c = 1.8 × 10[sup]12[/sup] furlongs/fortnight.
Chronos, I’m lost.
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Of course saying c = 1 is not the same thing as saying 3 inches = 3 cm. That’s exactly what I said. What I said is that units matter, and c = 1 in systems with the right units.
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3e8 meters = 1 second? Oh, I get it now. Had to think that one through.
You are correct that 1/90,000,000 is wrong. But you made the same mistake the op and I did. Unit of measure is important!
C is not 300,000 or 30,000,000,000 or 186,000 or 1.
C is 300,000 km/sec or 30,000,000,000 m/sec or 186,000 miles/sec or 1 light year/year.
3e8 m does not equal 1 second. You can add some context to make a true statement.
Just to be clear energy and mass are not synonyms, but there is nothing wrong with viewing mass as a type of energy. In terms of expressing the one in the units of the other, this perfectly fine: for example saying the Earth has 3 million tons of kinetic energy, may sound a little odd, but it is totally unambiguous and no additional context is required (except frame of reference, as is always required for KE). Requiring that energy and mass have different units can be done, but it is not a necessary requirement.
I perhaps wouldn’t make the point of c=1 quite as strongly as Chronos, but it certainly has deeper meaning than simply selecting units to make c=1. If we take spacetime as our starting point (which is not a novel approach at all), then c=1 is buried in the way that the physics relate to the quasi-inner product space structure of the tangent spaces. Or stated more simply, when you have a bunch of quantities that you can add and subtract unambiguously, it is natural to express them in the same units.
These young kids just aren’t being taught their algebra like our fathers before us learned it. (That’s a criticism of how algebra is taught these days, not necessarily of today’s young kids.)
I have a College Algebra textbook (Britton & Snively) published in 1948, making it older than I am, that devotes an entire chapter to the topic of “Ratio, Proportion, and Variation”.
The rock-bottom basic formula for proportional variables is: y = kx where x and y are the variables in a linear (i.e., proportional) relation, and k is the constant of proportion. This itself is beyond the conceptual abilities of many algebra students these days. (Ask me how I know: I used to tutor some of those. Just mention the word “proportion” and they freak.)
If you graph that, it’s a straight line, passing through the origin, with a slope of k.
If x and y are physical measures expressed in some units, you can convert them to any other units, simply by adjusting k, the constant of proportion. Why, if you choose your units carefully for x and y, you can even make the constant equal to 1, reducing the relation to simply y=x
The equation E = mc[sup]2[/sup] is exactly none other than an equation of just this sort. E and m are the proportionally related variables, in a simple linear relation, and c[sup]2[/sup] is the constant of proportionality.
As Chronos and others point out, if you choose your units wisely, you can reduce the constant c[sup]2[/sup] factor to be exactly 1, reducing the equation to simply E=m.
(Unfortunately, SDStaff Karen’s explanation of this was, I thought, thoroughly obscure.)
Actually, square rods seem to be common enough. See here.
c is not 300,000 or 30,000,000,000 or 186,000. It is 1.
Think of it this way: Suppose, for some reason, the tradition developed that we would measure all vertical heights in inches (or feet or miles), and all horizontal distances in centimeters (or meters or kilometers). We could do things like putting a 3 inch block on top of a 4 inch block, and measure that the whole stack was 7 inches tall. Likewise, we could put a 2 cm block next to a 3 cm block, and measure that the total width was 5 cm. We couldn’t add a height to a width, but that’s OK, because how often do you do that anyway?
But what we can do, is tilt things over at an angle. And we would notice that when we do so, their heights and widths change. With some work, we could figure out the formulas to describe just how much heights and widths would change, and those formulas would have this ratio of 2.54 cm/inch all over the place in them. And we would further notice that, aside from that funny ratio, our formulas would look an awful lot like the formulas you use to describe something rotated horizontally, where all of the measurements are in cm.
Does this mean that there’s some fundamental constant whose value is 2.54 cm/inch (or 0.3048 m/foot, or 1.60934 km/mile), and which must be expressed in metric units per American units? No, of course not. All it means is that we made a silly choice of units, and that number is showing up all over the place because, in fact, 2.54 cm is one inch, and therefore 2.54 cm/inch = 1.
In nearly exactly the same way, though historical happenstance, we traditionally measure space and time in different units. But we find that we can rotate things through spacetime, and when we do, their lengths and durations change. And the formulas for those changes have c all over the place in them, but otherwise look very much like the formulas for rotation. Should we say that c must be expressed in length units divided by time units, or should we just use the same units for length and time, and say that c is thus 1?
I completely screwed that up in the translation from my thought to posting plus I used km/second instead of m/second. Trying again…
e = m * c^2 is meaningless without context. It is so famous that average person knows it has something to do with “science” and I suspect a physicist has a very specific context.
If I say e = m * c^2 and by the way, e is for elephants, m is for milk and c is for chalk then I’m wrong, e is not m * c^2.
If the most common context is energy in joules, mass in kilograms and c in meters per second, e = m * c^2
If I were to say e = m * c^2 and energy is in joules, mass is in stones and c is in meters per second, I have to convert units before I multiply.
In that sense, if you want to express c in light years per year and mass in kilograms and still get a computed result in joules, you have to convert units. One way to do that is to divide both sides of the equation by 9e16.
Or you can express c as 1, mass in kilograms, and energy in kilograms.
You could do that…
In the interest of fighting ignorance (mine) can someone translate “Chronos” to “tim-n-va”. I’m missing something.
Thank, Chronos, that explanation is very helpful.
tim-n-va, are you familiar with the concept in English customary (or American) units of g[sub]c[/sub]?
Briefly, whenever one wishes to use pounds to express both force and mass, you run into this interesting relationship. The standard method used is to use lbf and lbm to express pounds-force and pounds-mass.
If you revert back to the basic F=ma relationship, and plug in units, you get
lbf = lbm x ft/sec[sup]2[/sup]
Except people want the lbf value to equal the lbm value, so that under standard gravity, 1 lb of material = 1 lb of material. Because the standard acceleration of gravity is taken to be 32.2 ft/sec[sup]2[/sup], you are required to add a correction factor to the relationship. Thus,
lbf = lbm x a(ft/sec[sup]2[/sup]) x g[sub]c[/sub]
where g[sub]c[/sub] = 32.2 (ft lbm)/(lbf sec[sup]2[/sup])
Anytime you wish to use the unit lbm, you find g[sub]c[/sub] sneaking into all the relationships.
I actually had an instructor once tell me that in English customary units, the equation for Force was F = m x a x g[sub]c[/sub] . This certainly confused some of my classmates and made for a struggle to clear up the misconception.
g[sub]c[/sub] is just a correction factor required for choosing to use goofy units.
With respect to what Chronos is saying, it seems to me that you can express energy in kilograms, but you’re just burying the conversion factor. If you don’t want to redefine meters and seconds, you will require inserting a c somewhere in the process.
If I understand what Chronos is saying, if we take spacetime to be a thing, then at some fundamental level time and length are the same stuff. Therefore, there is a relationship between length and time. If we define that relationship by 3e8 meters = 1 second, that conversion factor converts to 1, and we can express energy in kilograms.
Irishman - Thank you
That holds only if E is defined in terms of, say, kg x lightseconds^2 / sec^2.
Actually, what it is saying is that they are proportional; E/M is a constant. The specific constant depends on the types of measurement used; IIRC, the original used energy in ergs, mass in grams, and c was in cm/sec.
Aha- I’ve seen you say this many times before, but I could not quite grok it. Thanks- the analogy (which I mostly snipped out) helped a lot.
Can you describe a relatively simple example of when we might rotate something through space-time?
Anytime someone is looking at relativistic effects, such as proposed space travel, you are, in effect, rotating between axes. That is what causes the apparent foreshortening of relativistic travel. Distances get smaller in the direction of travel because one is rotating from a space to time dimension with respect to the original (viewing) dimension.
c is constant in all reference frames, but the different reference frames are rotated with respect to each other.
Simple. Every time someone is moving at a non-zero velocity relative to you (down the x-axis of some coordinate system, say). In that moment, that person’s x-axis is rotated a little relative to yours so that it is pointing a little bit in your time direction. And his time axis is rotated so it is pointing a little bit in your x-direction.
If the velocity is small, then the rotation is very slight, which is why we don’t really notice it. This could also be phrased as typical human-scale distances in space, like a metre, are a lot smaller than typical human-scale distances in time, like a second = 299792458 metres.