Yes, the watch does get heavier when you wind it. Potential energy is mass.
There was some discussion lately that the mass of the earth is measurably less than the sum of all its parts. That is, if you start with a huge number of rocks floating in space and build a planet out of it, you can extract energy from this process. (falling rocks release energy) So the finished planet weighs less.
To put things in perspective, I’ll calculate the mass gained by heating up water for coffee, since that’s the easiest example you mentioned. I don’t drink coffee, but I understand that people drink it pretty hot, like 55°C maybe? Heating 400 mL of water from 20°C to 55°C requires adding 58600 Joules to the system. Using E=mc², that’s equal to a mere 0.65 nanograms. Of course, it takes much less Energy to wind a watch, so you can imagine how little mass that adds.
It’s much more complicated than that. E=mc^2 means that a certain amount of mass CAN be turned into a certain amount of energy. If you take a certain amount of energy you can turn it into a certain mass of molecules. That doesn’t mean giving something that energy makes it that much heavier.
By heating up coffee you are making it more massive because the molecules move around faster so they gain mass from relativistic effects. To calculate how much heavier you’d have to figure out how much speed every single molecule gains and from that add up the total mass gained. (There is a statistical way of doing it that probably would be accurate.) At the speeds we’re dealing with the gain in mass would be very, very small. Probably even smaller than nanograms.
Konrad, you’re right about the relativistic effects of speeding molecules, but the result is exactly the same. The relativistic form of the Kinetic Energy equation is this:
KE = (m - m[sub]0[/sub])c²
m - m[sub]0[/sub] is the change in mass you’d get. KE is the Kinetic Energy you add. So this is just another form of E=mc², which is what I used.
I know this isn’t extremely clear, but this is one part of Relativity that I think I do understand, so if you want a more in-depth explanation (with formulas and numbers and everything!) I can try to pull something together, or one of the more veteran Physicists can do it. But I’m pretty sure I’m right.
Er, I believe you are incorrect. E = mc[sup]2[/sup] is no mere equality, it’s an identity. There is no valid way of detecting a difference between mass and energy. Our insufficient senses appear to show us a difference, but that’s a problem with our senses, not the identity.
JonF is correct; mass is energy. Adding energy to a body in any form will make it more massive. In the identity E = mc[sup]2[/sup], the speed of light squared is the conversion factor. Since the speed of light is very large, c squared is even larger. A wound watch has more energy and hence more mass, but not really a measurable amount.
Not unless you call one large :D. (It’s a conversion factor, remember, and all conversion factors are equal to one). The problem is, one second is a lot longer than one meter, so we just think that c is large.
Yes, I know that we relativists are lazy, but things are so much easier with less units.
I know that the speed of light canbe set to one. But in practice you don’t. You don’t really measure distance in seconds do you? How tall are you in seconds? Or do you measure time in meters? How old are you in meters?
No. It may be an artifact of the way we are, but the speed of light is large.
Have you ever tried using sqrt(-1) for the speed of light. The formulas work out even nicer than using one.
I can make the speed of light as large as I want or as small – I can measure distance in light-years and time in seconds. These units do not go together naturally. Meters to measure distance and seconds to measure time much more natural together. A little change in mass yields a large change in energy using any standard set of units.
Konrad
Photons have no “rest mass”. It does seem silly to talk about the rest mass of a particle that is never at rest. I have seen rest mass refered to as invariant mass. Their energy is all energy of motion. This mass-energy has an effect on gravity just like any other mass.
Depends on what you’re practicing. You’d be hard-pressed to find a journal paper or textbook in relativity that doesn’t use the same units for length and duration, typically either meters or (if you’re trying to do quantum gravity) Planck units. I’ll be the first to admit that outside of relativity, this practice is almost nonexistant, but we’re not outside of relativity on this question.
And yes, I’ve used i as c, also, and it does work very well. Food for thought, though, does this mean that time or space is imaginary? In other words, is that i length units per 1 time unit, or 1 length unit per -i time units?
Using sqrt(-1) as the speed of light is something I came up with about 30 years ago. I haven’t really thought about it since. As I recall, you can set one light-year to sqrt(-1) years. (To me this is exactly the same as setting one year to -sqrt(-1) light-years). I think it makes velocity look just like a rotation about the plane formed by the time axis and one spacial dimension. Also the distance formula ds[sup]2[/sup] = dx[sup]2[/sup] + dy[sup]2[/sup] + dz[sup]2[/sup] - (c dt)[sup]2[/sup] becomes ds[sup]2[/sup] = dx[sup]2[/sup] + dy[sup]2[/sup] + dz[sup]2[/sup] + dt[sup]2[/sup]. I forget what else…I gotta go look up some of the formulas and see what happens. S. Hawking talked about imaginary time in A Brief History of Time. I think he was talking about something completely different, but I did not understand what he meant, so I dunno.
About what you said about setting c to unity. You are right; if you are talking about relativity (and we are) it makes sense to set c equal to a unit.
Hey, you came up with that yourself, no kidding? I didn’t realize it until I’d had rather a bit of exposure to SR in classes. Yes, it does indeed turn velocity into a simple rotation and time dilation and space contraction into projection effects-- That’s exactly the value of it. On the other hand, many folks seem to say that time is imaginary (and yes, that’s what Hawking was talking about), but it seems to me to make more sense to make time real, and space imaginary. My reason for this is that the imaginary axis, like the spatial axes, is symmetric (given a true equation, if you replace every instance of i with -i, the equation remains true), while the real axis, like the time axis, has asymmetries in it: The equation x[sup]2[/sup] = x is true for x=1, but not for x=-1. Furthermore, you can use quaternions instead of complex numbers (four distinct units 1,i,j,k such that i[sup]2[/sup] = j[sup]2[/sup] = k[sup]2[/sup] = -1, but i != j != k), and you have an analogy to our three symmetric spatial dimensions and one asymmetric time dimension.
Unfortunately, I’ve no clue what sort of experiment might distinguish between a real dimension and an imaginary one, if such is even meaningful, so the distinction must remain on paper for now.
The way I see it, it doesn’t matter whether you have real time with imag distance or imag time with real distance. Asking which one is really real, to me, is like asking how do we know that what we call matter isn’t really anti-matter and what we call anti-matter is really matter. It doesn’t matter. <sorry> If you want to take seconds, say, as your fundamental unit and define a light-second as i seconds, umm… OK, sure. I don’t see the difference. You said that, WRT equations, the imag axis is symmetric and the real is not. That is only true if you confine your coefficients to real values. If you allow complex coefficients, the asymmetry vanishes.
See if this makes any sense. Since complex numbers can be put in a one to one correspondence with points in a plane, every statement about complex numbers is a statement about points in a plane. Statements in complex algebra are statements in geometry and visa versa. Geometry shows no preferred direction while complex numbers seem to have an asymmetry about the real axis. This apparent asymmetry arises because, in order to go from geometry to algebra, we must choose some point and call it Origin, and some direction and call it real. From the perspective of geometry, the choice is arbitrary and there is no asymmetry. The asymmetry is created for you convenience – to allow a change in notation, but it is not real. Nothing in geometry can tell you which direction is real, so I would say no physical experiment could distinguish either.
I’ve seen quaternions mentioned several times. (Isn’t i j = k, j k = i, and k i = j, or something?) I never really looked at them though. Maybe I’ll play around with them.
Calculating in imaginary time is just a math trick towards calculating real time. Actually the real time may just be our perception of time and the imaginary time might indeed be the real time. Hawking was looking in applying Feynmann’s sum over histories to Einstien’s view on gravity. Taking curved space-time into imaginary/ rectangular time is/was just a trick to make the calculation of probability. Then again Hawking was looking at imaginary time as a different way to look at the (proposed Big Bang) inflationary model of the Universe.