Really?

by John Q on October 1, 2005

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 an arbitrary choice of units in the metric system. But maybe that’s wrong too.

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1

Aaron 10.01.05 at 2:29 am

It all depends on what you mean by ‘mass’. The main way to measure ‘mass’ is to weigh something and use gravity. But gravity couples to (stress-)energy, so you’re really measuring energy that way. You can define mass to be the length of the energy-momentum 4-vector, ie,

E^2 – p^2 c^2 = m^2 c^4

(it’s hard typing in all those c’s) but if you do that for a big system, you’re going to get mass = center of mass energy. I suppose you could try some other definition of mass for composite systems, but I’m not sure how useful it would be. What is being liberated in nuclear reactions is the nuclear binding energy. The strong nuclear force is really strong, even the residual part, so you get a lot of energy that way. By contrast, chemical binding energies are miniscule, so mass is conserved in those reactions to a very good approximation.

I’d think Leo Szilard’s letter had a lot more to do with the atomic bomb than E=mc^2, but what do I know?

2

Carlos 10.01.05 at 2:34 am

a) No, both chemical and nuclear reactions will convert mass into energy. However, the mass fraction converted in chemical reactions is very small, on the order of a millionth of that converted in nuclear reactions. In practice, it’s pretty much undetectable.

b) Yes, c = 1 is used in physics because c ~ 300,000 km/s is an accidental result of its units (depending on the accident of the earth’s radius, its period of rotation, the Babylonian number system, and the French Revolution).

HTH.

3

abb1 10.01.05 at 3:08 am

But even if c=1, don’t you need c^2 to end up with the correct units? Energy is measured in joules, not kilograms.

4

Ali Soleimani 10.01.05 at 3:25 am

Greene is correct. The measured mass of an object is equal to its total energy, so any way of adding or removing energy to it affects the measured mass. Heating up a box will cause it to become slightly heavier, for example.

A good way to think of it is that measured mass is really just another word for “total internal energy.” They’re not two separate quantities; you can’t change one into the other. You can change one ‘type’ of energy into another, and you can transfer energy around, thereby changing the measured mass.

So, eg, burning a quantity of gasoline in your car would not actually change its mass if you didn’t allow any of the reaction products (including heat!) to escape. It would just change chemical potential energy into thermal energy. If heat or light escape, then the mass of the car will decrease corresponding. And if you transform some of this energy into motion (ie kinetic energy), then the proper mass of your car will again decrease correspondingly.

There’s nothing intrinsically different about nuclear reactions. However, there’s a vast quantitative difference. In a typical chemical reaction, the energy change in an atom is a negligibly tiny fraction of the total energy (same thing as mass, remember) of that atom. A ballpark figure would be around 1 part in a trillion. So for *practical* purposes we can ignore it — this is why you see sometimes see books (erroneously) say that mass doesn’t change in chemical reactions. For nuclear reactions the energies involved aren’t so negligible compared to the total; the change in mass might be of the order of 1%.

I would agree that E=m is in some ways a better formulation. c^2 is just the constant that establishes the distance/time unit conversion ratio; the key idea is that mass is just another name for total energy. Indeed we high-energy physicists prefer to use units in which the speed of light is simply 1, and so the equation becomes simply E=m.

5

black iris dancer 10.01.05 at 3:29 am

I think what Greene is getting at is, in fact, the E = M relationship. That is, mass and energy are fundamentally measures of the same thing, namely (lacking as far as I’m aware a better term) mass-energy. Viewed this way, the c2 term is simply a constant, and E = mc2 is not categorically different from the relation dk = K dm, (where dk represents distance in kilometers, dm represents distance in miles, and K = 0.621…mi / km).

The fact that we measure mass in kilograms and energy in joules and treat them as separate quantities is itself an “accident of units,” albeit an understandable one. The existence of a relation of the form E = kM for some universal constant k implies that they are measures of the same physical quantity within the model.

6

black iris dancer 10.01.05 at 3:31 am

Lies! Damned lies!

Live preview told me that all my pretty superscripts and subscripts would show up, and they do not. I can only attribute this to the most far-reaching of conspiracies within the open source community to stifle free speech and the aesthetically agreeable representation of mathematical expressions.

7

andrew 10.01.05 at 4:15 am

E=mc^2 still contains a key insight, namely that the conversion constant just happens to be the speed of light, as well as the maximum possible speed. You can still redefine the conversion ratio to 1 but it still has physical implications. It’s the same with ma = GMm/r^2; the insight is that it’s the same m.

8

abb1 10.01.05 at 4:21 am

Mass can’t be just another name for total energy because mass with momentum has more energy than mc^2.

9

Kenny Easwaran 10.01.05 at 4:29 am

The fact that we need some constant to convert the units (as abb1 points out, even if that constant has a 1 attached to it) isn’t just an understandable accident of units as black iris dancer suggests. Unless I’m missing some basic insight of modern physics that lets us literally add distances and times, masses and energies, and otherwise whatever sorts of quantities we want. Units really are important.

This is because there’s a sense in which (many) laws of physics (at least under a Newtonian system) can be stated without reference to numbers at all, provided we always keep our units separate, as shown by Hartry Field in Science without Numbers. We can compare two distances by rotating and translating one onto the other, we can compare two time intervals by translating them, and we probably have similar operations for other quantities like mass. We can compare the product of d1 and t1 with the product of d2 and t2 by doing something like “drawing little parallelograms in spacetime”, describing a set of points that define a grid in spacetime, and counting which parallelogram contains more vertices of the grid. However, note that we don’t seem to be able to guarantee anything about the relation of the spatial separation of points of the grid and the temporal separation – fixing such a grid corresponds to choosing certain units for distance and time. A similar maneuver lets us compare products of masses and times, or masses and distances, or products or ratios of these complex quantities. But at each stage, we’ve got such a grid, so there’s always a difference between spatial separation, temporal separation, and mass-density separation.

Of course, a lot of this probably changes with relativity, so that maybe distance and time really are interchangeable (we can require that the grid’s temporal spacing and spatial spacing are related by the speed of light), and maybe mass and energy are as well (using c^2). But I suspect that even if these equations are directly possible with no unit conversion, we’ll always have to have some sort of multiplier to convert between masses and distances, or energies and times.

10

Chris W. 10.01.05 at 4:39 am

Another comment to chime in and agree with your first two commenters. Brian Greene is perfectly right.

For a comparison between chemical binding energy and nuclear binding energy contributing to mass, here’s a remark from a Wikipedia Talk page:

Strictly speaking, you are correct that the chemical binding energy contributes mass. However, this mass is very small compared to the mass due to the nuclear binding energy and so can usually be neglected. For example, the chemical binding energy in a hydrogen molecule contributes a mass of ~2.4E-9 g/mol. By contrast the nuclear binding energy in Deuterium contributes a mass of ~0.0029 g/mol (over six orders of magnitude greater than the chemical binding energy).

It’s of course the same for the mass loss via changes in chemical binding energy. A quick over-the-thumb calculation for the mass equivalent of the energy needed to accelerate a car from zero to about 50 km/h, neglecting friction (bad thing…) gives me a mass loss of the order of 10(-9)g for what also amounts to a mole or a fraction of a mole of molecules (hydrogen, water, other) produced in the combustion process — the mass equivalent to the breaking of chemical bonds.

As for setting c=1, so-called natural units, that’s fine because Einstein’s formula is universal, meaning that mass and energy are fundamentally the same thing and can be measured in the same units. But then you need to measure mass in eV/c². If you prefer giving c in miles/hour, better leave c in the formula.

(As an aside, I’m aghast that someone would teach that E=mc² is only valid for nuclear reactions. This is an obstacle to the students’ understanding of the difference between [what are considered] universal equations, i.e. the principles of the physical world, and ad-hoc, approximate formulae that are only valid under certain conditions.)

11

John Quiggin 10.01.05 at 6:20 am

To defend my teachers , the claim as I understood it wasn’t that “E=mc² is only valid for nuclear reactions”, but that, in chemical reactions both mass and energy, considered separately, are both conserved, whereas in nuclear reactions mass-energy is conserved.

As I read Greene, he’s claiming that when you burn gasoline, mass decreases and energy increases:
“when 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”
I can accept that the decrease in mass is too small to be measured, but why isn’t the extra energy detected in experiments testing conservation of energy.

If what he means is that mass and energy are the same thing, it doesn’t seem to me that he is being too clear about it.

Or is the claim that what I was taught to call chemical energy actually counts as mass?

12

Jeremy Osner 10.01.05 at 6:41 am

Thanks for asking the question, John — the same thing occurred to me while reading the article, I wanted to ask somebody but the number of high-energy physicists who read my blog is far far fewer than the corresponding number for CT — so it is more useful for you to ask the question.

13

almostinfamous 10.01.05 at 6:49 am

re: the units. doesn’t dimensional analysis not make sense if you could just equate mass to energy? that would be M=ML^2^T^-2^

i realize we aren’t talking about billiard balls anymore, but i thought dimensional analysis held up in most cases. unless, of course something has changed in the way physicists think about these things since early 2002, the last date at which i took a class in physics.

14

phil 10.01.05 at 6:58 am

John: all energy counts as mass as you put it. So if for instance you burn two molecules of hydrogen using a molecule of oxygen, the resulting 2 molecules of water will weigh a bit less than the original 3 molecules. The energy released during the reaction in the form of heat & light will be equal to the difference in mass using the E=mc^2 equation.

abb1: Of course it isn’t. The E=mc^2 equation is the energy of mass when the speed is 0. This page has the formula of relativistic kinetic energy. E=mc^2 is the result when you set v = 0.

15

abb1 10.01.05 at 7:00 am

Einstein said:

…mass and energy are both but different manifestations of the same thing…

which clearly is quite different from saying (above) that …mass and energy are fundamentally the same thing. Mass and energy are different things.

I’m quite sure that a chemical reaction merely converts energy from one form to another without converting mass to energy.

16

abb1 10.01.05 at 7:16 am

Phil, if that’s true that 2 molecules of hydrogen and 1 molecule of oxygen have greater mass than 1 molecule water (if that’s what you’re saying), then I learned something new here. Thanks.

17

jeb 10.01.05 at 7:22 am

One possible reason for a teacher to conceal the mass effects in chemical reactions is the onion theory of science teaching. The secrets of science are, according to this theory, best revealed by removing successive layers of mystery at a rate that a class can assimilate. Note the word class instead of the word pupil.

When I was first taught Latent Heat of Evaporation I remember asking if when water was finely sprayed its temperature dropped due to the many separations. I was told “No”. The teacher was deliberately lying, of course, but he did so because he needed to manage the education of a class to a schedule. When I was a science teacher, I sometimes found myself doing the same regretable but necessary thing.

18

jw 10.01.05 at 8:02 am

Units are important, but there are different systems of units. While freshmen use the metric system of units, working physicists typically use a system more appropriate to their work, such as natural units. Unlike the metric system, where you have different named units, there is only one unit in natural units and it’s nameless.

We begin defining natural units by setting c=1. This means that our unit of length is the distance travelled by light in one second. You can still do dimensional analysis in natural units. Distance quantities have one unit and time quantities also have one unit, but velocity as distance/time has zero units. Acceleration is distance/time^2 and so has -1 units.

We continue defining natural units by setting Planck’s constant, hbar = 1. As h = mass*length*length/time, we find that mass quantities have -1 units. That’s basically all you need to start doing dimensional analysis in natural units. As you can imagine, it’s much easier to perform calculations without the large number of c’s and h’s that you get in typical relativistic quantum mechanics calculations. You can apply a conversion factor to your natural value at the end of your calculation to get a metric value if you need one.

Einstein’s full energy equation in natural units is E^2 = m^2 + p^2.

19

abb1 10.01.05 at 11:00 am

One possible reason for a teacher to conceal the mass effects in chemical reactions is the onion theory of science teaching.

Not only they ‘conceal’, but I seem to remember some kind of a fundamental law that says that mass doesn’t change in a chemical reaction. Total mass of all the components before the reaction is equal to the total mass of all the products after the reaction – that’s what I was told anyway. They actually put a lot effort to emphasize this point.

20

Eric H 10.01.05 at 11:17 am

E=mc^2 is a specific form of the longer equation to which it collapses under certain circumstances far too remote in my memory of Quantum Physics class to recall. However, c is the speed of light. It is always the speed of light. It is a constant. It does not come out of that equation (it was measured by Roemer in 1676, far in advance of Einstein). 3 x 10^8 m/s is a good estimate.

With tongue slightly planted in cheek, someone recently calculated that people who commute at speeds considerably slower than light will tend to live slightly longer than people who don’t. A few seconds over the course of a lifetime, I think it came out to be. Sorry, I can’t seem to find a link (wouldn’t surprise me if it was The Onion, though).

21

Ben M 10.01.05 at 11:40 am

A comment. The fact that a high-energy molecule has *more mass* than a low-energy molecule can be measured directly. Dave Pritchard’s group at MIT has the world’s most accurate mass spectrometer, a two-ion trap in which they measure an atom’s cyclotron frequency, and thus its mass, to 1 part in 10^11. In some situations, they can see their trapped ion’s mass increase, stay at a higher level for a few minutes, and then decrease again. This mass change is interpreted as the ion entering an excited state due to a spin flip. Here is an news article on it. So yes, all forms of energy—nuclear, chemical, atomic, kinetic—contribute to inertial and gravitational masses.

22

rich 10.01.05 at 12:00 pm

Like the difference between high school chemistry and P-chem.
Like the difference between Newtonian and Einsteinian physics.
Like the difference between an experiment conducted in an open Erhlenmeyer flask or a study performed in a bomb calorimeter.

23

Aaron 10.01.05 at 12:02 pm

Something very odd is going on with subscripts here….

Aha. It’s Textile. The formula in my original post should be

E^2^ – p^2^c^2^ = m^2^c^4^

I hope.

24

Aaron 10.01.05 at 12:12 pm

Er, superscripts, that is. But the formula worked…. The trick is to use the carat character to surround what you want superscripted. Sort of like

E^2
^

but all on one line — I can’t figure out how to make Textile not turn carats into superscripts

25

Sean 10.01.05 at 12:15 pm

I just commented over at Cosmic Variance, but trackbacks over here baffle me.

I think the point is exactly what abb1 says in comment 16: one water molecule has less mass than the combined mass of two hydrogen atoms and an oxygen atom. The mass is just the rest-energy, and the tiny negative binding energy of the molecule makes the mass you’d measure on a scale be lower. Too tiny to measure, but it’s there.

26

abb1 10.01.05 at 12:29 pm

Suppose I took elevator to the second floor. Now I have more potential gravitational energy than I had on the first floor, just like 2xH2 + O2 have more potential chemical energy than 2xH2O. Correct? Do I have larger mass on the second floor?

27

Michael Meo 10.01.05 at 1:02 pm

I would just like to confirm what rich has said: We do not bring in the Special Theory of Relativity when we start teaching Newtonian mechanics to high-school sophomores: it’s difficult enough to convey the fact centrifugal force is a pseudo-force, and different from centripetal force.
Greene may be correct in one way of looking at it, but he is certainly confusing: we can go along just fine with quite precise measurements of the mass of gasoline in our car, assuming every bit of it is conserved in the course of the chemical transformation.

We do not measure the mass to thirteen orders of magnitude. Questions of more subtle definition arise when we do.

28

Jason G. Williscroft 10.01.05 at 1:39 pm

This is what we get when we attempt to debate science in what is intrinsically a liberal arts forum. We all use the same words, but to some of us–the scientists–those words have concrete meaning reflecting underlying mathematical constructs, while to everybody else, the meaning of words is fluid and capricious.

For the record, ALL formulations of Einstein’s famous relationship are equivalent, by definition. Exactly so, a priori, forever amen. Period.

That this relationship takes different forms on paper is entirely a matter of notational convenience, no more significant to the operation of the universe than it matters to the playgoer whether the actors’ copies of Shakespeare’s script are rendered in a serif or a sans-serif font.

Energy IS mass, and vice versa. That they appear distinct to us is casual life reflects an idiosyncratic coincidence driven by our physical context and the arrangement of our sensory apparatus, nothing more.

It probably sounds like I’m advising all the liberal-arts types out there to go talk about something else. I’m not. Instead, I’m suggesting that the concepts discussed in this conversation are ones that only retain their entire content when expressed in the language of mathematics. Learn the math, and the personal rewards will be beyond what you are currently equipped to imagine.

Don’t bother, and your experience in these sorts of discussions will amount to simple mental masturbation.

29

giles l 10.01.05 at 2:26 pm

Jason, I don’t see anyone here claiming that scientific terms are “capricious” in their use: rather, the concern seems to be pin down a single correct understanding of the terms “energy” and “mass”, which implies exactly the converse assumption.

As far as fluidity goes – while the meaning of scientific terms depends on mathematical constructs, it also depends, as do those constructs, on empirical relations and experimental conventions, so that they are not concrete and immutable. Scientific terms undergo almost constant revision as our understanding grows (“gene” is an excellent example that is still undergoing radical change).

30

Redshift 10.01.05 at 2:43 pm

John:
I can accept that the decrease in mass is too small to be measured, but why isn’t the extra energy detected in experiments testing conservation of energy.

It’s not too small to measure. In my high school chemistry class, lo these many years ago, one of the experiments we did measured the difference in mass before and after an exothermic reaction, and compared it to the mass-energy calculation.

Or is the claim that what I was taught to call chemical energy actually counts as mass?

Yep, it not only counts as mass, it actually is mass. Pretty freaky, but it’s true.

31

Chris W. 10.01.05 at 2:48 pm

John et al., I wrote that in my previous comment, and others have: the mass is not preserved in chemical reactions because chemical bond energy is energy and therefore ipso facto mass. That’s one of the most fundamental ideas of Einstein’s special relativity.

Now, the conservation of mass, separate from the conservation of energy, is a pre-Einsteinian notion, and perfectly fine as an approximation. Heck, you have to look real hard to find any difference in any case (if you don’t happen to have a nuclear bomb or reactor available). It’s also fine to teach the conservation of mass, if — and that’s a big if — it is pointed out that 20th century physics has shown that this is not precisely true. It’s not a fundamental principle (but what considered one for a long time).

32

Al Cantworth 10.01.05 at 3:23 pm

I always regarded the “c^2?” part of the equation to be a placeholder for the phrase “an unbelievably huge number”. Why the heck should the speed of light – squared – have any particular significance? I think that the equation that Einstein used, which solves to e=mc^2, is intended to convey the idea that a given mass would convert to “an unbelieveably huge” amount of energy.

33

eudoxis 10.01.05 at 3:29 pm

chris w. the mass is not preserved in chemical reactions because chemical bond energy is energy and therefore ipso facto mass.When chemists talk about mass conservation in chemical reaction they are talking about the sum of the masses of atoms, not molecules on one side and educts on the other side of the reaction.

In general, the change in resting mass due to chemical reactions is negligibly small enough to be zero for all practical purposes, including ordinary high school physics. It differs by about 10-12 orders of magnitude from nuclear reactions.

34

Chris W. 10.01.05 at 4:13 pm

@eudoxis: That’s what I wrote myself, in comment nr. 10, as well.

But we were talking physical principles here, not reasonable approximations or, indeed, counting atoms of all types.

35

Matt Austern 10.01.05 at 4:15 pm

As others have already said, it’s conventional in particle physics (and perhaps in other parts of physics too) to use “natural units” in which not just the speed of light, but also Plank’s constant and Boltzmann’s constant, are taken to be the dimensionless number 1. In this way of looking at things, choosing between GeV and MHz is sort of like choosing between inches and centimeters: you use whatever is most convenient for the problem at hand.

But just getting back to “E = mc2: I think most physicists would agree that the modern way to write it is “E0 = m”. That is, an object’s mass is defined to be its rest energy, i.e. the energy in the object’s rest frame. (This isn’t the only possible way to define the notion of mass, but it’s the conventional definition today and it’s the most convenient one.)

36

rilkefan 10.01.05 at 4:47 pm

This discussion needs the term “rest mass”. If you consider a chemical reaction as mess_of_quarks_and_electrons -> different_arrangements_of_the_mess you can see that the rest mass stays the same – that is, the individual particles in isolation are unchanged hence the total rest mass is unchanged. What happens is that the binding energy of the mess changes, and the total mass if you exclude the radiated energy.

In nuclear reactions you have mess_0 -> mess_1, with different particles on either side, plus different binding energies.

37

Xmas 10.01.05 at 4:55 pm

It’s the other way around for water. 2 H and 1 0 have less mass that H2O. The chemical bonds between the atoms are stored potential energy, therefore the bonds themselves have mass.

But the amount of mass would be neglible (since the energy of the bonds is tiny.)

As for the moving car example. Lorentz equations account for the addition mass of a moving object. The faster an object moves the more mass it gains. Though you any see a noticable gain in mass at speeds close to c (the speed of light).

38

eudoxis 10.01.05 at 7:06 pm

Considering the difference in mass for one the examples that Greene uses, take the combustion of methane (in the same range as Greene’s gasoline engine example). The enthalpy change = -890 kJ/mol. From there (handwave) we calculate a relative change in mass for that reaction of 9.8 * 10^-12 kg. That’s out of a total mass of 80g.

For a 10kg of methane car tank, the change in relative mass after burning is about 1*10^-8 kg. That’s a trivial change and one that can be ignored for ordinary chemical reactions.

What Greene is getting at, in my view, is that E=mc^2 is true – even for different scales of practical measurability.

39

eudoxis 10.01.05 at 7:12 pm

Strike?
Again.
The enthalpy change = negative 890 kJ/mol. From there a relative change in mass for that reaction is 9.8 * 10^12 kg.

40

eudoxis 10.01.05 at 7:13 pm

For the moving car example and the elevator question, the relative change in mass is even smaller. Something on the order of 10^16 kg.

41

eudoxis 10.01.05 at 7:23 pm

That is 10 to the negative 12 and negative 16 kg respectively

42

Wolfgang 10.01.05 at 8:45 pm

By the way, Einstein never wrote the famous equation E = mc2. Albert wrote L/V2 as I explain on my blog.

43

JR 10.01.05 at 9:21 pm

In their eagerness to show how smart they are, no one is answering John’s questions. John, if you’re still here (and I wouldn’t blame you if you’re not):

1) A car engine is powered by a chemical reaction. The energy is released when bonds between atoms are severed — that is, large molecules are broken up — and new, less complex molecules are formed. No atoms are destroyed or changed. It is true that the resulting products have a tiny, tiny amount less mass than the startig products. But this is not taught in elementary chemistry because it is much more important for the students to understand that when, e.g. a piece of coal burns, the resulting ash plus the gases that are emitted contain all the atoms that the coal had to begin with. If you teach high school students that mass is converted to energy during burning, they will conclude that when stuff burns, what’s left over weighs less than what they started with because the rest turned into energy. They will never understand the chemical reaction at issue.

2) Yes, c^2 is a constant, so if you define c=1 you can say E=M. But for a beginning student, the mind-boggling implications of E=Mc^2 come from the fact that c^2 is a really big number. So even a dullard can understand that a little itty bit of mass can turn into a huge amount of energy – enough to destroy a city, just to pull an example out of a hat.

44

eudoxis 10.01.05 at 9:26 pm

Chris W. “But we were talking physical principles here, not reasonable approximations or, indeed, counting atoms of all types.”

Why is this directed at me? I didn’t say anything about counting atoms, and, yes, indeed, most of the comments are about reasonable approximations, especially ones that exclude miniscule changes in relative mass.

45

Darren 10.01.05 at 11:20 pm

You are correct. These are all chemical reactions and can be (and were) completely described without E=MC^2. Totally different from nuclear reactions.

46

abb1 10.02.05 at 4:48 am

So, eudoxis, you’re saying that when I’m lifted to the second floor my mass does increase, is this correct?

This sounds like quite an extraordinary claim, because while in the chemical reaction example you could confuse me with quarks and electrons – here’s no change in the matter taking place whatsoever, only in the location of the object. If my mass on the second floor of the building is different from my mass on the first floor of the same bulding, doesn’t it make the concept of ‘mass’ completely meaningless?

Thanks.

47

Matt Daws 10.02.05 at 6:49 am

abb1, That’s (general) relatively for you! If you are willing to believe that an object moving at high speed has more mass (which it does from special relatively, as xmas in 38 notes) then why have problems with an object being moved out of a gravity-well gaining mass? After all, the spaceship travelling at relativistic speeds is still the same ship composed of the same atoms. I’m afraid it’s just one of the correct but rather counter-intuitive facts to come out of relativity theory.

As for whether the concept of “mass” is meaningless? Well, I guess it’s all (ahem) relative. What do you mean by “mass”? Well, presumably you have to mean something to do with measurement, and then we come down to different frames of reference etc. An interesting question for those who know more physics than I do: if I move out of a gravity-well, then I gain mass from the perspective of an observer still in the well. I presume I don’t gain mass from my own perspective, as I *think* this is what happens in the fast-moving spaceship example?

48

abb1 10.02.05 at 7:39 am

So, Matt, once again: is my mass on the second floor (from my point of view) the same as my mass on the first floor (from my point of view) or is it different? I am the observer.

Once I’m on the second floor, I can now generate some energy by sliding down to the first floor. And if ‘energy’ and ‘mass’ are the same, then I’m bound to loose some mass in the process, correct?

Since I’m the observer, there shouldn’t be any relativity involved here, correct?

49

Matt Daws 10.02.05 at 8:57 am

Abb1, Sorry, apparently mass effects are only special relativity. I recommend everyone to go and read e.g. Wiki article

Okay, I will admit defeat here: I don’t really understand what is happening in the gravity example. I think general relativity might be needed: I cannot find of a related thought experiment which doesn’t use gravity. My guess would be that as the geometry of spacetime changes as you fall towards the Earth, this might explain away the apparent problem. Anyone an expert here?

50

eudoxis 10.02.05 at 9:19 am

abb1: It is the relativemass that increases and only while you are moving in the elevator. You will have the same rest mass when you come to a stop on the second floor as you did on the first floor. It’s useful to talk about relative mass only as it relates to energy.

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Matt Daws 10.02.05 at 9:59 am

Eudoxis, I think abb1 is talking about “gravitational potential energy” though. I’ve ask a theoretical physicist, and he says the following. When you climb in the lift, the lift then raises you up through the gravity well: it does work against the geometry of spacetime, if you will. Go don’t gain any energy: potential energy isn’t real, it’s just a useful fiction. Similarly, when you, say, jump down from a wall to the ground, your mass does increase as you accelerate (from my fixed reference point) but then decreases again as you stop against the ground: the energy gets absorbed the ground as it does work to stop you moving.

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Eli Rabett 10.02.05 at 11:04 am

What all this misses is that sciences are nests of models. Generally speaking the simplest models are easy to apply but incomplete and the most general are much more difficult to apply.

What a good scientist does is select and use the simplest possible model that describes a situation and seek to extend the most general models. Selection of the simple model requires knowledge of the limits of both models.

On that basis, if you want to understand chemistry, you can ignore mass changes during chemical reactions. You are thus ignoring the equivalence between mass and energy and treating each separately. To the limits of our ability to measure this is the case, so the retreat to a simpler model is fully justified. On the otherhand we recognize that as an absolute matter there will be a change, and that it is calculable, just not measureable.

We also recognize that for nuclear reactions we must fully accout for mass-energy equivalence and again, this is absolutely clear within the range of applicability of both models.

Someone will reply, what are six orders of magnitude among friends.

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Matt Daws 10.02.05 at 4:27 pm

I’m even more confused than when I started now. Take back everything I said. But I’d really love to know the answer!

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phil 10.02.05 at 5:59 pm

xmas: You are wrong. When hydrogen reacts with oxygen energy is released. The gravitational equivalent is an object falling: it’s potential energy is converted to kinetic energy.

When you perform electrolysis to separate water into hydrogen and oxygen you supply energy to break the bonds. The equivalent is lifting an object where you have to supply energy, increasing its potential energy. Hydrogen and oxygen molecules have greater potential energy than water.

abb1: However I wouldn’t extend the gravitational analogy to say that two identical objects at different heights have a different mass. As soon as you start talking about acceleration and/or gravity you enter the realm of general relativity and I don’t know anything about that.

Regarding the constant c in the E=mc^2 equation you need to bear in mind that even though it is a constant, it has dimensions (like the gravitational constant G, and unlike Pi which is just a number). In any equation the dimensions (M mass, L length, T time) on one side must be the same as the dimensions on the other. On the left had of the equation you have energy (dimensions ML^2T^-2), and on the right hand of the equation you have mass (M) times a constant (L^2T-2 or (LT^-1)^2). So the square root of the constant has the dimension of length / time, i.e. speed. And regardless of what units of measurement you use, the square root of the constant (by constant I mean the whole c^2 part, and by square root just c) is the speed of light.

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phil 10.02.05 at 6:03 pm

Apologies for the formatting, I didn’t mean to use super/subscripts at all. In my post above “2” should be read in all cases as “to the power of 2”, not as a subscript.

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Donald Johnson 10.02.05 at 8:29 pm

People have already covered this, but the difference between chemical and nuclear reactions is one of degree. Exothermic chemical reactions typically convert about one part in billions or so of the mass of the reactants into energy. In particular, exploding TNT converts one part in 20 billion (10*9) of its mass into energy.. A kilogram of TNT exploding releases about 4 x 10*6 Joules. A kilogram of uranium that undergoes fission releases about 20 million times as much energy, which means about one part in one thousand of its mass is converted into energy.
One kg of uranium undergoing fission therefore equals 20 kilotons of TNT.
The complete conversion of 1 kilogram of mass into energy would be 9 x 10*16 Joules, or roughly 20 Megatons of TNT.

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Christopher M 10.02.05 at 8:35 pm

A really excellent educated-layman’s introduction to this subject and others in modern physics is, in fact, Brian Greene’s The Fabric of the Cosmos.

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abb1 10.03.05 at 2:34 am

OK, forget gravity. Spring. You stressed a metal spring, coil of wire, and you keep it stressed. Has its mass increased? Then you release it, it generates energy – does it mean it lost some of its mass now?

Thanks.

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abb1 10.03.05 at 3:40 am

OK, a guy at work says that what we might be missing here (unless someone mentioned it before in this thread) is that there are particles with zero mass and non-zero energy (because of their infinite momentum). In a chemical reaction atoms are not split, matter is not converted into energy; the only thing that happens is electrons jumping from one energy level to another – producing or consuming a bunch of these zero-mass particles – thus not affecting the total mass.

How does this sound?

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agm 10.03.05 at 3:44 am

abb1, pretty much yes. Let’s shed a little light on the discussion. Rather, let’s muddle it up by bringing in photons, and quantum mechanics to boot.

First off, mass represents inertial response. It starts off getting interpreted in Newtonian mechanics as the resistance of an object to change in motion as a force is applied. The more mass a particle has, the bigger a force one has to apply to get a certain acceleration. At this point, one is assuming that mass is a constant. But to start to see what is really happening, one must use Newton’s form of the relevant equation,

F = dp/dt.

He actually used rate of change in momentum, not F = m*a. Now we jump to relativity using the spatial part of the momentum 4-vector, which is just plain old Newtonian momentum. One can calculate a mass from relativistic momentum, and this momentum varies depending on velocity. One can also calculate energy, which it related to the time part of the momentum 4-vector and also depends on velocity.

Photons are massless particles (or rather, they have zero rest mass), but one can calculate an inertial response based on a photon’s energy calculated from quantum mechanics and the relationships relating momentum and energy in relativity and photon energy on quantum mechanics (think gravitational lensing). One can see pretty quickly that energy corresponds to inertia, i.e., energy is related to mass. The same sorts of considerations apply to particles with non-zero rest mass.

Special relativity is, among other things, about showing that energy is related to the environment the particle lives in. Generaln relativity is, in this context, about saying that there are more environments that behave this way that we thought special relativity applied to.

All of which is to say, mass is essentially a form of energy (the case of the photon illustrates which is more “fundamental”, thought really, the longer you are in physics, the more you think that momentum is the most fundamental quantity), and that yes, adding or removing energy changes the mass of a system; this includes chemical binding energy. One of the traditional introductory modern physics problems is to calculate how much energy is involved in going from a collection of unbound particles to a collection of bound particles and back (don’t remember if it was an atom or a molecule); go look at the problem set in Alfred Beiser’s Survey of Modern Physics book (chapter 1, I think) for it. It asks you to calculate the sum of the masses of the constituent particles, compare that to the measured mass of the aggregate, and calculate the energy associated with the difference of the two. This energy, stored in the arranging, is precisely the chemical energy that our cells get from turning ATP to ADP or we get from burning gasoline or …

Hope this helps.

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agm 10.03.05 at 3:50 am

abb1, nope, the springs you ask about have indeed gained a bit of “mass” from the energy stored by stretching the springs. It’s a tiny, tiny amount, but it’s there. The same for chemical bonding. If we were still dealing in a Newtonian universe but relaxed the assumption of constant mass, then the energy added by setting up the situation itself represents an increase in inertail response, hence an increase in mass. Think about where the mass comes from in pair production: energy in a photon, under an interaction that conserves momentum, can become the mass and kinetic energy of two electron-sized particles. Energy and mass are the same thing, but we see them as different things because we are looking at them differently.

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Matt Daws 10.03.05 at 4:17 am

Abb1, Okay, I’ve asked around, and I’m assured from physicists that, if you carry a ball up, say, the Empire State Building, then you are, in a tiny way, adding to its mass. This does seem rather counter-intuitive to me, but apparently it’s correct. I’m tempted not to trust my intuition, given how many other strange things happen in G.R. (I mean, time seems for absolute than mass, and yet we know that’s a fiction).

Of course, you yourself will weight a bit less after walking up the Empire State building, as you’ll be using energy. My friend was undecided on the point of a spaceship leaving orbit around the sun and travelling into clear space: he thought that maybe the loss of mass from the ship’s engines would cancel out any possible gain in mass. The problem is that it’s impossible to think of an isolated system in G.R. as you cannot remove everything, or you remove the gravity as well!

One can explain the spring more easily: ultimately, it’s a change in the atomic behaviour of the lattice of atoms which makes up the metal which makes the spring: if molecules can have less mass than their constituate atoms, then it doesn’t seem too surprising that stressing a spring can change its mass a small amount.

I think your collegue is wrong: mass most certainly *is* converted into energy in a chemical reaction. A water molecule literally has (a tiny tiny amount) less mass than 2 hydrogen atoms and an oxygen atom. If you read the Wikipedia on S.R. you’ll find that there is a difference between mass and momentum: a photon is massless but it has momentum; I have mass, and if I speed up, I gain momentum! An electron has mass though, so I’m not quite sure what you collegue meant.

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abb1 10.03.05 at 5:20 am

I know that an electron has mass.

I think the explanation goes something like this: chemical reaction 2H2 + O2 = 2H2O can be interpreted as electrons jumping from higher energy levels to lower ones to form a stable molecule of water.

Energy that these electrons lose is released as a bunch of zero-mass-moving-at-a-speed-of-light particles (photons?) with no change in mass for the whole thing.

IOW, in a very-very rough analogy: electrons in H2 and O2 are moving fast; they then slow themselves down by shooting out zero-mass particles which (slowing down that is) allows them to form an H2O molecule. In H2O electrons are moving slow; the total number of electrons, neutrons and protons is the same, the total mass is the same.

That’s my story anyway.

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Matt Daws 10.03.05 at 6:36 am

Abb1, Okay, yes, but mass *is* lost: others have said this, and I’m certain it’s true: when the electrons move to lower energy orbits, they also loose a small amount of mass. Read again what Ali Soleimani says in #4.

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phil 10.03.05 at 6:43 am

abb1, there is no conservation of mass. Energy and momentum are conserved. So when these zero-mass-but-nonzero-energy-and-nonzero-momentum-moving-at-speed-of-light particles (photons) are released, the mass deacreases.

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abb1 10.03.05 at 7:31 am

I refuse to believe this, dammit. And I refuse to believe that a wind-up toy is heavier wound-up than unwound. This is a hoax.

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eudoxis 10.03.05 at 9:06 am

But, photons have a gravitational mass (energy/c^2)…

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phil 10.03.05 at 9:16 am

eudoxis, there is no such thing as gravitational mass. There is just mass, and photons don’t have any.

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Matt Daws 10.03.05 at 9:27 am

Abb1, I’m not sure if you’re being humourous or not… But why is mass being variable any weirder than time or length contraction. We know for a fact that atomic clocks up in orbit *do* run slower than an atomic clock on Earth. That seems just as odd to me, and it’s beyond doubt. Intuition and common sense are remarkably bad guides when it comes to science…

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abb1 10.03.05 at 10:11 am

Matt, I have no problem with time; I can explain it to myself as something roughly analogous to doppler effect, which is perfectly obvious. But this mass thing is really really counter-intuitive.

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Matt Daws 10.03.05 at 11:03 am

Abb1, How weird, I’m the exact opposite. Time to me is a very odd concept: how can it run slow? Would it be possible for time to run in reverse? I find slowing clocks and so on very, very counter intuitive. Mass less so, but there we go!

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abb1 10.03.05 at 11:29 am

Suppose you’re flying away from earth on a spaceship and looking at a clock left on earth. As you’re moving away and it takes time for the light reflected off the clock on earth to reach your eye, it’ll take you a while to observe the minute hand on the clock jumping from 6:24 to 6:25, it’ll take longer to observe it jumping from 6:25 to 6:26, yet longer to see it jumping from 6:26 to 6:27 and so on. Meanwhile the clock in your spaceship is ticking as usual. This is how I imagine it.

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eudoxis 10.03.05 at 11:46 am

phil, there are recent papers (PRL)that indicate the presence of a non-zero lower limit on the mass of a photon. A simple declaration of ‘it is not so’ is open to argument but such an argument is beyond the scope of my knowledge or, for that matter, of this thread. I’m simply referring to the frequency shift of a photon as demonstrated by Mossbauer. How to explain such a shift, whether in terms of gravitational mass or a “coupling constant” or curvature of space is really a semantic one.

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Donald Johnson 10.03.05 at 11:47 am

One of the standard derivations (going back to Einstein, I think) of E = mc*2 is the flashlight in a box example. You’ve got a box floating in space with a flashlight buried in the wall on one side of the box. Turn the flashlight on. It’s facing inside, so the light leaves the wall, heading to the other side. The momentum of the light is E/c. The recoil momentum of the box is E/c. The box is moving slowly enough so that one can use the classical momentum for it, so it travels at E/Mc, where capital M is the mass of the box. The light reaches the other side in a time L/c (the box is moving so slowly we can neglect its speed in calculating this). The light is absorbed on the other side and the box stops.

The overall center of mass of the system doesn’t change, because all this happened internally, yet the box shifted ever so slightly to the left. In fact it shifted (E/Mc) times (L/c) or EL /M c*2).
To counter this leftward motion of the box, the energy of the light was moved a distance L, so if the overall center of mass didn’t change, we’d have

mass of box times distance it moved = mass associated with E times the distance it moved or

M times EL/(M c*2) = m L, where m is the mass associated with the energy of the light that moved. Solve for m and you get m = E/c*2.

The energy that went into making the light beam was originally stored as chemical energy in the battery and now it is thermal energy added to the opposite side of the box, but in-between it was the energy of the photons. But anyway, when in its chemical or thermal energy form, it added to the mass of the box by the amount E/c*2. That mass was transferred to the other side of the box.

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Wayne Urban Wasserman 10.03.05 at 11:12 pm

It might be fun, or even useful, to see what Einstein and Infeld say about related matters in their brilliant, equation-free “The Evolution of Physics: From Early Concepts to Relativity and Quanta”:

Classical physics introduced two substances: matter and energy…in classical physics we had two conservation laws: one for matter, the other for energy. We have already asked whether modern physics still holds this view of two substabnces and the conservation laws. The answer is: “No.” According to the theory of relativity, there is no essential distinction between mass and energy…Instead of two conservation laws we have only one, that of mass-energy…How is it that this fact of energy having mass and mass representing energy remained for so long obscured?…The reason for this lack of immediate evidence is the very small rate of exchange between matter and energy…An example will make this clear. The quantity of heat able to convert thirty thousand tons of water into steam would weight about one gram! (Simon and Schuster, 1966; pp. 197-198)

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