units of measurement unimportant? not so!

[Arnold Winkelried]

Referring to
In E=mc2, what units of measurement was Einstein using? (10-May-2000) (http://www.straightdope.com/mailbag/memc2.html)

The column says "Equations such as E=mc2 are independent of units of measurement.". I don't think that tells the whole story! The column should be modified to say that if you want to plug in values, and obtain correct results, you must use the correct units of measurement.

i.e. if I measure mass in slugs, and speed of light in rods per hour, I'm not going to get the right number for energy in electronvolts.

To get a correct conversion, one way would be (using the SI - International System of Units (http://physics.nist.gov/cuu/Units/units.html)):

for mass, use kilogram (kg)
speed of light in metres per second (m s-1)
then the energy value will be in joule (J = kg m2 s-2)

[Karen Too]

I disagree. If you use mass in slugs, and speed of light in rods per hour, you will come up with the perfectly correct energy units of slug-rods^2/hour^2. Those units are energy units. Totally useless, unintuitive energy units, but energy units nonetheless. As I said, the choice of units is determined by personal preference. If you don't like ending up with energy in slug-rod^2/hour^2, then convert it to electronvolts or whatever. (Just like you can convert miles/hour into meters/sec without knowing the distance in meters and the time in seconds to begin with.)

Things make a lot more sense if you understand that the units are wedded to the values, and you carry the units with the values during your calculations. This is where freshman physics students go wrong every day. Physics problems are not arithmetic problems.

[Billdo]

This question came up recently in General Questions, as the Einstein's equation and units (http://boards.straightdope.com/sdmb/showthread.php?threadid=15507) thread.

In that thread, we came upon the interesting fact that light travels at 7.2x1013 rods per fortnight.

[Philbuck]

Of course, the most glaring error is that the rest mass of an electron is 0.511 MeV/c^2, or 5.11x10^5 eV/c^2, not 511 eV/c^2.

What a coincidence that I used that in a mass-equivalence relation on a problem set five minutes ago, then got up to check the SD page...

Freshman physics students need to get the units right. Juniors try for the constants, too. :)

[Irishman]

Arnold, Karen's point is that the units are part of the values, and conversion factors can be added to clear up the units. But the relationship is expressed without inherent units, because the units are arbitrary from the physical relationship point of view.

An example: Force = mass x acceleration

If you use mass in kg and accel in m/s2, then you get a force in kg-m/s2. In the SI system, this is defined to be a Newton, or you could say the conversion factor is 1 Newton per kg-m/s2.

However, you could measure the mass in lbs mass (lbm) and the accel in ft/s2. Then you get the force in
lbm-ft/s2. Now that term is "funky", so engineers use a conversion factor

32.2 ft-lbm/lbf-s2 . That's in units of foot-pound mass per pound force second squared. That's because technically pounds is a measure of weight, not mass, but it has also been used as a measure of mass with the assumed gravity field (32.2 ft/s2), thus the conversion factor.

So you get

Force = mass (lbm) x accel (ft/s2) x conversion factor (32.2 ft-lbm/lbf-s2)

So although the 32.2 conversion factor is required, it is an artifact of the units chosen. Therefore it is not an inherent part of the force/mass/acceleration relationship.

Thats the same thing as Karen's point with Einstein's relativity equation. The inherent relationship is E = m x c2. Conversion factors are added to make the units work out, depending on the values plugged in.

Does that make it more clear?

[C K Dexter Haven]

In ref the comment that there was a GQ question on this topic a few weeks ago:

Generally, the Mailbag items are prepared several weeks ahead of their actual publication. We live in a fast-moving age, and often we often see in GQ the same (or similar) question that the Straight Dope Staff is dealing with in the Mailbag. We're sorry about that, but there's not much help for it.

[Arnold Winkelried]

Irishman, Karen, I'm not saying that the mailbag item is wrong, I was just saying that it's incomplete.

As Karen said in her post in this thread,

Things make a lot more sense if you understand that the units are wedded to the values, and you carry the units with the values during your calculations. This is where freshman physics students go wrong every day. Physics problems are not arithmetic problems.

What my original post was trying to say is that I personally would have included something like that in the mailbag answer. Right now I could just see some high school student multiplying a mass in pounds by a speed in miles per second and expecting to get an energy value in calories.

[Chronos]

Another way of looking at the famous equation, is that an object has rest energy equal to twice the kinetic energy that Newtonian physics would predict for the object, if travelling at the speed of light. All clear now?

[moriah]

Another reason for not expressing equations of pure physics in specific values is that the constants in the equations may change over time* as the universe expands and/or contracts.

If the speed of light (or the gravitational constant or Plank's constant) ever changed, it is unlikely that the formula would change. So, if the speed of light (in a vacuum) ever became 100 meters per second, E=mc2 would still hold true (resulting in an atomic explosion looking like a cap gun).

Peace.
_____________
*a purely theoretical hypothesis at this point in time

[Boris B]

I comes from not understanding that the units we created for measuring energy are derived directly from mass and velocity. I mean, the process of going from a numeric mass and velocity to a numeric energy is what the energy units were created for in the first place.

Most of us non-physics types like the Mailbag questioner, and myself a couple of weeks ago, would not have guessed that there was an inherent relationship between joules on the one hand, and kilograms, meters, and seconds on the other. Sure, they're all metric, but calories are metric too. So when people say, "Units don't matter" it doesn't mean a whole lot to us, since we have never conceived of a joule, calorie, or BTU as nothing more than a couple of square meter kilograms per square second. Being "consistent" doesn't tell us much either, since "joule" seems like kind of its own thing.

So basically, when explaining this sort of thing to a scientifically-challenged person like myself, it's best to steer clear of glittering generalities on the lines of "Einstein's equation is so cosmic that units don't matter" which sounds perilously close to "joules = calories". Just explain that the energy units already have mass\time\distance equivalence, and you should fall back on conversions previously established from basic kinetic energy equations.

The real coincidence of Einstein's equation, the one with cosmic significance (not really a casual coincidence) is the one Chronos pointed out: disregarding relativistic effects, the energy you get from converting mass to energy is twice what you'd have in the same mass if it were travelling at light speed.

[DrMatrix]

moriah pondered what would happen:
If the speed of light (or the gravitational constant or Plank's constant) ever changed, ...
[/B]
Well,... There is no way the speed of light can change. The meter is defined by using the speed of light. In other words the speed of light is constant by definition.

[DSYoungEsq]

Arnold, no offense, but Karen did say it was important to keep track of units of measurement. The whole sideswipe at the NASA Mars probe was making precisely that point.

What Karen meant (and, to my mind, managed to actually say) is that you can measure any such equation in any units you prefer; the units don't change the equation. Keeping track of the units while you plug in the values to replace the variables in the equation becomes a concern only for someone actually doing the arithmetic.

In short, Karen SAID you have to keep track.

[JonF]

There is no way the speed of light can change. The meter is defined by using the speed of light.

I don't agree ... If light started travelling at a different speed, the speed of light in meters per second would remain constatn. We'd either have to redefine the meter or recall a lot of meter sticks {grin}. However, it may be physically possible for the speed of light to change. I wouldn't bet on it.

[DrMatrix]

JonF
What you just said seems to contradict what you said here (http://boards.straightdope.com/sdmb/showthread.php?threadid=24204).

The units of time and distance are defined concepts. If I use as my unit of length the distance between the ends of a piece of elastic, then I can change the speed of light by streching or releasing my standard unit. But if the speed of light is measured in meters/sec, and the meter is defined as the distance light travels in 1/299 792 458 sec, then in what sense can you say that the speed of light could change?

[JonF]

What you just said seems to contradict what you said here.

I don't see a contradiction; I think it's a matter of semantics. What I said eleswhere relates to the current situation. What we're discussing here is a hypothetical future situation.

We have defined various tools used for measuring and quantifying the universe. The definition we have chosen for the meter (which is the ultimate basis of all our tools for measuring length) is such that it makes no sense to try to measure the speed of light with the tools we have chosen; it's always going to come out 299,792,458 m/sec. So there's no point in measuring the speed of light unless and until we change the definition of the meter.

But the universe doesn't care about our definitions. The universe does what it does, no matter how we define a meter. The speed of light is the distance traveled divided by the time elapsed no matter how you define the units.

It may be possible for the speed of light to change in this latter sense, so it takes a different time to cover the same distance. There are people who have proposed that it has changed or is changing or will change (IMHO, these people are cranks). But, just as a thought experiment, consider the possibility that light starts moving slower, so it now takes 10 minutes for light to reach us from the Sun instead of the time it takes now (8-something minutes?). In this circumstance, we would have two options:

1. Maintain the same definition of the meter; the speed of light would still be 299,792,458 m/sec, but the numerical value of all our measurements of length would change.

2. Change our definition of the meter so that our current measurements of length agree with our previous measurments of length, but the speed of light would come out to a different number in m/sec.

My guess is that, in the incredibly remote possibility that we actually face this choice, we would pick door #2. Depending on how we chose to redefine the meter, it might or might not be sensible to measure the speed of light after the change in definition. We probably wouldn't want to base the meter on the speed if light if the speed of light proved to be inconstant.

[DrMatrix]

JonF said:
But the universe doesn't care about our definitions. The universe does what it does, no matter how we define a meter. The speed of light is the distance traveled divided by the time elapsed no matter how you define the units.
I agree. I still maintain units of length and time are coordinate definitions. If it took light 10 mins to arrive from the sun, I would have to conclude that the distance from earth to sun had increased. If we noticed that all (or almost) all material objects also increased by apporox the same amount, I think we might wish to reconsider how we defined our unit of length. But as our units of time and distance are currently defined, the speed of light cannot change. And as the definitions stand your door #2 is not an option.

This is not just semantics. How we decide wheither two intervals seperated in space or time are equal is, by necessity, a definition. We must decide if the (arbitrary) standard we have chosen is useful or not. There is no sense in asking wheither the standard is "real" or not. You cannot say the speed of light really changes without resorting to some definition of units. The units we have chose do not allow the speed of light to change. For you to say the speed of light has changed, you must use a different standard.

[JonF]

I find it difficult to debate this subject, perhaps partly because I think that the speed of light is a fundamental characteristic of the universe. We have some pretty good evidence for that and, were it to change in some fundamental manner, the universe itself would be very different.

However, what the hell, this is an interesting question. I agree that the units we have chosen do not allow the speed of light to change. I also agree that, to say that the speed of light has changed, we must use a different standard. But I don't think we completely agree ...

If it took light 10 minutes to arrive from the Sun then, by definition, the distance to the sun in meters would have changed. But would it be necessarily true that the physical distance had changed? Or might not time itself have changed? Or perhaps something more fundamental, of which we do not know now, could have changed?

And, if we noticed that suddenly it took light 10 minutes to arrive from the Sun and we noticed that all light-speed based measurements had changed by the same proportion and we decided to redefine our unit of length so the tape measure in my pocket was still 10 feet long, would you not say then that the speed of light had changed? The number in m/sec would be different ...

[Chronos]

Ok, so suppose we use some other definition for the meter-- Suppose, for instance, that we go back to the good ol' scratches on that platinum-iridium rod. The speed of light still couldn't change. Why? Because the length of that rod depends on a number of things, specifically the separation of the atoms of platinum and iridium in that alloy, which separation, in turn, depends on a number of other factors, including the speed of light. Likewise, the frequency of the light emitted by that particular transition of cesium (the current standard for the definition of the second, IIRC) also depends on the speed of light. Furthermore, the dependence on the speed of light in both cases is just such that, if the speed of light did change, our definitions of the meter and the second would change in exactly just such a way that the measured value, in meters/second, would be exactly the same. Any possible definitions of units based on physical artifacts (be those artifacts a rod of precious metal, the Solar System, an atom, or anything else) would give the same result.

[JonF]

Chronos, I agre but I don't agree ...

I can't conceive of any way in which the speed of light could change without changing all sorts of other things, and it certainly seems likely to me that all those other changes would add up to a universe in which we couldn't detect that the speed of light had changed. Hell, I can't even conceive of a way in which the speed of light could change at all.

But I've learned over the course of many years that my inability to conceive something doesn't affect the universe much; the universe does what it does whether or not I can conceive it.

It's difficult to discuss changes in the speed of light without a mechanism, without a thorough exploration of the consequences and forced related changes ... but that seems to be what we're stuck with here.

[Arnold Winkelried]

DSYoungEsq is right, Karen did say in her mailbag item "You've got to attach units to make the numbers meaningful."

My apologies to Karen. This thread should be deleted and the storage device that was used to keep it should be demagnetized and sown with salt.

[grg]

There's a few disconcerting statement in the E=mc2 article that really rub me the wrong way. I don't know much physics, but I think these statements are misleading at best:

(1) "The speed of light of course is a very large number, so E=mc2 tells you mass contains an large amount of energy. The atomic bomb is testimony to that."

Large number?? It can be a small a number as you like,
if eexpressed in the right unit. It's really small if expressed in googol-KM per picosecond. it's really big if
expressed in angstroms per Century. Neither figure tells you anything basic about the speed of light.

(1.5) "...The atomic bomb is testimony to that."
An impressive blast, but it's not a very good example
of mass to energy. Fission only converts a small
percent of the mass of the nucleus to energy.

(2) "In most of your daily, mundane interactions, mass is conserved, so when you add up all the forms of energy, you have an mc2 on each side of the equation and they cancel each other out."

No, No, No. EVERY reaction follows the rule, even chemical
ones. So when you burn wood, the heat comes from a loss of mass. A very small loss, as chemical reactions are relatively weak mass to energy converters. But it's still there. E = mc2 is universal, not jsut for nuclear reactions.

[Chronos]

grg, I was actually going to raise some of those points myself, but I didn't want to seem to seem too picky about it. You're right, it's not meaningful to refer to an amount as large or small if it has units attached. In fact, it would be more accurate to say that the speed of light is 1, and that our typical units for measuring spatial separation are much smaller than our typical units for temporal separation.
On your first and a half point, atomic bombs do, indeed, only convert a fraction of a percent of their mass to energy. This only reinforces what Karen is saying: Even a fraction of a percent of the mass of an atomic bomb still makes enough bang to flatten a city.
As for your second point, you're technically correct, all reactions are a mass-energy conversion of a sort. What Karen was trying to get at is that such a small percentage of the mass gets converted that we can consider the mass, per se, to be constant, to a good approximation. The amount of energy (in the form of what we commonly call energy) that changes from one form to another, however, is significant. Therefore, for ordinary reactions, it's reasonable to use two separate conservation laws for mass and energy.

[Boris B]

I must admit I never learned that in physical science. I was taught that a simple chemical reaction does conserve mass, and that thermal energy produced is simply converted out of chemical potential energy.

Chronos and grg, are you saying that if you burn a piece of wood, you don't conserve mass? Obviously the ashes weigh less than the wood, but adding in all the smoke results than less than the original hunk of wood? Does burning wood really convert mass to energy? Or does it merely convert chemical potential energy to Brownian motion?

I'm just wondering exactly how much of my science education I am supposed to forget....

[Chronos]

In the context of chemical reactions, it's easier to think of it as potetial energy being a form of mass. If you have a sealed container and a really sensitive scale, and you burn a piece of wood, say, in the sealed container, its weight will, in fact, decrease by a very slight amount-- theoretically, at least. I'm not sure that this has ever actually been tested... Anyone know?

[Karen Too]

The statements that grg has labeled "(1)" were inserted by the editor. Alas, they do cloud the point I was trying to make. Let's cross out the word "number" and change it to something like "The speed of light is of course very large". Since "large" is a relative term, the statement is hunky dorey.

Secondly, look at the particle physics interaction where a B meson (mass = 5,280 MeV) decays into two pions (mass = 140 MeV each). Lots of mass has vanished (but total energy is conserved.) My point was that most interactions are more like dropping balls from leaning towers and such. When you drop a ball you usually write your equation like this: mgh = mv^2/2, whereas you could write it like this: mgh + mc^2 = mv^2/2 + mc^2. Where mass is conserved, the mc^2 terms cancel, so you usually leave them out. But if mass is not conserved (like in particle physics) you have to include mc^2 along with kinetic and potential energy.

[ZenBeam]

If you have a sealed container and a really sensitive scale, and you burn a piece of wood, say, in the sealed container, its weight will, in fact, decrease by a very slight amount-- theoretically, at least.

This is a nit-pick, and I apologize in advance to Chronos and everyone else who thinks "Well, duh.", but I think this is too important a point to leave as is:

The sealed container will only weigh less if heat or light from the fire escapes. If the box is opaque and completely insulated, burning the piece of wood won't affect the weight.

[Boris B]

Actually, ZenBeam, I don't think you're nitpicking. I think you're disagreeing with Chronos head on. I think he's saying that potential energy not only has mass equivalence, but also has just plain old mass.

Correct me if I'm wrong, Chronos.

Here's a test. Forget the burning wood. Let's deal with four blocks. If gravitational potential energy has mass, the four blocks should weigh more stacked up than they do if they're all sitting on the ground.

I'm not saying I believe this - I'm way too confused on the whole mass-energy conversion thing to believe anything outright. I'm just saying, that's my reading of Chronos' statement, If you have a sealed container and a really sensitive scale, and you burn a piece of wood, say, in the sealed container, its weight will, in fact, decrease by a very slight amount, and grg's So when you burn wood, the heat comes from a loss of mass. A very small loss, as chemical reactions are relatively weak mass to energy converters. But it's still there. E = mc2 is universal, not jsut for nuclear reactions. That is, Chronos and grg are saying that conservation of mass is an approximation, and no more. They very idea makes my scales itch, but then I'm kind of itchy about a lot physics in the first place.

[Chronos]

Right, ZenBeam, sorry I neglected that. If the energy from the burning wood stays in the box, the total mass stays the same.
Boris, excellent example with the stacked bricks. That's actually one way of approaching General Relativity, saying that gravity is Newtonian, but including the gravitational potential energy itself as part of what's gravitating.
As for conservation of mass being an approximation, conservation of energy is exact, and mass is one form of energy. It happens that in everyday situations, only a very small percentage of mass ever gets converted, so one can usually be pretty safe in saying that mass is conserved. If, however, you have a kilogram each of matter and antimatter, and you react them, the approximation fails big time-- You have 2 kg before, and zero kg after.

[Boris B]

Sorry. I thought ZenBeam had read Chronos' post wrong, and was talking about smoke escaping. Why Z.B. would have written "heat or light" if he meant "smoke", well, I don't know what I was thinking. Like I said, my brain itches. It looks like you two agree after all.

So, if I burn wood in a sealed container, and the temperature returns to normal after the fire has gone out, the container and contents will weigh slightly less afterwards. Anybody know how to get topical antihistamines into cerebral tissue?

[NicholasD]

Hello all brainbusters,

I'm the guy who originally posted the question "In e=mc2, what units of measurement was Einstein using?".

Firstly, I'd like to thank Karen for taking the time to write me such a well considered and informative reply (I was very excited to have received a published reply from Straight Dope!) .

I fully understand Karen's point that the equation exists independently of units of measurement, because it represents a fundamental relationship in nature which is true regardless of how we measure it, and units of measure are constructs of man. However, this explanation left me curiously unsatisfied and I felt that in general, my question was left unanswered.

As I thought about it, however, I realized it was the complete stupidity of the wording of my question which might have led the thing off in the wrong direction, because it was Arnold Winkelried's first reply which came closest to giving me a satisfactory answer. He seemed to be able to intuit what I was really asking through the haze. I guess the basis of my curiosity was a much more mundane matter, and therefore my question, should have been:

"Putting theoretical physics and the actual meaning of the equation aside, in practical real world applications of the equation, what standard units of measurement are often plugged into those variables in order to get usable mass-to-energy yield figures? So for example, if I wanted to know how many joules could be released from a croquet ball (yes I know that's dumb), what other 2 units of measurement could be used in the other variables to give a correct answer?"

Karen might argue that it doesn't matter what units you use, because conversion factors can always be applied to jump from one unit to another. However, I can't imagine that certain, related, standard units of measurement are not plugged into the variables in various practical applications for which the equation is useful, in order to give an answer that is intuitive and easily comprehended.

Thank you Karen! and Arnold! and all of you who contributed to this most interesting thread and took the time to try and answer my question.

NicholasD

[Chronos]

For a croquet ball, you'd probably want to use kilograms, meters, seconds, and Joules (the energy unit for the mks system). I'm not sure what the mass of a croquet ball is, but let's say .5 kg. Since c = 3*108 m/s, the energy released by the annihilation of a croquet ball would then be 4.5*1016 J. Of course, you could, if you really wanted, measure the mass in atomic mass units, and the speed of light in cubits per U238 half-life, and you'd still get a valid energy unit, but in practice, physicists usually use either the mks (meters, kilograms, seconds) or the cgs (centimeters, grams, seconds) system.

[C K Dexter Haven]

The league, stone, and fortnight measurement was good enough for me grandpap, and I ain't about to change to one of these new-fangled systems.