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.
for mass, use kilogram (kg)
speed of light in metres per second (m s[sup]-1[/sup])
then the energy value will be in joule (J = kg m[sup]2[/sup] s[sup]-2[/sup])
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.
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/s[sup]2[/sup], then you get a force in kg-m/s[sup]2[/sup]. In the SI system, this is defined to be a Newton, or you could say the conversion factor is 1 Newton per kg-m/s[sup]2[/sup].
However, you could measure the mass in lbs mass (lbm) and the accel in ft/s[sup]2[/sup]. Then you get the force in
lbm-ft/s[sup]2[/sup]. Now that term is “funky”, so engineers use a conversion factor
32.2 ft-lbm/lbf-s[sup]2[/sup] . 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/s[sup]2[/sup]), thus the conversion factor.
So you get
Force = mass (lbm) x accel (ft/s[sup]2[/sup]) x conversion factor (32.2 ft-lbm/lbf-s[sup]2[/sup])
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 c[sup]2[/sup]. Conversion factors are added to make the units work out, depending on the values plugged in.
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.
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,
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.
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?
Another reason for not expressing equations of pure physics in specific values is that the constants in the equations may change over time[sup]*[/sup] 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=mc[sup]2[/sup] would still hold true (resulting in an atomic explosion looking like a cap gun).
Peace.
[sup][/sup]a purely theoretical hypothesis at this point in time*
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 ime\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.
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.
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.
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.
JonF
What you just said seems to contradict what you said here.
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?
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:
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.
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.
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.
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 …
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.
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.
