The New York Times has an article by Brian Greene, a professor of physics and mathematics at Columbia about Einstein’s famous equation E=mc². In it he says:
The standard illustrations of Einstein’s equation – bombs and power stations – have perpetuated a belief that E = mc² has a special association with nuclear reactions and is thus removed from ordinary activity.
This isn’t true. When you drive your car, E = mc² is at work. As the engine burns gasoline to produce energy in the form of motion, it does so by converting some of the gasoline’s mass into energy, in accord with Einstein’s formula. When you use your MP3 player, E = mc² is at work. As the player drains the battery to produce energy in the form of sound waves, it does so by converting some of the battery’s mass into energy, as dictated by Einstein’s formula. As you read this text, E = mc² is at work. The processes in the eye and brain, underlying perception and thought, rely on chemical reactions that interchange mass and energy, once again in accord with Einstein’s formula.
I only did high school science, but I’m sure I remember learning the exact opposite of this claim, that chemical reactions like combustion leave mass and energy unchanged, only converting some of the chemical energy in the fuel into kinetic energy, and some into heat, with a net increase in entropy. Only nuclear reactions, I was taught, converted mass to energy. Wikipedia seems to back this up, though it isn’t absolutely unambiguous.
Can anyone set me (or, less plausibly, Greene) straight here?
fn1. As an aside, I also remember reading that a more correct version would be E=M. The term in c² just reflects a poor choice of units in the metric system. But maybe that’s wrong too.
John Quiggin 
Greene is a famous string theorist (who is erudite and articulate, both in person and in his books). Who knows what strange ideas string theorists believe!
The comment in your footnote about units of measurement only makes sense if you assume the speed of light c is a constant. This assumption has been seriously challenged recently on theoretical grounds. Empirically, JH Rush, in a paper in “Scientific American” in 1955 (vol. 193, pp. 62-67) presented a graph which appears to indicate that the speed of light changed around 1906. This could have been an experimental artefact, or it may indicate a true change.
John, I did a sub-major in Physics, but I don’t pretend to remember things like this. A couple of URLs, particularly this Stanford one, reveal that it’s more accurate to consider conservation of the combination of mass and energy. The amount of energy is tiny for chemical reactions, which is why the relationship can be simplified to conservation of mass for school science.
By the way, that is a brilliant piece of writing by Brian Greene. Explaining relativity in a newspaper piece is very ambitious.
I believe Greene is right.
The difference between chemical reactions and nuclear reactions is that chemical reactions derive their energy from the electromagnetic force, whereas nuclear reactions derive their energy from the strong force.
A chemical reaction that releases energy does so by rearranging the constituent atoms of the molecules involved in the reaction into a lower energy state (ie, the molecules that come out of the reaction have less configuration energy than the molecules that go into the reaction).
So even though the atoms haven’t changed in the reaction, the molecules have, and their energy is lower. If you could weigh them, they’d have lower mass.
Nuclear reactions work the same way, but instead of rearranging the atoms in molecules, they rearrange quarks in atoms. The quarks don’t change but the atoms do, and the atoms that come out the end of a nuclear reactions are in a lower energy state than the atoms that go in to start with. Again, if you weighed them, they’d weigh less by an amount equal to the energy difference.
The reason nuclear reactions generate so much more energy than chemical reactions is that the energy difference between the atoms before and after a nuclear reaction is huge compared with the energy difference between the molecules before and after a chemical reaction. And that is because the strong force (which binds atoms together) is so much stronger than the electromagnetic force that binds molecules together – it’s not called the strong force for nothing.
I cannot add anything sensible here – but reading that article is highly recommended.
eg The speed of light for me has, ’til now, been exactly equal to unimaginably fast. Greene’s measure that the speed of light is to travel around the earth seven times in one second is, I now see, far more accurate.
It might be worth adding that with nuclear reactions you can look up the atomic weights of the atoms involved and calculate the loss of mass in the reaction. The loss of mass is normally so large that this calculation easily shows a definite loss. I remember seeing this calculation done in a chemistry text book for some particular reaction. Of course, as a previous poster mentioned, in an (exothermic) chemical reaction there is a loss of molecular weight but it’s extremely small. (I remember seeing this in a chemistry text book too.) It’s not exactly correct to calculate the molecular weight of a molecule by adding up the atomic weights of its component atoms but it’s extremely accurate.
GDP – you are right, except that nuclear reactions rely (in the main) for the weak force for their strength. Fission involves releasing the energy used to bind the protons and neutrons in the nucleus of the atom, not breaking down the protons and neutrons themselves, which would be a strong force reaction.
The weak force is stronger than electromagnetism, but weaker than the strong force.
There will be some strong force liberated in a nuclear detonation, but that is a side effect, not the main point. A full scale liberation of the strong force would be a truly mighty energy release.
Greene is, therefore, right, but the amount of mass lost is really tiny as the amount of energy released is small in a chemical reaction that it would be easy to miss the loss entirely.
Four momentum is conserved. That’s a tensor with the three ordinary dimensions holding momentum, and the fourth holding kinetic energy, adjusted for the right units.
That adjusting for units is why you can’t get rid of c by changing your basis – you might get a constant of unity that way, but you would need to change the dimensions (which is what the tensor manipulations will do for you when you change the frame of reference). Economists know about differential equations (sort of), so it might help to think of homogenous equations to see how the dimensions matter.
As for losing mass when burning chemical fuel – of course you do. It’s just not the rest mass of atomic particles, though. Sitting there as chemical energy, it showed up as distortions of the energy fields associated with electron orbitals, and so affected the measured mass.
When you go all out nuclear, you find the same proportions of energy to mass released – only this time, it comes from the rest mass unless you are measuring in a different frame of reference. But the tensors sort that out for you. One result of conserving four momentum is that when an electron and a positron annihilate, you don’t get just one photon – that would breach the conservation laws involved.
PML, That sounded good. I just hope it is blinding with science rather than baffling with ….
AR: I am pretty sure nuclear fission relies on the strong force. The weak force is, well, too weak and short-ranged (it is on a par with the electromagnetic force. In fact they are coupled – and are described together as the electroweak force).
Wiki has a good discussion on binding energy:
http://en.wikipedia.org/wiki/Binding_energy
And the standard model if you want to wade through it (the standard model describes three of the four major forces: strong, weak and electromagnetic – there’s a nobel prize waiting for the first person to work out how to get gravity in there):
http://en.wikipedia.org/wiki/Standard_model
GDP Says: October 1st, 2005 at 7:55 pm
I believe that GDP is right, although my physics and chemistry is only high-school level and not very good at that.
Greene is correct to imply that chemical (ie molecular) reactions are on a par with nuclear reactions in that they entail mass-energy equivalence, with relativistic effects at the limit. But it was not very helpful of Greene to invoke Einsteinian relativity to explain mundane events.
Most activity that goes on at terrestial scales – small masses using chemical energies at low speeds – can be explained using Newton-Maxwell-Faraday’s theories. That is why astronomers, chemists and engineers were able to get on with the job well before Einstein came on the scene.
Einstein theory of relativity was a more general explanation of gravity than Newton’s celestial mechanics. Since gravity explains how the whole universe, including nuclear and molecular entities, hangs together it is trivially true that Einstein’s theory explains everyday physical activity. Thus Green’s use of the phrase “in accord with Einstein’s formula” is somewhat wiggly .
As GDP says, the effects of Einsteian relativity can still be detected in even in these events. But they are of negligible significance at low speeds, small masses etc. Obviously at such small masses the force of gravity will be extremely weak.
Roughly speaking, the Strong Nuclear (SN) force governs activity within the atom whereas the Electro-Magnetic (EM) force regulates activity between atoms. The SN force is much stronger than the EM force because it requires more energy (per unit of mass?) to get a given atom to hold itself together than it does to get it to hang out with another atom.
Green is right.
Anything that emits energy loses mass. Anything that accepts energy gains mass. If I burn gasoline in oxygen, the system emits energy as the various bonds break and reform between its constituents. It loses mass. If however I add up all the atoms in a can of gasoline and roomfull of air that number will not change. It will be the same before and after the fire; every atom, be it now in smoke or soot, will be accounted for and will weigh the same as it did before. This is because the mass to energy conversions in this reaction is not concerned with the atoms themselves rather it takes place in the bonds between atoms.
In a typical nuclear reaction however, the number of atoms will change or else the mass of these atoms will change (isotopes).
There is several orders of magnitude more energy tied up as mass within atoms than there is within the bonds between atoms. Hence destroying those atoms in a nuclear reaction typically produces several orders of magnitude more energy than destroying and reforming the bonds as is done in a chemical reaction.
Suggested reading ‘E=mc2, Biography of the worlds most famous equation’ by David Bodanis, a UK academic.
Bodanis goes through the development of the understanding of Mass (Lavoisier) and Energy (Faraday) and their conservation as separate urelated concepts, and their linkage through Einstein’s thinking on relativity and the speed of light.
GDP,
If you were using the strong force it would not be nuclear fission, but protonic and/or neutronic fission. Because it is the nucleus breaking up I believe it must be the weak force that is being liberated.
The weak force is a lot stronger than electromagnetism, but of longer range than the strong force.
I do not think I will win that Nobel – but you never know…
http://davidbodanis.com/books/emc2/index.html
Link to the website for the above mentioned book …….. highly recommended for interested ‘non-scientists’ …… an all time personal favorite.
I’d also add that E=mc^2 has to be dimensionally consistent, and so the formular cannot be E=M as the units of E and M are different. I havnt checked this carefully but I’m confident from my feel of physics (from ages ago!) that it must be a valid remark.
Also I want congratulate Q for posing the topic in such a modest inviting way that itled to a facinating thread.
Maybe Ill go back to Feyman’s Red books to find some other Gems.
Suppose you have a wind-up toy. Is it heavier wound-up than unwound?
I think Greene is winding up someones toy.
My degree is in Electrical Engineering so I did entry level physics at university (plus loads and loads of abstract maths).
My vote is with Quiggin. When mass actually converts to energy we are talking nuclear.
Terje Petersen Says: October 4th, 2005 at 6:39 am
I dont think so. With respect to Terje Petersen, his qualifications, though worthy, look pretty shabby in comparison to Brian Greene’s. In any case Argumentum ad Qualification is not conclusive.
James Newton-Thomas argues that chemical-molecular (as well as physical-nuclear) reactions lead to the conversion of mass to energy in accordance with Einstein-ian principles. I think that this is the correct answer (disclosure: I have personal and professional associations with JNT).
abb1 Says: October 4th, 2005 at 4:50 am also correctly implies that gross changes in the energy state of an entity can have subtle effects on its rest mass, per Einstein.
This is because Einstein’s formula implies that the (energetic) EM force that bonds molecular entities to each other is equivalent to mass. These molecular bonds are destroyed and reformed in chemical reactions that can result in the system losing gross measurable mass, as described by E=mc2.
This result could not have been predicted by Netwton, Faraday, Maxwell, Boltzmann, Helmholtz et al because of the very tiny scale of activity at which these effects are noticeable. And if these molecular transformations did not occur according to Einstein-ian principles then we would be living in a very different universe from the one we observe with our Newtonian-Helmholtzian spectacles, or not alive at all.
Therefore Greene is correct to say that chemical reactions follow Einstein-ian principles.
PS I have no formal training in physical theory beyond Year 10. So these arguments should be swallowed with a grain of salt, as when a self-taught flyer is winging it he is bound to crash and burn at some stage.
I don’t think there was any diirect empirical evidence for E=mc2 at the time Einstein derived it. Part of Einstein’s genius was that he arrived at the result quite indirectly by assuming that the speed of light was independent of the reference frame of the observer (ie, the speed of light is the same for you and me, even if we are travelling in different directions at great speed. Think about it, it is very unintuitive.)
There _was_ empirical evidence for the constancy of the velocity of light, namely the the Michelson Morley experiment which attempted to measure the velocity of the earth with respect the “aether” (the medium through which light was supposed to propogate). The experiment established that the earth did not move at all relative to the aether, which allowed the aether to be discarded altogether in Einstein’s theory.
Even then, Einstein did not need the Michelson Morley experiment. The constancy of light follows from his more fundamental assumption – the principle of “special relativity” – that the laws of physics are the same regardless of the velocity of the observer (provided you are in “uniform” motion – ie not accelerating. To handle accleration you need general relativity). Since the laws of physics are the same for all observers regardless of motion, the laws of electromagnetism (Maxwell’s equations) must therefore be independent of motion. But you can derive the speed of light from Maxwell’s equations; hence the speed of light is constant.
So I would disagree that the result could not have been predicted by earlier physicsts – in fact Lorentz worked out the transformations of special relativity before Einstein, but he did not take the extra step of discarding the aether and formulating the principle of special relativity, which explained why the transformations had to be that way. Those observations required Einstein’s unique brand of pure physics genius.
AR: the atomic nucleus is held together by the strong force not the weak force. The quarks in the different protons and neutrons still bind to one another even though they are not in the same particle.
abb1: correct – the wind up toy is heavier by an amount E / c2, where E is the energy you use to wind it up. To get some perspective, the spring in the toy probably stores on the order of a joule. c2 is approximately 10^17, so the toy will weigh about 0.00000000000001 grams more (10^-14) wound up than unwound. Probably not going to be noticable on your bathroom scales 🙂
Some of the explanations in response to JQ’s first question leave me vaguely unsatisfied. The scientist in me prefers to cite some names and numbers.
Einstein himself, in the second of his 1905 papers, had looked to the possibility of testing his theory using certain “…bodies whose energy content is variable to a high degree (e.g., radium salts)â€?. However, according to Abraham Pais (in Subtle is the Lord: The Science and the Life of Albert Einstein), the first person to “remark that the mass-energy relation bears on binding energy was Planckâ€? and that in 1907 Planck “…estimated the molecular binding energy for a mole of water. This amount (about 10^-8 g) was of course too small to be observed but at least it could be calculated. A quarter of a century had to pass before a similar estimate could be made for nuclear binding energy.â€? I.e., the first consideration was done in terms of chemical processes.
The outstanding Soviet physicist Lev Landau co-authored a small non-mathematical book (L. D. Landau and Y. Rumer, What is the Theory of Relativity, Peace Publishers, Moscow, 1965), presumably intended either for schoolchildren or interested adults, in which the authors explained mass-energy equivalence as follows. “All force applied to [a] body, and increment of body energy, increases its mass. This is exactly why a body has greater mass when heated, why a spring has greater mass when it is compressed. True, the coefficient of proportionality between the change of mass and change of energy is insignificant: to add a gram of mass of [sic] a body we should have to apply 25,000,000 kWh of energy.� The authors also gave the following numerical example of the mass change during a chemical process: “If we burn a ton of coal in a closed furnace, the products of combustion will have a mass one three thousandth of a gram less than the original coal and oxygen. This missing mass is carried away by the heat generated in the process of burning.�
By the way, “E=mc^2� is not quite the correct formulation, though this form of the equation is so ubiquitous in mass culture that complaints about its accuracy are going to seem a waste of time. The version used by Einstein in the second of his 1905 papers on special relativity in the Annalen der Physik (vol. 18, p.639) actually had the following form: (delta)E(nought) = (delta)mc^2. That is, the change in rest energy equals the change in the product of mass and c squared. “Mass� here is Newtonian mass. The problems that even great physicists can introduce for the rest of us and our understanding of “mass� by using the incorrect popular version of the formula are well, and amusingly, summarized by Lev Okun in an issue of Physics Today (June, 1989).
Oops…. sorry…the tiny type size is what can happen when you format in Word with a superscript and copy & paste into the browser window.
When you lift a tennis ball from the floor and place it on your desktop, does it increase its mass?
Thanks.
Excellent question. And I have to confess I am not sure of the answer.
You can’t answer the question with special relativity alone, because now gravity is involved and special relativity only applies to bodies in uniform motion (no acceleration and no gravity). Special relativity is what gets you E=mc2.
So you need general relativity, which incorporates gravity and acceleration.
But here things are not so obvious to me (my memory of undergrad physics is somewhat fuzzy). But I’ll have a crack anyway.
The key insight of general relativity is that gravity is caused by mass bending space(time), not by masses actiing on other masses at a distance (as in the Newtonian theory). Now general relativity reduces to special relativity as the gravitational force goes to zero, so since mass and energy are equivalent in special relativity, it must be the case that energy bends spacetime just like mass does.
So far so good. But the energy of the ball sitting on the desk is caused by gravity itself. It is gravitational energy if you like. This is different from the energy in the wind up spring of the toy, which at bottom is electromagnetic energy (it is caused by the electromagnetic forces between the atoms in the steel spring).
So it seems that to answer abb1’s question we need to answer whether gravitaional energy acts like the “E” in “E=mc2”. And it seems to me that sometimes gravity probably does act that way – eg gravitational waves – and sometimes it may not (general relativity would be a highly nonlinear theory if all gravity acted back on itself through the gravitational energy).
So I guess I am stuck. Anyone else know? Am I missing something obvious?
Are you saying that gravitational force is fundamentally different from electromagnetic force? Why would it be, isn’t there something called ‘graviton’, at least in theory?
Back to the wind-up toy.
Do all observed energy emissions have to be acounted for by a conversion of matter into energy? What about the fact that energy can change in form? Just because the elastic potential energy can be calculated, then substituted into E/c2=m, does that mean there had to be an actual conversion of matter into energy when this energy was tranferred to the spring? The equation gives an energy-matter relationship. Is that relationship necessarily a mechanism as well?
What’s wrong with scenario? I compress a spring in the wind-up toy with my fingers. The elastic potential energy stored in the spring is the result of a series of energy tranformations that began with a nuclear reaction in the sun in which matter IS converted to electromagnetic energy. That energy is transformed into chemical potential energy when it is stored in the bonds of a glucose molecule during photosynthesis. This glucose then undergoes cellular respiration in the muscle-cells of my finger and the chemical energy is then transformed into mechanical energy when the muscles in my finger contract and wind the spring.
Also, it occurs to me that any carbon atom (even one of the ones in that ton of coal mentioned in psdoidge’s post) would have been involved in “zillions” of reactions since the big bang. If every exothermic reaction involves a loss of mass, and since electrons are not very massive, wouldn’t we be seeing some “worn down” carbon atoms or carbon atoms with “sluggish” electrons in the universe by now??
abb1: to be precise, gravitational _energy_ is different from electromagnetic _energy_ (and all other forms of energy). The reason is that gravity is the backdrop that defines how things move through space and time – it is the fundamental substrate of the universe. Or put another way, before you can discuss the laws of physics relating to the movement of bodies, you have to define what they are moving through; what it means to “move”. Gravity, which defines the shape of space and time, does that for you. This is at the heart of why gravity is yet to be unified with all the other forces.
Get thinking: that Nobel prize is still available 🙂
kam: an exothermic chemical reaction transorms molecules, not atoms. That is, the atoms in all the products of an exothermic reaction are the same as the atoms that went in. What has changed is the molecules – the atoms are bolted together in different (and lower energy) configurations afterwards. So those “worn out” carbon atoms haven’t changed, they’re just living in different molecules than they were originally.
My doubts are similar to what Kam said, only I feel that kinetic energy must be playing a role here.
Take this example: every website I’ve read in the past few days says that when you apply heat to an object its mass increases. When I went to school they told me something different: when you heat an objects the molecules in it start moving faster and this is how energy is accumulated. There’s no need for mass change – momentum is changed.
E^2 = (MC^2)^2 + (PC)^2
P is momentum, proprtional to speed.
Wouldn’t it make more sense to define ‘mass’ as constant for the object – whether it’s heated or squeezed or not, and attribute changes in potential energy to changes of momenta of particles inside the object? Same thing for the chemical reactions.
The only occurrence of mass being converted to energy I can imagine is when a particle is hit by an anti-particle and both of them disappear; or when two opposite particles are created from nothing but energy. I don’t think anything like this happens in chemical reactions or mechanical transformations.
abb1:
It’s erroneous to think in terms of mass being converted into energy. That’s not what happens. Take an electron and a positron. In this instance, two particles each of mass me, get converted into a two photons, which also each have a mass of me.
The total mass of the system is unchanged. It was 2me beforehand, and 2me afterward. Say you had the electron/positron pair inside a sealed container, the mass of which you could very, very accurately measure. You would find that the mass was x before they annnihilated, it was still x after they annihilated and the photons were bouncing around inside the container, and it was still x after the photons were absorbed by the container and converted into the kinetic energy of particles in the container.
Regarding the earlier question about gravitational potential energy: when you put the ball on the table, the mass of the ball doesn’t change, because the potential energy is not stored in the ball. It’s stored in the gravitational field. The inertial mass of the gravitational field would be the thing that changed.
Two analagous situations:
a) there’s no gravity, but the ball is attached to the floor by a spring. The mass of the spring changes, because that’s where the energy is stored, not the ball.
b) there’s no gravity, but the ball is magnetic and is attracted to a magnetic floor. The mass of the magnetic field changes, not that of the ball.
The subscripts appeared properly in the preview window, but not in the post.
Test: me
The reason I used the word “atoms” is because I was referring to the “ton of coal” in a previous post. Either way, if a molecule is composed of atoms and the atoms have not chnged except in of which molecule they are a part, then how can mass be gained or lost?
More importantly, the question for me is: If matter is being converted to energy in an exothermic process, where does that matter come from to begin with? Or, from the perspective of an endothermic process, where absorbing energy causes the mass of something to increase, what is the source of that matter?
Are we saying that simply because E=mc2 provides us the ability to calculate a change in mass from a measured energy change, that EVERY energy change MUST be a mass change?
This is an irrelevant question. Particles with rest mass (e.g. atoms) can turn into particles with no rest mass (i.e. photons), but which still have exactly the same mass. The process can also work in reverse. Asking where the mass came from originally is a distraction.
No. We are saying that the things that we call energy and mass are essentially the same thing. The mass never changes. The energy never changes. The energy/mass can move from one place to another, say from my muscles to the spring in a wind-up toy, or from an atom bomb into Nagasaki, but the total mass and the total energy remains unchanged.
SJ,
but photons have no mass:
Photons are pure energy. Why are you saying that mass is the same after the annihilation?
No, that wikipedia article refers to “intrinsic mass”, which is usually referred to as “rest mass”, i.e. the mass that the photon would have if it was not moving.
But all of the photons we know about move. They move at exactly the same speed, c, the speed of light. They posess energy which is independent of their speed of movement. A photon’s energy depends on something else.
I’m saying that simply because that’s the way things are. In the sealed container example I gave above, you cannot tell by external measurement whether the annihilation has taken place or not.
You say: The energy/mass can move from one place to another, say from my muscles to the spring in a wind-up toy, or from an atom bomb into Nagasaki, but the total mass and the total energy remains unchanged.
I’m not arguing that the total energy/mass of a system is not conserved. I’m asking, once the spring is wound, does it contain more matter than it did before, i.e. does it actually weigh more? More generally, are there any real, documented experiments involving matter that has increased or decreased in mass because it has undergone any energy change (i.e. been heated up or cooled down) or is this a mathematical exercise without experimental verification?
The spring contains no more matter after it is wound than before. But it contains more energy. That is, assuming the spring is completely sealed and doesn’t leak atoms, winding it does not add any material (atoms) to or remove any material from the spring. But it does move the atoms of the Spring into a higher energy state. That energy is stored in the electromagnetic fields between the atoms. And therefore the spring weighs more by an amount equal to E / (speed of light squared).
When you read that the atomic weight of an atom is a particular value, that is what the atom weighs when it is “unbound” or “free” – ie not interacting with any other atoms. Once atoms interact to make molecules, crystals, metal alloys etc, they are no longer free – they interact with all the other atoms in the molecule, crystal, metal alloy, etc.
That interaction is mediated by the electromagnetic fields of the atoms, and the energy of the interacting group of atoms can be more or less than the sum of the energies of all the free atoms. That’s because the electromagnetic fields of the atoms when they interact are different from when they are free. Roughly, the difference in energy between the interacting and free states is stored in (or subtracted from) the electromagnetic fields. The difference between the energy of the interacting group and the sum of the free energies is usually called the binding energy. And the interacting group’s mass weighs more or less than the free group by an amount equal to the binding energy / (speed of light squared).
Now, if you split all the atoms apart, they’ll still have the same free energy/mass. But if the binding energy is negative (that is the interacting atoms have less energy than the free atoms), you’ll need to add energy to the interacting group to split them apart (an endothermic chemical reaction if you like). Alternatively, if the binding energy is positive, you’ll get energy out by splitting them apart.
Unwinding a spring does not involve splitting its atoms apart, but it does involve the atoms transitioning from a higher energy wound state to a lower energy unwound state. That energy difference comes from the difference in the electromagnetic fields between the atoms in the two states.
The principle is exactly the same for nuclear reactions, except now the protons and neutrons within the atoms get rearranged into lower energy configurations – ie different atoms – and the strong force stores the energy difference, not the electromagnetic force. Because the strong force is so strong, the energy lost in nuclear fission is around 0.1% of the mass, and even more in Hydrogen fusion (H bombs).
I’ll be damned.
the m in E=mc^2 is matter not mass. In fact mass in educated layman’s english is a property of matter, that is to say it is a trait that gets measured.
So talking about chemical reactions as a conversion described by E=mc^2 is wrong. Saying the equivalence described in the equation affects chemical reactions may be technically right – just like saying the entire Earth falls towards a dropped pencil is technically right, though negligible compared to the pencil falling towards the Earth.
You are wrong, Lucas.
The m in the equation is indeed mass. Photons bouncing off a mirror impart momentum to the mirror, exactly as if they had the mass derived from the equation. But photons aren’t “matter” by any stretch.
This is the principle involved in solar sails.
It’s nonsense to say that “chemical reactions as a conversion described by E=mc^2 is wrong” but “technically right”. It’s technically right, and, um, it’s right.
If the chemical reaction doesn’t convert matter into energy it is not what is described by the equation. Greene’s examples are not a conversion. The energy is staying energy and the matter is staying matter. Greene is using mass for both sides of the equation the E, and the M. In that sense chemical reactions may be affected by the equivalence but are not conversions because the energy “mass” is staying energy after the reactions. A nuclear reaction is a conversion between the m side of the equation to the E side of the equation and Greene is wrong to imply thats the case in chemical reation by switching from newtonian mass (matter) that nearly all of the readers will think to a more technical one where energy counts as mass.
In Greene’s example the energy is staying as energy, so they do represent genuine conversions of matter to energy.
For the car, the energy is in the car’s motion, and once the car stops, in the heat transferred from the car’s brakes to the surrounding air, which is also motion (hot air is hot because the air molecules are moving faster than those in cold air).
For the MP3 player, the sound waves are energy, again in the form of motion of air molecules.
Just like a computer, your brain uses energy while processing this text. That energy is dissipated as heat, again ending up as motion in the air molecules around you.
To go somewhere slightly different. Whilst on the bus the other day a schoolboy stated that one would be weightless at the centre of the earth. I corrected him, suggesting that, one’s mass would be the same but that one would have some weight because of the gravitational effects of the sun and, perhaps, the universe. Was I correct?
No.
What does it mean to feel weight? That there’s a force acting upon you with respect to your surroundings. If you’re in orbit in the space shuttle, the earth is exerting a considerable force on you, but it’s also exerting a corresponding force on the space shuttle, so you don’t feel any weight with respect to the space shuttle. There’s nothing preventing either you or the shuttle moving, so you and the shuttle follow exactly the same trajectory and experience no force between one another.
Similarly with the centre of the earth thing. Yes, you would be subjected to the gravitational effects of the sun, and indeed, everything else in the universe. But that would not cause you to experience weight, because the earth would be subjected to the same gravitational effects. The forces pulling you toward, say, the sun, are pulling the earth toward the sun at exactly the same rate.
Lucas Bachmann Says:
This analysis is incorrect. GDP has already pointed out that in a fission or fusion reaction, the “conversion between the m side of the equation to the E side of the equation” involves nuclear binding energy. The thing that you want to call “matter” is illusory.
If you combine say two deuterium nucleii into a helium nucleus (note that this reaction does not occur in practice), the helium nucleus has less mass than its constituent parts. The energy released carries away the extra mass. This energy is called nuclear binding energy.
Which is “matter”? The constituent parts, or the resulting product? Note that higher up in the periodic table, the products have more mass than the constituent parts. How would you define “matter” in these circumstances?
A chemical reaction is no different. If you combine say two hydrogen atoms and an oxygen atom, the resulting water molecule has less mass than its constituent parts. The energy released carries away the extra mass. This energy is called chemical binding energy.
Thanks SJ. However, I would ahve thought that it depends on how you define weight. Is it with relative to Earth, to the sun or, to the universe? If it’s relative to the last two then surely one has weight with respect to them even though one may be weightless relative to Earth.
The “m” in “E=mc2” is mass, not weight. They are not the same thing. You weigh different amounts on different planets depending on the gravitational pull of the planet, but your mass stays the same.
Mass is the amount an object “pushes back” when you try to acclerate it. Massive objects (eg cars) push back a lot, non-massive objects (eg feathers) don’t push back much at all. More precisely, this kind of mass is “inertial mass”.
There is another kind of mass: gravitational mass. That’s the amount by which two bodies attract each other gravitationally. Double the gravitational mass and you double the gravitational force between two objects.
Now here is something that seems obvious, but is actually quite remarkable: gravitational mass and inertial mass are the same. That is, an object that exerts twice the gravitational force (at a given distance), also has twice the inertial mass.
This is certainly not true for other forces, eg electromagnetism: two charged bodies will attract (or repel) one another twice as much if you double the charge on one of them, but their inertial mass remains (more or less) the same.
In other words, you cannot alter the gravitational attraction between two objects without also proportionally altering the inertia of the objects. There is no fundamental reason why this must be so (eg – it is not so for the forces between charged bodies), hence it gets its own special name: “Einstein’s Equivalence Principle”, after the guy who first realized its significance.
The equivalence principle fundamentally underpins general relativity, and from it you can derive all kinds of nifty things, like the fact that light must get bent in gravitational fields (so the earth bends light as it goes by), and that clocks run at different rates in gravity than they do in free space.
Neophyte Says:
I gave the only sensible definition of weight in this context:
The sun exerts a force on you, right now. You cannot feel that force as weight, because the sun is also exerting a corresponding force upon everything around you. That’s not “weight”.
If you jump off a building, for a time you experience weightlessness. You do not experience “weight” with respect to the earth, even though it’s obviously the earth’s gravitational field that’s accelerating you toward the ground.
You won’t feel weight until “there’s a force acting upon you with respect to your surroundings”. In the jumping off the building scenario, it can occur when you hit the ground. If the building is high enough, it can start to occur once your velocity is great enough that you start to feel significant wind resistance.
The “weight” thing only applies to your immediate surroundings.