only has a time component (the "rest frame" of the object), the two are exactly equal. The fact that the total momentum 4-vector for a composite object may be a sum of vectors pointing in different directions (and which may have space components even when their sum doesn't) will not change any of that.

 

[Dr. Trintignant]

What (mostly) cleared up the concept for me was the realization that invariant masses are not additive: the invariant mass of a system is not the sum of its component invariant masses. The energy of the system must also be included.

- Dr. Trintignant

 

[Ziggurat]

What (mostly) cleared up the concept for me was the realization that invariant masses are not additive

Indeed. This is equivalent to saying that the length of a vector is not the sum of the length of its components. In Euclidean geometry it can be less, but with the Minkowski metric it can be more.